Substrates for cellular metabolism within the parenchymal cells of an organ must undergo convective, diffusive and permeative transport, no matter what further reactions occur. To obtain physiologically meaningful estimates of the parameters for the exchanges, the models used for analyses of data must be obedient to the anatomy and to the recognized processes for exchange, and not provide merely descriptions of kinetics. Though the processes of faciltated transport and enzymatic reaction are inherently non-linear, much can be learned from steady state tracer experiments by using a series of studies at different states or by abandonning steady states in favor of fast chemical transients. Examples of the tradeoffs between these approaches is illustrated by studies on RBC/endothelial/myocyte exchanges for purine nucleosides and nucleotides in the heart, using Ringer-perfused and blood-perfused hearts. Oxygen and carbon dioxide levels, requiring complex models themselves for hemoglobin binding, influence the cellular and capillary purine levels. These multimodel combinations represent an approach to complex biological situations.
The regulation of tissue/organ perfusion is a multi-factorial process wherein a diverse array of contributors produces an integrated outcome. Among these are those that can be considered to be part of the extrinsic regulation of vascular tone through systems including neural control of vascular diameter, the presence of humoral mediators of vascular tone and vasoactive mediators which arise from the metabolic activity of the surrounding parencyhmal tissues. Additionally, a wide array of vasoactive processes arise from pathways intrinsic to microvessels themselves and can include those leading to vasodilation (e.g., wall shear stresses), vasoconstriction (e.g., myogenic or pressure-induced effects) or those that can lead to activation of either dilator or constrictor signaling pathways (e.g., conducted responses). Additionally, the physical structure of individual microvessels, including incremental distensibility and stress versus strain relationships, and the structure of the microvessel networks (i.e., microvessel density), can also have profound implications for the regulation of tissue or organ perfusion and the properties of mass transport and exchange therein.
Of increasing importance is the impact of pathological conditions such as peripheral vascular disease (PVD) on the integration of these contributors for the regulation of tissue perfusion. Notably, with the evolution of pathologies which increase predisposition to the development of PVD (e.g., obesity, insulin-resistance), most identified contributors to the regulation of perfusion can be profoundly impacted, leading to an impairment of vascular function and the development of an ischemic condition. The challenge facing investigators is how to incorporate the myriad information that is being produced into a conceptual framework for an improved understanding of PVD and for the development of more informative, targeted hypotheses.
The importance of the microcirculation in delivering oxygen to tissue is well known; however, many details of this process remain to be understood. The small scale and spatiotemporal complexity of microvascular oxygen delivery present challenges to experimentation that have motivated mathematical and computational studies. Beginning with the work of Krogh and Erlang, experiment-based theoretical models have been used to obtain a number of new insights into the basic transport process. One of the most important aspects of these models is their ability to use intravascular oxygen transport data to estimate details of tissue oxygenation, which are relatively difficult to obtain experimentally. This has enabled the study of structure-function relationships and shed light on issues such as the heterogeneity of tissue oxygen delivery, the development of localized hypoxia, and the interpretation of larger-scale measures of oxygenation. Although the role of spatial complexity has been studied extensively, the microcirculation is a dynamic system and the importance of observed temporal variations remains largely unknown. Currently, computational studies are being performed of oxygen transport during spontaneous physiological oscillations in blood flow (vasomotion) and during the (irregular) time progression of microvascular injury caused by sepsis. The effect of imposed oscillations on the tissue oxygen environment, which may or may not stimulate blood flow regulation, is also being studied computationally. In this work, tissue oxygen distributions are being calculated based on intravascular measurements, as a first step toward better understanding of how oxygen is delivered by the microcirculation, how oxygen delivery depends on tissue oxygenation, and how defects in microvascular oxygen delivery can affect the surrounding tissue.
Neovascularization is the process by which microcirculatory networks are expanded within a tissue to deliver more blood to a tissue. Successful neovascularization includes not just angiogenesis, but a number of vascular activities effectively coordinated into a contiguous process. The transition from angiogenesis to a stable microcirculation involves differentiation of neovessels into functional vessel types and organization of these vessels into a mature architecture (e.g. a vascular tree). This maturation of the network involves continued refinement of vessel elements into larger and smaller caliber vessels, longer and smaller vessel segment lengths and vessel removal or "pruning." Ultimately, these structural changes in vessels lead to long-term adjustments in blood flow resistances and flow pathways within the network. In disease conditions, this coordination is perturbed resulting in a dysfunctional microvasculature and microcirculatory insufficiency. Although much is known concerning the determinants of angiogenesis and vessel caliber, less is known about the mechanisms driving these processes. Even less is known about the control mechanisms that regulate progression from angiogenesis to network maturity.
The branching pattern and vascular geometry of biological tree structure is complex. Here we show that the design of all vascular trees for which there exists morphometric data in the literature (e.g., coronary, pulmonary; vessels of various skeletal muscles, mesentery, omentum and conjunctiva) obeys a set of scaling laws which are based on the hypothesis that the cost of construction of the tree structure and operation of fluid conduction are minimized. The laws consist of scaling relationships between 1) length and vascular volume of tree, 2) lumen diameter and blood flow rate in each branch and 3) diameter and length of vessel branches. The exponent of the diameter-flow rate relation is not necessarily equal to 3.0 as required by Murray's law but depends on the ratio of metabolic-to-viscous power dissipation of the tree of interest. The major significance of the present analysis is to show that the design of various vascular trees of different organs and species can be deduced on the basis of minimum energy hypothesis and conservation of energy under steady state conditions. The present study reveals the similarity of nature's scaling laws that dictate the design of various vascular trees and the underlying physical and physiological principles.
