Structural adaptation of microvascular networks: functional roles of adaptive responses

Terminal vascular beds continually adapt to changing demands. A theoretical model is used to simulate structural diameter changes in response to hemodynamic and metabolic stimuli in microvascular networks. Increased wall shear stress and decreased intravascular pressure are assumed to stimulate diameter increase. Intravascular partial pressure of oxygen (PO(2)) is estimated for each segment. Decreasing PO(2) is assumed to generate a metabolic stimulus for diameter increase, which acts locally, upstream via conduction along vessel walls, and downstream via metabolite convection. By adjusting the sensitivities to these stimuli, good agreement is achieved between predicted network characteristics and experimental data from microvascular networks in rat mesentery. Reduced pressure sensitivity leads to increased capillary pressure with reduced viscous energy dissipation and little change in tissue oxygenation. Dissipation decreases strongly with decreased metabolic response. Below a threshold level of metabolic response flow shifts to shorter pathways through the network, and oxygen supply efficiency decreases sharply. In summary, the distribution of vessel diameters generated by the simulated adaptive process allows the network to meet the functional demands of tissue while avoiding excessive viscous energy dissipation.     

An evolutionary hybrid cellular automaton model of solid tumour growth

We propose a cellular automaton model of solid tumour growth, in which each cell is equipped with a micro-environment response network. This network is modelled using a feed-forward artificial neural network, that takes environmental variables as an input and from these determines the cellular behaviour as the output. The response of the network is determined by connection weights and thresholds in the network, which are subject to mutations when the cells divide. As both available space and nutrients are limited resources for the tumour, this gives rise to clonal evolution where only the fittest cells survive. Using this approach we have investigated the impact of the tissue oxygen concentration on the growth and evolutionary dynamics of the tumour. The results show that the oxygen concentration affects the selection pressure, cell population diversity and morphology of the tumour. A low oxygen concentration in the tissue gives rise to a tumour with a fingered morphology that contains aggressive phenotypes with a small apoptotic potential, while a high oxygen concentration in the tissue gives rise to a tumour with a round morphology containing less evolved phenotypes. The tissue oxygen concentration thus affects the tumour at both the morphological level and on the phenotype level.     

Tubular fluid flow and distal NaCl delivery mediated by tubuloglomerular feedback in the rat kidney

The glomerular filtration rate in the kidney is controlled, in part, by the tubuloglomerular feedback (TGF) system, which is a negative feedback loop that mediates oscillations in tubular fluid flow and in fluid NaCl concentration of the loop of Henle. In this study, we developed a mathematical model of the TGF system that represents NaCl transport along a short loop of Henle with compliant walls. The proximal tubule and the outer-stripe segment of the descending limb are assumed to be highly water permeable; the thick ascending limb (TAL) is assumed to be water impermeable and have active NaCl transport. A bifurcation analysis of the TGF model equations was performed by computing parameter boundaries, as functions of TGF gain and delay, that separate differing model behaviors. The analysis revealed a complex parameter region that allows a variety of qualitatively different model equations: a regime having one stable, time-independent steady-state solution and regimes having stable oscillatory solutions of different frequencies. A comparison with a previous model, which represents only the TAL explicitly and other segments using phenomenological relations, indicates that explicit representation of the proximal tubule and descending limb of the loop of Henle lowers the stability of the TGF system. Model simulations also suggest that the onset of limit-cycle oscillations results in increases in the time-averaged distal NaCl delivery, whereas distal fluid delivery is not much affected.     

Thermal behavior of biological tissues—A general analysis

A general analysis is presented for the thermal behavior of a biological tissue. Energy transport by the circulatory system is assumed to be represented by a modified Fick's law. General boundary conditions are assumed for the two-dimensional model and solutions are obtained for rectangular, cylindrical, and spherical geometries. The effects of blood perfusion rate, metabolic rate, arterial temperature and heat exchange with the environment are considered. Results indicate a region of almost constant temperature in the deeper layers of the tissue and reaffirm the important role which blood flow plays in maintaining homeostasis.     

Analysis of oxygen exchange between arterioles and surrounding capillary-perfused tissue.

