Mathematical modeling of oxygen transport in skeletal muscle

Inhomogeneous perfusion of capillary beds can result in large-scale diffusion of oxygen between distant portions of an organ. The conceptual model of a single capillary supplying oxygen to a surrounding concentric cylinder of tissue is not applicable to a consideration of such processes. An entirely different approach to the modeling of oxygen transport to tissue, with specific reference to the capillary beds of skeletal muscle, is presented here. This approach is intended to replace the theoretical Krogh cylinder model of capillary-tissue oxygen transport with a much more realistic model that takes into account inhomogeneities of capillary density, blood flow velocity, and oxygen concentration inherent in the micro-vasculature. The oxygen distribution in inhomogeneously perfused skeletal muscle is analyzed mathematically by defining an averaged concentration profile that neglects the fine-scale variation from capillary to capillary.     

Axial diffusion and wall permeability effects in perfused capillary-tissue structures

Attempts to experimentally examine oxygen supply and distribution in the isolated perfused heart and brain have renewed interest in mathematical models of artificially perfused capillary-tissue structures. The need to understand histograms of PO2 measurements from these isolated-perfused organ studies (modified Lagendorf preparations) has required that existing mathematical models and their boundary conditions be re-examined in the context of these experiments. A unifying system of equations and boundary conditions are examined here for the purpose of studying the effects of anisotropic diffusion, and capillary vessel wall permeability on both the capillary and tissue substrate supply. The mathematical models are explored for parameters of physiologic interest, and some comparisons are made with experimental determinations. The comparisons with data suggest an anisotropic transport of oxygen in the tissue that is unexplained by known physiologic mechanisms.     

Simulation study of pO2 distribution in induced tumour masses and normal tissues within a microcirculation environment

The biological microenvironment is interrupted when tumour masses are introduced because of the strong competition for oxygen. During the period of avascular growth of tumours, capillaries that existed play a crucial role in supplying oxygen to both tumourous and healthy cells. Due to limitations of oxygen supply from capillaries, healthy cells have to compete for oxygen with tumourous cells. In this study, an improved Krogh's cylinder model which is more realistic than the previously reported assumption that oxygen is homogeneously distributed in a microenvironment, is proposed to describe the process of the oxygen diffusion from a capillary to its surrounding environment. The capillary wall permeability is also taken into account. The simulation study is conducted and the results show that when tumour masses are implanted at the upstream part of a capillary and followed by normal tissues, the whole normal tissues suffer from hypoxia. In contrast, when normal tissues are ahead of tumour masses, their pO₂ is sufficient. In both situations, the pO₂ in the whole normal tissues drops significantly due to the axial diffusion at the interface of normal tissues and tumourous cells. As the existence of the axial oxygen diffusion cannot supply the whole tumour masses, only these tumourous cells that are near the interface can be partially supplied, and have a small chance to survive.     

利用大型蒸渗仪模拟土壤-植物-大气连续体水分蒸散

在农田水量转化各分量中,蒸散与潜水蒸发是最难测定的。在地下水浅埋地区,地下水通过毛管上升而补给包气带土壤水的作用十分明显,对作物生长意义重大。利用大型蒸渗仪、波文比、水力蒸发器等仪器,获得了大量水平衡因子的试验数据和土壤植物大气连续体(SPAC)模型中的有关参数。以大型蒸渗仪实测值为基准,验证了农田土壤植物大气连续体模型的模拟值,并主要就蒸散和潜水蒸发量,对实测与模拟值作了比较分析,探讨了导致两者差异的原因。     

Analytic solution to steady-state radial diffusion of a substrate with first-order reaction kinetics in the tissue of a Krogh's cylinder

It is often useful to calculate the concentration profile for a substrate undergoing reaction in the tissue surrounding a capillary. In this paper, we consider a model geometry consisting of a long straight cylinder of tissue surrounding a capillary. Substrate diffuses radially out of the capillary through the tissue, with consumption of substrate in the tissue directly proportional to substrate concentration (i.e., first-order reaction kinetics). The model is extended to include the case where a cylinder of necrotic tissue surrounds a metabolically active inner tissue cylinder. A simple analytic solution is derived, and concentration profiles are generated for various combinations of parameters. Compared to the case where substrate consumption is independent of concentration, this model predicts much more rapid depletion of substrate near the capillary interface. This can have significant implications for the calculation of the hypoxic fraction (e.g., tissue with pO(2)<0.5-5 mmHg) when tumor oxygenation is modeled. The model also permits calculation of the limiting substrate concentration for cell viability when the reaction rate constant is known and vice versa.     

