Timothy W Secomb

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Chapters

• Secomb, T. W. (2022). Simulation of blood flow and oxygen transport in vascular networks. In The Vasculome: From Many, One(pp 173-179). San Diego: Elsevier Inc./Academic Press.
• Secomb, T. W., Pries, A. R., & Gaehtgens, P. (1995). Architecture and Hemodynamics of Microvascular Networks. In Biological Flows", ed. M.Y. Jaffrin and C.G. Caro. Plenum, New York. Springer, Boston, MA. doi:10.1007/978-1-4757-9471-7_9
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  The main function of the circulation is to transport materials between different parts of the body. Transport over large distances is accomplished by convection, in blood flowing through large vessels. Exchange of materials between blood and tissues occurs mainly over short distances in the peripheral vascular beds, which consist of numerous very small vessels (the microcirculation). These microvessels provide a large surface area for exchange, and bring blood into close proximity to nearly all parts of most organs. Transport at this microscopic level occurs by diffusion, by active cellular transport, or by convective motion of water through microvessel walls.
• Secomb, T. W., Fleischman, G. J., Papenfuss, H. D., Intaglietta, M., & Gross, J. F. (1988). Der Einfluß von verminderter Strömung und erniedrigtem Hämatokrit auf die Blutverteilung in kapillaren Netzwerken. In In "Mikrozirkulation und Entzündung: Beziehungen zwischen Gefäßwand, Entzündungszellen and Mediatoren," ed. K. Meßmer and F. Hammersen, Karger, Basel. Karger Publishers. doi:10.1159/000415129
• Skalak, R., Ozkaya, N., & Secomb, T. W. (1986). Biomechanics of Capillary Blood Flow. In Frontiers in Biomechanics" (Schmid Schonbein, G.W., Woo, S.L Y., and Zweifach, B.W. eds.) Springer, New York. Springer, New York, NY. doi:10.1007/978-1-4612-4866-8_21
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  The quantitative study of capillary flow dates back to the work of Poiseuille (1840). Because of the difficulties with the clotting of blood, Poiseuille used homogeneous fluids in his famous tests, which established empirically that the discharge through a capillary varies directly with the pressure drop per unit length and the fourth power of the diameter. The formula, which is usually called Poiseuille’s Law, is: $$Q = P\pi {D^4}/128\mu L,$$ (21.1) where Q is the discharge; P is the pressure drop in the length of the capillary, L; D is the diameter of the capillary; and µ is the fluid viscosity. Pouiseuille’s Law does not apply directly to capillary blood flow because the blood cells are of the same order of magnitude in diameter as the capillaries themselves. Equation (21.1) nevertheless, is useful to define an apparent viscosity of blood in capillaries. If Q, P, D, and L are all measured, then Equation (21.1) may be solved for the apparent viscosity, µ a . The ratio of µ a /µ., where µ is the viscosity of the suspending fluid, is defined as the relative apparent viscosity,η

Journals/Publications

• Djurich, S., & Secomb, T. W. (2024). Analysis of potassium ion diffusion from neurons to capillaries: Effects of astrocyte endfeet geometry. The European journal of neuroscience, 59(3), 323-332.
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  Neurovascular coupling (NVC) refers to a local increase in cerebral blood flow in response to increased neuronal activity. Mechanisms of communication between neurons and blood vessels remain unclear. Astrocyte endfeet almost completely cover cerebral capillaries, suggesting that astrocytes play a role in NVC by releasing vasoactive substances near capillaries. An alternative hypothesis is that direct diffusion through the extracellular space of potassium ions (K ) released by neurons contributes to NVC. Here, the goal is to determine whether astrocyte endfeet present a barrier to K diffusion from neurons to capillaries. Two simplified 2D geometries of extracellular space, clefts between endfeet, and perivascular space are used: (i) a source 1 μm from a capillary; (ii) a neuron 15 μm from a capillary. K release is modelled as a step increase in [K ] at the outer boundary of the extracellular space. The time-dependent diffusion equation is solved numerically. In the first geometry, perivascular [K ] approaches its final value within 0.05 s. Decreasing endfeet cleft width or increasing perivascular space width slows the rise in [K ]. In the second geometry, the increase in perivascular [K ] occurs within 0.5 s and is insensitive to changes in cleft width or perivascular space width. Predicted levels of perivascular [K ] are sufficient to cause vasodilation, and the rise time is within the time for flow increase in NVC. These results suggest that direct diffusion of K through the extracellular space is a possible NVC signalling mechanism.
• Roy, T. K., Joyner, M. J., Senefeld, J. W., Wiggins, C. C., & Secomb, T. W. (2024). An empirical model for world record running speeds with distance, age, and sex: anaerobic and aerobic contributions to performance. Journal of applied physiology (Bethesda, Md. : 1985), 137(2), 357-363.
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  The objective of this study is to derive mathematical equations that closely describe published data on world record running speed as a function of distance, age, and sex. Running speed declines with increasing distance and age. Over long distances, where aerobic metabolism is dominant, speed declines in proportion to the logarithm of distance. Over short distances, anaerobic metabolism contributes significantly to performance, and speed is increased relative to the trend of the long-distance data. Equations are derived that explicitly represent these effects. The decline in speed with age is represented by an age-dependent multiplicative factor, which exhibits increasing sensitivity to age as age increases. Using these equations, data are analyzed separately for males and females, and close fits to published data are demonstrated, particularly for younger age groups. These equations provide insight into the contributions of aerobic and anaerobic components of metabolism to athletic performance and a framework for comparisons of performance across wide ranges of distance and age. World record speeds at different distances for men and women in different age categories are used to develop a model to predict running performance as a function of race distance, age, and sex. This empirical model quantifies the decline in running speed with distance and age in a way that provides insight into the aerobic and anaerobic contributions to running speed and may help with developing training strategies for different age groups at various distances.
• Alberding, J. P., & Secomb, T. W. (2023). Simulation of Angiogenesis in Three Dimensions: Development of the Retinal Circulation. Bulletin of mathematical biology, 85(4), 27.
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  A theoretical model is used to describe the three-dimensional development of the retinal circulation in the human eye, which occurs after the initial spread of vasculature across the inner surface of the retina. In the model, random sprouting angiogenesis is driven by a growth factor that is produced in tissue at a rate dependent on oxygen level and diffuses to existing vessels. Vessel sprouts connect to form pathways for blood flow and undergo remodeling and pruning. These processes are controlled by known or hypothesized vascular responses to hemodynamic and biochemical stimuli, including conducted responses along vessel walls. The model shows regression of arterio-venous connections on the surface of the retina, allowing perfusion of the underlying tissue. A striking feature of the retinal circulation is the formation of two vascular plexuses located at the inner and outer surfaces of the inner nuclear layer within the retina. The model is used to test hypotheses regarding the formation of these structures. A mechanism based on local production and diffusion of growth factor is shown to be ineffective. However, sprout guidance by localized structures on the boundaries of the inner nuclear layer can account for plexus formation. The resulting networks have vascular density, perfusion and oxygen transport characteristics consistent with observed properties. The model shows how stochastic generation of vascular sprouts combined with a set of biologically based response mechanisms can lead to the generation of a specialized three-dimensional vascular structure with a high degree of organization.
• Delmoe, M., & Secomb, T. W. (2023). Conditions for Kir-induced bistability of membrane potential in capillary endothelial cells. Mathematical biosciences, 355, 108955.
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  A simplified model for electrophysiology of endothelial cells is used to examine the conditions that can lead to bistability of membrane resting potential. The model includes the effects of inward-rectifying potassium (Kir) ion channels, whose current-voltage relationship shows an interval of negative slope and whose maximum conductance is dependent on the extracellular potassium concentration. The background current resulting from other types of channels is assumed to be linearly related to membrane potential. A method is presented for identifying the boundaries in the parameter space for the background currents of the regions of bistability. It is shown that these regions are relatively narrow and depend on extracellular potassium concentration. The results are used to define conditions leading to transitions between depolarized and hyperpolarized membrane states. These behaviors can influence the properties of conducted responses, in which changes in membrane potential are propagated along blood vessel walls. Conducted responses are important in the local regulation of blood flow in the brain and other tissues.
• Hu, N. W., Lomel, B. M., Rice, E. W., Hossain, M. M., Sarntinoranont, M., Secomb, T. W., Murfee, W. L., & Balogh, P. (2023). Estimation of shear stress heterogeneity along capillary segments in angiogenic rat mesenteric microvascular networks. Microcirculation (New York, N.Y. : 1994), 30(8), e12830.
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  Fluid shear stress is thought to be a regulator of endothelial cell behavior during angiogenesis. The link, however, requires an understanding of stress values at the capillary level in angiogenic microvascular networks. Critical questions remain. What are the stresses? Do capillaries experience similar stress magnitudes? Can variations explain vessel-specific behavior? The objective of this study was to estimate segment-specific shear stresses in angiogenic networks.
• Moulton, M. J., & Secomb, T. W. (2023). A fast computational model for circulatory dynamics: effects of left ventricle-aorta coupling. Biomechanics and modeling in mechanobiology, 22(3), 947-959.
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  The course of diseases such as hypertension, systolic heart failure and heart failure with a preserved ejection fraction is affected by interactions between the left ventricle (LV) and the vasculature. To study these interactions, a computationally efficient, biophysically based mathematical model for the circulatory system is presented. In a four-chamber model of the heart, the LV is represented by a previously described low-order, wall volume-preserving model that includes torsion and base-to-apex and circumferential wall shortening and lengthening, and the other chambers are represented using spherical geometries. Active and passive myocardial mechanics of all four chambers are included. The cardiac model is coupled with a wave propagation model for the aorta and a closed lumped-parameter circulation model. Parameters for the normal heart and aorta are determined by fitting to experimental data. Changes in the timing and magnitude of pulse wave reflections by the aorta are demonstrated with changes in compliance and taper of the aorta as seen in aging (decreased compliance, increased diameter and length), and resulting effects on LV pressure-volume loops and LV fiber stress and sarcomere shortening are predicted. Effects of aging of the aorta combined with reduced LV contractile force (failing heart) are examined. In the failing heart, changes in aortic properties with aging affect stroke volume and sarcomere shortening without appreciable augmentation of aortic pressure, and the reflected pressure wave contributes an increased proportion of aortic pressure.
• Pian, Q., Alfadhel, M., Tang, J., Lee, G. V., Li, B., Fu, B., Ayata, Y., Yaseen, M. A., Boas, D. A., Secomb, T. W., & Sakadzic, S. (2023). Cortical microvascular blood flow velocity mapping by combining dynamic light scattering optical coherence tomography and two-photon microscopy. Journal of biomedical optics, 28(7), 076003.
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  The accurate large-scale mapping of cerebral microvascular blood flow velocity is crucial for a better understanding of cerebral blood flow (CBF) regulation. Although optical imaging techniques enable both high-resolution microvascular angiography and fast absolute CBF velocity measurements in the mouse cortex, they usually require different imaging techniques with independent system configurations to maximize their performances. Consequently, it is still a challenge to accurately combine functional and morphological measurements to co-register CBF speed distribution from hundreds of microvessels with high-resolution microvascular angiograms.
• Webb, K. L., Joyner, M. J., Wiggins, C. C., Secomb, T. W., & Roy, T. K. (2023). The dependence of maximum oxygen uptake and utilization (V̇O max) on hemoglobin-oxygen affinity and altitude. Physiological reports, 11(17), e15806.
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  Oxygen transport from the lungs to peripheral tissue is dependent on the affinity of hemoglobin for oxygen. Recent experimental data have suggested that the maximum human capacity for oxygen uptake and utilization (V̇O max) at sea level and altitude (~3000 m) is sensitive to alterations in hemoglobin-oxygen affinity. However, the effect of such alterations on V̇O max at extreme altitudes remains largely unknown due to the rarity of mutations affecting hemoglobin-oxygen affinity. This work uses a mathematical model that couples pulmonary oxygen uptake with systemic oxygen utilization under conditions of high metabolic demand to investigate the effect of hemoglobin-oxygen affinity on V̇O max as a function of altitude. The model includes the effects of both diffusive and convective limitations on oxygen transport. Pulmonary oxygen uptake is calculated using a spatially-distributed model that accounts for the effects of hematocrit and hemoglobin-oxygen affinity. Systemic oxygen utilization is calculated assuming Michaelis-Menten kinetics. The pulmonary and systemic model components are solved iteratively to compute predicted arterial and venous oxygen levels. Values of V̇O max are predicted for several values of hemoglobin-oxygen affinity and hemoglobin concentration based on data from humans with hemoglobin mutations. The model predicts that increased hemoglobin-oxygen affinity leads to increased V̇O max at altitudes above ~4500 m.
• Williams, K. S., Secomb, T. W., & El-Kareh, A. W. (2023). An autonomous mathematical model for the mammalian cell cycle. Journal of theoretical biology, 569, 111533.
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  A mathematical model for the mammalian cell cycle is developed as a system of 13 coupled nonlinear ordinary differential equations. The variables and interactions included in the model are based on detailed consideration of available experimental data. A novel feature of the model is inclusion of cycle tasks such as origin licensing and initiation, nuclear envelope breakdown and kinetochore attachment, and their interactions with controllers (molecular complexes involved in cycle control). Other key features are that the model is autonomous, except for a dependence on external growth factors; the variables are continuous in time, without instantaneous resets at phase boundaries; mechanisms to prevent rereplication are included; and cycle progression is independent of cell size. Eight variables represent cell cycle controllers: the Cyclin D1-Cdk4/6 complex, APC, SCF, Cdc25A, MPF, NuMA, the securin-separase complex, and separase. Five variables represent task completion, with four for the status of origins and one for kinetochore attachment. The model predicts distinct behaviors corresponding to the main phases of the cell cycle, showing that the principal features of the mammalian cell cycle, including restriction point behavior, can be accounted for in a quantitative mechanistic way based on known interactions among cycle controllers and their coupling to tasks. The model is robust to parameter changes, in that cycling is maintained over at least a five-fold range of each parameter when varied individually. The model is suitable for exploring how extracellular factors affect cell cycle progression, including responses to metabolic conditions and to anti-cancer therapies.
• Dewhirst, M. W., Oleson, J. R., Kirkpatrick, J., & Secomb, T. W. (2022). Accurate Three-Dimensional Thermal Dosimetry and Assessment of Physiologic Response Are Essential for Optimizing Thermoradiotherapy. Cancers, 14(7).
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  Numerous randomized trials have revealed that hyperthermia (HT) + radiotherapy or chemotherapy improves local tumor control, progression free and overall survival vs. radiotherapy or chemotherapy alone. Despite these successes, however, some individuals fail combination therapy; not every patient will obtain maximal benefit from HT. There are many potential reasons for failure. In this paper, we focus on how HT influences tumor hypoxia, since hypoxia negatively influences radiotherapy and chemotherapy response as well as immune surveillance. Pre-clinically, it is well established that reoxygenation of tumors in response to HT is related to the time and temperature of exposure. In most pre-clinical studies, reoxygenation occurs only during or shortly after a HT treatment. If this were the case clinically, then it would be challenging to take advantage of HT induced reoxygenation. An important question, therefore, is whether HT induced reoxygenation occurs in the clinic that is of radiobiological significance. In this review, we will discuss the influence of thermal history on reoxygenation in both human and canine cancers treated with thermoradiotherapy. Results of several clinical series show that reoxygenation is observed and persists for 24-48 h after HT. Further, reoxygenation is associated with treatment outcome in thermoradiotherapy trials as assessed by: (1) a doubling of pathologic complete response (pCR) in human soft tissue sarcomas, (2) a 14 mmHg increase in pO2 of locally advanced breast cancers achieving a clinical response vs. a 9 mmHg decrease in pO2 of locally advanced breast cancers that did not respond and (3) a significant correlation between extent of reoxygenation (as assessed by pO2 probes and hypoxia marker drug immunohistochemistry) and duration of local tumor control in canine soft tissue sarcomas. The persistence of reoxygenation out to 24-48 h post HT is distinctly different from most reported rodent studies. In these clinical series, comparison of thermal data with physiologic response shows that within the same tumor, temperatures at the higher end of the temperature distribution likely kill cells, resulting in reduced oxygen consumption rate, while lower temperatures in the same tumor improve perfusion. However, reoxygenation does not occur in all subjects, leading to significant uncertainty about the thermal-physiologic relationship. This uncertainty stems from limited knowledge about the spatiotemporal characteristics of temperature and physiologic response. We conclude with recommendations for future research with emphasis on retrieving co-registered thermal and physiologic data before and after HT in order to begin to unravel complex thermophysiologic interactions that appear to occur with thermoradiotherapy.
• Roy, T. K., & Secomb, T. W. (2022). Functional implications of microvascular heterogeneity for oxygen uptake and utilization. Physiological reports, 10(10), e15303.
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  In the vascular system, an extensive network structure provides convective and diffusive transport of oxygen to tissue. In the microcirculation, parameters describing network structure, blood flow, and oxygen transport are highly heterogeneous. This heterogeneity can strongly affect oxygen supply and organ function, including reduced oxygen uptake in the lung and decreased oxygen delivery to tissue. The causes of heterogeneity can be classified as extrinsic or intrinsic. Extrinsic heterogeneity refers to variations in oxygen demand in the systemic circulation or oxygen supply in the lungs. Intrinsic heterogeneity refers to structural heterogeneity due to stochastic growth of blood vessels and variability in flow pathways due to geometric constraints, and resulting variations in blood flow and hematocrit. Mechanisms have evolved to compensate for heterogeneity and thereby improve oxygen uptake in the lung and delivery to tissue. These mechanisms, which involve long-term structural adaptation and short-term flow regulation, depend on upstream responses conducted along vessel walls, and work to redistribute flow and maintain blood and tissue oxygenation. Mathematically, the variance of a functional quantity such as oxygen delivery that depends on two or more heterogeneous variables can be reduced if one of the underlying variables is controlled by an appropriate compensatory mechanism. Ineffective regulatory mechanisms can result in poor oxygen delivery even in the presence of adequate overall tissue perfusion. Restoration of endothelial function, and specifically conducted responses, should be considered when addressing tissue hypoxemia and organ failure in clinical settings.
• Secomb, T. W., Oleson, J. R., Kirkpatrick, J., & Dewhirst, M. W. (2022). Accurate Three-Dimensional Thermal Dosimetry and Assessment of Physiologic Response Are Essential for Optimizing Thermoradiotherapy.. Cancers, 14(7), 1701. doi:10.3390/cancers14071701
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  Numerous randomized trials have revealed that hyperthermia (HT) + radiotherapy or chemotherapy improves local tumor control, progression free and overall survival vs. radiotherapy or chemotherapy alone. Despite these successes, however, some individuals fail combination therapy; not every patient will obtain maximal benefit from HT. There are many potential reasons for failure. In this paper, we focus on how HT influences tumor hypoxia, since hypoxia negatively influences radiotherapy and chemotherapy response as well as immune surveillance. Pre-clinically, it is well established that reoxygenation of tumors in response to HT is related to the time and temperature of exposure. In most pre-clinical studies, reoxygenation occurs only during or shortly after a HT treatment. If this were the case clinically, then it would be challenging to take advantage of HT induced reoxygenation. An important question, therefore, is whether HT induced reoxygenation occurs in the clinic that is of radiobiological significance. In this review, we will discuss the influence of thermal history on reoxygenation in both human and canine cancers treated with thermoradiotherapy. Results of several clinical series show that reoxygenation is observed and persists for 24-48 h after HT. Further, reoxygenation is associated with treatment outcome in thermoradiotherapy trials as assessed by: (1) a doubling of pathologic complete response (pCR) in human soft tissue sarcomas, (2) a 14 mmHg increase in pO2 of locally advanced breast cancers achieving a clinical response vs. a 9 mmHg decrease in pO2 of locally advanced breast cancers that did not respond and (3) a significant correlation between extent of reoxygenation (as assessed by pO2 probes and hypoxia marker drug immunohistochemistry) and duration of local tumor control in canine soft tissue sarcomas. The persistence of reoxygenation out to 24-48 h post HT is distinctly different from most reported rodent studies. In these clinical series, comparison of thermal data with physiologic response shows that within the same tumor, temperatures at the higher end of the temperature distribution likely kill cells, resulting in reduced oxygen consumption rate, while lower temperatures in the same tumor improve perfusion. However, reoxygenation does not occur in all subjects, leading to significant uncertainty about the thermal-physiologic relationship. This uncertainty stems from limited knowledge about the spatiotemporal characteristics of temperature and physiologic response. We conclude with recommendations for future research with emphasis on retrieving co-registered thermal and physiologic data before and after HT in order to begin to unravel complex thermophysiologic interactions that appear to occur with thermoradiotherapy.
• Secomb, T. W., Scarpa, F., Rong, W. W., Pries, A. R., Nitzsche, B., Kuebler, W. M., Hoffmann, B., & Goede, A. (2022). Coalescent angiogenesis-evidence for a novel concept of vascular network maturation.. Angiogenesis, 25(1), 35-45.
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  Angiogenesis describes the formation of new blood vessels from pre-existing vascular structures. While the most studied mode of angiogenesis is vascular sprouting, specific conditions or organs favor intussusception, i.e., the division or splitting of an existing vessel, as preferential mode of new vessel formation. In the present study, sustained (33-h) intravital microscopy of the vasculature in the chick chorioallantoic membrane (CAM) led to the hypothesis of a novel non-sprouting mode for vessel generation, which we termed "coalescent angiogenesis." In this process, preferential flow pathways evolve from isotropic capillary meshes enclosing tissue islands. These preferential flow pathways progressively enlarge by coalescence of capillaries and elimination of internal tissue pillars, in a process that is the reverse of intussusception. Concomitantly, less perfused segments regress. In this way, an initially mesh-like capillary network is remodeled into a tree structure, while conserving vascular wall components and maintaining blood flow. Coalescent angiogenesis, thus, describes the remodeling of an initial, hemodynamically inefficient mesh structure, into a hierarchical tree structure that provides efficient convective transport, allowing for the rapid expansion of the vasculature with maintained blood supply and function during development.