When standing up, blood is pooled in the legs due to the effect of gravity resulting in a drop in systemic arterial pressure and widening of the blood flow velocity. This can be modeled by increasing the blood pressure in the compartments representing the lower body. To restore blood pressure and blood flow velocity a number of regulatory mechanisms are activated. The most important mechanisms are autonomic reflexes mediated by the sympathetic nervous system and cerebral autoregulation mediated by changes in concentrations of oxygen and carbon dioxide. The response to standing is an increase in nervous activity, which results in increased heart rate and cardiac contractility, vasoconstriction of the systemic arterioles, and changes in unstressed volume and venous compliance. The response by the cerebral autoregulation is to dilate arterioles in the cerebral vascular bed. It is not clear how the autonomic and autoregulation interacts; one theory suggests that vasoconstriction, resulting from increased sympathetic activity, has an effect throughout the body, but that cerebral vasoconstriction gets overridden (possibly with a significant delay) by autoregulation resulting in a net vasodilatation of the cerebral vascular bed. In this work we demonstrate how mathematical modeling can be used to predict the interaction between autonomic and autoregulation, and how methods from optimal control theory can be used to identify model parameters to make the model patient specific.
Microvascular growth and adaptations in response to physiological and pathological stimuli involve a cascade of different molecular signals and cellular behaviors, which give rise to patterns of vascular networks exhibiting spatial and temporal heterogeneity. To capture these interactions and to study the emergent properties of this complex biological system, it is necessary to consider the interactions of multiple cells with one another and with their environment within the context of the whole tissue. We have employed agent-based modeling (ABM) to compute vascular cell and vascular-associated cell behaviors in space and time in response to their environmental signals, such as growth factors, extracellular matrix interactions, and hemodynamic forces. An empirically-derived rule set dictates how cells will 'behave' in certain settings, and the individualized behaviors of thousands of cells in the simulation give rise to the aggregate patterning response: the growth of new microvessels (angiogenesis), the maturation of existing microvessels (arterialization), or the accumulation and distribution of vascular-associated cell types, including macrophages and pericytes. We have applied the ABM approach to study various phenomena related to disease states and therapies impacting the microcirculation. Examples include the effect of exogenous growth factor delivery on angiogenesis, the distribution of 'injected' pericyte precursor cells, and the impact of hemodynamic forces on the leukocyte adhesion cascade. A central aspect of this approach is its intimate pairing with in vivo experimental work. In this way the ABM informs the experiments by facilitating systematic and efficient hypothesis testing, and the experimental work informs the computational model by providing independent validation of the predictions and rule set. The long term goals of this combined approach are to expedite the discovery of fundamental mechanisms underlying microvascular growth and remodeling, enable faster and more effective drug discovery, and advance regenerative medicine strategies.
An adequate supply of oxygen is critical to the survival of every cell in mammalian organisms. The cardiovascular system accomplishes this task by the coordinated action of convection of oxygen through the large supply arteries and the subsequent diffusion of oxygen from the blood to the tissues across the walls of the much smaller vessels, comprising the microcirculation.
Computational modeling, based on established physical principles and observed architecture of the circulation to create a realistic representation of the complex network of microvessels and the associated parenchymal cells, is an important tool to enhance our understanding of oxygen transport in health and disease. In order for theoretical /computational approaches to enjoy the greatest success, there should be a close connection between theory/computation and experiment - a marriage of the two, in which each side of the relationship communicates with clarity and the two components work in concert to approach the truth.
The basic elements of oxygen transport are well known by now. The erythrocytes or red blood cells (RBCs) are the vehicles that carry virtually all of the oxygen, and in recent years studies have been carried out where the oxygen carriage by the hemoglobin in RBCs is supplemented by artificial oxygen carriers, such as HBOCs and PFCs. The blood is conducted to the periphery by a complex, branching network of vessels, with an increasing number of smaller and smaller elements in the network. The distributions of blood flow and RBC concentration (hematocrit) within the network are key factors that determine the supply of oxygen to the tissues. When the RBCs, loaded with oxygen, reach the smallest vessels (i.e., arterioles, capillaries and venules), the conditions are ripe for oxygen to be released from the hemoglobin in the RBCs and for its diffusion out of these vessels to the mitochondria inside the cells, where it is consumed in the process of oxidative phosphorylation - the creation of ATP from which cells derive energy for their myriad functions.
Comparison of the predictions of theoretical/computational models with the results of experiments is the ultimate test of both. However, when a disagreement arises - in some cases, a substantial one - this signals an opportunity to improve understanding and possibly discover new phenomena. When there is a discrepancy, what could be the culprit? Some obvious possibilities are, but not limited to: Theory (invalid or unrealistic assumptions in the model) and/or Experiment (poorly executed experiment, unexpected artifacts, misinterpretation of raw signals).
The basics of measuring convective and diffusive oxygen transport will be described, as well as the simple interpretation often given to such measurements. In cases where there are substantial discrepancies between theoretical predictions and experimental results, it is only when we dig a little deeper that the truth and a keener understanding emerge. Two case studies involving the measurement of oxygen transport will be discussed to illustrate the care with which experimental methods and the data they produce need to be examined: the measurement of hemoglobin oxygen saturation (SO2) and the partial pressure of oxygen (PO2). A greater appreciation of the difficulties in the interpretation of experimental data should result from this examination.
