Workshop 4: Blood Flow in the Microcirculation: Function, Regulation, and Adaptation

(January 22,2007 - January 26,2007 )

Organizers


Daniel Beard
Physiology, Medical College of Wisconsin
Timothy Secomb
Department of Physiology, University of Arizona

The function of the systemic circulatory system is to distribute and remove materials and heat as needed throughout the body. Transport is achieved by convection in the blood and diffusive exchange with surrounding tissue. Because diffusion is effective only over short distances, blood must be brought close to every point in the tissue. To make this possible, the peripheral circulation consists of a highly branched system of blood vessels, containing more than 109 segments, ranging in diameter from about 1 cm down to a few µm. The vessels of diameter about 100 µm or less are referred to as the microcirculation.

This workshop will focus on three areas:

  • blood flow and mass transport in the microcirculation;
  • short-term regulation of blood flow, including vascular smooth muscle behavior;
  • structural adaptation of blood vessels, including angiogenesis.

 

Mathematical and computational approaches can make important contributions in all these areas. Continuum and multiphase models can be applied to study blood flow. Simulations of mass and heat transport also typically require solution of partial differential equations. Consideration of network properties is critical to understanding short and long-term control of blood flow. The network can be regarded as a dynamic system in which the properties of each segment (diameter, etc.) evolve with time. Simulations of angiogenesis can use a variety of approaches, including deterministic and stochastic models and cellular automata.

Accepted Speakers

James Bassingthwaighte
Bioengineering, Radiology, University of Washington
Daniel Beard
Physiology, Medical College of Wisconsin
Daniel Goldman
Department of Medical Biophysics, University of Western Ontario
James Hoying
Regenerative Medicine/BIO5 Institute , University of Arizona
Ghassan Kassab
Biomedical Engineering&MedicineCardiology , University of California, Irvine
Mette Sofie Olufsen
Mathematics, North Carolina State University
Shayn Peirce-Cottler
Biomedical Engineering , University of Virginia
Roland Pittman
Physiology , Virginia Commonwealth University
Aleksander Popel
Biomedical Engineering, Johns Hopkins University
Axel Pries
Dept. of Physiology, Charité - Universitätsmedizin Berlin
Timothy Secomb
Department of Physiology, University of Arizona
Nikolaos Tsoukias
Biomedical Engineering, Florida International University
Monday, January 22, 2007
Time Session
09:15 AM
10:00 AM
Timothy Secomb - Microcirculation and mathematical modeling: A brief history

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.

10:30 AM
11:30 AM
Erik Ritman - Measuring intact Vascular Trees by 3D Imaging Methods

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.

01:30 PM
02:30 PM
Ghassan Kassab - Scaling laws of vascular trees

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.

03:00 PM
04:00 PM
James Bassingthwaighte - Transport and Exchange in the Microcirculation: Interactions amongst RBC, Endothelial Cells and Cardiomyocytes

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.

Tuesday, January 23, 2007
Time Session
09:00 AM
10:00 AM
Roland Pittman - An Overview of Oxygen Transport in the Microcirculation: Experimental Considerations

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.

10:30 AM
11:30 AM
Daniel Goldman - Computational modeling of blood-tissue oxygen transport

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.

01:30 PM
02:30 PM
Nikolaos Tsoukias - Integrative Models of Nitric Oxide Biotransport and Calcium Dynamics in the Microcirculation

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.

06:30 PM
07:30 PM
James Hoying - Public Lecure - Tissue Engineering and Repair: A Vascular Problem