• Stepien, T. L., & Secomb, T. W. (2022). Spreading mechanics and differentiation of astrocytes during retinal development. Journal of theoretical biology, 549, 111208.
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  The retinal vasculature supplies oxygen to the inner layers of the retina, the light-sensitive tissue in the eye. During development, formation of the retinal vasculature depends on prior establishment of a mesh of astrocytes, a type of glial cell, which guide the growth of the vascular network. Astrocytes emerge from the optic nerve head and proliferate and spread, forming a mesh-like layer over the retinal surface. The initially formed cells are termed astrocyte precursor cells (APCs), which differentiate into immature perinatal astrocytes (IPAs) during the prenatal period. A continuum model is developed to describe the proliferation, differentiation, and migration these cells. Effects of oxygen and growth factor levels on proliferation and differentiation are included. Cell migration is driven by gradients in tension in the astrocyte mesh, which varies inversely with total density. The resulting governing equations have the form of a nonlinear diffusion-like equation. The model can account for the observed radial spread over time of the astrocyte disk. Experimental observations show that the APCs form a narrow rim around the edge of this disk, with IPAs in the interior. The model predicts this behavior if the mobility of the APCs is assumed to be higher than that of the IPAs under a given tension gradient. Thus, the model shows how tension-driven cell motions can account for separation of cell types in a cell layer spreading over a substrate.
• Wright, S. H., & Secomb, T. W. (2022). Novel method for kinetic analysis applied to transport by the uniporter OCT2. American journal of physiology. Renal physiology, 323(3), F370-F387.
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  The kinetics of solute transport shed light on the roles these processes play in cellular physiology, and the absolute values of the kinetic parameters that quantitatively describe transport are increasingly used to model its impact on drug clearance. However, accurate assessment of transport kinetics is challenging. Although most carrier-mediated transport is adequately described by the Michaelis-Menten equation, its use presupposes that the rates of uptake used in the analysis of maximal rates of transport () and half-saturation constants () reflect true unidirectional rates of influx from known concentrations of substrate. Most experimental protocols estimate the initial rate of transport from net substrate accumulation determined at a single time point (typically between 0.5 and 5 min) and assume it reflects unidirectional influx. However, this approach generally results in systematic underestimates of and overestimates of ; the former primarily due to the unaccounted impact of efflux of accumulated substrate, and the latter due to the influence of unstirred water layers. Here, we describe the bases of these time-dependent effects and introduce a computational model that analyzes the time course of net substrate uptake at several concentrations to calculate and for unidirectional influx, taking into account the influence of unstirred water layers and mediated efflux. This method was then applied to calculate the kinetics of transport of 1-methyl-4-phenylpryridinium and metformin by renal organic cation transporter 2 as expressed in cultured Chinese hamster ovary cells. Here, we describe a mathematical model that uses the time course of net substrate uptake into cells from several increasing concentrations to calculate unique kinetic parameters [maximal rates of transport () and half-saturation constants ()] of the process. The method is the first to take into consideration the common complicating factors of unstirred layers and carrier-mediated efflux in the experimental determination of transport kinetics.
• Alberding, J. P., & Secomb, T. W. (2021). Simulation of angiogenesis in three dimensions: Application to cerebral cortex. PLoS computational biology, 17(6), e1009164.
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  The vasculature is a dynamic structure, growing and regressing in response to embryonic development, growth, changing physiological demands, wound healing, tumor growth and other stimuli. At the microvascular level, network geometry is not predetermined, but emerges as a result of biological responses of each vessel to the stimuli that it receives. These responses may be summarized as angiogenesis, remodeling and pruning. Previous theoretical simulations have shown how two-dimensional vascular patterns generated by these processes in the mesentery are consistent with experimental observations. During early development of the brain, a mesh-like network of vessels is formed on the surface of the cerebral cortex. This network then forms branches into the cortex, forming a three-dimensional network throughout its thickness. Here, a theoretical model is presented for this process, based on known or hypothesized vascular response mechanisms together with experimentally obtained information on the structure and hemodynamics of the mouse cerebral cortex. According to this model, essential components of the system include sensing of oxygen levels in the midrange of partial pressures and conducted responses in vessel walls that propagate information about metabolic needs of the tissue to upstream segments of the network. The model provides insights into the effects of deficits in vascular response mechanisms, and can be used to generate physiologically realistic microvascular network structures.
• Celaya-Alcala, J. T., Lee, G. V., Smith, A. F., Li, B., Sakadžić, S., Boas, D. A., & Secomb, T. W. (2021). Simulation of oxygen transport and estimation of tissue perfusion in extensive microvascular networks: Application to cerebral cortex. Journal of cerebral blood flow and metabolism : official journal of the International Society of Cerebral Blood Flow and Metabolism, 41, 656-669.
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  Advanced imaging techniques have made available extensive three-dimensional microvascular network structures. Simulation of oxygen transport by such networks requires information on blood flow rates and oxygen levels in vessels crossing boundaries of the imaged region, which is difficult to obtain experimentally. Here, a computational method is presented for estimating blood flow rates, oxygen levels, tissue perfusion and oxygen extraction, based on incomplete boundary conditions. Flow rates in all segments are estimated using a previously published method. Vessels crossing the region boundary are classified as arterioles, capillaries or venules. Oxygen levels in inflowing capillaries are assigned based on values in outflowing capillaries, and similarly for venules. Convective and diffusive oxygen transport is simulated. Contributions of each vessel to perfusion are computed in proportion to the decline in oxygen concentration along that vessel. For a vascular network in the mouse cerebral cortex, predicted tissue oxygen levels show a broad distribution, with 99% of tissue in the range of 20 to 80 mmHg under reference conditions, and steep gradients near arterioles. Perfusion and extraction estimates are consistent with experimental values. A 30% reduction in perfusion or a 30% increase in oxygen demand, relative to reference levels, is predicted to result in tissue hypoxia.
• Dominelli, P. B., Wiggins, C. C., Roy, T. K., Secomb, T. W., Curry, T. B., & Joyner, M. J. (2021). The Oxygen Cascade During Exercise in Health and Disease. Mayo Clinic proceedings, 96(4), 1017-1032.
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  The oxygen transport cascade describes the physiological steps that bring atmospheric oxygen into the body where it is delivered and consumed by metabolically active tissue. As such, the oxygen cascade is fundamental to our understanding of exercise in health and disease. Our narrative review will highlight each step of the oxygen transport cascade from inspiration of atmospheric oxygen down to mitochondrial consumption in both healthy active males and females along with clinical conditions. We will focus on how different steps interact along with principles of homeostasis, physiological redundancies, and adaptation. In particular, we highlight some of the parallels between elite athletes and clinical conditions in terms of the oxygen cascade.
• Fry, B. C., & Secomb, T. W. (2021). Distinct roles of red-blood-cell-derived and wall-derived mechanisms in metabolic regulation of blood flow. Microcirculation (New York, N.Y. : 1994), 28(5), e12690.
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  A theoretical model is used to analyze combinations of RBC-derived and wall-derived (RBC-independent) mechanisms for metabolic blood flow regulation, with regard to their oxygen transport properties.
• Johnson, D. W., Roy, T. K., & Secomb, T. W. (2021). Analysis of flow resistance in the pulmonary arterial circulation: implications for hypoxic pulmonary vasoconstriction. Journal of applied physiology (Bethesda, Md. : 1985), 131(4), 1211-1218.
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  Hypoxic pulmonary vasoconstriction (HPV) plays an essential role in distributing blood in the lung to enhance ventilation-perfusion matching and blood oxygenation. In this study, a theoretical model of the pulmonary vasculature is used to predict the effects of vasoconstriction over specified ranges of vessel diameters on pulmonary vascular resistance (PVR). The model is used to evaluate the ability of hypothesized mechanisms of HPV to account for observed levels of PVR elevation during hypoxia. The vascular structure from pulmonary arteries to capillaries is represented using scaling laws. Vessel segments are modeled as resistive elements and blood flow rates are computed from physical principles. Direct vascular responses to intravascular oxygen levels have been proposed as a mechanism of HPV. In the lung, significant changes in oxygen level occur only in vessels less than 60 μm in diameter. The model shows that observed levels of hypoxic vasoconstriction in these vessels alone cannot account for the elevation of PVR associated with HPV. However, the elevation in PVR associated with HPV can be accounted for if larger upstream vessels also constrict. These results imply that upstream signaling by conducted responses to engage constriction of arterioles plays an essential role in the elevation of PVR during HPV. A theoretical model of the pulmonary vasculature is used to predict the effects of vasoconstriction over specified ranges of vessel diameters on pulmonary vascular resistance (PVR). The model shows that observed levels of hypoxic vasoconstriction in terminal vessels cannot account for the elevation of PVR associated with hypoxic pulmonary vasoconstriction (HPV). Upstream signaling by conducted responses to engage constriction of arterioles, therefore, plays an essential role in the elevation of PVR during HPV.
• Nitzsche, B., Rong, W. W., Goede, A., Hoffmann, B., Scarpa, F., Kuebler, W. M., Secomb, T. W., & Pries, A. R. (2021). Coalescent angiogenesis-evidence for a novel concept of vascular network maturation. Angiogenesis.
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  Angiogenesis describes the formation of new blood vessels from pre-existing vascular structures. While the most studied mode of angiogenesis is vascular sprouting, specific conditions or organs favor intussusception, i.e., the division or splitting of an existing vessel, as preferential mode of new vessel formation. In the present study, sustained (33-h) intravital microscopy of the vasculature in the chick chorioallantoic membrane (CAM) led to the hypothesis of a novel non-sprouting mode for vessel generation, which we termed "coalescent angiogenesis." In this process, preferential flow pathways evolve from isotropic capillary meshes enclosing tissue islands. These preferential flow pathways progressively enlarge by coalescence of capillaries and elimination of internal tissue pillars, in a process that is the reverse of intussusception. Concomitantly, less perfused segments regress. In this way, an initially mesh-like capillary network is remodeled into a tree structure, while conserving vascular wall components and maintaining blood flow. Coalescent angiogenesis, thus, describes the remodeling of an initial, hemodynamically inefficient mesh structure, into a hierarchical tree structure that provides efficient convective transport, allowing for the rapid expansion of the vasculature with maintained blood supply and function during development.
• Maibier, M., Bintig, W., Goede, A., Hoepfner, M., Kuebler, W. M., Secomb, T. W., Nitzsche, B., & Pries, A. R. (2020). Gap junctions regulate vessel diameter in chick chorioallantoic membrane vasculature by both tone-dependent and structural mechanisms. MICROCIRCULATION, 27(1).
• Maibier, M., Bintig, W., Goede, A., Höpfner, M., Kuebler, W. M., Secomb, T. W., Nitzsche, B., & Pries, A. R. (2020). Gap junctions regulate vessel diameter in chick chorioallantoic membrane vasculature by both tone-dependent and structural mechanisms. Microcirculation (New York, N.Y. : 1994), 27(1), e12590.
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  In this study, we examined the impact of gap junction blockade on chick chorioallantoic membrane microvessels.
• Roy, T. K., & Secomb, T. W. (2020). Effects of impaired microvascular flow regulation on metabolism-perfusion matching and organ function. Microcirculation (New York, N.Y. : 1994), e12673.
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  Impaired tissue oxygen delivery is a major cause of organ damage and failure in critically ill patients, which can occur even when systemic parameters, including cardiac output and arterial hemoglobin saturation, are close to normal. This review addresses oxygen transport mechanisms at the microcirculatory scale, and how hypoxia may occur in spite of adequate convective oxygen supply. The structure of the microcirculation is intrinsically heterogeneous, with wide variations in vessel diameters and flow pathway lengths, and consequently also in blood flow rates and oxygen levels. The dynamic processes of structural adaptation and flow regulation continually adjust microvessel diameters to compensate for heterogeneity, redistributing flow according to metabolic needs to ensure adequate tissue oxygenation. A key role in flow regulation is played by conducted responses, which are generated and propagated by endothelial cells and signal upstream arterioles to dilate in response to local hypoxia. Several pathophysiological conditions can impair local flow regulation, causing hypoxia and tissue damage leading to organ failure. Therapeutic measures targeted to systemic parameters may not address or may even worsen tissue oxygenation at the microvascular level. Restoration of tissue oxygenation in critically ill patients may depend on restoration of endothelial cell function, including conducted responses.
• Secomb, T. W., & Roy, T. K. (2020). Characterizing the Effect of the Hypoxic Vasoconstriction Response Curve on Blood Flow Distribution and Oxygenation in the Lung. The FASEB Journal, 34(S1), 1-1. doi:10.1096/fasebj.2020.34.s1.04215
• Secomb, T. W., Bullock, K. V., Boas, D. A., & Sakadžić, S. (2020). The mass transfer coefficient for oxygen transport from blood to tissue in cerebral cortex. Journal of cerebral blood flow and metabolism : official journal of the International Society of Cerebral Blood Flow and Metabolism, 40(8), 1634-1646.
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  The functioning of cerebral cortex depends on adequate tissue oxygenation. MRI-based techniques allow estimation of blood oxygen levels, tissue perfusion, and oxygen consumption rate (CMRO), but do not directly measure partial pressure of oxygen (PO) in tissue. To address the estimation of tissue PO, the oxygen mass transfer coefficient (KTO) is here defined as the CMRO divided by the difference in spatially averaged PO between blood and tissue, and is estimated by analyzing Krogh-cylinder type models. Resistance to radial diffusion of oxygen from microvessels to tissue is distributed within vessels and in the extravascular tissue. The value of KTO is shown to depend strongly on vascular length density and also on microvessel tube hematocrits and diameters, but to be insensitive to blood flow rate and to transient changes in flow or oxygen consumption. Estimated values of KTO are higher than implied by previous studies, implying smaller declines in PO from blood to tissue. Average tissue PO can be estimated from MRI-based measurements as average blood PO minus the product of KTO and CMRO. For oxygen consumption rates and vascular densities typical of mouse cortex, the predicted difference between average blood and tissue PO is about 10 mmHg.
• Sweeney, P. W., Smith, A. F., Shipley, R. J., Secomb, T. W., & Pries, A. R. (2020). A hybrid discrete-continuum approach for modelling microcirculatory blood flow.. Mathematical medicine and biology : a journal of the IMA, 37(1), 40-57. doi:10.1093/imammb/dqz006
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  In recent years, biological imaging techniques have advanced significantly and it is now possible to digitally reconstruct microvascular network structures in detail, identifying the smallest capillaries at sub-micron resolution and generating large 3D structural data sets of size >106 vessel segments. However, this relies on ex vivo imaging; corresponding in vivo measures of microvascular structure and flow are limited to larger branching vessels and are not achievable in three dimensions for the smallest vessels. This suggests the use of computational modelling to combine in vivo measures of branching vessel architecture and flows with ex vivo data on complete microvascular structures to predict effective flow and pressures distributions. In this paper, a hybrid discrete-continuum model to predict microcirculatory blood flow based on structural information is developed and compared with existing models for flow and pressure in individual vessels. A continuum-based Darcy model for transport in the capillary bed is coupled via point sources of flux to flows in individual arteriolar vessels, which are described explicitly using Poiseuille's law. The venular drainage is represented as a spatially uniform flow sink. The resulting discrete-continuum framework is parameterized using structural data from the capillary network and compared with a fully discrete flow and pressure solution in three networks derived from observations of the rat mesentery. The discrete-continuum approach is feasible and effective, providing a promising tool for extracting functional transport properties in situations where vascular branching structures are well defined.
• Şencan, ., Esipova, T., Kılıç, K., Li, B., Desjardins, M., Yaseen, M. A., Wang, H., Porter, J. E., Kura, S., Fu, B., Secomb, T. W., Boas, D. A., Vinogradov, S. A., Devor, A., & Sakadžić, S. (2020). Optical measurement of microvascular oxygenation and blood flow responses in awake mouse cortex during functional activation. Journal of cerebral blood flow and metabolism : official journal of the International Society of Cerebral Blood Flow and Metabolism, 271678X20928011.
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  The cerebral cortex has a number of conserved morphological and functional characteristics across brain regions and species. Among them, the laminar differences in microvascular density and mitochondrial cytochrome c oxidase staining suggest potential laminar variability in the baseline O metabolism and/or laminar variability in both O demand and hemodynamic response. Here, we investigate the laminar profile of stimulus-induced intravascular partial pressure of O (pO2) transients to stimulus-induced neuronal activation in fully awake mice using two-photon phosphorescence lifetime microscopy. Our results demonstrate that stimulus-induced changes in intravascular pO are conserved across cortical layers I-IV, suggesting a tightly controlled neurovascular response to provide adequate O supply across cortical depth. In addition, we observed a larger change in venular O saturation (ΔsO) compared to arterioles, a gradual increase in venular ΔsO response towards the cortical surface, and absence of the intravascular "initial dip" previously reported under anesthesia. This study paves the way for quantification of layer-specific cerebral O metabolic responses, facilitating investigation of brain energetics in health and disease and informed interpretation of laminar blood oxygen level dependent functional magnetic resonance imaging signals.
• Dewhirst, M. W., Mowery, Y. M., Mitchell, J. B., Cherukuri, M. K., & Secomb, T. W. (2019). Rationale for hypoxia assessment and amelioration for precision therapy and immunotherapy studies. JOURNAL OF CLINICAL INVESTIGATION, 129(2), 489-491.
• Hong, B. D., Moulton, M. J., & Secomb, T. W. (2019). Modeling left ventricular dynamics with characteristic deformation modes. BIOMECHANICS AND MODELING IN MECHANOBIOLOGY, 18(6), 1683-1696.
• Roy, T. K., & Secomb, T. W. (2019). Effects of pulmonary flow heterogeneity on oxygen transport parameters in exercise. RESPIRATORY PHYSIOLOGY & NEUROBIOLOGY, 261, 75-79.
• Sandoval, P. J., Morales, M., Secomb, T. W., & Wright, S. H. (2019). Kinetic basis of metformin-MPP interactions with organic cation transporter OCT2. AMERICAN JOURNAL OF PHYSIOLOGY-RENAL PHYSIOLOGY, 317(3), E720-E734.
• Secomb, T. W., Bullock, K. V., Boas, D. A., & Sakadzic, S. (2019). The mass transfer coefficient for oxygen transport from blood to tissue in cerebral cortex. JOURNAL OF CEREBRAL BLOOD FLOW AND METABOLISM.
• Secomb, T. W., Burton, J. K., & Bottino, D. (2019). A Systems Pharmacology Model for Drug Delivery to Solid Tumors by Antibody-Drug Conjugates: Implications for Bystander Effects.. The AAPS journal, 22(1), 12. doi:10.1208/s12248-019-0390-2
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  Antibody-drug conjugates (ADCs) are cancer drugs composed of a humanized antibody linked to a cytotoxic payload, allowing preferential release of payload in cancer cells expressing the antibody-targeted antigen. Here, a systems pharmacology model is used to simulate ADC transport from blood to tumor tissue and ADC uptake by tumor cells. The model includes effects of spatial gradients in drug concentration in a three-dimensional network of tumor blood vessels with realistic geometry and accounts for diffusion of ADC in the tumor extracellular space, binding to antigen, internalization, intracellular processing, and payload efflux from cells. Cells that process an internalized ADC-antigen complex may release payload that can be taken up by other "bystander" cells. Such bystander effects are included in the model. The model is used to simulate conditions in previous experiments, showing good agreement with experimental results. Simulations are used to analyze the relationship of bystander effects to payload properties and single-dose administrations. The model indicates that exposure of payload to cells distant from vessels is sensitive to the free payload diffusivity in the extracellular space. When antigen expression is heterogeneous, the model indicates that the amount of payload accumulating in non-antigen-expressing cells increases linearly with dose but depends only weakly on the percentage of antigen-expressing cells. The model provides an integrated mechanistic framework for understanding the effects of spatial gradients on drug distribution using ADCs and for designing ADCs to achieve more effective payload distribution in solid tumors, thereby increasing the therapeutic index of the ADC.
• Shepherd, J., Dominelli, P. B., Roy, T. K., Secomb, T. W., Hoyer, J. D., Oliveira, J. L., & Joyner, M. J. (2019). Modelling the relationships between haemoglobin oxygen affinity and the oxygen cascade in humans. JOURNAL OF PHYSIOLOGY-LONDON, 597(16), 4193-4202.
• Lucker, A., Secomb, T. W., Barrett, M., Weber, B., & Jenny, P. (2018). The Relation Between Capillary Transit Times and Hemoglobin Saturation Heterogeneity. Part 2: Capillary Networks. FRONTIERS IN PHYSIOLOGY, 9.
• Lucker, A., Secomb, T. W., Weber, B., & Jenny, P. (2018). The Relation Between Capillary Transit Times and Hemoglobin Saturation Heterogeneity. Part 1: Theoretical Models. FRONTIERS IN PHYSIOLOGY, 9.
• Rasmussen, P. M., Secomb, T. W., & Pries, A. R. (2018). Modeling the hematocrit distribution in microcirculatory networks: A quantitative evaluation of a phase separation model. MICROCIRCULATION, 25(3).
• Williams, K. S., Secomb, T. W., & El-Kareh, A. W. (2018). Additive Damage Models for Cellular Pharmacodynamics of Radiation-Chemotherapy Combinations. BULLETIN OF MATHEMATICAL BIOLOGY, 80(5), 1236-1258.
• Dewhirst, M. W., & Secomb, T. W. (2017). Transport of drugs from blood vessels to tumour tissue. NATURE REVIEWS CANCER, 17(12), 738-750.
• Lucker, A., Secomb, T. W., Weber, B., & Jenny, P. (2017). The relative influence of hematocrit and red blood cell velocity on oxygen transport from capillaries to tissue. MICROCIRCULATION, 24(3).
• Moulton, M. J., Hong, B. D., & Secomb, T. W. (2017). Simulation of Left Ventricular Dynamics Using a Low-Order Mathematical Model. CARDIOVASCULAR ENGINEERING AND TECHNOLOGY, 8(4), 480-494.