Angiogenesis is the growth of new microvessels from pre-existing vessels. Angiogenesis is important under physiological and pathological conditions (e.g., exercise, cancer, age-related macular degeneration, rheumatoid arthritis, myocardial ischemia, peripheral arterial disease). Over 70 diseases have been identified as angiogenesis dependent. Angiogenesis involves numerous processes such as: cell sensing of oxygen during hypoxia; upregulation of vascular endothelial growth factor (VEGF), and of matrix metalloproteinases (MMPs); extracellular matrix (ECM) proteolysis and release of matrix-binding growth factors; endothelial cell migration, proliferation and differentiation; tubulogenesis or formation of capillary tubes; network morphogenesis or formation of capillary networks; and vessel maturation that involves recruitment of supporting cells such as pericytes and smooth muscle cells. We have developed several molecular-based computational models that will serve as modules in multi-scale integrative models. These include a model of Hypoxia-Inducible Factor HIF1, a transcription factor largely responsible for upregulation of VEGF in hypoxia; a model of interactions of VEGF splice isoforms with their receptors VEGFR1, VEGFR2, Neuropilin-1 and heparan sulfate proteoglycans; and a model of ECM proteolysis by MMPs, specifically MMP2, MMP9 and membrane-type MT1-MMP, in the presence of tissue inhibitors of metalloproteinases (TIMPs). In addition to these molecular-level models, a framework will be described for incorporating these models into multi-scale rule-based models, thus spanning the levels from the molecular to microvascular. Several therapeutic applications to disease conditions will be presented including pro-angiogenic approaches to peripheral arterial disease and anti-angiogenic approaches to breast cancer.
The basic outline of the vascular system is determined during development by complex genetic programming guided by the unique temporal and spatial patterns of structural and molecular features available in the embryo. With establishment of blood flow, control of vascular development is increasingly taken over by feedback signals derived from vascular function including blood flow (shear stress) blood pressure (circumferential wall stress) and tissue metabolic state. Such signals also govern the postnatal structural adaptation of vascular beds with respect to vessel diameter and wall thickness, vessels length and vessel number in response to functional requirements (angioadaptation). In angioadaptation, the properties of peripheral vascular beds are determined by the interplay between vascular and cellular reactions to signals related to functional stimuli and the functional implications of these reactions. Under physiological conditions including growth and physical exercise, angioadaptive responses lead to adequate adjustment of the properties of vascular beds. However, pathophysiological changes of vascular response characteristics or environmental conditions may lead to vascular mal-adaptation, e.g. inward remodeling and rarefaction in hypertension.
Many components and mechanisms of angioadaptation have been described. However, the complex interaction of functional stimuli, molecular mediators, cellular reactions and resulting functional properties of vascular beds is still poorly understood. Integrative approaches, including the analysis and extrapolation of experimental findings by mathematical models are thus needed. For the structural adaptation of existing vessels (remodeling), mathematical models have been presented which allow prediction of realistic vascular properties based on a generic set of adaptation characteristics. These models allow a quantitative analysis of the relation between vascular reaction patterns to mechanical stimuli and properties of terminal vascular beds including situations with aberrant adaptive properties or systemic conditions. They also show that very different combinations of reaction patterns of vessel diameter and wall thickness, resp., to shear stress and wall stress can lead to identical structural network properties, rendering robustness to the biological system. However, an number of relevant areas are still underrepresented in such models, including the representation of longitudinal stretch or pulsatile effects, and the representation of the molecular layer mediating between local stimuli and vascular responses.
Mathematical model simulations of angiogenic processes which include the molecular level will be useful not only to understand the involved mechanisms in a quantitative fashion, but also to define possible targets for effective therapeutic interventions and to predict the corresponding effects.
Measurement of vascular trees involves measuring the interbranch segment lengths, lumen diameters , branching-angles with its daughter branches, the volume of tissue perfused by it and the segments' hierarchical locations within the tree. Various methodologies for preparation of the vascular tree for imaging introduces some artifacts and the non-destructive three-dimensional imaging methods (such as multi-slice computed tomography, confocal-type microscopy and destructive methods such as progressive serial section histology) have their strengths and deficiencies in terms of the tree's distortion, the spatial resolution and volume of tree that can be imaged. The dimensional analysis of the three-dimensional images, which are essentially identical regardless of the image-generation method, introduces limits to accuracy due to the above-mentioned sources of artifact and inaccuracies.
Although many attempts have and are being made to overcome these sources of inaccuracy, the inevitable trade-offs between effort spent on preparing , imaging and analyzing vascular trees are greatly affected by the biological question(s) being addressed , many of these image processing developments are not attuned to the various biological questions being addressed. Indeed, the questions that can be effectively addressed with three-dimensional imaging methods are constrained by the measurement capabilities of the various methodologies. The presentation will provide an overview of these issues.
The discovery of the circulation of blood is generally credited to William Harvey (1578-1657). Previously, it was believed that venous and arterial blood had different origins and functions, as stated by the ancient Greek physician Galen (129-200 AD). From his anatomical observations, Harvey deduced that blood leaving the heart in the arteries returned by the veins, although he could not see how they were connected. Marcello Malpighi (1628-1694) used a microscope to observe the capillaries, thus initiating the study of microcirculation. As well as being a physician and physiologist, Jean Louis Marie Poiseuille (1799-1869) was trained in physics and mathematics. His interest in the flow of human blood in narrow tubes led him to discover the fourth-power relationship between flow rate and diameter in tubes. August Krogh (1874-1949) was a professor of zoophysiology at the University of Copenhagen. He was awarded the Nobel Prize for his work on regulation of blood flow. With the help of his friend, the mathematician Erlang, he developed a mathematical model for oxygen delivery to skeletal muscle, first published in 1919. In the late 1960s, work by James Lighthill and David Hellums on red blood cell motion in capillaries initiated an era in which theoretical approaches have been applied to many aspects of microcirculation, including blood rheology, red blood cell mechanics, oxygen transport, exchange of water and solutes through vessel walls, the mechanics of vessel walls, analysis of network structure, blood flow and red blood cell distribution in networks, white blood cell mechanics and adhesion, regulation of blood flow, structural adaptation of blood vessels, and growth of blood vessels.