Tissue engineering and related cell-based therapies promise to not only facilitate tissue repair but also functionally replace damaged and diseased tissues. With tissue engineering, the goal is to fabricate tissue constructs, comprised of cells in a supportive environment, which mimic the function and/or architecture of the target tissue. The source of cells used in these constructs is the subject of considerable scientific discussion (and, in the case of stem cells, public discussion). However, regardless of the source and types of the cells incorporated into these engineered constructs, there remains a significant challenge in providing sufficient nutrients to the cells during fabrication and following implantation. Any tissue implant greater in dimension than a few millimeters is too big for nutrients to efficiently diffuse to the construct's cells from outside the construct. This is why the first successfully engineered tissues have been thin, sheets of cells (e.g. a simple skin). As advances give rise to more complicated, 3-dimensional tissue designs, the need for a strategy to support the health of these constructs becomes more urgent. In the body, the cardiovascular system serves to effectively deliver nutrients to any tissue. Therefore, the ability to form and incorporate blood vessels (particularly microvessels) into the constructs is critically important for construct health and function. We will discuss the particular challenges related to providing proper nutrition to constructed tissues and the strategies being employed to build vessels and vessel networks in the laboratory.

Wednesday, January 24, 2007
Time Session
09:00 AM
10:00 AM
Jefferson Frisbee - Overview of blood flow regulation

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.

10:30 AM
11:30 AM
Timothy Secomb - Microcirculation and mathematical modeling: A brief history

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.

01:30 PM
02:30 PM
Mette Sofie Olufsen - Dynamics of blood flow regulation

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.

Thursday, January 25, 2007
Time Session
09:00 AM
10:00 AM
James Hoying - Public Lecure - Tissue Engineering and Repair: A Vascular Problem

Tissue engineering and related cell-based therapies promise to not only facilitate tissue repair but also functionally replace damaged and diseased tissues. With tissue engineering, the goal is to fabricate tissue constructs, comprised of cells in a supportive environment, which mimic the function and/or architecture of the target tissue. The source of cells used in these constructs is the subject of considerable scientific discussion (and, in the case of stem cells, public discussion). However, regardless of the source and types of the cells incorporated into these engineered constructs, there remains a significant challenge in providing sufficient nutrients to the cells during fabrication and following implantation. Any tissue implant greater in dimension than a few millimeters is too big for nutrients to efficiently diffuse to the construct's cells from outside the construct. This is why the first successfully engineered tissues have been thin, sheets of cells (e.g. a simple skin). As advances give rise to more complicated, 3-dimensional tissue designs, the need for a strategy to support the health of these constructs becomes more urgent. In the body, the cardiovascular system serves to effectively deliver nutrients to any tissue. Therefore, the ability to form and incorporate blood vessels (particularly microvessels) into the constructs is critically important for construct health and function. We will discuss the particular challenges related to providing proper nutrition to constructed tissues and the strategies being employed to build vessels and vessel networks in the laboratory.

10:30 AM
11:30 AM
Shayn Peirce-Cottler - Multicellular Simulations of Microvascular Patterning

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.

01:30 PM
02:30 PM
Axel Pries - Structural vascular adaptation in the microcirculation

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.

Friday, January 26, 2007
Time Session
09:00 AM
10:00 AM
Aleksander Popel - Systems Biology of Angiogenesis: From Molecules to Therapy

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.

10:30 AM
11:30 AM
Daniel Beard - Integrating cell metabolism, flow, and tissue function