• Rasmussen, P. M., Smith, A. F., Sakadzic, S., Boas, D. A., Pries, A. R., Secomb, T. W., & Ostergaard, L. (2017). Model-based inference from microvascular measurements: Combining experimental measurements and model predictions using a Bayesian probabilistic approach. MICROCIRCULATION, 24(4).
• Reglin, B., Secomb, T. W., & Pries, A. R. (2017). Structural Control of Microvessel Diameters: Origins of Metabolic Signals. FRONTIERS IN PHYSIOLOGY, 8.
• Secomb, T. W. (2017). Blood Flow in the Microcirculation. ANNUAL REVIEW OF FLUID MECHANICS, VOL 49, 49, 443-461.
• Xiang, W., Reglin, B., Nitzsche, B., Maibier, M., Rong, W. W., Hoffmann, B., Ruggeri, A., Guimaraes, P., Secomb, T. W., & Pries, A. R. (2017). Dynamic remodeling of arteriolar collaterals after acute occlusion in chick chorioallantoic membrane. MICROCIRCULATION, 24(4).
• Xiang, W., Reglin, B., Nitzsche, B., Maibier, M., Rong, W., Hoffmann, B., Ruggeri, A., Guimarães, P., Secomb, T. W., & Pries, A. R. (2017). Dynamic remodeling of arteriolar collaterals after acute occlusion in chick chorioallantoic membrane. Microcirculation (New York, N.Y. : 1994).
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  After arteriolar occlusion, collaterals enlarge and initially elevated wall shear stress (WSS) normalizes. While most previous studies focused on endpoints of such adaptive changes in larger collaterals the present investigation aimed to continuously determine the relation between WSS and diameter in microvascular collaterals during adaptive reactions METHODS: In Hamburger-Hamilton stage 40 chick chorioallantoic membranes (CAMs), junction points between arteriolar segments were identified and the third upstream segment on one side was occluded. Intravital microscopy recordings were taken for 24h post-occlusion. Segment diameter and blood velocity were measured, WSS and capillary density were calculated RESULTS: After occlusion, vascular diameters exhibited an immediate decrease, then increased with a time constant of 2.5 ± 0.8h and reached a plateau of up to 60% above baseline after about 7h. Vascular tone showed no significant change. WSS exhibited an immediate increase post-occlusion and linearly returned to baseline after about 12h. Local WSS change and diameter change rate showed similar patterns during the initial but not the later phase of post-occlusive adaptation CONCLUSIONS: CAM collaterals undergo fast structural remodeling within 24h post-occlusion. This remodeling might be driven by local WSS and by other regulators within the vascular network. This article is protected by copyright. All rights reserved.
• Gagnon, L., Smith, A. F., Boas, D. A., Devor, A., Secomb, T. W., & Sakadžić, S. (2016). Modeling of Cerebral Oxygen Transport Based on In vivo Microscopic Imaging of Microvascular Network Structure, Blood Flow, and Oxygenation. Frontiers in computational neuroscience, 10, 82.
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  Oxygen is delivered to brain tissue by a dense network of microvessels, which actively control cerebral blood flow (CBF) through vasodilation and contraction in response to changing levels of neural activity. Understanding these network-level processes is immediately relevant for (1) interpretation of functional Magnetic Resonance Imaging (fMRI) signals, and (2) investigation of neurological diseases in which a deterioration of neurovascular and neuro-metabolic physiology contributes to motor and cognitive decline. Experimental data on the structure, flow and oxygen levels of microvascular networks are needed, together with theoretical methods to integrate this information and predict physiologically relevant properties that are not directly measurable. Recent progress in optical imaging technologies for high-resolution in vivo measurement of the cerebral microvascular architecture, blood flow, and oxygenation enables construction of detailed computational models of cerebral hemodynamics and oxygen transport based on realistic three-dimensional microvascular networks. In this article, we review state-of-the-art optical microscopy technologies for quantitative in vivo imaging of cerebral microvascular structure, blood flow and oxygenation, and theoretical methods that utilize such data to generate spatially resolved models for blood flow and oxygen transport. These "bottom-up" models are essential for the understanding of the processes governing brain oxygenation in normal and disease states and for eventual translation of the lessons learned from animal studies to humans.
• Lücker, A., Secomb, T. W., Weber, B., & Jenny, P. (2016). The relative influence of hematocrit and red blood cell velocity on oxygen transport from capillaries to tissue. Microcirculation (New York, N.Y. : 1994).
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  Oxygen transport to parenchymal cells occurs mainly at the microvascular level, and depends on convective red blood cell (RBC) flux, which is proportional in an individual capillary to the product of capillary hematocrit and red blood cell velocity. This study investigates the relative influence of these two factors on tissue oxygen partial pressure (PO2 ).
• Maibier, M., Reglin, B., Nitzsche, B., Xiang, W., Rong, W. W., Hoffmann, B., Djonov, V., Secomb, T. W., & Pries, A. R. (2016). Structure and hemodynamics of vascular networks in the chorioallantoic membrane of the chicken. American journal of physiology. Heart and circulatory physiology, 311(4), H913-H926.
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  The chick chorioallantoic membrane (CAM) is extensively used as an in vivo model. Here, structure and hemodynamics of CAM vessel trees were analyzed and compared with predictions of Murray's law. CAM microvascular networks of Hamburger-Hamilton stage 40 chick embryos were scanned by videomicroscopy. Three networks with ∼3,800, 580, and 480 segments were digitally reconstructed, neglecting the capillary mesh. Vessel diameters (D) and segment lengths were measured, and generation numbers and junctional exponents at bifurcations were derived. In selected vessels, flow velocities (v) and hematocrit were measured. Hemodynamic simulations, incorporating the branching of capillaries from preterminal vessels, were used to estimate v, volume flow, shear stress (τ), and pressure for all segments of the largest network. For individual arteriovenous flow pathways, terminal arterial and venous generation numbers are negatively correlated, leading to low variability of total topological and morphological pathway lengths. Arteriolar velocity is proportional to diameter (v∝D(1.03) measured, v∝D(0.93) modeling), giving nearly uniform τ levels (τ∝D(0.05)). Venular trees exhibit slightly higher exponents (v∝D(1.3), τ∝D(0.38)). Junctional exponents at divergent and convergent bifurcations were 2.05 ± 1.13 and 1.97 ± 0.95 (mean ± SD) in contrast to the value 3 predicted by Murray's law. In accordance with Murray's law, τ levels are (nearly) maintained in CAM arterial (venular) trees, suggesting vascular adaptation to shear stress. Arterial and venous trees show an interdigitating arrangement providing homogeneous flow pathway properties and have preterminal capillary branches. These properties may facilitate efficient oxygen exchange in the CAM during rapid embryonic growth.
• Rasmussen, P. M., Smith, A. F., Sakadžić, S., Boas, D. A., Pries, A. R., Secomb, T. W., & Østergaard, L. (2016). Model based inference from microvascular measurements: Combining experimental measurements and model predictions using a Bayesian probabilistic approach. Microcirculation (New York, N.Y. : 1994).
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  In vivo imaging of the microcirculation and network-oriented modeling have emerged as powerful means of studying microvascular function and understanding its physiological significance. Network-oriented modeling may provide the means of summarizing vast amounts of data produced by high-throughput imaging techniques in terms of key, physiological indices. To estimate such indices with sufficient certainty, however, network-oriented analysis must be robust to the inevitable presence of uncertainty due to measurement errors as well as model errors METHODS: We propose the Bayesian probabilistic data analysis framework as a means of integrating experimental measurements and network model simulations into a combined and statistically coherent analysis. The framework naturally handles noisy measurements and provides posterior distributions of model parameters as well as physiological indices associated with uncertainty RESULTS: We applied the analysis framework to experimental data from three rat mesentery networks and one mouse brain cortex network. We inferred distributions for more than five hundred unknown pressure and hematocrit boundary conditions. Model predictions were consistent with previous analyses, and remained robust when measurements were omitted from model calibration CONCLUSION: Our Bayesian probabilistic approach may be suitable for optimizing data acquisition and for analyzing and reporting large datasets acquired as part of microvascular imaging studies. This article is protected by copyright. All rights reserved.
• Secomb, T. W. (2016). A Green's function method for simulation of time-dependent solute transport and reaction in realistic microvascular geometries. Mathematical medicine and biology : a journal of the IMA, 33(4), 475-494.
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  A novel theoretical method is presented for simulating the spatially resolved convective and diffusive transport of reacting solutes between microvascular networks and the surrounding tissues. The method allows for efficient computational solution of problems involving convection and non-linear binding of solutes in blood flowing through microvascular networks with realistic 3D geometries, coupled with transvascular exchange and diffusion and reaction in the surrounding tissue space. The method is based on a Green's function approach, in which the solute concentration distribution in the tissue is expressed as a sum of fields generated by time-varying distributions of discrete sources and sinks. As an example of the application of the method, the washout of an inert diffusible tracer substance from a tissue region perfused by a network of microvessels is simulated, showing its dependence on the solute's transvascular permeability and tissue diffusivity. Exponential decay of the washout concentration is predicted, with rate constants that are about 10-30% lower than the rate constants for a tissue cylinder model with the same vessel length, vessel surface area and blood flow rate per tissue volume.
• Secomb, T. W. (2016). Hemodynamics. Comprehensive Physiology, 6(2), 975-1003.
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  A review is presented of the physical principles governing the distribution of blood flow and blood pressure in the vascular system. The main factors involved are the pulsatile driving pressure generated by the heart, the flow characteristics of blood, and the geometric structure and mechanical properties of the vessels. The relationship between driving pressure and flow in a given vessel can be understood by considering the viscous and inertial forces acting on the blood. Depending on the vessel diameter and other physical parameters, a wide variety of flow phenomena can occur. In large arteries, the propagation of the pressure pulse depends on the elastic properties of the artery walls. In the microcirculation, the fact that blood is a suspension of cells strongly influences its flow properties and leads to a nonuniform distribution of hematocrit among microvessels. The forces acting on vessel walls include shear stress resulting from blood flow and circumferential stress resulting from blood pressure. Biological responses to these forces are important in the control of blood flow and the structural remodeling of vessels, and also play a role in major disease processes including hypertension and atherosclerosis. Consideration of hemodynamics is essential for a comprehensive understanding of the functioning of the circulatory system.
• Secomb, T. W., & Pries, A. R. (2016). Microvascular Plasticity: Angiogenesis in Health and Disease--Preface. Microcirculation (New York, N.Y. : 1994), 23(2), 93-4.
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  This Special Topic Issue is concerned with the mechanisms that determine the structure of microvascular networks. The vast number of vessels and the highly plastic character of the microcirculation give evidence that microvascular network structures emerge as a result of responses of individual vessels and cells to the local stimuli that they experience, through a combination of angiogenesis, remodeling and pruning. The articles in this issue of Microcirculation address a range of cellular and molecular mechanisms involved in these processes.
• Smith, A. F., Nitzsche, B., Maibier, M., Pries, A. R., & Secomb, T. W. (2016). Microvascular hemodynamics in the chick chorioallantoic membrane. Microcirculation (New York, N.Y. : 1994), 23(7), 512-522.
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  The microvasculature of the CAM in the developing chick embryo is characterized by interdigitating arteriolar and venular trees, connected at multiple points along their lengths to a mesh-like capillary plexus. Theoretical modeling techniques were employed to investigate the resulting hemodynamic characteristics of the CAM.
• Smith, S. P., Secomb, T. W., Hong, B. D., & Moulton, M. J. (2016). Time-Dependent Regional Myocardial Strains in Patients with Heart Failure with a Preserved Ejection Fraction. BioMed research international, 2016, 8957307.
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  To better understand the etiology of HFpEF in a controlled human population, regional time-varying strains were computed using echocardiography speckle tracking in patients with heart failure with a preserved ejection fraction and normal subjects.
• Eberson, L. S., Sanchez, P. A., Majeed, B. A., Tawinwung, S., Secomb, T. W., & Larson, D. F. (2015). Effect of Lysyl Oxidase Inhibition on Angiotensin II-Induced Arterial Hypertension, Remodeling, and Stiffness. PLOS ONE, 10(4).
• Hariprasad, D. S., & Secomb, T. W. (2015). Prediction of noninertial focusing of red blood cells in Poiseuille flow. PHYSICAL REVIEW E, 92(3).
• Majeed, B. A., Eberson, L. S., Tawinwung, S., Larmonier, N., Secomb, T. W., & Larson, D. F. (2015). Functional aortic stiffness: role of CD4(+) T lymphocytes. FRONTIERS IN PHYSIOLOGY, 6.
• Secomb, T. W. (2015). Krogh-Cylinder and Infinite-Domain Models for Washout of an Inert Diffusible Solute from Tissue. MICROCIRCULATION, 22(1), 91-98.
• Shibayama, T., Morales, M., Zhang, X., Martinez-Guerrero, L. J., Berteloot, A., Secomb, T. W., & Wright, S. H. (2015). Unstirred Water Layers and the Kinetics of Organic Cation Transport. PHARMACEUTICAL RESEARCH, 32(9), 2937-2949.
• Smith, A. F., Secomb, T. W., Pries, A. R., Smith, N. P., & Shipley, R. J. (2015). Structure-Based Algorithms for Microvessel Classification. MICROCIRCULATION, 22(2), 99-108.
• Buerk, D. G., Hirai, D. M., Roseguini, B. T., Silva, B. M., Vagula, M. C., Roy, T. K., & Secomb, T. W. (2014). Commentaries on viewpoint: A paradigm shift for local blood flow regulation. Journal of applied physiology (Bethesda, Md. : 1985), 116(6), 706-7.
• El-kareh, A. W., Secomb, T. W., Secomb, T. W., & El-kareh, A. W. (2014). Abstract 5333: A mathematical model for the kinetics of metastasis formation based on SEER data. Cancer Research, 74, 5333-5333. doi:10.1158/1538-7445.am2014-5333
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  Proceedings: AACR Annual Meeting 2014; April 5-9, 2014; San Diego, CA Metastasis is responsible for most cancer mortality, yet many questions remain about how best to describe the kinetics of metastasis formation mathematically. Because the initial step of metastasis is the detachment of cells from a tumor mass, the rate of formation of new metastases is assumed to depend on the size of the original tumor mass. A power-law dependence, rather than linear proportionality, has been assumed, since access to blood vessels and lymphatics becomes more limited as tumor mass increases. Under this assumption, the cumulative distribution of fraction of patients not showing detectable metastases at diagnosis should decrease monotonically as tumor size at detection increases. Using data from the NCI's SEER (Surveillance, Epidemiology, and End Results) 1973-2010 database, it is shown here that this relation in fact shows non-monotonic behavior at larger detection sizes. A mathematical model for tumor growth and metastasis kinetics is developed to show that this implies that proliferation rates and metastatic rates are not independently distributed across the patient population, but rather, show some correlation. The model is further applied to deduce quantitative information about the size-dependence of the rate of metastasis. Kaplan-Meier curves stratified by tumor size, also computed from SEER 1973-2010 data, provide an additional data source for determining model parameters. It is found that accounting for age dependence is necessary for fitting these curves. Two possible age-related effects: a lower lethal tumor burden threshold; and an increased coefficient of metastatic rate due to declining immunity; are considered in the model, and a comparison is made of their ability to describe the observed age-dependence. In sum, the model shows how SEER data can be used to deduce information about the kinetics of metastasis formation, and is expected to have application in optimization of the treatment of metastatic tumors as well as screening schedules. Citation Format: Ardith W. El-Kareh, Timothy W. Secomb. A mathematical model for the kinetics of metastasis formation based on SEER data. [abstract]. In: Proceedings of the 105th Annual Meeting of the American Association for Cancer Research; 2014 Apr 5-9; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2014;74(19 Suppl):Abstract nr 5333. doi:10.1158/1538-7445.AM2014-5333
• El-kareh, A. W., Secomb, T. W., Williams, K. S., Secomb, T. W., & El-kareh, A. W. (2014). Abstract 5336: Mathematical models for cellular response to radiochemotherapy. Cancer Research, 74, 5336-5336. doi:10.1158/1538-7445.am2014-5336
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  Proceedings: AACR Annual Meeting 2014; April 5-9, 2014; San Diego, CA To optimize radiochemotherapy, several clinically adjustable factors must be considered: sequence order (drug administered before, during, or after radiation exposure); time interval between the agents; dose of each agent; and fractionation. Because the combined effects of all these factors are complex, mathematical modeling can provide a useful framework for discerning the role of each individual factor and can be applied to predict response to future experimental protocols. To date, there has been scant effort devoted to mathematical models for cellular response to radiochemotherapy. One widely-used model is based on the assumption that the two modalities act completely independently; the total cell survival fraction is the product of survival fractions for each agent acting alone. However, an examination of a number of dose-response data sets for paclitaxel followed by radiation exposure shows that the independent cell kill model consistently underestimates cell kill. The additive damage model, which is based on the assumption that cell kill depends on a quantity called cellular damage, is constructed as the sum of terms from each agent and is shown to better describe some data sets for ascorbate and paclitaxel. Cellular dose-response data sets for radiochemotherapy have shown complex interactions between therapies, with strong dose-dependent antagonism in some cases, and significant, non-monotonic dependence on the time interval between the modalities in others. We present generalizations of the additive damage model, which include either dose-dependent cell-cycle blocking or transport kinetics, and show that they can describe some of these experimentally-observed complex behaviors. In conclusion, the generalized additive damage model is a promising tool to describe cellular pharmacodynamics of radiochemotherapy and is expected to have application to the optimization of dosage and schedule. Citation Format: Katherine S. Williams, Ardith W. El-Kareh, Timothy W. Secomb. Mathematical models for cellular response to radiochemotherapy. [abstract]. In: Proceedings of the 105th Annual Meeting of the American Association for Cancer Research; 2014 Apr 5-9; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2014;74(19 Suppl):Abstract nr 5336. doi:10.1158/1538-7445.AM2014-5336
• Hariprasad, D. S., & Secomb, T. W. (2014). Two-dimensional simulation of red blood cell motion near a wall under a lateral force. Physical review. E, Statistical, nonlinear, and soft matter physics, 90(5-1), 053014.
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  The motion of a red blood cell suspended in a linear shear flow adjacent to a fixed boundary subject to an applied lateral force directed toward the boundary is simulated. A two-dimensional model is used that represents the viscous and elastic properties of normal red blood cells. Shear rates in the range of 100 to 600 s^{-1} are considered, and the suspending medium viscosity is 1 cP. In the absence of a lateral force, the cell executes a tumbling motion. With increasing lateral force, a transition from tumbling to tank-treading is predicted. The minimum force required to ensure tank-treading increases nonlinearly with the shear rate. Transient swinging motions occur when the force is slightly larger than the transition value. The applied lateral force is balanced by a hydrodynamic lift force resulting from the positive orientation of the long axis of the cell with respect to the wall. In the case of cyclic tumbling motions, the orientation angle takes positive values through most of the cycle, resulting in lift generation. These results are used to predict the motion of a cell close to the outer edge of the cell-rich core region that is generated when blood flows in a narrow tube. In this case, the lateral force is generated by shear-induced dispersion, resulting from cell-cell interactions in a region with a concentration gradient. This force is estimated using previous data on shear-induced dispersion. The cell is predicted to execute tank-treading motions at normal physiological hematocrit levels, with the possibility of tumbling at lower hematocrit levels.
• Jones, L. B., Secomb, T. W., Dewhirst, M. W., & El-Kareh, A. W. (2014). The additive damage model: a mathematical model for cellular responses to drug combinations. Journal of theoretical biology, 357, 10-20.
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  Mathematical models to describe dose-dependent cellular responses to drug combinations are an essential component of computational simulations for predicting therapeutic responses. Here, a new model, the additive damage model, is introduced and tested in cases where varying concentrations of two drugs are applied with a fixed exposure schedule. In the model, cell survival is determined by whether cellular damage, which depends on the concentrations of the drugs, exceeds a lethal threshold, which varies randomly in the cell population with a prescribed statistical distribution. Cellular damage is assumed to be additive, and is expressed as a sum of separate terms for each drug. Each term has a saturable dependence on drug concentration. The model has appropriate behavior over the entire range of drug concentrations, and is predictive, given single-agent dose-response data for each drug. The proposed model is compared with several other models, by testing their ability to fit 24 data sets for platinum-taxane combinations and 21 data sets for various other combinations. The Akaike Information Criterion is used to assess goodness of fit, taking into account the number of unknown parameters in each model. Overall, the additive damage model provides a better fit to the data sets than any previous model. The proposed model provides a basis for computational simulations of therapeutic responses. It predicts responses to drug combinations based on data for each drug acting as a single agent, and can be used as an improved null reference model for assessing synergy in the action of drug combinations.
• Majeed, B., Tawinwung, S., Eberson, L. S., Secomb, T. W., Larmonier, N., & Larson, D. F. (2014). Interleukin-2/Anti-Interleukin-2 Immune Complex Expands Regulatory T Cells and Reduces Angiotensin II-Induced Aortic Stiffening. International journal of hypertension, 2014, 126365.
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  Adaptive immune function is implicated in the pathogenesis of vascular disease. Inhibition of T-lymphocyte function has been shown to reduce hypertension, target-organ damage, and vascular stiffness. To study the role of immune inhibitory cells, CD4(+)CD25(+)Foxp3(+) regulatory T cells (Tregs), on vascular stiffness, we stimulated the proliferation of Treg lymphocytes in vivo using a novel cytokine immune complex of Interleukin-2 (IL-2) and anti-IL-2 monoclonal antibody clone JES6-1 (mAbCD25). Three-month-old male C57BL/6J mice were treated with IL-2/mAbCD25 concomitantly with continuous infusion of angiotensin type 1 receptor agonist, [Val(5)]angiotensin II. Our results indicate that the IL-2/mAbCD25 complex effectively induced Treg phenotype expansion by 5-fold in the spleens with minimal effects on total CD4(+) and CD8(+) T-lymphocyte numbers. The IL-2/mAbCD25 complex inhibited angiotensin II-mediated aortic collagen remodeling and the resulting stiffening, analyzed with in vivo pulse wave velocity and effective Young's modulus. Furthermore, the IL-2/mAbCD25 complex suppressed angiotensin II-mediated Th17 responses in the lymphoid organs and reduced gene expression of IL-17 as well as T cell and macrophage infiltrates in the aortic tissue. This study provides data that support the protective roles of Tregs in vascular stiffening and highlights the use of the IL-2/mAbCD25 complex as a new potential therapy in angiotensin II-related vascular diseases.