Short-term regulation of blood flow in response to changing conditions is achieved by active contraction and dilation of smooth muscle cells in the arterioles. These changes in vascular tone occur in response to several stimuli, including tension in vessel walls, wall shear stress, and levels of metabolites including oxygen, potassium ions, adenosine triphosphate (ATP) and nitric oxide. Vessel walls act not only as conduits for blood flow but also as a communication system. In conducted responses, signals are transmitted along vessel walls by electrical coupling of the endothelial and smooth muscle cells. Blood flow is regulated in response to changes of oxygen consumption, particularly in skeletal muscle where oxygen demand increases almost two orders of magnitude between rest and maximal exercise. Experimental studies have shown that ATP release by red blood cells plays an important role in this process. The rate of ATP release increases with decreasing hemoglobin saturation in the red blood cell, allowing the red blood cell to act as a sensor as well as a carrier of oxygen. Increased ATP levels in venules initiate conducted responses that are propagated upstream and cause arteriolar dilation. We have developed a theoretical model for analyzing these processes, taking into account the mechanics of vascular smooth muscle and the responses of microvessels to wall shear stress and wall tension. The model shows how responses to the multiple stimuli mentioned above can account for the variation of flow with changing oxygen demand, and also the maintenance of almost constant flow despite changes in arterial blood pressure.
Work done in collaboration with Julia C. Arciero and Brian E. Carlson.
The intracellular concentration of free Ca2+ in smooth muscle cells is the main determinant of vascular tone and regional blood flow. An elaborate network of signaling pathways exists that regulates [Ca2+]i in smooth muscle cells. This network includes intracellular signaling as well as cell-to-cell communication with paracrine factors or diffusion of species through homo- and hetero- cellular gap junctions. This multitude of signaling pathways create multiple feedback loops that tightly regulate Ca2+ homeostasis. Over the last twenty years NO has emerged as the key signaling molecule involved in the regulation of vascular tone. In response to hemodynamic or agonist stimuli vascular endothelial cells produce NO which can diffuse freely across cell membranes to the adjacent smooth muscle where it activates the enzyme soluble guanylate cyclase (sGC) leading to smooth muscle relaxation. The close proximity of the red blood cells to the site of NO production and the fast consumption of NO by hemoglobin (Hb) suggest however that a significant amount of endothelium derived NO will be scavenged by the blood, leading to what is often referred as the "NO paradox". Thus, despite significant scientific contributions over the last few years, fundamental questions about basic physiological functions of NO and its role in the regulation of vascular tone remain unanswered. Experimentation continuously provides new insights about the physiology of blood vessels and the mechanisms that regulate tone and blood flow. Mathematical modeling offers a systematic approach for system analysis and can assist in this effort both as a tool for data analysis and for guiding new experimental studies.
A proposed mechanism for metabolic flow regulation involves the oxygen-dependent release of ATP by red blood cells, which triggers an upstream conducted response signal and arteriolar vasodilation. To analyze this mechanism, a theoretical model is used to simulate the variation of oxygen and ATP levels along a pathway of seven representative segments, including two vasoactive arteriolar segments. An expression for the conducted response signal is defined by integrating the ATP concentration along the vascular pathway, taking into account exponential decay of the signal in the upstream direction. Arteriolar tone depends on the conducted metabolic signal and on local wall shear stress and wall tension. Arteriolar diameters vary according to changes in the passive characteristics and active responses of the vessel wall. In the model, tone and diameter are treated as time-dependent variables and are described by a system of ordinary differential equations. For some pressure and oxygen demand levels, vessel tone and diameter oscillate with time. This model prediction corresponds to vasomotion, a biological phenomenon in which arterioles show spontaneous rhythmic variations in diameter. The model also predicts that the combined effects of the conducted, myogenic, and shear-dependent responses can account for a nearly 10-fold increase in perfusion in response to a 20-fold increase in oxygen demand.
In collaboration with Brian Carlson and Timothy Secomb.
A physiological model of regulatory pathways in the cerebral circulation is described. While the model is incomplete, and contains many simplified and caricatured processes, it has been constructed in a modular way, allowing for easy alteration and expansion in response to increased knowledge. Constructing large models in situations where experimental data is hard to come by raises several theoretical issues: For example, to what extent can qualitative predictions about the input-output response of the model be made on the basis of model structure alone? Do errors and inaccuracies in the model qualitatively affect the responses of model outputs to inputs? When the behaviour of submodels is simple and well characterised, to what extent does this allow us to make predictions when the submodels are combined? These theoretical issues are discussed.
Closed loop lumped parameter circulatory models have been developed previously by many groups (Neal and Bassingthwaighte, Lu and Clark, Noodergraaf et al.) but none of these models take into account the active response to local conditions in microcirculatory vessels. We have developed a model of a microcirculatory vessel compartment that actively responds to local pressure and flow. The response characteristics take into account the passive components of the vessel wall and a variable degree of vascular smooth muscle (VSM) activation. The sigmoidal VSM activation response is a function of the vessel wall tension and the shear stress on the vessel wall. Closed loop circulatory models with and without local vasoactive response in the microcirculation will be compared. Developed in collaboration with Maxwell L. Neal and James B. Bassingthwaighte.