Parameter estimation and change-point detection from Dynamic Contrast Enhanced MRI data using stochastic differential equations

Dynamic Contrast Enhanced imaging (DCE-imaging) following a contrast agent bolus allows the extraction of information on tissue micro-vascularization. The dynamic signals obtained from DCE-imaging are modeled by pharmacokinetic compartmental models which integrate the Arterial Input Function. These models use ordinary differential equations (ODEs) to describe the exchanges between the arterial and capillary plasma and the extravascular-extracellular space. Their least squares fitting takes into account measurement noises but fails to deal with unpredictable fluctuations due to external/internal sources of variations (patients’ anxiety, time-varying parameters, measurement errors in the input function, etc.). Adding Brownian components to the ODEs leads to stochastic differential equations (SDEs). In DCE-imaging, SDEs are discretely observed with an additional measurement noise. We propose to estimate the parameters of these noisy SDEs by maximum likelihood, using the Kalman filter. In DCE-imaging, the contrast agent injected in vein arrives in plasma with an unknown time delay. The delay parameter induces a change-point in the drift of the SDE and ODE models, which is estimated also. Estimations based on the SDE and ODE pharmacokinetic models are compared to real DCE-MRI data. They show that the use of SDE provides robustness in the estimation results. A simulation study confirms these results.     

The Vital Role of Blood Flow-Induced Proliferation and Migration in Capillary Network Formation in a Multiscale Model of Angiogenesis

Sprouting angiogenesis and capillary network formation are tissue scale phenomena. There are also sub-scale phenomena involved in angiogenesis including at the cellular and intracellular (molecular) scales. In this work, a multiscale model of angiogenesis spanning intracellular, cellular, and tissue scales is developed in detail. The key events that are considered at the tissue scale are formation of closed flow path (that is called loop in this article) and blood flow initiation in the loop. At the cellular scale, growth, migration, and anastomosis of endothelial cells (ECs) are important. At the intracellular scale, cell phenotype determination as well as alteration due to blood flow is included, having pivotal roles in the model. The main feature of the model is to obtain the physical behavior of a closed loop at the tissue scale, relying on the events at the cellular and intracellular scales, and not by imposing physical behavior upon it. Results show that, when blood flow is considered in the loop, the anastomosed sprouts stabilize and elongate. By contrast, when the loop is modeled without consideration of blood flow, the loop collapses. The results obtained in this work show that proper determination of EC phenotype is the key for its survival.     

Estimation of contrast agent concentration in intra- and extra-vascular spaces of brain tissue

This article presents a new method for estimating the leakage of a contrast agent out of a vessel. The proposed method is developed based on tissue homogeneity (TH) model, modified Patlak model, and Monte Carlo simulation. The analytical methods published in the literature estimate the contrast agent leakage by solving the coupled differential equations associated with the TH model under adiabatic conditions. These methods employ unrealistic simplifying assumptions and become intractable in their applications to the vessels that have a non-uniform permeability. Without making any unrealistic assumptions, our approach simply tracks the passage of the contrast agent through the capillary and its crossing of the vessel walls based on the blood flow in the vessel, the vessel's permeability, and the condition of the blood-brain barrier (BBB). These are treated as statistical processes that can be modeled reasonably well using the Monte Carlo method. In the proposed approach, the intra- and extra-vascular spaces are divided into multiple compartments, similar to the Patlak model. A real, measured arterial input function (AIF) is used as the capillary input and the concentration of the contrast agent is found as a function of time and distance, inside and outside of the capillary. This is done for normal and abnormal capillaries with uniform and non-uniform permeability. The proposed method generates concentration curves similar to those of the analytical method for simple AIF models. It also generates reasonable concentration curves for a real AIF. The proposed method does not fit a mathematical function to the measured AIF and does not make unrealistic simplifying assumptions. It is not therefore prone to the fitting errors and generates more realistic and more accurate results than the analytical methods.     