N/A

Name Affiliation
Aguda, Baltazar bdaguda@gmail.com MBI - Long Term Visitor, Bioinformatics Institute, Singapore
Arciero, Julia jarciero@pitt.edu Mathematics, University of Arizona
Banaji , Murad mbanaji@medphys.ucl.ac.uk Medical Physics and Bioengineering, University of London
Bassingthwaighte, James jbb@bioeng.washington.edu Bioengineering, Radiology, University of Washington
Bateman, Ryon rbateman@mrl.ubc.ca Department of Medicine , University of British Columbia
Bauer, Amy albauer@umich.edu Department of Mathematics, University of Michigan
Beard, Daniel dbeard@mcw.edu Physiology, Medical College of Wisconsin
Best, Janet jbest@mbi.osu.edu
Cardinal, Trevor trc@email.arizona.edu Physiological Sciences, University of Arizona
Carlson, Brian carlsonb@u.washington.edu Bioengineering, University of Washington
Chen, Kejing kchen21@jhu.edu Department of Biomedical Engineering, Johns Hopkins University
Cornelissen, Annemiek annemiek.cornelissen@univ-rennes1.fr Groupe de Matière Condensée et Matériaux, Equipe de Biophysique; Université Rennes1
Dembele, Bassidy bdembele@mbi.osu.edu MBI - Long Term Visitor, Howard University
Djordjevic, Marko mdjordjevic@mbi.osu.edu Mathematical Biosciences Institute (MBI), The Ohio State University
Ellwein, Laura lmellwei@ncsu.edu Mathematics, Virginia Commonwealth University
Enciso, German German_Enciso@hms.harvard.edu Mathematical Biosciences Institute (MBI), The Ohio State University
Frisbee, Jefferson jfrisbee@hsc.wvu.edu Physiology and Pharmacology, West Virginia University
Geddes, John john.geddes@olin.edu unknown, Franklin W. Olin College of Engineering
Goldman, Daniel dgoldma2@uwo.ca Department of Medical Biophysics, University of Western Ontario
Goriely, Alain goriely@math.arizona.edu Mathematics, University of Arizona
Grajdeanu, Paula pgrajdeanu@mbi.osu.edu Mathematical Biosciences Institute (MBI), The Ohio State University
Green, Edward egreen@mbi.osu.edu Mathematical Biosciences Institute (MBI), The Ohio State University
Gruionu, Gabriel ggruionu@wlgore.com W.L. Gore & Associates, Inc., Medical Products Division
Hart, Richard hart.322@osu.edu Department of Biomedical Engineering, The Ohio State University
Hartvigsen, Greg MBI-Long Term Visitor, University at Buffalo (SUNY)
Hoying, James jhoying@email.arizona.edu Regenerative Medicine/BIO5 Institute , University of Arizona
Jacobsen, Jens jcbrings@mfi.ku.dk The Panum Institute, University of Copenhagen
Kao, Chiu-Yen kao.71@osu.edu MBI - Long Term Visitor, The Ohio State University
Kassab, Ghassan gkassab@iupui.edu Biomedical Engineering&MedicineCardiology , University of California, Irvine
Kavdia, Mahendra mkavdia@uark.edu Engineering, University of Arkansas
Kim, Yangjin ykim@mbi.osu.edu Mathematical Biosciences Institute (MBI), The Ohio State University
Leiderman, Karin karin@math.utah.edu Mathematics, University of Utah
Lou, Yuan lou@math.ohio-state.edu MBI - Long Term Visitor, The Ohio State University
Lubkin, Sharon lubkin@eos.ncsu.edu Mathematics, North Carolina State University
Mac Gabhann, Feilim feilim@jhu.edu Biomedical Engineering , Johns Hopkins University
Moldovan, Nicanor moldovan.6@osu.edu Internal Medicine/Cardiology, The Ohio State University
Mukdadi, Sam sam.mukdadi@mail.wvu.edu Mechanical and Aerospace Engineering, West Virginia University
Nevai, Andrew anevai@mbi.osu.edu Mathematical Biosciences Institute (MBI), The Ohio State University
Olufsen, Mette Sofie msolufse@unity.ncsu.edu Mathematics, North Carolina State University
Oster, Andrew aoester@mbi.osu.edu Mathematical Biosciences Institute (MBI), The Ohio State University
Peirce-Cottler, Shayn smp6p@virginia.edu Biomedical Engineering , University of Virginia
Pittman, Roland pittman@hsc.vcu.edu Physiology , Virginia Commonwealth University
Popel, Aleksander apopel@jhu.edu Biomedical Engineering, Johns Hopkins University
Pries, Axel axel.pries@charite.de Dept. of Physiology, Charité - Universitätsmedizin Berlin
Qutub, Amina aminaq@jhu.edu Department of Biomedical Engineering, Johns Hopkins University
Rempe, Michael mrempe@mbi.osu.edu Mathematical Biosciences Institute (MBI), The Ohio State University
Ritman , Erik elran@mayo.edu Physiology and Biomedical Engineering, Mayo Clinic College of Medicine
Roy, Tuhin Roy.TK@mayo.edu Department of Anesthesiology, Mayo Clinic
Schugart, Richard richard.schugart@wku.edu Mathematical Biosciences Institute (MBI), The Ohio State University
Secomb, Timothy Department of Physiology, University of Arizona
Smith, Nicolas np.smith@auckland.ac.nz Bioengineering Institute, unknown
Srinivasan, Partha p.srinivasan35@csuohio.edu Mathematical Biosciences Institute (MBI), The Ohio State University
Stigler, Brandy bstigler@mbi.osu.edu Mathematical Biosciences Institute (MBI), The Ohio State University
Sun, Shuying ssun@mbi.osu.edu Mathematical Biosciences Institute (MBI), The Ohio State University
Szomolay, Barbara b.szomolay@imperial.ac.uk Mathematical Biosciences Institute (MBI), The Ohio State University
Thomas, Evelyn ethomas@mbi.osu.edu MBI - Long Term Visitor, Howard University
Tian, Paul tianjj@mbi.osu.edu Mathematical Biosciences Institute (MBI), The Ohio State University
Tsoukias, Nikolaos tsoukias@fiu.edu Biomedical Engineering, Florida International University
Walton , Jay jwalton@math.tamu.edu Mathematics, Texas A & M University
Wang, Ying wang@math.ohio-state.edu Department of Mathematics, The Ohio State University
Wu, Fan fwu@mcw.edu physiology, Medical College of Wisconsin
Yang, Jin jyang@lanl.gov Theoretical Biology and Biophysics, Los Alamos National Laboratory
Zhu, Luoding lzhu@math.iupui.edu Math , Indiana University--Purdue University
Transport and Exchange in the Microcirculation: Interactions amongst RBC, Endothelial Cells and Cardiomyocytes