Nitric oxide (NO) derived from nitric oxide synthase (NOS) is an important paracrine effector maintaining vascular tone. NO release through NOS isozymes under different O2 conditions critically determines the NO bioavailability in tissues. In this study, we used computational models based on the analysis of biochemical pathways of enzymatic NO synthesis and the availability of NOS isozymes to quantify the NO production by neuronal NOS (NOS1) and endothelial NOS (NOS3). We compared the catalytic activities of NOS1 and NOS3 and their different sensitivities to the concentration of substrate O2. Based on the predicted NO release rates from the kinetic models, the topographic distribution of NO sources and mass balance analysis, we predicted the NO concentration profiles around an arteriole under different O2 conditions. Results showed that NO production by NOS1 was significantly more sensitive to ambient O2 concentration than that by NOS3. Also, the high sensitivity of NOS1 catalytic activity to O2 makes NO production and therefore NO concentration significantly reduced upon hypoxia. Moreover, NOS1 abundantly located in the nerve fibers and mast cells close to arterioles, rather than NOS3 in the endothelium, is the major source determining the NO concentration distribution. Finally, the perivascular NO concentration under normoxia predicted by the models was at least an order of magnitude lower than a number of experimental measurements, suggesting either a higher abundance of NOS1 or NOS3 distribution and/or other enzymatic or non-enzymatic NO sources in the microvasculature.
Nitric oxide (NO) is a potent vasodilator on the microvasculature. We have previously constructed computational models based on biochemical pathway analysis of different nitric oxide synthase isoforms and found a large discrepancy between our predictions and perivascular NO measurements, suggesting the existence of non-enzymatic sources of NO. One potential source is red blood cells (RBCs), which have been hypothesized to preserve NO bioactivity. S-nitrosohemoglobin (SNOHb) in RBCs has been put forward as a major contributor to NO-induced hypoxic vasodilation; however, the amount of NO delivered by intraerythrocytic SNOHb to smooth muscle has not been quantified experimentally or calculated using mathematical models. In the present study, we have formulated a multicellular computational model to quantify NO exposure in arteriolar smooth muscle when the NO released by SNOHb is the sole NO source in the vasculature. Our calculations show an NO exposure of ~6 pM in the smooth muscle region. This amount is far below the measured values for the perivascular NO concentration, which are generally reported as several hundred nM, and it does not account for the large discrepancy that we encountered. We found that the amount of NO delivered by SNOHb to smooth muscle strongly depends on the SNOHb concentration and half-life, which further determine the rate of NO release, as well as on the membrane permeability of RBC to NO. In conclusion, our mathematical model predicts that picomolar amounts of NO can be delivered to the vascular smooth muscle by intraerythrocytic SNOHb; how this amount of NO alone might induce hypoxic vasodilation requires further investigation.
Work done in collaboration with Roland N. Pittman and Aleksander S. Popel.
By Newton's law vascular development, in more general tissue morphogenesis, requires mechanical forces to position cells and shape the developing organs. In a simplified picture, the addition of all incremental pushes of growing cells generates a stress gradient inside the growing tissue. Such stress gradient can on the one hand serve as a signal for cellular mechanotransducers resulting in gene activation. On the other hand, the dynamically changing landscape of stress guides the displacing, newly created, growing cells in an auto-organizing way.
To address the auto-organizing role of stress gradients in vascular development we imaged the developing vasculature in the chick embryo yolk-sac in vivo. As soon as the vessels appeared, we measured the displacement of characteristic features in the vasculature. We show that the expansion speed inside the yolk-sac is higher at the edge than internally, with a linear variation of 0.016 h-1. Additionally, we isolated the yolk-sac from the yolk and mapped the compliance, a measure for in-plane stress, with a newly developed instrument: a scanning air puff tonometer. We observed a more than linear decrease in stress along the growth direction of the yolk-sac.
A simple mathematical model is used to interpret the observations. The growth of the yolk-sac is modeled as a radial displacement of a viscous fluid between two fixed parallel plates, while fluid is injected uniformly into the yolk-sac. The injection represents the local additional flow due to cell dilation or mitosis. The model reveals that only uniform expansion of all cells in the yolk-sac can explain our observations.
In conclusion, our observations suggest that the in-tissue stress gradient that correspond with softer regions at the edge of the yolk-sac is closely associated with the paths taken by the blood vessels.
Work done in collaboration with Thi-Hanh Nguyen, Mathieu Unbekandt, Loïc Leroy, Alia Al-Kilani, Ferdinand Le Noble and Vincent Fleury.
Short term cardiovascular responses to postural change from sitting to standing involve complex interactions between the autonomic nervous system, which regulates blood pressure, and cerebral autoregulation, which maintains cerebral perfusion. We have developed a model which can predict dynamic changes in beat-to-beat arterial blood pressure and middle cerebral artery blood flow velocity during postural change from sitting to standing. This model uses an electrical circuit analogy, predicting changes in blood pressure (voltage) and blood flow (current) as functions of resistance and compliance (capacitance). The base model has more than 100 parameters that must be identified to predict regulatory response for individual subjects. In preliminary work, an inverse least-squares problem was formulated to estimate all 100 parameters to minimize the difference between observed data and computed values. This optimization process is time-consuming and not feasible if the model is to be validated against multiple datasets. We have used sensitivity analysis to identify a small number (approximately 20) of sensitive parameters. Furthermore, we have shown, using sensitivity analysis, that it is possible to reduce the structure of that model and that with additional data it is possible to identify more parameters. Finally, we have developed a physiological model for autoregulation that predicts cerebrovascular resistance as a function of blood flow and blood pressure. It is not well understood how autoregulation is mediated; it is known that several components play a role including myogenic, metabolic, and neural contributions. We plan to use our mathematical model to understand dynamics of each of these components.