• Marki, A., Ermilov, E., Zakrzewicz, A., Koller, A., Secomb, T. W., & Pries, A. R. (2014). Tracking of fluorescence nanoparticles with nanometre resolution in a biological system: assessing local viscosity and microrheology. Biomechanics and modeling in mechanobiology, 13(2), 275-88.
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  The aim of the study was to establish a user-friendly approach for single fluorescence particle 3D localization and tracking with nanometre precision in a standard fluorescence microscope using a point spread function (PSF) approach, and to evaluate validity and precision for different analysis methods and optical conditions with particular application to microcirculatory flow dynamics and cell biology. Images of fluorescent particles were obtained with a standard fluorescence microscope equipped with a piezo positioner for the objective. Whole pattern (WP) comparison with a PSF recorded for the specific set-up and measurement of the outermost ring radius (ORR) were used for analysis. Images of fluorescent particles were recorded over a large range (about 7μm) of vertical positions, with and without distortion by overlapping particles as well as in the presence of cultured endothelial cells. For a vertical range of 6.5μm the standard deviation (SD) from the predicted value, indicating validity, was 9.3/8.7 nm (WP/ORR) in the vertical and 8.2/11.7 nm in the horizontal direction. The precision, determined by repeated measurements, was 5.1/3.8 nm in the vertical and 2.9/3.7 nm in the horizontal direction. WP was more robust with respect to underexposure or overlapping images. On the surface of cultured endothelial cells, a layer with 2.5 times increased viscosity and a thickness of about 0.8μm was detected. With a validity in the range of 10 nm and a precision down to about 3-5 nm obtained by standard fluorescent microscopy, the PSF approach offers a valuable tool for a variety of experimental investigations of particle localizations, including the assessment of endothelial cell microenvironment.
• Pries, A. R., & Secomb, T. W. (2014). Making microvascular networks work: angiogenesis, remodeling, and pruning. Physiology (Bethesda, Md.), 29(6), 446-55.
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  The adequate and efficient functioning of the microcirculation requires not only numerous vessels providing a large surface area for transport but also a structure that provides short diffusion distances from capillaries to tissue and efficient distribution of convective blood flow. Theoretical models show how a combination of angiogenesis, remodeling, and pruning in response to hemodynamic and metabolic stimuli, termed "angioadaptation," generates well organized, functional networks.
• Roy, T. K., & Secomb, T. W. (2014). Functional sympatholysis and sympathetic escape in a theoretical model for blood flow regulation. Frontiers in physiology, 5, 192.
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  A mathematical simulation of flow regulation in vascular networks is used to investigate the interaction between arteriolar vasoconstriction due to sympathetic nerve activity (SNA) and vasodilation due to increased oxygen demand. A network with 13 vessel segments in series is used, each segment representing a different size range of arterioles or venules. The network includes five actively regulating arteriolar segments with time-dependent diameters influenced by shear stress, wall tension, metabolic regulation, and SNA. Metabolic signals are assumed to be propagated upstream along vessel walls via a conducted response. The model exhibits functional sympatholysis, in which sympathetic vasoconstriction is partially abrogated by increases in metabolic demand, and sympathetic escape, in which SNA elicits an initial vasoconstriction followed by vasodilation. In accordance with experimental observations, these phenomena are more prominent in small arterioles than in larger arterioles when SNA is assumed to act equally on arterioles of all sizes. The results imply that a mechanism based on the competing effects on arteriolar tone of SNA and conducted metabolic signals can account for several observed characteristics of functional sympatholysis, including the different responses of large and small arterioles.
• Roy, T. K., & Secomb, T. W. (2014). Theoretical analysis of the determinants of lung oxygen diffusing capacity. Journal of theoretical biology, 351, 1-8.
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  The process of pulmonary oxygen uptake is analyzed to obtain an explicit equation for lung oxygen diffusing capacity in terms of hematocrit and pulmonary capillary diameter. An axisymmetric model with discrete cylindrical erythrocytes is used to represent radial diffusion of oxygen from alveoli through the alveolar-capillary membrane into pulmonary capillaries, through the plasma, and into erythrocytes. Analysis of unsteady diffusion due to the passage of the erythrocytes shows that transport of oxygen through the alveolar-capillary membrane occurs mainly in the regions adjacent to erythrocytes, and that oxygen transport through regions adjacent to plasma gaps can be neglected. The model leads to an explicit formula for diffusing capacity as a function of geometric and oxygen transport parameters. For normal hematocrit and a capillary diameter of 6.75 μm, the predicted diffusing capacity is 102 ml O₂ min⁻¹ mmHg⁻¹. This value is 30-40% lower than values estimated previously by the morphometric method, which considers the total membrane area and the specific uptake rate of erythrocytes. Diffusing capacity is shown to increase with increasing hematocrit and decrease with increasing capillary diameter and increasing thickness of the membrane. Simulations of pulmonary oxygen uptake in humans under conditions of exercise or hypoxia based show closer agreement with experimental data than previous models, but still overestimate oxygen uptake. The remaining discrepancy may reflect effects of heterogeneity of perfusion and ventilation in the lung.
• Secomb, T. W., & Moulton, M. J. (2014). A Low-Order Parametric Description of Left Ventricular Kinematics. Cardiovascular Engineering and Technology, 5(4), 348-358. doi:10.1007/s13239-014-0191-9
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  An approximate description for the deformation of the left ventricle (LV) throughout the cardiac cycle is developed in terms of three time-dependent parameters. The reference configuration, corresponding to end-diastole, is represented as a thick-walled prolate spheroid. By using prolate spheroidal coordinates, a three-parameter family of mappings is defined to represent the deformed shapes of the LV wall, while identically conserving wall volume. The three parameters represent lengthening with constant internal volume, contraction with reduction of internal volume, and torsion. Feasibility is illustrated using echocardiography data from a healthy subject. The reference configuration was defined by fitting observed points on LV endocardial and epicardial surfaces in long-axis images at end diastole. Time-courses of parameters defining LV kinematics were obtained for best fit to longitudinal strains, circumferential strains and rotations (LS, CS and R) obtained from speckle tracking echocardiography at 36 LV regions. Fitted versus echocardiography-measured CS, LS and R were compared at the LV base, mid-wall and apex at the endocardium and epicardium. The RMS deviation between fitted and measured peak strains was 0.06 (LS) and 0.08 (CS). Fitted and measured LV volume changes during contraction agreed closely. Circumferential variations in strain, which may be significant in normal and pathological hearts, are not represented here. The results demonstrate feasibility of a low-order description of the deformation field in the LV myocardium, based on echocardiographic imaging data. Such a description provides a basis for low-order models of LV mechanics and eventual real-time patient-specific analysis of LV mechanical parameters.
• Ackermann, M., Tsuda, A., Secomb, T. W., Mentzer, S. J., & Konerding, M. A. (2013). Intussusceptive remodeling of vascular branch angles in chemically-induced murine colitis. Microvascular research, 87, 75-82.
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  Intussusceptive angiogenesis is a developmental process linked to both blood vessel replication and remodeling in development. To investigate the prediction that the process of intussusceptive angiogenesis is associated with vessel angle remodeling in adult mice, we systematically evaluated corrosion casts of the mucosal plexus in mice with trinitrobenzesulfonic acid (TNBS)-induced and dextran sodium sulfate (DSS)-induced colitis. The mice demonstrated a significant decrease in vessel angles in both TNBS-induced and DSS-induced colitis within 4 weeks of the onset of colitis (p
• Dewhirst, M. W., Ong, E. T., Rosner, G. L., Rehmus, S. W., Shan, S., Braun, R. D., Brizel, D. M., & Secomb, T. W. (2013). Arteriolar oxygenation in tumour and subcutaneous arterioles: Effects of inspired air oxygen content. BRITISH JOURNAL OF CANCER, 74, S241-S246.
• El-kareh, A. W., Secomb, T. W., Williams, K. S., Secomb, T. W., & El-kareh, A. W. (2013). Abstract 439: Mathematical modeling of cellular dose-response for radiation and radiation-drug combinations including cell cycle effects.. Cancer Research, 73, 439-439. doi:10.1158/1538-7445.am2013-439
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  Proceedings: AACR 104th Annual Meeting 2013; Apr 6-10, 2013; Washington, DC Thirty-six in vitro radiation dose-response data sets having 7 or more points, covering a variety of cell lines, were used to test four proposed models as alternatives to the linear-quadratic equation: the single-mechanism lognormal, the two-mechanism lognormal, the single-mechanism Hill and the two-mechanism Hill. All four models are forms of our general “damage model,” developed originally for drug dose-response. The lognormal models relate survival to damage using the cumulative lognormal density function, whereas the Hill models use a Hill equation. The models were ranked according to the Akaike Information Criterion, with the single-mechanism lognormal performing the best overall. A subset of the data showing 2-mechanism behavior was identified using the F-test method; for this subset, the two-mechanism lognormal model was superior. The lognormal model is therefore proposed as an alternative to the linear-quadratic. The lognormal damage model is then extended to the case of radiochemotherapy using the concept of additive damage, which we previously developed for drugs with two mechanisms, and then applied to combination exposure. Previously, it was found that fixed-schedule, varying-dose exposures to drug and radiation could be described by the additive damage model. Here, the case of varying schedule is considered, for gemcitabine, which showed schedule-dependent radiosensitization in studies by Pauwels et al (2003, 2009). Cell cycle effects are included through a kinetic equation describing the evolution of the population distribution over the cell cycle. In contrast to some previous models where discrete phases of the cell cycle are treated as compartments, here the distribution is a continuous density function. Gemcitabine is assumed to cause a block at a single point, with dose-dependent duration. The cell cycle phase population density distribution then enters into the radiation damage term. The model thus includes radiation-drug interaction in two ways: through additive damage and through cell cycle effects. The case of fractionated doses of gemcitabine alone is also considered. Overall, the model provides a framework for understanding the complex effects of schedule and dose, as well as a method to potentially predict combination or fractionated dose-response from knowledge of a drug's perturbations to the cell cycle phase distribution. It is expected that these models will have application in preclinical optimization of combination therapy, as well as in computational simulation of anticancer therapy. Citation Format: Katherine S. Williams, Ardith W. El-Kareh, Timothy W. Secomb. Mathematical modeling of cellular dose-response for radiation and radiation-drug combinations including cell cycle effects. [abstract]. In: Proceedings of the 104th Annual Meeting of the American Association for Cancer Research; 2013 Apr 6-10; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2013;73(8 Suppl):Abstract nr 439. doi:10.1158/1538-7445.AM2013-439
• Foehrenbacher, A., Patel, K., Abbattista, M. R., Guise, C. P., Secomb, T. W., Wilson, W. R., & Hicks, K. O. (2013). The Role of Bystander Effects in the Antitumor Activity of the Hypoxia-Activated Prodrug PR-104. Frontiers in oncology, 3, 263.
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  Activation of prodrugs in tumors (e.g., by bioreduction in hypoxic zones) has the potential to generate active metabolites that can diffuse within the tumor microenvironment. Such "bystander effects" may offset spatial heterogeneity in prodrug activation but the relative importance of this effect is not understood. Here, we quantify the contribution of bystander effects to antitumor activity for the first time, by developing a spatially resolved pharmacokinetic/pharmacodynamic (SR-PK/PD) model for PR-104, a phosphate ester pre-prodrug that is converted systemically to the hypoxia-activated prodrug PR-104A. Using Green's function methods we calculated concentrations of oxygen, PR-104A and its active metabolites, and resultant cell killing, at each point of a mapped three-dimensional tumor microregion. Model parameters were determined in vitro, using single cell suspensions to determine relationships between PR-104A metabolism and clonogenic cell killing, and multicellular layer (MCL) cultures to measure tissue diffusion coefficients. LC-MS/MS detection of active metabolites in the extracellular medium following exposure of anoxic single cell suspensions and MCLs to PR-104A confirmed that metabolites can diffuse out of cells and through a tissue-like environment. The SR-PK/PD model estimated that bystander effects contribute 30 and 50% of PR-104 activity in SiHa and HCT116 tumors, respectively. Testing the model by modulating PR-104A-activating reductases and hypoxia in tumor xenografts showed overall clonogenic killing broadly consistent with model predictions. Overall, our data suggest that bystander effects are important in PR-104 antitumor activity, although their reach may be limited by macroregional heterogeneity in hypoxia and reductase expression in tumors. The reported computational and experimental techniques are broadly applicable to all targeted anticancer prodrugs and could be used to identify strategies for rational prodrug optimization.
• Foehrenbacher, A., Secomb, T. W., Wilson, W. R., & Hicks, K. O. (2013). Design of optimized hypoxia-activated prodrugs using pharmacokinetic/pharmacodynamic modeling. Frontiers in oncology, 3, 314.
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  Hypoxia contributes to resistance of tumors to some cytotoxic drugs and to radiotherapy, but can in principle be exploited with hypoxia-activated prodrugs (HAP). HAP in clinical development fall into two broad groups. Class I HAP (like the benzotriazine N-oxides tirapazamine and SN30000), are activated under relatively mild hypoxia. In contrast, Class II HAP (such as the nitro compounds PR-104A or TH-302) are maximally activated only under extreme hypoxia, but their active metabolites (effectors) diffuse to cells at intermediate O2 and thus also eliminate moderately hypoxic cells. Here, we use a spatially resolved pharmacokinetic/pharmacodynamic (SR-PK/PD) model to compare these two strategies and to identify the features required in an optimal Class II HAP. The model uses a Green's function approach to calculate spatial and longitudinal gradients of O2, prodrug, and effector concentrations, and resulting killing in a digitized 3D tumor microregion to estimate activity as monotherapy and in combination with radiotherapy. An analogous model for a normal tissue with mild hypoxia and short intervessel distances (based on a cremaster muscle microvessel network) was used to estimate tumor selectivity of cell killing. This showed that Class II HAP offer advantages over Class I including higher tumor selectivity and greater freedom to vary prodrug diffusibility and rate of metabolic activation. The model suggests that the largest gains in class II HAP antitumor activity could be realized by optimizing effector stability and prodrug activation rates. We also use the model to show that diffusion of effector into blood vessels is unlikely to materially increase systemic exposure for realistic tumor burdens and effector clearances. However, we show that the tumor selectivity achievable by hypoxia-dependent prodrug activation alone is limited if dose-limiting normal tissues are even mildly hypoxic.
• Fontanella, A. N., Schroeder, T., Hochman, D. W., Chen, R. E., Hanna, G., Haglund, M. M., Secomb, T. W., Palmer, G. M., & Dewhirst, M. W. (2013). Quantitative mapping of hemodynamics in the lung, brain, and dorsal window chamber-grown tumors using a novel, automated algorithm. Microcirculation (New York, N.Y. : 1994), 20(8), 724-35.
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  Hemodynamic properties of vascular beds are of great interest in a variety of clinical and laboratory settings. However, there presently exists no automated, accurate, technically simple method for generating blood velocity maps of complex microvessel networks.
• Secomb, T. W., & Pries, A. R. (2013). Blood viscosity in microvessels: experiment and theory. Comptes rendus. Physique, 14(6), 470-478.
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  The apparent viscosity of blood flowing through narrow glass tubes decreases strongly with decreasing tube diameter over the range from about 300 μm to about 10 μm. This phenomenon, known as the Fåhraeus-Lindqvist effect, occurs because blood is a concentrated suspension of deformable red blood cells with a typical dimension of about 8 μm. Most of the resistance to blood flow through the circulatory system resides in microvessels with diameters in this range. Apparent viscosity of blood in microvessels in vivo has been found to be significantly higher than in glass tubes with corresponding diameters. Here we review experimental observations of blood's apparent viscosity in vitro and in vivo, and progress towards a quantitative theoretical understanding of the mechanisms involved.
• Secomb, T. W., Alberding, J. P., Hsu, R., Dewhirst, M. W., & Pries, A. R. (2013). Angiogenesis: an adaptive dynamic biological patterning problem. PLoS computational biology, 9(3), e1002983.
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  Formation of functionally adequate vascular networks by angiogenesis presents a problem in biological patterning. Generated without predetermined spatial patterns, networks must develop hierarchical tree-like structures for efficient convective transport over large distances, combined with dense space-filling meshes for short diffusion distances to every point in the tissue. Moreover, networks must be capable of restructuring in response to changing functional demands without interruption of blood flow. Here, theoretical simulations based on experimental data are used to demonstrate that this patterning problem can be solved through over-abundant stochastic generation of vessels in response to a growth factor generated in hypoxic tissue regions, in parallel with refinement by structural adaptation and pruning. Essential biological mechanisms for generation of adequate and efficient vascular patterns are identified and impairments in vascular properties resulting from defects in these mechanisms are predicted. The results provide a framework for understanding vascular network formation in normal or pathological conditions and for predicting effects of therapies targeting angiogenesis.
• Secomb, T., Fry, B. C., Roy, T. K., & Secomb, T. W. (2013). Capillary recruitment in a theoretical model for blood flow regulation in heterogeneous microvessel networks. Physiological reports, 1(3).
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  In striated muscle, the number of capillaries containing moving red blood cells increases with increasing metabolic demand. This phenomenon, termed capillary recruitment, has long been recognized but its mechanism has been unclear. Here, a theoretical model for metabolic blood flow regulation in a heterogeneous network is used to test the hypothesis that capillary recruitment occurs as a result of active control of arteriolar diameters, combined with unequal partition of hematocrit at diverging microvascular bifurcations. The network structure is derived from published observations of hamster cremaster muscle in control and dilated states. The model for modulation of arteriolar diameters includes length-tension characteristics of vascular smooth muscle and responses of smooth muscle tone to myogenic, shear-dependent, and metabolic stimuli. Blood flow is simulated including non-uniform hematocrit distribution. Convective and diffusive oxygen transport in the network is simulated. Oxygen-dependent metabolic signals are assumed to be conducted upstream from distal vessels to arterioles. With increasing oxygen demand, arterioles dilate, blood flow increases, and the numbers of flowing arterioles and capillaries, as defined by red blood cell flux above a small threshold value, increase. Unequal hematocrit partition at diverging bifurcations contributes to recruitment and enhances tissue oxygenation. The results imply that capillary recruitment, as observed in the hamster cremaster preparations, can occur as a consequence of local control of arteriolar tone and the resulting non-uniform changes in red blood cell fluxes, and provide an explanation for observations of sequential recruitment of individual capillaries in response to modulation of terminal arteriolar diameter.
• Secomb, T., Moulton, M. J., & Secomb, T. W. (2013). A low-order model for left ventricle dynamics throughout the cardiac cycle. Mathematical medicine and biology : a journal of the IMA, 30(1).
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  A theoretical model is used to simulate the dynamics of the left ventricle (LV) through all phases of the cardiac cycle, including interactions between myocardial contractility and ventricular pressure generation and effects of preload and afterload. The ventricle is represented as a cylinder containing helical muscle fibres with non-linear passive and active material properties, embedded in a uniform viscoelastic matrix. The dynamics of the ventricle are represented by a system of differential algebraic equations, whose numerical solution yields pressure-volume loops over successive cardiac cycles. Predicted time-dependent torsional, circumferential and longitudinal strains in the LV are consistent with experimental observations. The model is used to examine the effects of changes in underlying properties of the heart, including myocardial contractility, fibre orientation, passive stiffness, atrial pressure and peripheral resistance, on observable parameters such as stroke work, ejection fraction and end-systolic pressure-volume relationship. Stroke work is shown to be linearly dependent on end-diastolic volume but also to depend on afterload. Diastolic suction and its effect on filling are demonstrated. In this modelling approach, the dynamics of the heart are represented using a low-order dynamical system, and simulations can be carried out much faster than real time. Such a model could potentially be used to deduce patient-specific parameters of ventricular performance on-line from clinically available measurements.
• Arciero, J. C., & Secomb, T. W. (2012). Spontaneous oscillations in a model for active control of microvessel diameters. Mathematical medicine and biology : a journal of the IMA, 29(2), 163-80.
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  A new theory is presented for the origin of spontaneous oscillations in blood vessel diameters that are observed experimentally in the microcirculation. These oscillations, known as vasomotion, involve timevarying contractions of the vascular smooth muscle in the walls of arterioles. It is shown that such oscillations can arise as a result of interactions between the mechanics of the vessel wall and the dynamics of the active contraction of smooth muscle cells in response to circumferential tension in the wall. A theoretical model is developed in which the diameter and the degree of activation in a vessel are dynamic variables. The model includes effects of wall shear stress and oxygen-dependent metabolic signals on smooth muscle activation and is applied to a single vessel and to simplified network structures. The model equations predict limit cycle oscillations for certain ranges of parameters such as wall shear stress, arterial pressure and oxygen consumption rate. Predicted characteristics of the oscillations, including their sensitivity to arterial pressure, are consistent with experimental observations.