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.

Integrating cell metabolism, flow, and tissue function

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Overview of blood flow regulation

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.

Computational modeling of blood-tissue oxygen transport

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.

Public Lecure - Tissue Engineering and Repair: A Vascular Problem

Tissue engineering and related cell-based therapies promise to not only facilitate tissue repair but also functionally replace damaged and diseased tissues. With tissue engineering, the goal is to fabricate tissue constructs, comprised of cells in a supportive environment, which mimic the function and/or architecture of the target tissue. The source of cells used in these constructs is the subject of considerable scientific discussion (and, in the case of stem cells, public discussion). However, regardless of the source and types of the cells incorporated into these engineered constructs, there remains a significant challenge in providing sufficient nutrients to the cells during fabrication and following implantation. Any tissue implant greater in dimension than a few millimeters is too big for nutrients to efficiently diffuse to the construct's cells from outside the construct. This is why the first successfully engineered tissues have been thin, sheets of cells (e.g. a simple skin). As advances give rise to more complicated, 3-dimensional tissue designs, the need for a strategy to support the health of these constructs becomes more urgent. In the body, the cardiovascular system serves to effectively deliver nutrients to any tissue. Therefore, the ability to form and incorporate blood vessels (particularly microvessels) into the constructs is critically important for construct health and function. We will discuss the particular challenges related to providing proper nutrition to constructed tissues and the strategies being employed to build vessels and vessel networks in the laboratory.

Angiogenesis, vascular maturation, and microvessel network remodeling

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.

Scaling laws of vascular trees

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.

Dynamics of blood flow regulation

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.

Multicellular Simulations of Microvascular Patterning

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 Overview of Oxygen Transport in the Microcirculation: Experimental Considerations

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.

Systems Biology of Angiogenesis: From Molecules to Therapy

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.

Structural vascular adaptation in the microcirculation

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.

Measuring intact Vascular Trees by 3D Imaging Methods

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.

Microcirculation and mathematical modeling: A brief history

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.

Models for microvascular regulation of blood flow

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.

Integrative Models of Nitric Oxide Biotransport and Calcium Dynamics in the Microcirculation

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.