Multiple solutions of a model describing cancerous growth

Multiple solutions of a model describing immune surveillance against cancer are studied. The model was proposed by R. Lefever and co-workers. A limit cycle solution and two kinds of wave front solutions are worked out by means of a perturbation method. The problem of nucleation concerning this model is studied by a singular-perturbation technique. In addition, the pulse wave solution is also found by numerical simulations.     

Neural and vascular provisions of rat interscapular brown adipose tissue

The innervation of rat interscapular brown adipose tissue has been studied by light and fluorescence microscopy and electron microscopy after treatment with "false" adrenergic neurotransmitters 5- and 6-hydroxydopamine. The vascular markers neoprene latex and thioflavin S were used to define the blood vascular arrangements within the around the tissue. Catecholaminergic innervation was revealed by fluorescence microscopy at both parenchymal and vasomotor sites. In animals injected with 6-hydroxydopamine, this catecholaminergic fluorescence was extinguished in the parenchymal nerve distribution and markedly reduced in the vasomotor plexus. Identification of an extensive network of noradrenergic vasomotor and parenchymal nerve terminals was established by electron microscopy after 5- and 6-hydroxydopamine administration, but unmarked terminals were also observed in both distributions. These unmarked terminals might represent an additional nonnoradrenergic nerve supply to interscapular brown adipose tissue. The thoracodorsal veins draining the fat pads are directly tributary to a large median perforating vein, which joins the azygos vein, and are also continuous with the axillary vein. In addition to the recognized vascular distribution pattern of lobular arteries supplying an abundant capillary plexus drained by lobular veins, direct arteriovenous anastomoses were observed within the interscapular brown fat pad. It is postulated that these additional vascular arrangements are determinant in the phenomenal increase in blood flow through brown adipose tissue during metabolic stimulation.     

A model of oxygen exchange between an arteriole or venule and the surrounding tissue

A mathematical model is developed to study the effect of capillary convection on oxygen transport around segments of arterioles and venules that are surrounded by capillaries. These capillaries carry unidirectional flow perpendicular to the vessel. The discrete capillary structure is distributed in a manner determined by the capillary blood flow and capillary density. A nonlinear oxyhemoglobin dissociation curve described by the Hill equation is used in the analysis. Oxygen flux from the vessel is expressed as a relationship between Sherwood and Peclet numbers, as well as other dimensionless combinations involving parameters of the capillary bed. A numerical solution is obtained with a finite difference method. The numerical results obtained within the physiological range of parameters allow the prediction of longitudinal gradients of hemoglobin-oxygen saturation along the arterioles and venules.     

A computational model of hemodynamic parameters in cortical capillary networks

The analysis of hemodynamic parameters and functional reactivity of cerebral capillaries is still controversial. To assess the hemodynamic parameters in the cortical capillary network, a generic model was created using 2D voronoi tessellation in which each edge represents a capillary segment. This method is capable of creating an appropriate generic model of cerebral capillary network relating to each part of the brain cortex because the geometric model is able to vary the capillary density. The modeling presented here is based on morphometric parameters extracted from physiological data of the human cortex. The pertinent hemodynamic parameters were obtained by numerical simulation based on effective blood viscosity as a function of hematocrit and microvessel diameter, phase separation and plasma skimming effects. The hemodynamic parameters of capillary networks with two different densities (consistent with the variation of the morphometric data in the human cortical capillary network) were analyzed. The results show pertinent hemodynamic parameters for each model. The heterogeneity (coefficient variation) and the mean value of hematocrits, flow rates and velocities of the both network models were specified. The distributions of blood flow throughout the both models seem to confirm the hypothesis in which all capillaries in a cortical network are recruited at rest (normal condition). The results also demonstrate a discrepancy of the network resistance between two models, which are derived from the difference in the number density of capillary segments between the models.     

Mechanisms of Vascular Damage by Hemorrhagic Snake Venom Metalloproteinases: Tissue Distribution and In Situ Hydrolysis

Background

Envenoming by viper snakes constitutes an important public health problem in Brazil and other developing countries. Local hemorrhage is an important symptom of these accidents and is correlated with the action of snake venom metalloproteinases (SVMPs). The degradation of vascular basement membrane has been proposed as a key event for the capillary vessel disruption. However, SVMPs that present similar catalytic activity towards extracellular matrix proteins differ in their hemorrhagic activity, suggesting that other mechanisms might be contributing to the accumulation of SVMPs at the snakebite area allowing capillary disruption.