• El-kareh, A. W., Secomb, T. W., Secomb, T. W., Jones, L. B., & El-kareh, A. W. (2012). Abstract 3962: Mathematical models for cellular pharmacodynamics of fractionated radiation and radiochemotherapy. Cancer Research, 72, 3962-3962. doi:10.1158/1538-7445.am2012-3962
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  Proceedings: AACR 103rd Annual Meeting 2012‐‐ Mar 31‐Apr 4, 2012; Chicago, IL Cellular response to radiation is generally described by the linear-quadratic equation, but the extension to complicated fractionated schedules, and to combined exposure with chemotherapeutic drugs is not well-established. Mathematical models for cellular response are needed for predictive quantitative modeling of treatment. Here, such models are formulated using the previously developed concepts of peak damage models and additive damage, which in prior studies proved successful in describing cellular pharmacodynamics of single-agent and combination chemotherapy. The exponential form of the linear-quadratic equation is replaced by Hill or cumulative lognormal functions of the peak value over time of a quantity termed “cellular damage.” Damage is a saturating function of radiation dose. Cell survival depends on whether the peak value of cellular damage ever exceeds a threshold value during treatment. This model is tested for a range of data sets for single-dose radiation, and compares favorably with the linear-quadratic. In one data set (Park et al 2008) the dose range was sufficiently wide to show the significant overestimation of cell kill by the linear-quadratic model at high doses; the Hill/lognormal formulation behaves more satisfactorily in this limit. The model is then extended to the case of fractionated radiation through the assumption that damage decays between treatments according to first-order kinetics, and damage from different cycles is additive. The model gives a satisfactory fit to literature data with multiple fractionation schedules for the same drug and cell line. Finally, the principle of additive damage is applied to model cellular response to radiochemotherapy, with damage as a linear superposition of terms from radiation and drug. Overall, the peak cellular damage model with additive damage was found to fit in vitro dose-response data well, and provides a tool for understanding and predicting response as schedule and dose are varied in combination treatments. It is expected that these models will have application in assessment of preclinical cellular pharmacodynamics data, as well as computational simulations of treatment response. Citation Format: {Authors}. {Abstract title} [abstract]. In: Proceedings of the 103rd Annual Meeting of the American Association for Cancer Research; 2012 Mar 31-Apr 4; Chicago, IL. Philadelphia (PA): AACR; Cancer Res 2012;72(8 Suppl):Abstract nr 3962. doi:1538-7445.AM2012-3962
• Gruionu, G., Hoying, J. B., Pries, A. R., & Secomb, T. W. (2012). Structural remodeling of the mouse gracilis artery: coordinated changes in diameter and medial area maintain circumferential stress. Microcirculation (New York, N.Y. : 1994), 19(7), 610-8.
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  Vascular networks respond to chronic alterations in blood supply by structural remodeling. Previously, we showed that blood flow changes in the mouse GA lead to transient diameter increases, which can generate large increases in circumferential wall stress. Here, we examine the associated changes in the medial area of the arterial wall and the effects on circumferential wall stress.
• Roy, T. K., Pries, A. R., & Secomb, T. W. (2012). Theoretical comparison of wall-derived and erythrocyte-derived mechanisms for metabolic flow regulation in heterogeneous microvascular networks. American journal of physiology. Heart and circulatory physiology, 302(10), H1945-52.
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  The objective of this study is to compare the effectiveness of metabolic signals derived from erythrocytes and derived from the vessel wall for regulating blood flow in heterogeneous microvascular networks. A theoretical model is used to simulate blood flow, mass transport, and vascular responses. The model accounts for myogenic, shear-dependent, and metabolic flow regulation. Metabolic signals are assumed to be propagated upstream along vessel walls via a conducted response. Arteriolar tone is assumed to depend on the conducted metabolic signal as well as local wall shear stress and wall tension, and arteriolar diameters are calculated based on vascular smooth muscle mechanics. The model shows that under certain conditions metabolic regulation based on wall-derived signals can be more effective in matching perfusion to local oxygen demand relative to regulation based on erythrocyte-derived signals, resulting in higher extraction and lower oxygen deficit. The lower effectiveness of the erythrocyte-derived signal is shown to result in part from the unequal partition of hematocrit at diverging bifurcations, such that low-flow vessels tend to receive a reduced hematocrit and thereby experience a reduced erythrocyte-derived metabolic signal. The model simulations predict that metabolic signals independent of erythrocytes may play an important role in local metabolic regulation of vascular tone and flow distribution in heterogeneous microvessel networks.
• Russell, G., Harkins, K. D., Secomb, T. W., Galons, J., & Trouard, T. P. (2012). A finite difference method with periodic boundary conditions for simulations of diffusion-weighted magnetic resonance experiments in tissue. Physics in medicine and biology, 57(4), N35-46.
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  A new finite difference (FD) method for calculating the time evolution of complex transverse magnetization in diffusion-weighted magnetic resonance imaging and spectroscopy experiments is described that incorporates periodic boundary conditions. The new FD method relaxes restrictions on the allowable time step size employed in modeling which can significantly reduce computation time for simulations of large physical extent and allow for more complex, physiologically relevant, geometries to be simulated.
• Secomb, T. W., Dewhirst, M. W., & Pries, A. R. (2012). Structural adaptation of normal and tumour vascular networks. Basic & clinical pharmacology & toxicology, 110(1), 63-9.
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  Vascular networks are dynamic structures, adapting to changing conditions by structural remodelling of vessel diameters and by growth of new vessels and regression of existing vessels. The vast number of blood vessels in the circulatory system, more than 10⁹, implies that vessels' arrangement and structure are not under individual genetic control but emerge as a result of generic responses of each segment to the various stimuli that it experiences. To obtain insight into the types of response that are needed, a network-oriented approach has been used, in which theoretical models are used to simulate structural adaptation in vascular networks, and the results are compared with experimental observations. With regard to the structural control of vessel diameters, this approach shows that responses to both haemodynamic and metabolic stimuli are needed for the formation of functionally adequate and efficient network structures. Furthermore, information transfer in both upstream and downstream directions is essential for balancing flows between long and short flow pathways. Otherwise, functional shunting occurs, that is, short pathways become enlarged and flow bypasses longer pathways. Information transfer in the upstream direction is achieved by conducted responses communicated along vessel walls. Simulations of structural adaptation in tumour microvascular networks indicate that impaired vascular communication, resulting in functional shunting, may be an important factor causing the dysfunctional microcirculation and local hypoxia typically observed in tumours. Anti-angiogenic treatment of tumours may restore vascular communication and thereby improve or normalize flow distribution in tumour vasculature.
• Secomb, T., Fry, B. C., Lee, J., Smith, N. P., & Secomb, T. W. (2012). Estimation of blood flow rates in large microvascular networks. Microcirculation (New York, N.Y. : 1994), 19(6).
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  Recent methods for imaging microvascular structures provide geometrical data on networks containing thousands of segments. Prediction of functional properties, such as solute transport, requires information on blood flow rates also, but experimental measurement of many individual flows is difficult. Here, a method is presented for estimating flow rates in a microvascular network based on incomplete information on the flows in the boundary segments that feed and drain the network.
• Secomb, T., Hariprasad, D. S., & Secomb, T. W. (2012). Motion of red blood cells near microvessel walls: effects of a porous wall layer. Journal of fluid mechanics, 705.
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  A two-dimensional model is used to simulate the motion and deformation of a single mammalian red blood cell (RBC) flowing close to the wall of a microvessel, taking into account the effects of a porous endothelial surface layer (ESL) lining the vessel wall. Migration of RBCs away from the wall leads to the formation of a cell-depleted layer near the wall, which has a large effect on the resistance to blood flow in microvessels. The objective is to examine the mechanical factors causing this migration, including the effects of the ESL. The vessel is represented as a straight parallel-sided channel. The RBC is represented as a set of interconnected viscoelastic elements, suspended in plasma, a Newtonian fluid. The ESL is represented as a porous medium, and plasma flow in the layer is computed using the Brinkman approximation. It is shown that an initially circular cell positioned close to the ESL in a shear flow is deformed into an asymmetric shape. This breaking of symmetry leads to migration away from the wall. With increasing hydraulic resistivity of the layer, the rate of lateral migration increases. It is concluded that mechanical interactions of RBCs flowing in microvessels with a porous wall layer may reduce the rate of lateral migration and hence reduce the width of the cell-depleted zone external to the ESL, relative to the cell-depleted zone that would be formed if the interface between the ESL and free-flowing plasma were replaced by an impermeable boundary.
• Wilson, W. R., Secomb, T. W., Hicks, K. O., Foehrenbacher, A., & Abbattista, M. R. (2012). 983 The Role of Bystander Effects in the Anti-tumour Activity of the Hypoxia-activated Prodrug PR-104a. European Journal of Cancer, 48, S237. doi:10.1016/s0959-8049(12)71601-5
• Barber, J. O., Restrepo, J. M., & Secomb, T. W. (2011). Simulated Red Blood Cell Motion in Microvessel Bifurcations: Effects of Cell-Cell Interactions on Cell Partitioning. Cardiovascular engineering and technology, 2(4), 349-360.
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  Partitioning of red blood cell (RBC) fluxes between the branches of a diverging microvessel bifurcation is generally not proportional to the flow rates, as RBCs preferentially enter the higher-flow branch. A two-dimensional model for RBC motion and deformation is used to investigate the effects of cell-cell mechanical interactions on RBC partitioning in bifurcations. The RBC membrane and cytoplasm are represented by sets of viscoelastic elements immersed in a low Reynolds number flow. Several types of two-cell interactions that can affect partitioning are found. In the most frequent interactions, a `trade-off' occurs, in which a cell entering one branch causes a following cell to enter the other branch. Other types of interactions include `herding,' where the leading cell is caused to enter the same branch as the following cell, and `following,' where the trailing cell is caused to enter the same branch as the leading cell. The combined effect of these cell-cell interactions is a tendency towards more uniform partitioning, which results from the trade-off effect but is reduced by the herding and following effects. With increasing hematocrit, the frequency of interactions increases, and more uniform partitioning results. This prediction is consistent with experimental observations on how hematocrit affects RBC partitioning.
• DEWHIRST, M. W., VINUYA, R. Z., ONG, E. T., KLITZMAN, B., ROSNER, G., SECOMB, T. W., & GROSS, J. F. (2011). EFFECTS OF BRADYKININ ON THE HEMODYNAMICS OF TUMOR AND GRANULATING NORMAL TISSUE MICROVASCULATURE. RADIATION RESEARCH, 130(3), 345-354.
• El-kareh, A. W., Secomb, T. W., Secomb, T. W., & El-kareh, A. W. (2011). Abstract 39: A mathematical model for the dependence on tumor size of the rate of metastasis formation. Cancer Research, 71, 39-39. doi:10.1158/1538-7445.am2011-39
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  That the dependence of the rate of formation of new metastases on the existing tumor volume is weak has been demonstrated from data on the distribution of sizes of metastasic lesions, as well as data on the fraction of tumors that are metastatic as a function of tumor size at diagnosis. A mathematical model for the rate of metastasis formation as a function of tumor size was developed to explain this finding and is presented here. The model includes several stromal cell populations, in particular macrophages and fibroblasts, in addition to tumor cells, which are divided into epithelial- and mesenchymal-like populations. Epithelial-to-mesenchymal transitions are accounted for, as well as key growth factors that mediate tumor-stroma crosstalk and recruitment of stromal cells to the tumor mass. A key feature of the model is cooperativity in the steps involved in the establishment of metastases. Experimental findings of tumor cell-macrophage cooperativity in chemotaxis-driven migration towards microvessels, and EMT/non-EMT cooperativity in tumor cell escape, extravasation and intravasation are incorporated into the model. By involving the product of two or more rates, each of which depend on tumor volume raised to a power less than one, these cooperative processes result in a dependence of the metastatic rate on tumor size that is less than the 2/3 power expected if the rate were merely surface-area dependent. In addition to providing a framework for understanding experimental findings on the kinetics of metastasis, the model should prove useful for investigating the interplay between cell kill rates due to treatment and tumor regrowth kinetics in the most common case where the treated tumor consists of a distribution of metastatic lesions rather than a single mass. Citation Format: {Authors}. {Abstract title} [abstract]. In: Proceedings of the 102nd Annual Meeting of the American Association for Cancer Research; 2011 Apr 2-6; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2011;71(8 Suppl):Abstract nr 39. doi:10.1158/1538-7445.AM2011-39
• Pries, A. R., Reglin, B., & Secomb, T. W. (2011). Modeling of angioadaptation: insights for vascular development. The International journal of developmental biology, 55(4-5), 399-405.
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  Vascular beds are generated by vasculogenesis and sprouting angiogenesis, and these processes have strong stochastic components. As a result, vascular patterns exhibit significant heterogeneity with respect to the topological arrangement of the individual vessel segments and the characteristics (length, number of segments) of different arterio-venous pathways. This structural heterogeneity tends to cause heterogeneous distributions of flow and oxygen availability in tissue. However, these quantities must be maintained within tolerable ranges to allow normal tissue function. This is achieved largely through adjustment of vascular flow resistance by control of vessel diameters. While short-term diameter control by changes in vascular tone in arterioles and small arteries plays an important role, in the long term an even more important role is played by structural adaptation (angioadaptation), occurring in response to metabolic and hemodynamic signals. The effectiveness, stability and robustness of this angioadaptation depend sensitively on the nature and strength of the vascular responses involved and their interactions with the network structure. Mathematical models are helpful in understanding these complex interactions, and can be used to simulate the consequences of failures in sensing or signal transmission mechanisms. For the tumor microcirculation, this strategy of combining experimental observations with theoretical models, has led to the hypothesis that dysfunctional information transport via vascular connexins is a major cause of the observed vascular pathology and increased heterogeneity in oxygen distribution.
• Secomb, T. W., & Pries, A. R. (2011). Blood Flow in Microvascular Networks. Comprehensive Physiology, 3-36. doi:10.1016/b978-0-12-374530-9.00001-2
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  Publisher Summary This chapter focuses on the flow of blood through microvascular networks, with emphasis on biophysical aspects. The need for adequate exchange of materials between blood and tissue, and particularly for oxygen delivery, is met by large numbers of closely spaced microvessels with small diameters and large cumulative surface area, interconnected in intricate network structures. microvascular network is a system of conduits that distributes blood throughout tissues as needed. The distribution of flow and pressure can be analyzed by analogy to the distribution of current and voltage in a network of electrical resistances. The flow resistance of each segment depends on the segment geometry according to the Poiseuille relationship. The topological structure of microvascular networks is heterogeneous. The biophysical processes that govern distribution of blood flow within a microvascular network of given geometry have been studied for decades and, while areas of uncertainty remain, it may be claimed that a good overall level of understanding has been achieved. Much remains to be learned, however, about the active biological processes that control network geometry, including angiogenesis, structural adaptation, and control of vascular tone. The study of blood flow in microvascular networks thus represents a key step in the process of translating advances in molecular and cellular biology into improved understanding of cardiovascular function in health and disease. A discussion of the relationship of network structure and flow to physiological aspects of the microvasculature including transport functions, inflammatory and immune functions, regulation of blood flow, and structural adaptation is presented.
• Secomb, T. W., & Pries, A. R. (2011). The microcirculation: physiology at the mesoscale. The Journal of physiology, 589(Pt 5), 1047-52.
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  The microcirculation exemplifies the mesoscale in physiological systems, bridging larger and smaller scale phenomena. Microcirculatory research represents an example of a 'middle-out,' rather than 'top-down' or 'bottom-up,' approach to the study of biological function. Computational and mathematical approaches can be used to analyse the functioning of the microcirculation and to establish quantitative relationships between microvascular processes and phenomena occurring on larger and smaller scales, leading to insights which could not be obtained solely by reductionist biological experiments. Given its integrative approach to processes occurring on disparate scales and its emphasis on theoretical as well as experimental approaches, microcirculatory research belongs within current definitions of systems biology.
• Secomb, T., & Secomb, T. W. (2011). Mechanics and computational simulation of blood flow in microvessels. Medical engineering & physics, 33(7).
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  Blood is a concentrated suspension of red blood cells (RBCs). Motion and deformation of RBCs can be analyzed based on knowledge of their mechanical characteristics. Axisymmetric models for single-file motion of RBCs in capillaries yield predictions of apparent viscosity in good agreement with experimental results for diameters up to about 8 μm. Two-dimensional simulations, in which each RBC is represented as a set of interconnected viscoelastic elements, predict that off-centre RBCs in an 8-μm channel take asymmetric shapes and drift toward the centre-line. Predicted trajectories agree with observations in microvessels of the rat mesentery. An isolated RBC initially positioned near the wall of a 20-μm channel is deformed into an asymmetric shape, migrates away from the wall, and then enters a complex tumbling motion with continuous shape change. Realistic simulation of multiple interacting RBCs in microvessels remains as a major challenge.
• Waters, S. L., Alastruey, J., Beard, D. A., Bovendeerd, P. H., Davies, P. F., Jayaraman, G., Jensen, O. E., Lee, J., Parker, K. H., Popel, A. S., Secomb, T. W., Siebes, M., Sherwin, S. J., Shipley, R. J., Smith, N. P., & van de Vosse, F. N. (2011). Theoretical models for coronary vascular biomechanics: progress & challenges. Progress in biophysics and molecular biology, 104(1-3), 49-76.
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  A key aim of the cardiac Physiome Project is to develop theoretical models to simulate the functional behaviour of the heart under physiological and pathophysiological conditions. Heart function is critically dependent on the delivery of an adequate blood supply to the myocardium via the coronary vasculature. Key to this critical function of the coronary vasculature is system dynamics that emerge via the interactions of the numerous constituent components at a range of spatial and temporal scales. Here, we focus on several components for which theoretical approaches can be applied, including vascular structure and mechanics, blood flow and mass transport, flow regulation, angiogenesis and vascular remodelling, and vascular cellular mechanics. For each component, we summarise the current state of the art in model development, and discuss areas requiring further research. We highlight the major challenges associated with integrating the component models to develop a computational tool that can ultimately be used to simulate the responses of the coronary vascular system to changing demands and to diseases and therapies.
• El-kareh, A. W., Secomb, T. W., Secomb, T. W., & El-kareh, A. W. (2010). Abstract 102: Mathematical models for growth of metastatic tumors. Cancer Research, 70, 102-102. doi:10.1158/1538-7445.am10-102
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  Proceedings: AACR 101st Annual Meeting 2010‐‐ Apr 17‐21, 2010; Washington, DC The mathematical form best describing the growth of a single tumor mass has been found to be any of several models including the power law, logistic or Gompertzian, depending on the particular clinical or experimental tumor considered. All these models exhibit monotonically decreasing growth rate over time; some plateau at some finite volume. However, the kinetics of growth of the total mass of a metastatic tumor remains a separate question with relevance for understanding survival time, optimal treatment, and the development of symptoms that arise from total tumor burden. Here, exact analytical solutions for total tumor burden are derived under the assumption that both the primary tumor and each individual metastatic lesion grow with power law kinetics. Solutions are also derived for the case of metastatic growth after removal of the primary tumor. Two cases are considered: one where the rate of metastasis formation from an individual lesion grows with power law kinetics, and one where this rate is independent of size. Surprisingly, the latter assumption, which is consistent with only a small subpopulation of cells having metastatic potential, proves entirely adequate to describe data of Iwata et al on the cumulative distribution of metastases of a human hepatocellular carcinoma. For both cases, as time progresses, the growth ultimately becomes exponential, regardless of whether or not the primary tumor has been excised. However, it is shown that for the range of parameters describing observed tumors, this exponential limit is not reached before the tumor burden becomes lethal. The growth does not, however, show the trend of decreasing rate with increasing volume exhibited by single lesions, and appears in fact approximately exponential over the range of clinical detectability. This provides an explanation for experimental data showing near exponential growth of plasma tumor markers such as carcinoembryonic antigen in patients followed for metastatic colorectal or medullary thyroid carcinoma. Using the model, it can be seen that neither the growth rate of individual lesions, nor the rate of metastasis formation by itself causes tumor lethality; rather it is the ratio of these two rates that determines whether a large number of metastases form before the primary tumor reaches detectable size. The models derived here are expected to have application to mathematical modeling of optimal treatment and tumor-immune system interactions. Citation Format: {Authors}. {Abstract title} [abstract]. In: Proceedings of the 101st Annual Meeting of the American Association for Cancer Research; 2010 Apr 17-21; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2010;70(8 Suppl):Abstract nr 102.
• El-kareh, A. W., Secomb, T. W., Secomb, T. W., Jones, L. B., & El-kareh, A. W. (2010). Abstract 103: The additive damage model for dose-response of cancer drug combinations. Cancer Research, 70, 103-103. doi:10.1158/1538-7445.am10-103
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  Proceedings: AACR 101st Annual Meeting 2010‐‐ Apr 17‐21, 2010; Washington, DC New drug combinations are initially tested by generating dose-response data for their cytotoxic effect on cancer cell lines. Such data are then used to assess whether further development in animal models or clinical trials is merited. Several mathematical models have been proposed to describe and evaluate such data, including the Chou-Talalay median effect model, the Syracuse-Greco model, the isobologram model, and the White et al surface response model. Some of these models contain a parameter representing synergy or antagonism of the combination, while for others, deviations from the model fits are used as synergy tests. Here, the additive damage model, a new model proposed to describe drug combination cytotoxicity, is tested. Unlike previous models which are either empirical or derive drug effect in terms of fractional inhibition of enzyme-facilitated reaction rates, the current model is based on the concept of cellular damage, as measured by the maximum achieved concentration of some intracellular species of drug bound to target. Individual cells succumb at their threshold level of damage, assumed to be heterogeneous across a cell population. Survival relative to controls is therefore expressed in terms of the statistical distribution of the damage threshold over the population. In the case of two drugs acting simultaneously, total damage is a linear superposition of damage terms for each drug. The damage terms allow for saturation of cellular uptake or target binding. Here, two possible damage threshold distribution functions are considered, the cumulative lognormal function, and a power-law function (Hill equation). Two key features of the model that mathematically distinguish it from the median effect model are saturation effects in the drug concentration, and the difference in heterogeneity of the lethal damage threshold between the two drugs. The model was tested for 12 data sets for the cisplatin-paclitaxel combination, an additional 12 for other platinum-taxane combinations, and 15 data sets for a variety of drug combinations including the newer agents gefinitib and trastuzumab. The Hill form of the model was found in some cases to fit the data slightly better than the lognormal form. While a lognormal distribution would be expected as the result of linear superposition of a large number of cellular processes, each of which followed some statistical distribution across the cell cycle, nonlinearity and coupling can explain why power law forms may fit data better, a phenomenon already observed in other biological applications such as human memory. Overall, the model provided significantly superior fits to the data compared to previous models, by the Akaike Information Criterion. This suggests that deviations from the Chou-Talalay model previously interpreted as synergy or antagonism can equally well be explained in terms of saturation effects and differing heterogeneities in the threshold concentrations of two drugs. Citation Format: {Authors}. {Abstract title} [abstract]. In: Proceedings of the 101st Annual Meeting of the American Association for Cancer Research; 2010 Apr 17-21; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2010;70(8 Suppl):Abstract nr 103.