Oscillations in microvascular bloodflow have been observed by many investigators since August Krogh reported their existence in the web of frog feet in the 1920's. While most researchers suggest that biological control mechanisms, such as vasomotion, are responsible for fluctuations in bloodflow, there is some evidence that oscillations might occur spontaneously in the absence of biological control. We explore the dynamics of bloodflow in small networks of microvessels using a mathematical model which assumes that the flow can be described by a first-order wave equation in blood hematocrit. The model also includes two important rheological effects: the Fahraeus-Lindqvist effect which governs the viscosity of blood flow in a single vessel and the plasma skimming effect which describes the separation of red blood cells at diverging nodes. We demonstrate using a combination of analytical and numerical techniques that it is the relative strength of the Fahraeus-Lindqvist effect and the plasma skimming effect which determines the existence of a set of network parameter values which lead to spontaneous oscillations of flow and hematocrit in the absence of biological control.
In collaboration with Russell T. Carr and Fan Wu.
Vascular networks respond to chronic alterations in blood supply by structural remodeling. A previous study showed that the inner diameter of the mouse gracilis artery increased transiently following a permanent reduction in blood flow to one of its two blood supplies, starting at day 7, with a peak at 21 days followed by return to control values by day 56. The objective of the present study was to investigate whether the diameter increase is accompanied by an increase in vessel wall volume in the same experimental setting. Blood flow reduction was induced by removing a portion of the saphenous artery, one of the two blood supplies of the gracilis artery. After 7, 14, 21, 28 and 56 days, the vasculature was perfused with India ink for diameter measurements and then embedded in paraffin, sectioned and processed for immunocytochemistry using fluorescently labeled anti smooth muscle alpha-actin antibody and bisbenzimidazole (BBI) to mark cell nuclei. The remodeled vessel wall area was significantly larger at day 14 (7 days later than the diameter increase), but just like the diameter increase it stayed larger at days 21 and 28 and returned to control values by day 56. The wall remodeling was most significant in the middle and the region close to the removed blood supply (injury region). The number of smooth muscle cell nuclei was significantly increased only at day 21 in the middle region whereas the number of endothelial cell nuclei was significantly higher only at day 21 in the injury region. There was no significant change in the vessel tortuosity during remodeling. The index of the circumferential wall stress for a given pressure (estimated as the ratio of the inner vessel diameter and the vessel wall thickness) was significantly increased only at days 14 and 21. The results are consistent with the concept that outward remodeling causes increased circumferential wall stress, which then stimulates vessel wall adaptation (with a 7 day delay) to restore circumferential stress to near its initial level. The present study also suggests that the vessel adaptation is accomplished mainly by hypertrophy of the existing smooth muscle and endothelial cells rather than growth of new ones (hyperplasia). Supported by AHA grant 0010189Z and NIH grants HL34555 and HL63732.
Work done in collaboration with James B. Hoying and Timothy W. Secomb.
In unbranched segments of first order rat cremaster arterioles, structural diameter has been found to increase distally (1). In vivo circumferential wall stress is uniform along the vessel despite declining pressure. In contrast, wall shear stress declines by about one third. The reason for this pattern is unknown.
We present a simple mathematical model of a vessel containing passive elastic and active contractile elements. The wall is sensitive to circumferential stress and reacts acutely to a change in pressure with a change in tone (a myogenic response). Consistent with experimental findings, the model vessel is capable of reorganizing the wall constituents in response to a sustained change in tone. This response is eutrophic, i.e. without a change in the total amount of wall material and continues until the basal level of tone is restored at the new transmural pressure.
Simulations are performed using a 1 mm vessel segment with active and passive properties similar to those found in cremaster arterioles. As an approximation flow is calculated using Poiseuilles law. Initially the vessel has uniform structure and myogenic tone along its whole length. Pressure is then raised in one end and lowered in the other, to arrive at the in vivo up- and downstream pressures (1). The initial myogenic adjustment of tone is followed by a slow remodeling response over the following hours. As the simulation settles the experimentally observed pattern regarding structural diameter, circumferential wall stress, wall strain and wall shear stress emerges. Conclusion: in a stress sensitive vascular wall, the model suggests that adaptation towards a basal level of tone may enable the individual vascular smooth muscle cell to remain at a level wall stress, strain and activation optimal for short term flow regulation.
1. Bakker EN, Versluis JP, Sipkema P, VanTeeffelen JW, Rolf TM, Spaan JA and VanBavel E. Differential structural adaptation to haemodynamics along single rat cremaster arterioles. J Physiol 548: 549-555, 2003.
Work done in collaboration with Michael John Mulvany and Niels-Henrik Holstein-Rathlou.