Methodology/Principal Findings

In this work, we compared the tissue distribution and degradation of extracellular matrix proteins induced by jararhagin (highly hemorrhagic SVMP) and BnP1 (weakly hemorrhagic SVMP) using the mouse skin as experimental model. Jararhagin induced strong hemorrhage accompanied by hydrolysis of collagen fibers in the hypodermis and a marked degradation of type IV collagen at the vascular basement membrane. In contrast, BnP1 induced only a mild hemorrhage and did not disrupt collagen fibers or type IV collagen. Injection of Alexa488-labeled jararhagin revealed fluorescent staining around capillary vessels and co-localization with basement membrane type IV collagen. The same distribution pattern was detected with jararhagin-C (disintegrin-like/cysteine-rich domains of jararhagin). In opposition, BnP1 did not accumulate in the tissues.

Conclusions/Significance

These results show a particular tissue distribution of hemorrhagic toxins accumulating at the basement membrane. This probably occurs through binding to collagens, which are drastically hydrolyzed at the sites of hemorrhagic lesions. Toxin accumulation near blood vessels explains enhanced catalysis of basement membrane components, resulting in the strong hemorrhagic activity of SVMPs. This is a novel mechanism that underlies the difference between hemorrhagic and non-hemorrhagic SVMPs, improving the understanding of snakebite pathology.     

A study of nutritional transport in capillary-tissue exchange system

A two region flow model has been developed for a capillary-tissue exchange unit. Nutrient dissolved in plasma enters into the tissue from capillary through diffusion, filtration and osmosis. The governing non-linear coupled partial differential equations in the two regions (capillary and tissue) have been solved separately with suitable boundary and matching conditions. The results for the variation of Taylor's Diffusivity Coefficient and concentration of a nutrient in the tissue region have been brought out for various values of the parameters involved in the analysis and discussed. It has been particularly noted that the penetration depth in the tissue for the nutritional transport can be considered as an important diagnostic parameter for many cardiovascular diseases.     

A model of strain-dependent glomerular basement membrane maintenance and its potential ramifications in health and disease

A model is developed and analyzed for type IV collagen turnover in the kidney glomerular basement membrane (GBM), which is the primary structural element in the glomerular capillary wall. The model incorporates strain dependence in both deposition and removal of the GBM, leading to an equilibrium tissue strain at which deposition and removal are balanced. The GBM thickening decreases tissue strain per unit of transcapillary pressure drop according to the law of Laplace, but increases the transcapillary pressure drop required to maintain glomerular filtration. The model results are in agreement with the observed GBM alterations in Alport syndrome and thin basement membrane disease, and the model-predicted linear relation between the inverse capillary radius and inverse capillary thickness at equilibrium is consistent with published data on different mammals. In addition, the model predicts a minimum achievable strain in the GBM based on the geometry, properties, and mechanical environment; that is, an infinitely thick GBM would still experience a finite strain. Although the model assumptions would be invalid for an extremely thick GBM, the minimum achievable strain could be significant in diseases, such as Alport syndrome, characterized by focal GBM thickening. Finally, an examination of reasonable values for the model parameters suggests that the oncotic pressure drop-the osmotic pressure difference between the plasma and the filtrate due to large molecules-plays an important role in setting the GBM strain and, thus, leakage of protein into the urine may be protective against some GBM damage.     

Continuous versus conventional tissue expansion: experimental verification of a new technique

Historically, tissue expansion is a prolonged process, typically requiring at least 6 weeks to complete. Recently, interest has increased in shortening this time period. In the current study, a continuous infusion device maintaining constant expander pressure less than capillary filling pressure was used in a canine model in seven dogs to minimize the time period needed to achieve significant expansion. There were no complications, except one device malfunction, corrected by changes in design. The process was shown to be a safe and effective means of producing amounts of expansion similar to traditional methods in approximately 72 hours, with expansion of 28 percent (n = 6) for continuous tissue expansion (CTE) versus 34 percent (n = 6) for a 2-week rapid expansion protocol. This expansion was derived from either stretch of preexisting tissue (46 percent for CTE, 35 percent for 2-week expansion) or recruitment of adjacent tissue. The clinical application of continuous tissue expansion could permit the advantages of tissue expansion to be obtained in many more situations than are currently available to traditional tissue expansion techniques.     