• Konerding, M. A., Turhan, A., Ravnic, D. J., Lin, M., Fuchs, C., Secomb, T. W., Tsuda, A., & Mentzer, S. J. (2010). Inflammation-induced intussusceptive angiogenesis in murine colitis. Anatomical record (Hoboken, N.J. : 2007), 293(5), 849-57.
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  Intussusceptive angiogenesis is a morphogenetic process that forms new blood vessels by the division of a single blood vessel into two lumens. Here, we show that this process of intraluminal division participates in the inflammation-induced neovascularization associated with chemically induced murine colitis. In studies of both acute (4-7 days) and chronic (28-31 days) colitis, intravital microscopy of intravascular tracers demonstrated a twofold reduction in blood flow velocity. In the acute colitis model, the decreased velocity was associated with marked dilatation of the mucosal plexus. In contrast, chronic inflammation was associated with normal caliber vessels and duplication (and triplication) of the quasi-polygonal mucosal plexus. Scanning electron microscopy (SEM) of intravascular corrosion casts suggested that pillar formation and septation, previously linked to the morphogenetic process of intussusceptive angiogenesis, were present within days of the onset of inflammation. Four weeks after the onset of inflammation, SEM of vascular corrosion casts demonstrated replication of the mucosal plexus without significant evidence of sprouting angiogenesis. These data suggest that mucosal capillaries have comparable aggregate cross-sectional area in acute and chronic colitis; however, there is a significant increase in functional capillary density in chronic colitis. We conclude that intussusceptive angiogenesis is a fundamental mechanism of microvascular adaptation to prolonged inflammation.
• Pries, A. R., & Secomb, T. W. (2010). Microcirculatory network structures and models. ANNALS OF BIOMEDICAL ENGINEERING, 28(8), 916-921.
• Pries, A. R., & Secomb, T. W. (2010). Rheology of the microcirculation. CLINICAL HEMORHEOLOGY AND MICROCIRCULATION, 29(3-4), 143-148.
• Pries, A. R., Höpfner, M., le Noble, F., Dewhirst, M. W., & Secomb, T. W. (2010). The shunt problem: control of functional shunting in normal and tumour vasculature. Nature reviews. Cancer, 10(8), 587-93.
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  Networks of blood vessels in normal and tumour tissues have heterogeneous structures, with widely varying blood flow pathway lengths. To achieve efficient blood flow distribution, mechanisms for the structural adaptation of vessel diameters must be able to inhibit the formation of functional shunts (whereby short pathways become enlarged and flow bypasses long pathways). Such adaptation requires information about tissue metabolic status to be communicated upstream to feeding vessels, through conducted responses. We propose that impaired vascular communication in tumour microvascular networks, leading to functional shunting, is a primary cause of dysfunctional microcirculation and local hypoxia in cancer. We suggest that anti-angiogenic treatment of tumours may restore vascular communication and thereby improve or normalize flow distribution in tumour vasculature.
• DEWHIRST, M. W., SECOMB, T. W., ONG, E. T., HSU, R., & GROSS, J. F. (2009). DETERMINATION OF LOCAL OXYGEN-CONSUMPTION RATES IN TUMORS. CANCER RESEARCH, 54(13), 3333-3336.
• El-Kareh, A., & Secomb, T. W. (2009). A mathematical model for comparison of bolus injection, continuous infusion, and liposomal delivery of doxorubicin to tumor cells. NEOPLASIA, 2(4), 325-338.
• Kavanagh, B. D., Secomb, T. W., Hsu, R., Lin, P. S., Venitz, J., & Dewhirst, M. W. (2009). A theoretical model for the effects of reduced hemoglobin-oxygen affinity on tumor oxygenation. INTERNATIONAL JOURNAL OF RADIATION ONCOLOGY BIOLOGY PHYSICS, 53(1), 172-179.
• Secomb, T. W., Galons, J. P., Trouard, T. P., Secomb, T. W., Harkins, K. D., & Galons, J. P. (2009). Assessment of the effects of cellular tissue properties on ADC measurements by numerical simulation of water diffusion.. Magnetic resonance in medicine, 62(6), 1414-22. doi:10.1002/mrm.22155
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  The apparent diffusion coefficient (ADC), as measured by diffusion-weighted MRI, has proven useful in the diagnosis and evaluation of ischemic stroke. The ADC of tissue water is reduced by 30-50% following ischemia and provides excellent contrast between normal and affected tissue. Despite its clinical utility, there is no consensus on the biophysical mechanism underlying the reduction in ADC. In this work, a numerical simulation of water diffusion is used to predict the effects of cellular tissue properties on experimentally measured ADC. The model indicates that the biophysical mechanisms responsible for changes in ADC postischemia depend upon the time over which diffusion is measured. At short diffusion times, the ADC is dependent upon the intrinsic intracellular diffusivity, while at longer, clinically relevant diffusion times, the ADC is highly dependent upon the cell volume fraction. The model also predicts that at clinically relevant diffusion times, the 30-50% drop in ADC after ischemia can be accounted for by cell swelling alone when intracellular T(2) is allowed to be shorter than extracellular T(2).
• DEWHIRST, M. W., OLIVER, R., TSO, C. Y., GUSTAFSON, C., SECOMB, T., & GROSS, J. F. (2008). HETEROGENEITY IN TUMOR MICROVASCULAR RESPONSE TO RADIATION. INTERNATIONAL JOURNAL OF RADIATION ONCOLOGY BIOLOGY PHYSICS, 18(3), 559-568.
• El-kareh, A. W., Secomb, T. W., Secomb, T. W., Labes, R. E., & El-kareh, A. W. (2008). Cell cycle checkpoint models for cellular pharmacology of paclitaxel and platinum drugs.. The AAPS journal, 10(1), 15-34. doi:10.1208/s12248-007-9003-6
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  A pharmacokinetic-pharmacodynamic mathematical model is developed for cellular pharmacology of chemotherapeutic drugs for which the decisive step towards cell death occurs at a point in the cell cycle, presumably corresponding to a cell cycle checkpoint. For each cell, the model assumes a threshold level of some intracellular species at that checkpoint, beyond which the cell dies. The threshold level is assumed to have a log-normal distribution in the cell population. The kinetics of formation of the lethal intracellular species depends on the drug, and on the cellular pharmacokinetics and binding kinetics of the cell. Specific models are developed for paclitaxel and for platinum drugs (cisplatin, oxaliplatin and carboplatin). In the case of paclitaxel, two separate mechanisms of cell death necessitate a model that accounts for two checkpoints, with different intracellular species. The model was tested on a number of in vitro cytotoxicity data sets for these drugs, and found overall to give significantly better fits than previously proposed cellular pharmacodynamic models. It provides an explanation for the asymptotic convergence of dose-response curves as exposure time becomes long.
• HALPERN, D., & SECOMB, T. W. (2008). THE SQUEEZING OF RED BLOOD-CELLS THROUGH CAPILLARIES WITH NEAR-MINIMAL DIAMETERS. JOURNAL OF FLUID MECHANICS, 203, 381-400.
• SECOMB, T. W., & GROSS, J. F. (2008). FLOW OF RED-BLOOD-CELLS IN NARROW CAPILLARIES - ROLE OF MEMBRANE TENSION. INTERNATIONAL JOURNAL OF MICROCIRCULATION-CLINICAL AND EXPERIMENTAL, 2(3), 229-240.
• Sloot, A. A., Secomb, T. W., Pries, A. R., Hopfner, M., Hinkeldey, M., Dreher, M. R., Dewhirst, M. W., & Cornelissen, A. J. (2008). Changed microvascular adaptation characteristics may explain heterogeneity and hypoxia of tumor perfusion. The FASEB Journal, 22.
• WRIGHT, S. H., SECOMB, T. W., & BRADLEY, T. J. (2008). APICAL MEMBRANE-PERMEABILITY OF MYTILUS GILL INFLUENCE OF ULTRASTRUCTURE, SALINITY AND COMPETITIVE INHIBITORS ON AMINO-ACID FLUXES. JOURNAL OF EXPERIMENTAL BIOLOGY, 129, 205-230.
• Pries, A. R., Secomb, T. W., & Gaehtgens, P. (2007). Biophysical aspects of blood flow in the microvasculature. CARDIOVASCULAR RESEARCH, 32(4), 654-667.
• Secomb, T. W., Hsu, R., & Pries, A. R. (2007). Motion of red blood cells in a capillary with an endothelial surface layer: effect of flow velocity. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 281(2), H629-H636.
• Ulm, L., Styp-rekowska, B., Secomb, T. W., Reglin, B., Pries, A. R., Kuppe, H., & Disassa, N. M. (2007). An imaging spectroscopy approach for measurement of oxygen saturation and hematocrit during intravital microscopy.. Microcirculation (New York, N.Y. : 1994), 14(3), 207-21. doi:10.1080/10739680601139302
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  Oxygen supply and partial pressure are key determinants of tissue metabolic status and are also regulators of vascular function including production of reactive oxygen species, vascular remodeling, and angiogenesis. The objective of this study was to develop an approach for the determination of oxygen saturation and hematocrit for individual microvessels in trans- and epi-illumination intravital microscopy..A spectral approach was used, taking advantage of the availability of commercial imaging systems that allow digital recording of intravital images at a number of predetermined wavelengths within a relatively short time. The dependence of validity and precision of saturation measurements on critical experimental variables (reference spectra, number and selection of wavelengths, exposure time, analysis area, analysis model) was evaluated. In addition, a software approach for two-dimensional analysis of images was developed..Exposure times per wavelength of about 200 ms and use of up to 50 wavelengths evenly spaced from 500 to 598 nm allow automatic discrimination of microvessels from tissue background (segmentation) with reliable determination of oxygen saturation (in trans- and epi-illumination) and hematocrit (in transillumination)..The present imaging spectroscopy approach allows detailed assessment of oxygen transport and other functional parameters at the microcirculatory level.
• WRIGHT, S. H., & SECOMB, T. W. (2007). EPITHELIAL AMINO-ACID-TRANSPORT IN MARINE MUSSELS - ROLE IN NET EXCHANGE OF TAURINE BETWEEN GILLS AND SEA-WATER. JOURNAL OF EXPERIMENTAL BIOLOGY, 121, 251-270.
• PRIES, A. R., SECOMB, T. W., & GAEHTGENS, P. (2006). DESIGN PRINCIPLES OF VASCULAR BEDS. CIRCULATION RESEARCH, 77(5), 1017-1023.
• Secomb, T. W., Hsu, R., & Pries, A. R. (2006). A model for red blood cell motion in glycocalyx-lined capillaries. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 274(3), H1016-H1022.
• DEWHIRST, M. W., TSO, C. Y., OLIVER, R., GUSTAFSON, C. S., SECOMB, T. W., & GROSS, J. F. (2005). MORPHOLOGIC AND HEMODYNAMIC COMPARISON OF TUMOR AND HEALING NORMAL TISSUE MICROVASCULATURE. INTERNATIONAL JOURNAL OF RADIATION ONCOLOGY BIOLOGY PHYSICS, 17(1), 91-99.
• MURATA, T., & SECOMB, T. W. (2005). EFFECTS OF AGGREGATION ON THE FLOW PROPERTIES OF RED BLOOD-CELL SUSPENSIONS IN NARROW VERTICAL TUBES. BIORHEOLOGY, 26(2), 247-259.
• Pries, A. R., Reglin, B., & Secomb, T. W. (2005). Remodeling of blood vessels - Responses of diameter and wall thickness to hemodynamic and metabolic stimuli. HYPERTENSION, 46(4), 725-731.
• Pries, A. R., Secomb, T. W., & Gaehtgens, P. (2005). Relationship between structural and hemodynamic heterogeneity in microvascular networks. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 270(2), H545-H553.
• Secomb, T. W. (2005). Comments on Point:Counterpoint "Positive effects of intermittent hypoxia (live high:train low) on exercise performance are/are not mediated primarily by augmented red cell volume".. Journal of Applied Physiology, 99(6), 2454-2455.
• Secomb, T. W., & Carlson, B. E. (2005). A theoretical model for the myogenic response based on the length-tension characteristics of vascular smooth muscle.. Microcirculation (New York, N.Y. : 1994), 12(4), 327-38. doi:10.1080/10739680590934745
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  A theoretical model is developed to describe the myogenic response of resistance vessels to changes in intravascular pressure, based on a consideration of the active and passive length-tension characteristics of vascular smooth muscle (VSM). The dependence of model parameters on vessel diameter is examined..The vessel wall is represented mechanically as a nonlinear passive component in parallel with an active contractile component. The level of VSM tone is assumed to have a sigmoidal dependence on circumferential wall tension or stress. Model parameters are optimized for each of 18 independent experimental data sets previously obtained using pressure or wire myograph systems..Close fits between model predictions and experimental data are found in each case. An alternative formulation in which VSM tone depends on circumferential wall stress is found also to be consistent with available data. Significant trends in model parameters as a function of diameter are found..The results support the hypothesis that circumferential tension or stress in the wall provides the signal for myogenic responses. The model provides a basis for simulating steady-state myogenic responses in vascular networks containing a range of vessel diameters.
• Secomb, T. W., & Pries, A. R. (2005). Control of blood vessel structure: insights from theoretical models.. American journal of physiology. Heart and circulatory physiology, 288(3), H1010-5. doi:10.1152/ajpheart.00752.2004
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  Blood vessels are capable of continuous structural adaptation in response to changing local conditions and functional requirements. Theoretical modeling approaches have stimulated the development of new concepts in this area and have allowed investigation of the complex relations between adaptive responses to multiple stimuli and resulting functional properties of vascular networks. Early analyses based on a minimum-work principle predicted uniform wall shear stress in all segments of vascular networks and led to the concept that vessel diameter is controlled by a feedback system based on responses to wall shear stress. Vascular reactions to changes in transmural pressure suggested feedback control of circumferential wall stress. However, theoretical simulations of network adaptation showed that these two mechanisms cannot, by themselves, lead to stable and realistic network structures. Models combining reactions to fluid shear stress, circumferential stress, and metabolic status of tissue, with propagation of stimuli upstream and downstream along vascular segments, are needed to explain stable and functionally adequate adaptation of vascular structure. Such models provide a basis for predicting the response of vascular segments exposed to altered conditions, as, for example, in vascular grafts.
• Secomb, T. W., Hsu, R., Braun, R. D., Ross JR, ., Gross, J. F., & Dewhirst, M. W. (2005). Theoretical simulation of oxygen transport to tumors by three-dimensional networks of microvessels. OXYGEN TRANSPORT TO TISSUE XX, 454, 629-634.
• WRIGHT, S. H., & SECOMB, T. W. (2005). EPIDERMAL TAURINE TRANSPORT IN MARINE MUSSELS. AMERICAN JOURNAL OF PHYSIOLOGY, 247(2), R346-R355.
• El-kareh, A. W., Secomb, T. W., Secomb, T. W., & El-kareh, A. W. (2004). A theoretical model for intraperitoneal delivery of cisplatin and the effect of hyperthermia on drug penetration distance.. Neoplasia (New York, N.Y.), 6(2), 117-27. doi:10.1593/neo.03205
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  A theoretical model for the intraperitoneal (i.p.) delivery of cisplatin and heat to tumor metastases in tissues adjacent to the peritoneal cavity is presented. The penetration distance (the depth to which drug diffuses directly from the cavity into tissues) is predicted to be on the order of 0.5 mm. The model shows that exchange with the microvasculature has more effect than cellular uptake in limiting the penetration distance. Possible effects of hyperthermia are simulated, including increased cell uptake of drug, increased cell kill at a given level of intracellular drug, and decreased microvascular density. The model suggests that the experimental finding of elevated intracellular platinum levels up to a depth of 3 to 5 mm when drug is delivered i.p. by a heated infusion solution is due to penetration of heat to this distance, causing increased cell uptake of drug. Beyond a depth of about 0.5 mm, the drug is delivered mainly through the circulation. Use of sodium thiosulfate to deactivate systemic cisplatin may therefore be counterproductive when heat is delivered locally. The model suggests that i.p. delivery of heat, combined with systemic delivery of drug, may be as effective as i.p. delivery of heat and drug.
• McGuire, B. J., & Secomb, T. W. (2004). A theoretical model for oxygen transport in skeletal muscle under conditions of high oxygen demand. JOURNAL OF APPLIED PHYSIOLOGY, 91(5), 2255-2265.
• SECOMB, T. W., & SKALAK, R. (2004). SURFACE FLOW OF VISCOELASTIC MEMBRANES IN VISCOUS FLUIDS. QUARTERLY JOURNAL OF MECHANICS AND APPLIED MATHEMATICS, 35(MAY), 233-247.
• Secomb, T. W., & Carlson, B. E. (2004). A theoretical model for the myogenic response based on the mechanics of vascular smooth muscle. The FASEB Journal, 18.
• Secomb, T. W., Hsu, R., & Pries, A. R. (2004). Effect of the endothelial surface layer on transmission of fluid shear stress to endothelial cells. BIORHEOLOGY, 38(2-3), 143-150.
• DEWHIRST, M. W., ONG, E. T., KLITZMAN, B., SECOMB, T. W., VINUYA, R. Z., DODGE, R., BRIZEL, D., & GROSS, J. F. (2003). PERIVASCULAR OXYGEN-TENSIONS IN A TRANSPLANTABLE MAMMARY-TUMOR GROWING IN A DORSAL FLAP WINDOW CHAMBER. RADIATION RESEARCH, 130(2), 171-182.
• El-Kareh, A., & Secomb, T. W. (2003). A mathematical model for cisplatin cellular pharmacodynamics. NEOPLASIA, 5(2), 161-169.
• GOODMAN, P., & SECOMB, T. W. (2003). IDENTIFICATION OF ENANTIOMORPHOUSLY RELATED SPACE GROUPS BY ELECTRON-DIFFRACTION. ACTA CRYSTALLOGRAPHICA SECTION A, 33(JAN1), 126-&.
• Lo, A., Fuglevand, A. J., & Secomb, T. W. (2003). Oxygen delivery to skeletal muscle fibers: effects of microvascular unit structure and control mechanisms. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 285(3), H955-H963.
• McGuire, B. J., & Secomb, T. W. (2003). Estimation of capillary density in human skeletal muscle based on maximal oxygen consumption rates. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 285(6), H2382-H2391.
• SKALAK, R., KELLER SR, ., & SECOMB, T. W. (2003). MECHANICS OF BLOOD-FLOW. JOURNAL OF BIOMECHANICAL ENGINEERING-TRANSACTIONS OF THE ASME, 103(2), 102-115.
• Secomb, T. W., & Hsu, R. (2003). Motion of red blood cells in capillaries with variable cross sections. JOURNAL OF BIOMECHANICAL ENGINEERING-TRANSACTIONS OF THE ASME, 118(4), 538-544.
• Dewhirst, M. W., Navia, I. C., Brizel, D. M., Willett, C., & Secomb, T. W. (2002). Multiple etiologies of tumor hypoxia require multifaceted solutions. CLINICAL CANCER RESEARCH, 13(2), 375-377.
• Hsu, R., Secomb, T. W., Pries, A. R., & Hsu, R. (2002). Blood flow and red blood cell deformation in nonuniform capillaries: effects of the endothelial surface layer.. Microcirculation (New York, N.Y. : 1994), 9(3), 189-96. doi:10.1038/sj.mn.7800132
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  A theoretical model is used to examine the mechanics of red blood cell (RBC) motion in nonuniform capillaries. The model includes effects of the endothelial surface layer (ESL), which is a layer of macromolecules adjacent to the endothelium and which impedes plasma flow..The motion of an RBC traversing a capillary with diameter varying sinusoidally between 5.4 microm and 7.4 microm is simulated numerically. The ESL is assumed to be 0.7-microm wide and deformable. Axisymmetric RBC shapes are assumed. Lubrication theory is used to analyze the motion of plasma around the RBC and through the ESL..In a nonuniform capillary with no ESL, moving RBCs undergo large transient deformations and predicted flow resistance is substantially higher than in a uniform capillary with the same mean diameter. The presence of a deformable ESL reduces the transient fluid shear stresses and deformations experienced by RBCs traversing a nonuniform capillary. With an ESL, the increase in flow resistance resulting from nonuniformity is less than twofold versus three- to fourfold with no ESL in vessel geometries with the same ESL-free luminal region..The presence of the ESL reduces the impact of capillary irregularity on flow resistance and may protect RBCs traversing irregular capillaries from damage due to large, rapidly fluctuating external stresses.
• MURATA, T., & SECOMB, T. W. (2002). EFFECTS OF SHEAR RATE ON ROULEAU FORMATION IN SIMPLE SHEAR-FLOW. BIORHEOLOGY, 25(1-2), 113-122.
• Pries, A. R., Secomb, T. W., & Gaehtgens, P. (2002). Structural adaptation and stability of microvascular networks: theory and simulations. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 275(2), H349-H360.
• SECOMB, T. W., HSU, R., DEWHIRST, M. W., KLITZMAN, B., & GROSS, J. F. (2002). ANALYSIS OF OXYGEN-TRANSPORT TO TUMOR-TISSUE BY MICROVASCULAR NETWORKS. INTERNATIONAL JOURNAL OF RADIATION ONCOLOGY BIOLOGY PHYSICS, 25(3), 481-489.
• Secomb, T. W., & Pries, A. R. (2002). Information transfer in microvascular networks. MICROCIRCULATION, 9(5), 377-387.