A theoretical model is used to analyze oxygen transport in a three-dimensional tissue region containing an arteriole surrounded by an array of capillaries in planes perpendicular to the arteriole. Convective removal of oxygen from the vicinity of the arteriole by nearby capillaries is shown to increase diffusive oxygen loss from the arteriole. This effect depends on the locations of the capillaries, particularly those nearest to the arteriole. The arteriolar oxygen efflux is comparable to that predicted by a previous model which used a continuum approach, but the efflux does not increase with increasing perfusion as rapidly as predicted by the continuum model. Even a small capillary flow rate strongly influences the oxygen field surrounding the arteriole.     

Modeling of the pore domain of the GLUR1 channel: homology with K+ channel and binding of channel blockers

Molecular models of the M2 segments of the GluR1 channel have been elaborated using a molecular mechanics approach. The models are based on the homology between pore-lining segments of AMPA receptor channels and the KcsA K+ channel and on cyclic H bonds at the Q/R site of the AMPA receptor channel. The N-terminal region of an M2 segment of the channel is assumed, like that of the K+ channel, to adopt a helical conformation. Due to a deletion, the C-terminal end of the M2 segment of the AMPA receptor is more stretched than that of the K+ channel. As a result, only a single oxygen ring may be exposed to the AMPA receptor channel pore. Data on the block of AMPA receptor channels by dicationic adamantane derivatives have been used to select the most relevant model. The model with the oxygen of a Gly residue (position +2 from the Q/R site) exposed to the pore best fits the experimental data. This model also fits experimental data for another class of AMPA receptor antagonists, the polyamine amides. According to the model, the side-chains of the C-terminal residues are involved in intra-receptor interactions that stabilize the structure of the channel rather than in interactions with ions in the pore.     

Modeling and simulation of oxygen transfer in airlift fermentors

A mathematical model for simulation of oxygen transfer in airlift fermentors is presented. The airlift fermentor is represented by a number of interconnected compartments, each of which is assumed to be well mixed. In the annular region, the model includes both upflow and downflow for the gas phase. The model contains several adjustable parameters through which important hydrodynamic effects affecting oxygen transfer are incorporated. The effect of hydrostatic pressure is also included in the model. The model is simple enough to be used in design studies and it can be easily adapted to other airlift system configurations. The simulation results show good qualitative agreement with available experimental results.     

A computational model of oxygen transport in skeletal muscle for sprouting and splitting modes of angiogenesis

Oxygen transport from capillary networks in muscle at a high oxygen consumption rate was simulated using a computational model to assess the relative efficacies of sprouting and splitting modes of angiogenesis. Efficacy was characterized by the volumetric fraction of hypoxic tissue and overall heterogeneity of oxygen distribution at steady state. Oxygen transport was simulated for a three-dimensional vascular network using parameters for rat extensor digitorum longus (EDL) muscle when oxygen consumption by tissue reached 6, 12, and 18 times basal consumption. First, a control network was generated by using straight non-anastomosed capillaries to establish baseline capillarity. Two networks were then constructed simulating either abluminal lateral sprouting or intraluminal splitting angiogenesis such that capillary surface area was equal in both networks. The sprouting network was constructed by placing anastomosed capillaries between straight capillaries of the control network with a higher probability of placement near hypoxic tissue. The splitting network was constructed by splitting capillaries from the control network into two branches at randomly chosen branching points. Under conditions of moderate oxygen consumption (6 times basal), only minor differences in oxygen delivery resulted between the sprouting and splitting networks. At higher consumption levels (12 and 18 times basal), the splitting network had the lowest volume of hypoxic tissue of the three networks. However, when total blood flow in all three networks was made equal, the sprouting network had the lowest volume of hypoxic tissue. This study also shows that under the steady-state conditions the effect of myoglobin (Mb) on oxygen transport was small.     

Cell Proliferation and Oxygen Diffusion in a Vascularising Scaffold

The supply of oxygen to proliferating cells within a scaffold is a key factor for the successful building of new tissue in soft tissue engineering applications. A recent in vivo model, where an arteriovenous loop is placed in a scaffold, allows a vascularising network to form within a scaffold, establishing an oxygen source within, rather than external, to the scaffold. A one-dimensional model of oxygen concentration, cell proliferation and cell migration inside such a vascularising scaffold is developed and investigated. In addition, a vascularisation model is presented, which supports a vascularisation front which moves at a constant speed. The effects of vascular growth, homogenous and heterogenous seeding, diffusion of cells and critical hypoxic oxygen concentration are considered. For homogenous seeding, a relationship between the speed of the vascular front and a parameter defining the rate of oxygen diffusion relative to the rate of oxygen consumption determines whether a hypoxic region exists at some time. In particular, an estimate of the length of time that a fixed point in the scaffold will remain under hypoxic conditions is determined. For heterogenous seeding, a Fisher-like travelling wave of cells is established behind the vascular front. These findings provide a fundamental understanding of the important interplay between the parameters and allows for a theoretical assessment of a seeding strategy in a vascularising scaffold.     