Effects of Fiber Type and Size on the Heterogeneity of Oxygen Distribution in Exercising Skeletal Muscle

The process of oxygen delivery from capillary to muscle fiber is essential for a tissue with variable oxygen demand, such as skeletal muscle. Oxygen distribution in exercising skeletal muscle is regulated by convective oxygen transport in the blood vessels, oxygen diffusion and consumption in the tissue. Spatial heterogeneities in oxygen supply, such as microvascular architecture and hemodynamic variables, had been observed experimentally and their marked effects on oxygen exchange had been confirmed using mathematical models. In this study, we investigate the effects of heterogeneities in oxygen demand on tissue oxygenation distribution using a multiscale oxygen transport model. Muscles are composed of different ratios of the various fiber types. Each fiber type has characteristic values of several parameters, including fiber size, oxygen consumption, myoglobin concentration, and oxygen diffusivity. Using experimentally measured parameters for different fiber types and applying them to the rat extensor digitorum longus muscle, we evaluated the effects of heterogeneous fiber size and fiber type properties on the oxygen distribution profile. Our simulation results suggest a marked increase in spatial heterogeneity of oxygen due to fiber size distribution in a mixed muscle. Our simulations also suggest that the combined effects of fiber type properties, except size, do not contribute significantly to the tissue oxygen spatial heterogeneity. However, the incorporation of the difference in oxygen consumption rates of different fiber types alone causes higher oxygen heterogeneity compared to control cases with uniform fiber properties. In contrast, incorporating variation in other fiber type-specific properties, such as myoglobin concentration, causes little change in spatial tissue oxygenation profiles.     

Inert gas clearance from tissue by co-currently and counter-currently arranged microvessels

To elucidate the clearance of dissolved inert gas from tissues, we have developed numerical models of gas transport in a cylindrical block of tissue supplied by one or two capillaries. With two capillaries, attention is given to the effects of co-current and counter-current flow on tissue gas clearance. Clearance by counter-current flow is compared with clearance by a single capillary or by two co-currently arranged capillaries. Effects of the blood velocity, solubility, and diffusivity of the gas in the tissue are investigated using parameters with physiological values. It is found that under the conditions investigated, almost identical clearances are achieved by a single capillary as by a co-current pair when the total flow per tissue volume in each unit is the same (i.e., flow velocity in the single capillary is twice that in each co-current vessel). For both co-current and counter-current arrangements, approximate linear relations exist between the tissue gas clearance rate and tissue blood perfusion rate. However, the counter-current arrangement of capillaries results in less-efficient clearance of the inert gas from tissues. Furthermore, this difference in efficiency increases at higher blood flow rates. At a given blood flow, the simple conduction-capacitance model, which has been used to estimate tissue blood perfusion rate from inert gas clearance, underestimates gas clearance rates predicted by the numerical models for single vessel or for two vessels with co-current flow. This difference is accounted for in discussion, which also considers the choice of parameters and possible effects of microvascular architecture on the interpretation of tissue inert gas clearance.     

Model-Based Analysis of Mechanisms Responsible for Sleep-Induced Carbon Dioxide Differences

This work describes a comprehensive mathematical model of the human respiratory control system which incorporates the central mechanisms for predicting sleep-induced changes in chemical regulation of ventilation. The model integrates four individual compartments for gas storage and exchange, namely alveolar air, pulmonary blood, tissue capillary blood, body tissues, and gas transport between them. An essential mechanism in the carbon dioxide transport is its dissociation into bicarbonate and acid, where a buffering mechanism through hemoglobin is used to prevent harmfully low pH levels. In the current model, we assume high oxygen levels and consider intracellular hydrogen ion concentration as the principal respiratory control variable. The resulting system of delayed differential equations is solved numerically. With an appropriate choice of key parameters, such as velocity of blood flow and gain of a non-linear controller function, the model provides steady-state results consistent with our experimental observations measured in subjects across sleep onset. Dynamic predictions from the model give new insights into the behaviour of the system in subjects with different buffering capacities and suggest novel hypotheses for future experimental and clinical studies.