Endothelium-derived nitric oxide (NO) plays a key role in vascular functions including smooth muscle tone regulation, platelet activation and cell signaling. All cells of vascular tissue including endothelial cells, and smooth muscle cells have enzyme systems that produce reactive oxygen species (ROS). Important ROS generated in the vascular tissue are superoxide or hydrogen peroxide. Superoxide can react with endothelium-derived nitric oxide (NO) to form peroxynitrite. Antioxidant systems protect against damage from ROS in normal physiological conditions. When ROS generation overwhelms the antioxidant defense (known as oxidative stress), these radicals can alter cellular function by interacting with DNA, RNA, and fatty acids, contribute to impairment of endothelium-dependent vasodilation and lead to apoptosis. The effect of biochemical interactions of these reactive species on the respective bioavailability in the microcirculation remains unclear. We formulated a detailed computational model of NO, superoxide and peroxynitrite transport in a tissue containing an arteriolar blood vessel to quantify these species biochemical interactions and transport in the microcirculation. The model predictions include (a) NO interactions with oxygen, superoxide and peroxynitrite have relatively no effect on the NO level in the vascular smooth muscle under physiological conditions, (b) superoxide diffuses only few microns from its source, whereas peroxynitrite diffuses over a larger distance, and (c) reduced superoxide dismutase levels significantly increases superoxide and peroxynitrite concentrations and decreases NO concentration. Model results indicate that the reduced NO bioavailability and enhanced peroxynitrite formation may vary depending on the location of oxidative stress in the microcirculation, which occurs at diverse vascular cell locations in diabetes, aging, and cardiovascular diseases. The results will have significant implications for our understanding of these free radical interactions in physiological conditions and pathophysiological conditions resulting from endothelial dysfunction.
Endothelial cells line blood vessels in the body and are continuously exposed to blood flow, and thus, fluid mechanical forces such as shear stress. Variations in shear stress magnitude and distribution are known to affect many processes needed for proper vasoregulation. Such processes include, but are not limited to, permeability and hydraulic conductivity across vessel walls, gene and cell surface adhesion molecule expression, cytoskeletal rearrangement, and the release of vasodilators. The mechanism responsible for these changes is known as mechanotransduction and has three basic stages: stimulation of a mechanical sensor, transmission of stress through that sensor, and stress transduction which ultimately creates biochemical signals.
The endothelial glycocalyx, a dense matrix of membrane-bound macromolecules, is thought to be an important mechanical sensor for endothelial cells. We would like to find the shear stress patterns in and around the glycocalyx to gain a better understanding of the mechanotransduction process. Since the exact structure of this layer is not well understood, we use mathematical models to explore the effect of different matrix permeabilities and Reynolds number regimes on flow through a porous matrix to find the resulting exerted fluid stresses. We built a low Reynolds number flow tank and physical models of possible glycocalyx structures to compare our computational results with flow measurements over a range of Reynolds numbers.
Induction of angiogenesis to increase tissue perfusion is a promising therapeutic strategy for the amelioration of muscle ischemia in Peripheral Arterial Disease. Although the potent pro-angiogenic cytokines of the vascular endothelial growth factor (VEGF) family have been suggested as potential therapeutic agents, to date clinical trials have not been successful. Exercise training is currently the most effective treatment; expression of both VEGF and its receptors are altered by exercise. We use integrative multi-scale computational models to predict the effects of chronic exercise training on the signaling of the VEGF system in the rat extensor digitorum longus (EDL) following femoral ligation. We then compare this to two VEGF delivery strategies: viral VEGF gene delivery and VEGF cell-based therapy (injected myoblasts that overexpress VEGF). The computational model used to investigate these strategies is an integration of several components: an anatomical description of th e muscle geometry and cell types; microvascular blood flow; tissue oxygen distribution; oxygen-dependent VEGF secretion from muscle fibers; VEGF transport through interstitial space; and VEGF receptor binding on microvascular endothelial cells. The models thus incorporate tissue physiology, cell physiology and molecular biology into a computational framework. Exercise training was predicted to be more successful in increasing VEGF concentration gradients and VEGF receptor activation than the VEGF-only treatments. This suggests that where previous monotherapy clinical trials failed, combined therapy trials targeting both VEGF and its receptors (mimicking the effects of exercise training) may succeed. We tested this in simulations of human vastus lateralis muscle, confirming the predicted superiority of combined therapies over the VEGF-only approach.
Work done in collaboration with James W. Ji and Aleksander S. Popel.
Angiogenesis inhibition through the blockade of vascular endothelial growth factor VEGF signaling has long been a promising therapeutic strategy for diseases of hyper-vascularization, including cancer, and the first VEGF antagonists have recently been approved for cancer treatment. Neuropilin has been identified as a co-receptor for VEGF and is implicated as an enhancer of one isoform of VEGF in particular, VEGF165 . Using a computational model of VEGF transport and signaling in breast tumor tissue that incorporates anatomical geometry, multiple cell types including tumor cells and endothelial cells of the vasculature, extracellular matrix and basement membranes, we tested several methods of blocking VEGF signaling by targeting Neuropilin. For some of the drugs tested in silico, for example, those blocking VEGF-Neuropilin binding, or those inhibiting Neuropilin synthesis, adaptation of the VEGF-VEGF receptor system is predicted, resulting in only transient inhibition of VEGF signaling. However, one drug - specifically, an antibody to Neuropilin that does not directly interfere with VEGF-Neuropilin binding - is predicted to be resistant to adaptation and to inhibit VEGF signaling for as long as it is present in the system.
Work done in collaboration with Aleksander S. Popel.
Analysis of mouse hearts after experimental infarction (coronary ligation for one week), as well as a model of chronic inflammatory myocarditis, revealed deep penetration of the ischemic tissue by leukocytes. These cells were often accompanied by extravascular erythrocytes, in regions where the sprouting-based angiogenesis seemed to be missing. The findings raise the question of how the inflammatory cells perform and survive in severely hypoxic tissues.