Surface modeling of craniofacial form in human embryos with a limited graphics terminal

Three-dimensional morphology of the human embryo typically is visualized through computerized modeling techniques utilizing planar contours as the data base. Through this approach, tissue outlines are digitized, and contour lines are superimposed, providing a depth perspective. However, these techniques represent embryonic tissues as discontinuous surfaces and therefore ignore morphological information between sections. The purpose of this study was to develop a computerized routine for the three-dimensional surface modeling of craniofacial morphology in human embryos. Tissue outlines are digitized, thus converting contour information into x,y,z coordinate data. The three-dimensional reconstruction program BCSURF opens the data file and plots each tissue polygon. A center is determined for each contour, and this value is used to divide each polygon into four segments. Surface patches are generated by mapping each segment onto the corresponding segment of subsequent sections. A face table is constructed representing the surface patches and plane normals are generated for each patch. The normal and depth values are appended to the face table, and these measures determine the color intensity for each patch. Finally, patches are plotted providing a polygon mesh model, and each patch is filled with a dither pattern according to shading values. Three-dimensional reconstructions of the craniofacial region in Carnegie embryos (stages 15-17) are generated, and major morphological features are observed. Although bilevel shading capabilities cause discontinuous shading textures, this simple and inexpensive system can be easily upgraded for high-resolution graphics.     

Optimization of shunt placement for the Norwood surgery using multi-domain modeling

An idealized systemic-to-pulmonary shunt anatomy is parameterized and coupled to a closed loop, lumped parameter network (LPN) in a multidomain model of the Norwood surgical anatomy. The LPN approach is essential for obtaining information on global changes in cardiac output and oxygen delivery resulting from changes in local geometry and physiology. The LPN is fully coupled to a custom 3D finite element solver using a semi-implicit approach to model the heart and downstream circulation. This closed loop multidomain model is then integrated with a fully automated derivative-free optimization algorithm to obtain optimal shunt geometries with variable parameters of shunt diameter, anastomosis location, and angles. Three objective functions: (1) systemic; (2) coronary; and (3) combined systemic and coronary oxygen deliveries are maximized. Results show that a smaller shunt diameter with a distal shunt-brachiocephalic anastomosis is optimal for systemic oxygen delivery, whereas a more proximal anastomosis is optimal for coronary oxygen delivery and a shunt between these two anatomies is optimal for both systemic and coronary oxygen deliveries. Results are used to quantify the origin of blood flow going through the shunt and its relationship with shunt geometry. Results show that coronary artery flow is directly related to shunt position.     

Emergent vascular network inhomogeneities and resulting blood flow patterns in a growing tumor

Tumors acquire sufficient oxygen and nutrient supply by coopting host vessels and neovasculature created via angiogenesis, thereby transforming a highly ordered network into chaotic heterogeneous tumor specific vasculature. Vessel regression inside the tumor leads to large regions of necrotic tissue interspersed with isolated surviving vessels. We extend our recently introduced model to incorporate Fahraeus-Lindqvist- and phase separation effects, refined tissue oxygen level computation and drug flow computations. We find, unexpectedly, that collapse and regression accelerates rather than diminishes the perfusion and that a tracer substance flowing through the remodeled network reaches all parts of the tumor vasculature very well. The reason for decreased drug delivery well known in tumors should therefore be different from collapse and vessel regression. Implications for drug delivery in real tumors are discussed.     