• Venitz, J., Secomb, T. W., Lin, P. S., Kavanagh, B. D., Hsu, R., & Dewhirst, M. W. (2002). A theoretical model for the effects of reduced hemoglobin-oxygen affinity on tumor oxygenation.. International journal of radiation oncology, biology, physics, 53(1), 172-9. doi:10.1016/s0360-3016(02)02740-2
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  To develop a theoretical model for oxygen delivery to tumors, and to use the model to simulate the effects of changing the affinity of hemoglobin for oxygen on tumor oxygenation..Hemoglobin affinity is expressed in terms of P(50), the partial pressure of oxygen (Po(2)) at half saturation. Effects of changing P(50) on arterial Po(2) are predicted using an effective vessel approach to describe diffusive oxygen transport in the lungs, assuming fixed systemic oxygen demand and fixed blood flow rate. The decline in oxygen content of blood as it flows through normal tissue before entering the tumor region is assumed fixed. The hypoxic fraction of the tumor region is predicted using a three-dimensional simulation of diffusion from a network of vessels whose geometry is derived from observations of tumor microvasculature in the rat..In air-breathing rats, predicted hypoxic fraction decreases with moderate increases in P(50), but increases with further increases of P(50), in agreement with previous experimental results. In rats breathing hyperoxic gases, and in humans breathing either normoxic or hyperoxic gases, increased P(50) is predicted to improve tumor oxygenation..The results support the administration of synthetic agents to increase P(50) during radiation treatment of tumors.
• Yu, D., Yarmolenko, P., Secomb, T. W., Dewhirst, M. W., Cardenas-navia, L. I., & Braun, R. D. (2002). Characterizations of steady state and temporal variation in oxygenation of a rat fibrosarcoma (FSA) and 9L glioma. International Journal of Radiation Oncology Biology Physics, 54(2), 228. doi:10.1016/s0360-3016(02)03453-3
• Zakrzewicz, A., Zakrzewicz, A., Secomb, T. W., Secomb, T. W., Pries, A. R., & Pries, A. R. (2002). Angioadaptation: keeping the vascular system in shape.. News in physiological sciences : an international journal of physiology produced jointly by the International Union of Physiological Sciences and the American Physiological Society, 17(5), 197-201. doi:10.1152/nips.01395.2001
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  The development and maintenance of the vascular system requires not only the formation of new vessels (vasculogenesis, angiogenesis) but also the continuous adjustment of vessel and network structures in response to functional needs. This "angioadaptation" depends on the interplay of vascular responses to growth factors, to the metabolic status of the tissue, and to hemodynamic forces exerted by the flowing blood.
• Cardenas-Navia, L., Yu, D. H., Braun, R. D., Brizel, D. M., Secomb, T. W., & Dewhirst, M. W. (2001). Tumor-dependent kinetics of partial pressure of oxygen fluctuations during air and oxygen breathing. CANCER RESEARCH, 64(17), 6010-6017.
• HALPERN, D., & SECOMB, T. W. (2001). THE SQUEEZING OF RED-BLOOD-CELLS THROUGH PARALLEL-SIDED CHANNELS WITH NEAR-MINIMAL WIDTHS. JOURNAL OF FLUID MECHANICS, 244, 307-322.
• Pries, A. R., & Secomb, T. W. (2001). Microvascular blood viscosity in vivo and the endothelial surface layer. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 289(6), H2657-H2664.
• Pries, A. R., Cornelissen, A., Sloot, A. A., Hinkeldey, M., Dreher, M. R., Hoepfner, M., Dewhirst, M. W., & Secomb, T. W. (2001). Structural Adaptation and Heterogeneity of Normal and Tumor Microvascular Networks. PLOS COMPUTATIONAL BIOLOGY, 5(5).
• SECOMB, T. W., HSU, R., ONG, E. T., GROSS, J. F., & DEWHIRST, M. W. (2001). ANALYSIS OF THE EFFECTS OF OXYGEN-SUPPLY AND DEMAND ON HYPOXIC FRACTION IN TUMORS. ACTA ONCOLOGICA, 34(3), 313-316.
• Secomb, T. W., & Mcguire, B. J. (2001). A theoretical model for oxygen transport in skeletal muscle under conditions of high oxygen demand.. Journal of applied physiology (Bethesda, Md. : 1985), 91(5), 2255-65. doi:10.1152/jappl.2001.91.5.2255
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  Oxygen transport from capillaries to exercising skeletal muscle is studied by use of a Krogh-type cylinder model. The goal is to predict oxygen consumption under conditions of high demand, on the basis of a consideration of transport processes occurring at the microvascular level. Effects of the decline in oxygen content of blood flowing along capillaries, intravascular resistance to oxygen diffusion, and myoglobin-facilitated diffusion are included. Parameter values are based on human skeletal muscle. The dependence of oxygen consumption on oxygen demand, perfusion, and capillary density are examined. When demand is moderate, the tissue is well oxygenated and consumption is slightly less than demand. When demand is high, capillary oxygen content declines rapidly with axial distance and radial oxygen transport is limited by diffusion resistance within the capillary and the tissue. Under these conditions, much of the tissue is hypoxic, consumption is substantially less than demand, and consumption is strongly dependent on capillary density. Predicted consumption rates are comparable with experimentally observed maximal rates of oxygen consumption.
• Secomb, T. W., El-kareh, A. W., Secomb, T. W., & El-kareh, A. W. (2001). A theoretical model for the elastic properties of very soft tissues.. Biorheology, 38(4), 305-17.
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  A theoretical model is developed to predict the elastic properties of very soft tissues such as glands, tumors and brain. Tissues are represented as regular arrays of polyhedral (cubic or tetrakaidecahedral) cells, surrounded by extracellular spaces of uniform width. Cells are assumed to be incompressible, with very low resistance to shear deformation. Tissue shear rigidity is assumed to result mainly from the extracellular matrix, which is treated as a compressible elastic mesh of interconnected fibers. Small-strain elastic properties of tissue are predicted using a finite-element method and an analytical method. The model can be used to estimate the compressibility of a very soft tissue based on its Young's modulus and extracellular volume fraction.
• Secomb, T. W., Hsu, R., Park, E., & Dewhirst, M. W. (2001). Green's function methods for analysis of oxygen delivery to tissue by microvascular networks. ANNALS OF BIOMEDICAL ENGINEERING, 32(11), 1519-1529.
• Secomb, T. W., Laughlin, H. M., Gruionu, G., & Constantinescu, G. M. (2001). A hemodynamic study of the arteriolar arcades in the swine triceps brachii muscle. The FASEB Journal, 15(4).
• Carlson, B. E., Arciero, J. C., & Secomb, T. W. (2000). Theoretical model of blood flow autoregulation: roles of myogenic, shear-dependent, and metabolic responses. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 295(4), H1572-H1579.
• El-Kareh, A., & Secomb, T. W. (2000). Theoretical models for drug delivery to solid tumors. CRITICAL REVIEWS IN BIOMEDICAL ENGINEERING, 25(6), 503-571.
• El-kareh, A. W., Secomb, T. W., Secomb, T. W., & El-kareh, A. W. (2000). A mathematical model for comparison of bolus injection, continuous infusion, and liposomal delivery of doxorubicin to tumor cells.. Neoplasia (New York, N.Y.), 2(4), 325-38. doi:10.1038/sj.neo.7900096
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  Determining the optimal mode of delivery for doxorubicin is important given the wide use of the drug against many tumor types. The relative performances of bolus injection, continuous infusion, liposomal and thermoliposomal delivery are not yet definitely established from clinical trials. Here, a mathematical model is used to compare bolus injection, continuous infusion for various durations, liposomal and thermoliposomal delivery of doxorubicin. Effects of the relatively slow rate, and saturability, of doxorubicin uptake by cells are included. Peak concentrations attained in tumor cells are predicted and used as a measure of antitumor effectiveness. To measure toxicity, plasma area under the curve (AUC) and peak plasma concentrations of free doxorubicin are computed. For continuous infusion, the duration of infusion significantly affects predicted outcome. The optimal infusion duration increases with dose, and is in the range 1 to 3 hours at typical doses. The simulations suggest that continuous infusion for optimal durations is superior to the other protocols. Nonthermosensitive liposomes approach the efficacy of continuous infusion only if they release drug at optimal rates. Predictions for thermosensitive liposomes indicate a potential advantage at some doses, but only if hyperthermia is applied locally so that the blood is not significantly heated.
• El-kareh, A. W., Secomb, T. W., Secomb, T. W., & El-kareh, A. W. (2000). A model for red blood cell motion in bifurcating microvessels. International Journal of Multiphase Flow, 26(9), 1545-1564. doi:10.1016/s0301-9322(99)00096-8
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  Abstract A theoretical model is developed for red blood cell motion in a diverging microvessel bifurcation, where the downstream branches are equal in size but receive different flows. The model is used to study migration of red cells across streamlines of the underlying flow, due to particle shape and flow asymmetry. Effects of cell–cell interactions are neglected. Shapes of flowing red cells are approximated by rigid spherical caps. In uniform shear flows, such particles rotate periodically and oscillate about fluid streamlines with no net migration. However, net migration can occur in non-uniform flows due to the particles’ lack of fore-aft symmetry. A nonuniform flow field representative of a bifurcation is developed: flow bounded by two parallel plates, and divided by a cylindrical post. Significant migration is found to occur only with a nonuniform and asymmetric distribution of upstream orientations. The model suggests that the assumption made in previous models of bifurcations, that red cells follow fluid streamlines, is justified if cells approach the bifurcations with random orientations.
• Erickson, K., Braun, R. A., Yu, D. H., Lanzen, J., Wilson, D., Brizel, D. M., Secomb, T. W., Biaglow, J. E., & Dewhirst, M. W. (2000). Effect of longitudinal oxygen gradients on effectiveness of manipulation of tumor oxygenation. CANCER RESEARCH, 63(15), 4705-4712.
• Gruionu, G., Hoying, J. B., Pries, A. R., & Secomb, T. W. (2000). Structural remodeling of mouse gracilis artery after chronic alteration in blood supply. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 288(5), H2047-H2054.
• LILLIOJA, S., YOUNG, A. A., CULTER, C. L., IVY, J. L., ABBOTT, W., ZAWADZKI, J. K., YKIJARVINEN, H., CHRISTIN, L., SECOMB, T. W., & BOGARDUS, C. (2000). SKELETAL-MUSCLE CAPILLARY DENSITY AND FIBER TYPE ARE POSSIBLE DETERMINANTS OF INVIVO INSULIN RESISTANCE IN MAN. JOURNAL OF CLINICAL INVESTIGATION, 80(2), 415-424.
• Pries, A. R., & Secomb, T. W. (2000). Origins of heterogeneity in tissue perfusion and metabolism. CARDIOVASCULAR RESEARCH, 81(2), 328-335.
• HSU, R., & SECOMB, T. W. (1999). ANALYSIS OF OXYGEN-EXCHANGE BETWEEN ARTERIOLES AND SURROUNDING CAPILLARY-PERFUSED TISSUE. JOURNAL OF BIOMECHANICAL ENGINEERING-TRANSACTIONS OF THE ASME, 114(2), 227-231.
• Secomb, T. W., & Hsu, R. (1999). Resistance to blood flow in nonuniform capillaries. MICROCIRCULATION-LONDON, 4(4), 421-427.
• Secomb, T. W., Raghunand, N., Secomb, T. W., Schornack, P. A., Schomack, P. A., Raghunand, N., & Gillies, R. J. (1999). Causes and effects of heterogeneous perfusion in tumors.. Neoplasia (New York, N.Y.), 1(3), 197-207. doi:10.1038/sj.neo.7900037
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  A characteristic of solid tumors is their heterogeneous distribution of blood flow, with significant hypoxia and acidity in low-flow regions. We review effects of heterogeneous tumor perfusion are reviewed and propose a conceptual model for its cause. Hypoxic-acidic regions are resistant to chemo- and radiotherapy and may stimulate progression to a more metastatic phenotype. In normal tissues, hypoxia and acidity induce angiogenesis, which is expected to improve perfusion. However, aggressive tumors can have high local microvessel density simultaneously with significant regions of hypoxia and acidosis. A possible explanation for this apparent contradiction is that the mechanisms regulating growth and adaptation of vascular networks are impaired. According to a recent theory for structural adaptation of vascular networks, four interrelated adaptive responses can work as a self-regulating system to produce a mature and efficient blood distribution system in normal tissues. It is proposed that heterogeneous perfusion in tumors may result from perturbation of this system. Angiogenesis may increase perfusion heterogeneity in tumors by increasing the disparity between parallel low- and high-resistance flow pathways. This conceptual model provides a basis for future rational therapies. For example, it indicates that selective destruction of tumor vasculature may increase perfusion efficiency and improve therapeutic efficacy.
• El-Kareh, A., & Secomb, T. W. (1998). Two-mechanism peak concentration model for cellular pharmacodynamics of doxorubicin. NEOPLASIA, 7(7), 705-713.
• HSU, R., & SECOMB, T. W. (1998). A GREEN-FUNCTION METHOD FOR ANALYSIS OF OXYGEN DELIVERY TO TISSUE BY MICROVASCULAR NETWORKS. MATHEMATICAL BIOSCIENCES, 96(1), 61-78.
• Hsu, R., Secomb, T. W., Pries, A. R., & Hsu, R. (1998). A model for red blood cell motion in glycocalyx-lined capillaries.. The American journal of physiology, 274(3), H1016-22. doi:10.1152/ajpheart.1998.274.3.h1016
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  The interior surfaces of capillaries are lined with a layer (glycocalyx) of macromolecules bound or absorbed to the endothelium. Here, a theoretical model is used to analyze the effects of the glycocalyx on hematocrit and resistance to blood flow in capillaries. The glycocalyx is represented as a porous layer that resists penetration by red blood cells. Axisymmetric red blood cell shapes are assumed, and effects of cell membrane shear elasticity are included. Lubrication theory is used to compute the flow of plasma around the cell and within the glycocalyx. The effects of the glycocalyx on tube hematocrit (Fahraeus effect) and on flow resistance are predicted as functions of the width and hydraulic resistivity of the layer. A layer of width 1 micron and resistivity 10(8) dyn.s/cm4 leads to a relative apparent viscosity of approximately 10 in a 6-micron capillary at discharge hematocrit 45% and flow velocity of approximately 1 mm/s. This is consistent with experimental observations of increased flow resistance in microvessels in vivo, relative to glass tubes with the same diameters.
• Secomb, T. W. (1998). FLOW IN A CHANNEL WITH PULSATING WALLS. JOURNAL OF FLUID MECHANICS, 88(SEP), 273-288.
• Secomb, T. W., Hsu, R., & Pries, A. R. (1998). Blood flow and red blood cell deformation in nonuniform capillaries: Effects of the endothelial surface layer. MICROCIRCULATION, 9(3), 189-196.
• Sperandio, M., Secomb, T. W., Pries, A. R., & Gaehtgens, P. (1998). Blood flow resistance during hemodilution: effect of plasma composition.. Cardiovascular research, 37(1), 225-35. doi:10.1016/s0008-6363(97)00226-5
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  To investigate the causes of wide variations in reported effects of hemodilution on flow resistance of vascular beds..(a) In a meta-analysis of 28 prior studies, resistance values at hematocrits of zero (R0) and 0.45 (R0.45) were derived. Study design characteristics (presence of vasodilatory reserve or leukocytes, species, tissue, hemodiluent) were tested by ANOVA for their relation to the ratio R0/R0.45. (b) Experiments were performed to determine flow resistance during hemodilution in the rat mesentery with (n = 8) and without (n = 11) pretreatment with heparinase, which modifies the endothelial glycocalyx. (c) A mathematical flow simulation for mesenteric microvascular networks was used to predict resistance effects of hemodilution and of a hypothetical layer on the endothelial surface..(a) In prior studies using native plasma for hemodilution R0 averaged 50 +/- 8% of R0.45, while in studies using artificial solutions R0 averaged 32 +/- 12% of R0.45. The larger reduction of flow resistance upon dilution with artificial media is independent of viscosity and oncotic pressure. Other design characteristics did not show strong significant effects. (b) Present experiments showed large reductions of flow resistance with saline hemodilution which were nearly halved after heparinase pretreatment. (c) Resistance effects of hemodilution with plasma or after heparinase treatment agree with model predictions based on tube flow rheology of blood. The larger resistance effects of dilution with artificial media can be explained by the removal of an endothelial surface layer of approximately 1.5 microns thickness..The results imply that changes of plasma composition, due to use of artificial infusion media, influence peripheral resistance and tissue perfusion. They are consistent with the hypothesis that interactions between endothelial glycocalyx structures and plasma components lead to formation of a thick layer at the endothelial surface which increases flow resistance.
• Pries, A. R., Reglin, B., & Secomb, T. W. (1997). Structural response of microcirculatory networks to changes in demand: information transfer by shear stress. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 284(6), H2204-H2212.
• Secomb, T. W. (1997). FLOW-DEPENDENT RHEOLOGICAL PROPERTIES OF BLOOD IN CAPILLARIES. MICROVASCULAR RESEARCH, 34(1), 46-58.
• Secomb, T. W., Konerding, M. A., West, C. A., Su, M., Young, A. J., & Mentzer, S. J. (1997). Microangiectasias: Structural regulators of lymphocyte transmigration. PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, 100(12), 7231-7234.
• Carlson, B. E., & Secomb, T. W. (1996). A theoretical model for the myogenic response based on the length-tension characteristics of vascular smooth muscle. MICROCIRCULATION, 12(4), 327-338.
• ELKAREH, A. W., BRAUNSTEIN, S. L., & SECOMB, T. W. (1996). EFFECT OF CELL ARRANGEMENT AND INTERSTITIAL VOLUME FRACTION ON THE DIFFUSIVITY OF MONOCLONAL-ANTIBODIES IN TISSUE. BIOPHYSICAL JOURNAL, 64(5), 1638-1646.
• Hicks, K. O., Pruijn, F. B., Secomb, T. W., Hay, M. P., Hsu, R., Brown, J. M., Denny, W. A., Dewhirst, M. W., & Wilson, W. R. (1996). Use of three-dimensional tissue cultures to model extravascular transport and predict in vivo activity of hypoxia-targeted anticancer drugs. JOURNAL OF THE NATIONAL CANCER INSTITUTE, 98(16), 1118-1128.
• Pries, A. R., Hoepfner, M., le, N. F., Dewhirst, M. W., & Secomb, T. W. (1996). The shunt problem: control of functional shunting in normal and tumour vasculature. NATURE REVIEWS CANCER, 10(8), 587-593.
• Secomb, T. W. (1996). RED-BLOOD-CELL MECHANICS AND CAPILLARY BLOOD RHEOLOGY. CELL BIOPHYSICS, 18(3), 231-251.
• Secomb, T. W., & Hsu, R. (1996). Analysis of red blood cell motion through cylindrical micropores: Effects of cell properties. BIOPHYSICAL JOURNAL, 71(2), 1095-1101.
• Secomb, T. W., Alberding, J. P., Hsu, R., Dewhirst, M. W., & Pries, A. R. (1996). Angiogenesis: An Adaptive Dynamic Biological Patterning Problem. PLOS COMPUTATIONAL BIOLOGY, 9(3).
• Secomb, T. W., Pries, A. R., & Gaehtgens, P. (1996). Biophysical aspects of blood flow in the microvasculature. Cardiovascular Research, 32(4), 654-667. doi:10.1016/s0008-6363(96)00065-x
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  The main function of the microvasculature is transport of materials. Water and solutes are carried by blood through the microvessels and exchanged, through vessel walls, with the surrounding tissues. This transport function is highly dependent on the architecture of the microvasculature and on the biophysical behavior of blood flowing through it. For example, the hydrodynamic resistance of a microvascular network, which determines the overall blood flow for a given perfusion pressure, depends on the number, size and arrangement of microvessels, the passive and active mechanisms governing their diameters, and on the apparent viscosity of blood flowing in them. Suspended elements in blood, especially red blood cells, strongly influence the apparent viscosity, which varies with several factors, including vessel diameter, hematocrit and blood flow velocity. The distribution of blood flows and red cell fluxes within a network, which influences the spatial pattern of mass transport, is determined by the mechanics of red cell motion in individual diverging bifurcations. Here, our current understanding of the biophysical processes governing blood flow in the microvasculature is reviewed, and some directions for future research are indicated.
• Shan, S., Secomb, T. W., Rosner, G. L., Rehmus, S. W., Ong, E. T., Dewhirst, M. W., Brizel, D. M., & Braun, R. D. (1996). Arteriolar oxygenation in tumour and subcutaneous arterioles: effects of inspired air oxygen content.. The British journal of cancer. Supplement, 27, S241-6.
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  Carbogen is thought to be more effective than normobaric oxygen in reducing tumour hypoxia because it may reduce hyperoxic vasoconstriction. In this study, tumour and normal arteriolar diameters were measured simultaneously with perivascular pO2 during air breathing followed by either carbogen or 100% oxygen to determine whether the action of carbogen is the result of alterations in feeding vessel diameter. Fischer-344 rats bearing dorsal flap window chambers, with or without implanted R3230AC tumours, were the experimental subjects. Arteriolar diameters were measured using optical techniques and perivascular pO2 was measured using recessed-tip electrodes (3-6 microns tip diameter). Baseline arteriolar pO2 averaged 30-50% of blood gas pO2 (mean = 97 mmHg). Both hyperoxic gases increased blood gas pO2 by 4-to 5-fold, but relative improvements in arteriolar pO2 were < or = 2.5 for all arterioles studied. This means that these normobaric high O2 gases are not very efficient in increasing O2 delivery to tumours. In addition, improvements in tumour arteriolar pO2 were transient for both hyperoxic gases. Oxygen and carbogen caused no change and mild vasodilatory responses in tumour arterioles, respectively. Normal arterioles on the other hand, tended toward vasoconstriction by carbogen breathing. Peri-arteriolar pO2 in tumours increased within the first 5 min of breathing either hyperoxic gas, followed by a decline back toward values seen with air-breathing. These results suggest that temporal changes in tumour oxygenation after exposure to carbogen or O2 may not be due to changes in perfusion. Other factors, such as changes in O2 consumption rate may be involved.
• PRIES, A. R., SECOMB, T. W., GAEHTGENS, P., & GROSS, J. F. (1995). BLOOD-FLOW IN MICROVASCULAR NETWORKS - EXPERIMENTS AND SIMULATION. CIRCULATION RESEARCH, 67(4), 826-834.
• PRIES, A. R., SECOMB, T. W., GESSNER, T., SPERANDIO, M. B., GROSS, J. F., & GAEHTGENS, P. (1995). RESISTANCE TO BLOOD-FLOW IN MICROVESSELS IN-VIVO. CIRCULATION RESEARCH, 75(5), 904-915.