Here we suggest an alternative mechanism of oxygen transport in non-perfused tissues. It consists in facilitated diffusion of oxygen via extravascular erythrocytes pools, or through mixed leukocyte-erythrocyte cell columns formed by percolation. Our data indicates the metalloprotease-assisted penetration of extracellular matrix by monocytes/macrophages as a candidate mechanism of cell columns formation. This 'tunneling' activity, amply supported by in vitro experiments, could also assist progenitor cells engraftment, and thus neovascularization, during adult tissue regeneration (Stem Cells Dev., 2005; Am. J. Pathol., 2006).
Therefore, tissue percolation by erythrocytes might represent an emergency-type microcirculation, which precedes, facilitates and possibly conditions the second wave of definitive neovascularization. It also suggests novel interpretations of old puzzles (such residence of some tumors anti-angiogenic treatments, or the benefits of laser-drilled channels in the ischemic hearts), as well as new therapeutic options.
The purpose of this study is to quantitate the effects of the effect of variations in sympathetic nerve activity (SNA) on oxygen extraction in contracting skeletal muscle. The model utilizes a simplified arteriolar network consisting of compartments in series representing different vessel orders, starting with feed arteries (FA), progressing to first, second and third order arterioles (1A, 2A, 3A), and terminating in capillaries (C). Each compartment is assumed to consist of parallel conduits of a given vessel order supplying surrounding tissue regions. The model is used in conjunction with observations by vanTeeffelen and Segal [J Physiol 2003;52:563] in hamster retractor muscle with various combinations of SNA intensity and workload to investigate the net effect of changes in blood flow due to functional vasodilation and sympathetic vasoconstriction on oxygen extraction at steady state. Model predictions include the expected increase in oxygen extraction with increased SNA activity at rest and moderate workload; however, the effect of SNA relative to functional vasodilation is seen to be attenuated at higher workloads. Further development of the model is in progress to more accurately characterize the relationship between SNA, muscle contraction and passive vasodilation in determining flow and oxygen delivery.
Work done in collaboration with MJ Joyner.
Atherosclerosis is a disease of the cardiovascular system characterized by the formation and growth of fatty lesions inside the arterial wall. It is now widely accepted that atherosclerosis is a chronic disorder initiated by inflammatory processes taking place in the intimal layer of the arterial wall. The inflammatory processes, which include foam cell formation, lipid oxidation by free radicals, free radical production and anti-oxidant mitigation, can be modeled via a system of reaction-diffusion-chemotactic equations. These equations are sufficient to model atherogenesis, the initiation phase of the disease, as an instability from a "healthy state" which is defined to be an equilibrium state in which inflammatory markers are absent.
The central mathematical question pertaining to atherogenesis is the stability of this healthy state to small perturbations in the inflammatory markers. It is shown that the healthy state can be stabilized through diffusion processes and sufficiently strong anti-oxidant mitigation and destabilized through strong chemotactic effects which promote local lesion formation and growth and low anti-oxidant levels. An interesting feature of the model is that the principal source of two key component species in the inflammatory process is through the endothelial boundary between the lumen (where blood flows) and the intimal layer of the arterial wall. This results in coupled transport (third kind) boundary conditions on the inner intimal boundary that significantly affect the qualitative properties of the problem greatly complicate the stability analysis.
Modeling subsequent lesion growth past the initiation phase requires introduction of mechanical effects. Moreover, the reaction-diffusion-chemotactic system modeling the inflammatory processes must be augmented by inclusion of a new species, smooth muscle cells, migrating from the medial layer of the arterial wall (in response to chemical signals from the growing lesion) whose role is to form a "cap" around the lesion. Modeling lesion growth into the lumen results in a free boundary value problem with coupled mechanical and boundary transport effects providing the driving force for boundary evolution.
A physical theory explaining the anisotropic dispersion of water in biological tissues observed in so-called diffusion-weighted imaging (DWI) or diffusion-tensor magnetic resonance imaging (DTMRI) is introduced based on the phenomena of Taylor dispersion, in which highly diffusive solutes cycle between flowing and stagnant regions in the tissue, enhancing dispersion in the direction of microvascular flow. Failure to account for flow-mediated dispersion in vascular tissues has led to misinterpretations of imaging data and significant overestimates of directional bias in molecular diffusivity in biological tissues. An effective diffusion equation is derived, for which the coefficient of dispersion in the axial direction (direction of capillary orientation) depends on the molecular diffusion coefficient, tissue perfusion, and theory suggests a means of obtaining quantitative functional information on capillary vessel density from measurements of dispersion coefficients. It is shown that a measurement of the ratio of axial to transverse diffusivity-obtained by DTMRI-may be combined with an independent measurement of perfusion to provide an estimate of capillary vessel density in the tissue.
We present a mathematical model to investigate the underlying mechanisms of the Nitric Oxide/cGMP pathway in the vascular smooth muscle cell. The model describes the flow of NO-driven signal transduction: NO activation of soluble guanylate cyclase (sGC), sGC- and phosphodiesterase-catalyzed cGMP production and degradation, cGMP-mediated regulation of protein targets including the Ca2+-activated K+ channel, and the myosin-actin contractile mechanism. Model simulations reproduce NO/cGMP-induced VSMC relaxation effects, including intracellular Ca2+ concentration reduction and Ca2+ desensitization of myosin phosphorylation and force generation. We have examined several testable principles using this model.
Motivated by fragments of ruptured atherosclerosis plaque in flowing blood, we simulated flexible filaments interacting with a viscous pulsatile flow in two dimensions using the immersed boundary method. According to numerous simulation results, we found that 1) a horizontal position perpendicular to the mainstream flow is stable. 2) Hydrodynamical forces alone do not seem to cause aggregation of filaments in a 2D pulsatile viscous flow.