Urinary metabolic network analysis in trauma, hemorrhagic shock, and resuscitation

Hemorrhagic shock, often a result of traumatic injury, is a condition of reduced perfusion that results in diminished delivery of oxygen to tissues. The disruption in oxygen delivery induced by both ischemia (diminished oxygen delivery) and reperfusion (restoration of oxygen delivery) has profound consequences for cellular metabolism and the maintenance of homeostasis. The pathophysiologic state associated with traumatic injury and hemorrhagic shock was studied with a scale-invariant metabolic network. Urinary metabolic profiles were constructed from NMR spectra of urine samples collected at set timepoints in a porcine model of hemorrhagic shock that included a pulmonary contusion, a liver crush injury, and a 35 % controlled bleed. The network was constructed from these metabolic profiles. A partial least squares discriminant analysis (PLS-DA) model that discriminates by experimental timepoint was also constructed. Comparisons of the network (functional relationships among metabolites) and PLS-DA model (observable relationships to experimental time course) revealed complementary information. First, ischemia/reperfusion injury and evidence of cell death due to hemorrhage was associated with early resuscitation timepoints. Second, evidence of increased protein catabolism and traumatic injury was associated with late resuscitation timepoints. These results are concordant with generally accepted views of the metabolic progression of shock.     

Terminal and intermediate segment lenghts in neuronal trees with finite length

A basic but neglected property of neuronal trees is their finite length. This finite length restricts the length of a segment to a certain maximum. The implications of the finite length of the tree with respect to the segment length distributions of terminal and intermediate segments are shown by means of a stochastic model. In the model it is assumed that branching is governed by a Poisson process. The model shows that terminal segments are expected to be longer than intermediate segments. Terminal and intermediate segments are expected to decrease in length with incrasing centrifugal order. The results are compared with data fromin vivo pyramidal cells from rat brain and tissue cultured ganglion cells from chicken. A good agreement between data and model was found.     

A mathematical model of diffusional shunting of oxygen from arteries to veins in the kidney

To understand how arterial-to-venous (AV) oxygen shunting influences kidney oxygenation, a mathematical model of oxygen transport in the renal cortex was created. The model consists of a multiscale hierarchy of 11 countercurrent systems representing the various branch levels of the cortical vasculature. At each level, equations describing the reactive-advection-diffusion of oxygen are solved. Factors critical in renal oxygen transport incorporated into the model include the parallel geometry of arteries and veins and their respective sizes, variation in blood velocity in each vessel, oxygen transport (along the vessels, between the vessels and between vessel and parenchyma), nonlinear binding of oxygen to hemoglobin, and the consumption of oxygen by renal tissue. The model is calibrated using published measurements of cortical vascular geometry and microvascular Po(2). The model predicts that AV oxygen shunting is quantitatively significant and estimates how much kidney Vo(2) must change, in the face of altered renal blood flow, to maintain cortical tissue Po(2) at a stable level. It is demonstrated that oxygen shunting increases as renal Vo(2) or arterial Po(2) increases. Oxygen shunting also increases as renal blood flow is reduced within the physiological range or during mild hemodilution. In severe ischemia or anemia, or when kidney Vo(2) increases, AV oxygen shunting in proximal vascular elements may reduce the oxygen content of blood destined for the medullary circulation, thereby exacerbating the development of tissue hypoxia. That is, cortical ischemia could cause medullary hypoxia even when medullary perfusion is maintained. Cortical AV oxygen shunting limits the change in oxygen delivery to cortical tissue and stabilizes tissue Po(2) when arterial Po(2) changes, but renders the cortex and perhaps also the medulla susceptible to hypoxia when oxygen delivery falls or consumption increases.     

Constitutive formulation and numerical analysis of the heel pad region

The aim of this work is to provide a numerical approach for the investigation of the mechanical behaviour of the heel pad region. A visco-hyperelastic model is formulated with regard to fat pad tissue, while a fibre-reinforced hyperelastic model is considered for the heel skin tissue. Bone components are defined by means of an orthotropic linear elastic model. Particular attention is paid to the evaluation of constitutive parameters within different models adopted in consideration of experimental tests data. Preliminarily, indentation tests on a skinless cadaveric foot are considered with regard to fat pad tissue. Indentation tests on an intact heel pad of a cadaveric foot are subsequently adopted for the final identification of constitutive parameters of fat pad and skin tissues. A numerical model of the rear foot is defined and different loading conditions are assumed according to experimental data. A comparison between experimental and numerical data leads to the evaluation of the real capability of the procedure to interpret the actual response of the rear foot.     