• Pries, A. R., Reglin, B., & Secomb, T. W. (1995). Structural adaptation of microvascular networks: functional roles of adaptive responses. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 281(3), H1015-H1025.
• Secomb, T. W., Ong, E. T., Hsu, R., Gross, J. F., & Dewhirst, M. W. (1995). Analysis of the effects of oxygen supply and demand on hypoxic fraction in tumors.. Acta oncologica (Stockholm, Sweden), 34(3), 313-6. doi:10.3109/02841869509093981
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  The extent of hypoxic regions in a tumor tissue depends on the arrangement, blood flow rate and blood oxygen content of microvessels, and on the tissue's oxygen consumption rate. Here, the effects of blood flow rate, blood oxygen content and oxygen consumption on hypoxic fraction are simulated theoretically, for a region whose microvascular geometry was derived from observations of a transplanted mammary andenocarcinoma (R3230AC) in a rat dorsal skin flap preparation. In the control state, arterial PO2 is 100 mmHg, consumption rate is 2.4 cm3 O2/100 g/min, and hypoxic fraction (tissue with PO2 < 3 mmHg) is 30%. Hypoxia is abolished by a reduction in consumption rate of at least 30%, relative to control, or an increase in flow rate by a factor of 4 or more, or an increase in arterial PO2 by a factor of 11 or more. These results suggest that reducing oxygen consumption rate may be more effective than elevating blood flow or oxygen content as a method to reduce tumor hypoxia.
• Dewhirst, M. W., Kimura, H., Rehmus, S., Braun, R. D., Papahadjopoulos, D., Hong, K., & Secomb, T. W. (1994). Microvascular studies on the origins of perfusion-limited hypoxia. BRITISH JOURNAL OF CANCER, 74, S247-S251.
• Dewhirst, M. W., Klitzman, B., Braun, R. D., Brizel, D. M., Haroon, Z. A., & Secomb, T. W. (1994). Review of methods used to study oxygen transport at the microcirculatory level. INTERNATIONAL JOURNAL OF CANCER, 90(5), 237-255.
• PRIES, A. R., SECOMB, T. W., & GAEHTGENS, P. (1994). STRUCTURE AND HEMODYNAMICS OF MICROVASCULAR NETWORKS - HETEROGENEITY AND CORRELATIONS. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 269(5), H1713-H1722.
• Secomb, T. W., El-kareh, A. W., Secomb, T. W., & El-kareh, A. W. (1994). A model for motion and sedimentation of cylindrical red-cell aggregates during slow blood flow in narrow horizontal tubes.. Journal of biomechanical engineering, 116(3), 243-9. doi:10.1115/1.2895726
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  When blood flows slowly in a narrow tube, red-cell aggregation results in formation of an approximately cylindrical "core" of red cells, which moves as a rigid body. The core is denser than the surrounding fluid, and sedimentation is observed in horizontal tubes. To model this, the Stokes flow of a fluid surrounding a long solid cylinder (the core) contained in a long hollow cylinder (the tube) is considered. The cylinder axes are parallel but not coincident. An exact analytic expression for the resistance coefficient for motion perpendicular to the axes is given. This coefficient increases rapidly with the ratio of core radius to tube radius, and core eccentricity. The predicted rate of sedimentation is comparable to that observed experimentally. The apparent viscosity of a two-phase medium consisting of a core of aggregated particles and surrounding pure fluid is calculated. For a core radius corresponding to experimental conditions, the apparent viscosity increases rapidly with increasing eccentricity of the core.
• Hsu, R., Gross, J. F., Secomb, T. W., Klitzman, B., Hsu, R., Gross, J. F., & Dewhirst, M. W. (1993). Analysis of oxygen transport to tumor tissue by microvascular networks.. International journal of radiation oncology, biology, physics, 25(3), 481-9. doi:10.1016/0360-3016(93)90070-c
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  We present theoretical simulations of oxygen delivery to tumor tissues by networks of microvessels, based on in vivo observations of vascular geometry and blood flow in the tumor microcirculation. The aim of these studies is to investigate the impact of vascular geometry on the occurrence of tissue hypoxia. The observations were made in the tissue (thickness 200 microns) contained between two glass plates in a dorsal skin flap preparation in the rat. Mammary adenocarcinomas (R3230 AC) were introduced and allowed to grow, and networks of microvessels in the tumors were mapped, providing data on length, geometric orientation, diameter and blood velocity in each segment. Based on these data, simulations were made of a 1 mm x 1 mm region containing five unbranched vascular segments and a 0.25 mm x 0.35 mm region containing 22 segments. Generally, vessels were assumed to lie in the plane midway between the glass plates, at 100 microns depth. Flow rates in the vessels were based on measured velocities and diameters. The assumed rate of oxygen consumption in the tissue was varied over a range of values. Using a Green's function method, partial pressure of oxygen (PO2) was computed at each point in the tissue region. As oxygen consumption is increased, tissue PO2 falls, with hypoxia first appearing at points relatively distant from the nearest blood vessel. The width of the well-oxygenated region is comparable to that predicted by simpler analyses. Cumulative frequency distributions of tissue PO2 were compared with predictions of a Krogh-type model with the same vascular densities, and it was found that the latter approach, which assumes a uniform spacing of vessels, may underestimate the extent of the hypoxic tissue. Our estimates of the maximum consumption rate that can be sustained without tissue hypoxia were substantially lower than those obtained from the Krogh-type model. We conclude that the heterogeneous structure of tumor microcirculation can have a substantial effect on the occurrence of hypoxic micro-regions.
• Lanzen, J., Braun, R. D., Klitzman, B., Brizel, D., Secomb, T. W., & Dewhirst, M. W. (1993). Direct demonstration of instabilities in oxygen concentrations within the extravascular compartment of an experimental tumor. CANCER RESEARCH, 66(4), 2219-2223.
• Pries, A. R., Secomb, T. W., Sperandio, M., & Gaehtgens, P. (1993). Blood flow resistance during hemodilution: effect of plasma composition. CARDIOVASCULAR RESEARCH, 37(1), 225-235.
• Barber, J. O., Alberding, J. P., Restrepo, J. M., & Secomb, T. W. (1992). Simulated two-dimensional red blood cell motion, deformation, and partitioning in microvessel bifurcations. ANNALS OF BIOMEDICAL ENGINEERING, 36(10), 1690-1698.
• Dunn, T. J., Braun, R. D., Rhemus, W. E., Rosner, G. L., Secomb, T. W., Tozer, G. M., Chaplin, D. J., & Dewhirst, M. W. (1992). The effects of hyperoxic and hypercarbic gases on tumour blood flow. BRITISH JOURNAL OF CANCER, 80(1-2), 117-126.
• El-Kareh, A., & Secomb, T. W. (1992). A model for red blood cell motion in bifurcating microvessels. INTERNATIONAL JOURNAL OF MULTIPHASE FLOW, 26(9), 1545-1564.
• Secomb, T. W., & Hsu, R. (1992). Analysis of oxygen exchange between arterioles and surrounding capillary-perfused tissue.. Journal of biomechanical engineering, 114(2), 227-31. doi:10.1115/1.2891376
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  A theoretical model is used to analyze oxygen transport in a three-dimensional tissue region containing an arteriole surrounded by an array of capillaries in planes perpendicular to the arteriole. Convective removal of oxygen from the vicinity of the arteriole by nearby capillaries is shown to increase diffusive oxygen loss from the arteriole. This effect depends on the locations of the capillaries, particularly those nearest to the arteriole. The arteriolar oxygen efflux is comparable to that predicted by a previous model which used a continuum approach, but the efflux does not increase with increasing perfusion as rapidly as predicted by the continuum model. Even a small capillary flow rate strongly influences the oxygen field surrounding the arteriole.
• Secomb, T. W., Styp-Rekowska, B., & Pries, A. R. (1992). Two-dimensional simulation of red blood cell deformation and lateral migration in microvessels. ANNALS OF BIOMEDICAL ENGINEERING, 35(5), 755-765.
• Waters, S. L., Alastruey, J., Beard, D. A., Bovendeerd, P., Davies, P. F., Jayaraman, G., Jensen, O. E., Lee, J., Parker, K. H., Popel, A. S., Secomb, T. W., Siebes, M., Sherwin, S. J., Shipley, R. J., Smith, N. P., & van, d. (1991). Theoretical models for coronary vascular biomechanics: Progress & challenges. PROGRESS IN BIOPHYSICS & MOLECULAR BIOLOGY, 104(1-3), 49-76.
• Secomb, T. W., Pries, A. R., Gross, J. F., & Gaehtgens, P. (1990). Blood flow in microvascular networks. Experiments and simulation.. Circulation research, 67(4), 826-34. doi:10.1161/01.res.67.4.826
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  A theoretical model has been developed to simulate blood flow through large microcirculatory networks. The model takes into account the dependence of apparent viscosity of blood on vessel diameter and hematocrit (the Fahraeus-Lindqvist effect), the reduction of intravascular hematocrit relative to the inflow hematocrit of a vessel (the Fahraeus effect), and the disproportionate distribution of red blood cells and plasma at arteriolar bifurcations (phase separation). The model was used to simulate flow in three microvascular networks in the rat mesentery with 436,583, and 913 vessel segments, respectively, using experimental data (length, diameter, and topological organization) obtained from the same networks. Measurements of hematocrit and flow direction in all vessel segments of these networks tested the validity of model results. These tests demonstrate that the prediction of parameters for individual vessel segments in large networks exhibits a high degree of uncertainty; for example, the squared coefficient of correlation between predicted and measured hematocrit of single vessel segments ranges only between 0.15 and 0.33. In contrast, the simulation of integrated characteristics of the network hemodynamics, such as the mean segment hematocrit or the distribution of blood flow velocities, is very precise. In addition, the following conclusions were derived from the comparison of predicted and measured values: 1) The low capillary hematocrits found in mesenteric microcirculatory networks as well as their heterogeneity can be explained on the basis of the Fahraeus effect and phase-separation phenomena. 2) The apparent viscosity of blood in vessels of the investigated tissue with diameters less than 15 microns is substantially higher than expected compared with measurements in glass tubes with the same diameter.
• Skotheim, J. M., & Secomb, T. W. (1990). Red blood cells and other nonspherical capsules in shear flow: Oscillatory dynamics and the tank-treading-to-tumbling transition. PHYSICAL REVIEW LETTERS, 98(7).
• Zakrzewicz, A., Secomb, T. W., & Pries, A. R. (1990). Angioadaptation: Keeping the vascular system in shape. NEWS IN PHYSIOLOGICAL SCIENCES, 17, 197-201.
• Kimura, H., Braun, R. D., Ong, E. T., Hsu, R., Secomb, T. W., Papahadjopoulos, D., Hong, K. L., & Dewhirst, M. W. (1989). Fluctuations in red cell flux in tumor microvessels can lead to transient hypoxia and reoxygenation in tumor parenchyma. CANCER RESEARCH, 56(23), 5522-5528.
• Konerding, M. A., Turhan, A., Ravnic, D. J., Lin, M., Fuchs, C., Secomb, T. W., Tsuda, A., & Mentzer, S. J. (1989). Inflammation-Induced Intussusceptive Angiogenesis in Murine Colitis. ANATOMICAL RECORD-ADVANCES IN INTEGRATIVE ANATOMY AND EVOLUTIONARY BIOLOGY, 293(5), 849-857.
• Pries, A. R., & Secomb, T. W. (1989). Structural adaptation of microvascular networks and development of hypertension. MICROCIRCULATION, 9(4), 305-314.
• SECOMB, T. W., & HSU, R. (1989). SIMULATION OF O-2 TRANSPORT IN SKELETAL-MUSCLE - DIFFUSIVE EXCHANGE BETWEEN ARTERIOLES AND CAPILLARIES. AMERICAN JOURNAL OF PHYSIOLOGY, 267(3), H1214-H1221.
• SECOMB, T. W., INTAGLIETTA, M., & GROSS, J. F. (1989). EFFECTS OF VASOMOTION ON MICROCIRCULATORY MASS-TRANSPORT - THEORETICAL PREDICTIONS. VASOMOTION AND FLOW MODULATION IN THE MICROCIRCULATION, 15, 49-61.
• Secomb, T. W., & Hsu, R. (1989). A Green's function method for analysis of oxygen delivery to tissue by microvascular networks.. Mathematical biosciences, 96(1), 61-78. doi:10.1016/0025-5564(89)90083-7
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  A theoretical model is formulated for analyzing oxygen delivery from an arbitrary network configuration of cylindrical microvessels to a finite region of tissue. In contrast to models based on the classical Krogh cylinder approach, this model requires no a priori assumptions concerning the extent of the tissue region supplied with oxygen by each vessel segment. Steady-state conditions are assumed, and oxygen consumption in the tissue is assumed to be uniform. The nonlinear dissociation characteristics of oxyhemoglobin are taken into account. A computationally efficient Green's function approach is used, in which the tissue oxygen field is expressed in terms of the distribution of source strengths along each segment. The utility of the model is illustrated by analyses of oxygen delivery to a cuboidal tissue region by a single segment and by a six-segment network. It is found that the fractional contribution of the proximal segments to total oxygen delivery increases with decreasing flow rate and metabolic rate.
• Snyder, S. A., Lanzen, J. L., Braun, R. D., Rosner, G., Secomb, T. W., Biaglow, J., Brizel, D. M., & Dewhirst, M. W. (1989). Simultaneous administration of glucose and hyperoxic gas achieves greater improvement in tumor oxygenation than hyperoxic gas alone. INTERNATIONAL JOURNAL OF RADIATION ONCOLOGY BIOLOGY PHYSICS, 51(2), 494-506.
• Pries, A. R., & Secomb, T. W. (1988). Control of blood vessel structure: insights from theoretical models. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 288(3), H1010-H1015.
• Secomb, T. W., & Hsu, R. (1988). Analysis of oxygen delivery to tissue by microvascular networks.. Advances in experimental medicine and biology, 222, 95-103. doi:10.1007/978-1-4615-9510-6_11
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  Theoretical modeling of oxygen delivery to tissue provides a means to obtain information about oxygen concentration fields in tissue at the scale of individual capillaries. Such information is difficult or impossible to obtain with available experimental techniques. Theoretical models can provide insights into the relationship between perfusion and metabolic demand in tissue, and also contribute to the understanding of metabolic mechanisms of blood flow regulation. The value of theoretical analyses was recognized in the classic work of Krogh (1918) on oxygen diffusion from parallel arrays of capillaries, and since then most models of oxygen delivery to tissue have been modifications and extensions of the Krogh model (Middleman, 1972). Generally, these models have retained the assumption that each point in the tissue receives oxygen only from the nearest capillary. However, this assumption was relaxed, for the case of multiple parallel capillaries, in studies by Popel (1978, 1980), Salathe (1982) and Klitzman et al (1983).
• Pries, A. R., Secomb, T. W., & Gaehtgens, P. (1987). Structural autoregulation of terminal vascular beds - Vascular adaptation and development of hypertension. HYPERTENSION, 33(1), 153-161.
• Pries, A. R., Secomb, T. W., & Gaehtgens, P. (1987). The endothelial surface layer. PFLUGERS ARCHIV-EUROPEAN JOURNAL OF PHYSIOLOGY, 440(5), 653-666.
• SECOMB, T. W., SKALAK, R., OZKAYA, N., & GROSS, J. F. (1987). FLOW OF AXISYMMETRICAL RED-BLOOD-CELLS IN NARROW CAPILLARIES. JOURNAL OF FLUID MECHANICS, 163, 405-423.
• Secomb, T. W., Wright, S. H., Secomb, T. W., & Bradley, T. J. (1987). Apical Membrane Permeability of Mytilus Gill: Influence of Ultrastructure, Salinity and Competitive Inhibitors on Amino Acid Fluxes. The Journal of Experimental Biology, 129(1), 205-230. doi:10.1242/jeb.129.1.205
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  The apical membrane of gill integumental cells from the mussels Mytilus edulis and M. californianus serves as a permeability barrier separating sea water from a cytoplasm rich in amino acids and other small organic molecules. Morphometric analysis of transmission electronmicrographs indicates that the membrane area of these cells is increased between 10- and 18-fold by the presence of a microvillous brush border. The microvilli do not appear to influence the kinetics of solute transport across the cell apex, as determined using a mathematical model of the relationship between membrane structure and the kinetics of transport. Rates of amino acid loss from the integument were low, and estimates of the upper limit of the passive permeability of the apical membrane to amino acids ranged from 0.5 to 10×10−10 cm s−1. Abrupt exposure of intact mussels or isolated gill tissue to 60% sea water (19% salinity) resulted in a transient, 40- to 80-fold increase in the rate of loss of all amino acids from integumental tissues. Upon exposure to full-strength sea water, efflux rates returned to near control values. Exposure to 60% sea water also inhibited the carrier-mediated accumulation of amino acid: uptake of 0.5 μmol1−1 [14C]alanine and [14C]taurine was reduced by 80% compared to control uptake in 100% sea water. This inhibition was not adequate to account for the increase in net efflux of taurine from gill tissue into 60% artificial sea water (ASW), though the inhibition of alanine uptake may have contributed significantly to the increased loss of this amino acid. Efflux of discrete structural classes of amino acid occurred when integumental tissues were exposed to 50 μmoll−1 concentrations of structurally related analogues. It is concluded that the apical membrane of gill cells has a very low passive permeability to amino acids, and that the overall permeability of the gill can be increased in a reversible fashion by exposure to reduced salinity or to high external concentrations of amino acid.
• Arciero, J. C., Carlson, B. E., & Secomb, T. W. (1986). Theoretical model of metabolic blood flow regulation: roles of ATP release by red blood cells and conducted responses. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 295(4), H1562-H1571.
• Pries, A. R., Secomb, T. W., Jacobs, H., Sperandio, M., Osterloh, K., & Gaehtgens, P. (1986). Microvascular blood flow resistance: role of endothelial surface layer. AMERICAN JOURNAL OF PHYSIOLOGY-HEART AND CIRCULATORY PHYSIOLOGY, 273(5), H2272-H2279.
• Secomb, T. W., Hsu, R., Beamer, N. B., & Couli, B. M. (1986). Theoretical simulation of oxygen transport to brain by networks of microvessels: Effects of oxygen supply and demand on tissue hypoxia. MICROCIRCULATION, 7(4), 237-247.
• Harkins, K. D., Galons, J., Secomb, T. W., & Trouard, T. P. (1984). Assessment of the Effects of Cellular Tissue Properties on ADC Measurements by Numerical Simulation of Water Diffusion. MAGNETIC RESONANCE IN MEDICINE, 62(6), 1414-1422.
• Pries, A. R., & Secomb, T. W. (1983). Modeling Structural Adaptation of Microcirculation. MICROCIRCULATION, 15(8), 753-764.
• El-Kareh, A., & Secomb, T. W. (1982). A theoretical model for intraperitoneal delivery of cisplatin and the effect of hyperthermia on drug penetration distance. NEOPLASIA, 6(2), 117-127.
• SECOMB, T. W., & SKALAK, R. (1982). A TWO-DIMENSIONAL MODEL FOR CAPILLARY-FLOW OF AN ASYMMETRIC CELL. MICROVASCULAR RESEARCH, 24(2), 194-203.
• Skalak, R., & Secomb, T. W. (1982). A two-dimensional model for capillary flow of an asymmetric cell.. Microvascular research, 24(2), 194-203. doi:10.1016/0026-2862(82)90056-5
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  Abstract The effect of asymmetry of cell shape on capillary flow of tightly fitting red blood cells is examined, using a two-dimensional model. The cell membrane is assumed to be perfectly flexible and inextensible, and to contain a viscous fluid interior. Lubrication theory is used to model the flow in the narrow gaps between the cell and the wall. It is shown that asymmetric cell shapes lead to tank-treading motion of the membrane, and also to a reduction in the driving pressure required to sustain a given cell velocity.
• HSU, R., & SECOMB, T. W. (1981). MOTION OF NONAXISYMMETRIC RED-BLOOD-CELLS IN CYLINDRICAL CAPILLARIES. JOURNAL OF BIOMECHANICAL ENGINEERING-TRANSACTIONS OF THE ASME, 111(2), 147-151.
• Pries, A. R., Reglin, B., & Secomb, T. W. (1978). Structural adaptation of vascular networks - Role of the pressure response. HYPERTENSION, 38(6), 1476-1479.
• FLEISCHMAN, G. J., SECOMB, T. W., & GROSS, J. F. (1977). EFFECT OF EXTRAVASCULAR PRESSURE-GRADIENTS ON CAPILLARY FLUID EXCHANGE. MATHEMATICAL BIOSCIENCES, 81(2), 145-164.

Proceedings Publications

• Secomb, T. W., Dewhirst, M. W., Pries, A. R., & Sivaloganathan, S. (2012). Growth and Structural Adaptation of Blood Vessels in Normal and Tumor Tissues. In NEW PERSPECTIVES IN MATHEMATICAL BIOLOGY, 57, 21-35.
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  The circulatory system is a dynamic structure. Blood vessels grow or regress during development and in a variety of normal and disease states, over time scales of hours, days and longer. Under normal conditions, these structural changes ensure that all parts of the tissue are supplied with blood, and that the microvascular network structure is well organized and efficient with regard both to the volume of blood needed and the energy required to drive the flow. Theoretical models have been used to investigate how this is achieved through vessel responses to several stimuli, including wall shear stress, tension in vessel walls, metabolic needs, growth factors, and information transfer along vessel walls, and how perturbations of these processes lead to abnormal structural and functional characteristics in tumor tissues.

Poster Presentations

• Fox, A. E., Secomb, T. W., & Liong, M. (2021, Nov). Dominant mechanism for T cell killing of cancer cells is diffusible factor rather than direct contact kill. University of Arizona Cancer Center Retreat. Tucson AZ: University of Arizona Cancer Center.
• Secomb, T. W., & Fox, A. E. (2021, June). A mathematical model for S phase control. 11th Salk Institute Cell Cycle Meeting. La Jolla, California: Salk Institute, La Jolla CA.