Constitutive formulation and numerical analysis of the heel pad region

The aim of this work is to provide a numerical approach for the investigation of the mechanical behaviour of the heel pad region. A visco-hyperelastic model is formulated with regard to fat pad tissue, while a fibre-reinforced hyperelastic model is considered for the heel skin tissue. Bone components are defined by means of an orthotropic linear elastic model. Particular attention is paid to the evaluation of constitutive parameters within different models adopted in consideration of experimental tests data. Preliminarily, indentation tests on a skinless cadaveric foot are considered with regard to fat pad tissue. Indentation tests on an intact heel pad of a cadaveric foot are subsequently adopted for the final identification of constitutive parameters of fat pad and skin tissues. A numerical model of the rear foot is defined and different loading conditions are assumed according to experimental data. A comparison between experimental and numerical data leads to the evaluation of the real capability of the procedure to interpret the actual response of the rear foot.     

A multiphysics 3D model of tissue growth under interstitial perfusion in a tissue-engineering bioreactor

The main challenge in engineered cartilage consists in understanding and controlling the growth process towards a functional tissue. Mathematical and computational modelling can help in the optimal design of the bioreactor configuration and in a quantitative understanding of important culture parameters. In this work, we present a multiphysics computational model for the prediction of cartilage tissue growth in an interstitial perfusion bioreactor. The model consists of two separate sub-models, one two-dimensional (2D) sub-model and one three-dimensional (3D) sub-model, which are coupled between each other. These sub-models account both for the hydrodynamic microenvironment imposed by the bioreactor, using a model based on the Navier–Stokes equation, the mass transport equation and the biomass growth. The biomass, assumed as a phase comprising cells and the synthesised extracellular matrix, has been modelled by using a moving boundary approach. In particular, the boundary at the fluid–biomass interface is moving with a velocity depending from the local oxygen concentration and viscous stress. In this work, we show that all parameters predicted, such as oxygen concentration and wall shear stress, by the 2D sub-model with respect to the ones predicted by the 3D sub-model are systematically overestimated and thus the tissue growth, which directly depends on these parameters. This implies that further predictive models for tissue growth should take into account of the three dimensionality of the problem for any scaffold microarchitecture.     

O2 transport in cerebral microregions (mathematical simulation)

The effects of the circulation rate in capillaries, the intensity of O2 consumption by nerve cells and the capillary network density on the O2 tension distribution in the cerebral cortex have been studied, utilizing a mathematical model simulating actual neuron-capillary relationships. The model has been written as a system of equations in partial derivatives, its solution obtained by the net-point method. Regulatory variations of the capillary circulation rate in certain cerebral microregions have been shown to ensure similar changes in oxygen supply throughout the region. A drop of the pO2 level in a cerebral microregion with a rising O2 consumption by nerve cells is shown to be due, by 75 percent, to the increase of O2 consumption and by 25 percent, to the lower pO2 in the capillaries. Conversely, an increase in pO2 in microregions resulting from a lower O2 consumption by neurons is due by 75 percent, to a pO2 rise in capillaries and by 25 percent, at the expense of an O2 consumption decrease. In cerebral regions differing in capillary network density by 20 percent, changes in the conditions for oxygen supply to tissue are due by 1/3 to pO2 variations in the capillaries and by 2/3 to alterations in the diffusion distances.     

Modeling static and dynamic human cardiovascular responses to exercise.

A human performance model has been developed and described [9] which portrays the human circulatory, thermo regulatory and energy-exchange systems as an intercoupled set. In this model, steady state or static relationships are used to describe oxygen consumption and blood flow. For example, heart rate (HTRT) is calculated as a function of the oxygen and the thermo-regulatory requirements of each body compartment, using the steady state work values of cardiac output (CO, sum of all compartment blood flows) and stroke volume (SV, assumed maximal after 40% maximal oxygen consumption): HTRT=CO/SV. The steady state model has proven to be an acceptable first approximation, but the inclusion of transient characteristics are essential in describing the overall systems' adjustment to exercise stress. In the present study, the dynamic transient characteristics of heart rate, stroke volume and cardiac output were obtained from experiments utilizing step and sinusoidal forcing of work. The gain and phase relationships reveal a probable first order system with a six minute time constant, and are utilized to model the transient characteristics of these parameters. This approach leads to a more complex model but a more accurate representation of the physiology involved. The instrumentation and programming essential to these experiments are described.