Influence of tissue metabolism and capillary oxygen supply on arteriolar oxygen transport: a computational model

Reconfiguration and loss of peritubular capillaries in chronic kidney disease

Functional and structural alterations of peritubular capillaries (PTCs) are a major determinant of chronic kidney disease (CKD). Using a software-based algorithm for semiautomatic segmentation and morphometric quantification, this study analyzes alterations of PTC shape associated with chronic tubulointerstitial injury in three mouse models and in human biopsies. In normal kidney tissue PTC shape was predominantly elongated, whereas the majority of PTCs associated with chronic tubulointerstitial injury had a rounder shape. This was reflected by significantly reduced PTC luminal area, perimeter and diameters as well as by significantly increased circularity and roundness. These morphological alterations were consistent in all mouse models and human kidney biopsies. The mean circularity of PTCs correlated significantly with categorized glomerular filtration rates and the degree of interstitial fibrosis and tubular atrophy (IFTA) and classified the presence of CKD or IFTA. 3D reconstruction of renal capillaries revealed not only a significant reduction, but more importantly a substantial simplification and reconfiguration of the renal microvasculature in mice with chronic tubulointerstitial injury. Computational modelling predicted that round PTCs can deliver oxygen more homogeneously to the surrounding tissue. Our findings indicate that alterations of PTC shape represent a common and uniform reaction to chronic tubulointerstitial injury independent of the underlying kidney disease.

Sensitivity analysis of a multi-physics model for the vascular microenvironment

The vascular microenvironment is the scale at which microvascular transport, interstitial tissue properties and cell metabolism interact. The vascular microenvironment has been widely studied by means of quantitative approaches, including multi-physics mathematical models as it is a central system for the pathophysiology of many diseases, such as cancer. The microvascular architecture is a key factor for fluid balance and mass transfer in the vascular microenvironment, together with the physical parameters characterizing the vascular wall and the interstitial tissue. The scientific literature of this field has witnessed a long debate about which factor of this multifaceted system is the most relevant. The purpose of this work is to combine the interpretative power of an advanced multi-physics model of the vascular microenvironment with state of the art and robust sensitivity analysis methods, in order to determine the factors that most significantly impact quantities of interest, related in particular to cancer treatment. We are particularly interested in comparing the factors related to the microvascular architecture with the ones affecting the physics of microvascular transport. Ultimately, this work will provide further insight into how the vascular microenvironment affects cancer therapies, such as chemotherapy, radiotherapy or immunotherapy.

Computational Modeling and Imaging of the Intracellular Oxygen Gradient

Computational modeling can provide a mechanistic and quantitative framework for describing intracellular spatial heterogeneity of solutes such as oxygen partial pressure (pO₂). This study develops and evaluates a finite-element model of oxygen-consuming mitochondrial bioenergetics using the COMSOL Multiphysics program. The model derives steady-state oxygen (O₂) distributions from Fickian diffusion and Michaelis–Menten consumption kinetics in the mitochondria and cytoplasm. Intrinsic model parameters such as diffusivity and maximum consumption rate were estimated from previously published values for isolated and intact mitochondria. The model was compared with experimental data collected for the intracellular and mitochondrial pO₂ levels in human cervical cancer cells (HeLa) in different respiratory states and under different levels of imposed pO₂. Experimental pO₂ gradients were measured using lifetime imaging of a Förster resonance energy transfer (FRET)-based O₂ sensor, Myoglobin-mCherry, which offers in situ real-time and noninvasive measurements of subcellular pO₂ in living cells. On the basis of these results, the model qualitatively predicted (1) the integrated experimental data from mitochondria under diverse experimental conditions, and (2) the impact of changes in one or more mitochondrial processes on overall bioenergetics.

Modular microenvironment components reproduce vascular dynamics de novo in a multi-scale agent-based model

Cells do not exist in isolation; they continuously act within and react to their environment. And this environment is not static; it continuously adapts and responds to cells. Here, we investigate how vascular structure and function impact emergent cell population behavior using an agent-based model (ABM). Our ABM enables researchers to "mix and match" cell agents, subcellular modules, and microenvironment components ranging from simple nutrient sources to complex, realistic vascular architectures that accurately capture hemodynamics. We use this ABM to highlight the bilateral relationship between cells and nearby vasculature, demonstrate the effect of vascular structure on environmental heterogeneity, and emphasize the non-linear, non-intuitive relationship between vascular function and the behavior of cell populations over time. Our ABM is well suited to characterizing in vitro and in vivo studies, with applications from basic science to translational synthetic biology and medicine. The model is freely available at https://github.com/bagherilab/ARCADE. A record of this paper's transparent peer review process is included in the supplemental information.

The Bohr Effect Is Not a Likely Promoter of Renal Preglomerular Oxygen Shunting

The aim of this study was to evaluate whether possible preglomerular arterial-to-venous oxygen shunting is affected by the interaction between renal preglomerular carbon dioxide and oxygen transport. We hypothesized that a reverse (venous-to-arterial) shunting of carbon dioxide will increase partial pressure of carbon dioxide and decrease pH in the arteries and thereby lead to increased oxygen offloading and consequent oxygen shunting. To test this hypothesis, we employed a segment-wise three-dimensional computational model of coupled renal oxygen and carbon dioxide transport, wherein coupling is achieved by shifting the oxygen-hemoglobin dissociation curve in dependence of local changes in partial pressure of carbon dioxide and pH. The model suggests that primarily due to the high buffering capacity of blood, there is only marginally increased acidity in the preglomerular vasculature compared to systemic arterial blood caused by carbon dioxide shunting. Furthermore, effects of carbon dioxide transport do not promote but rather impair preglomerular oxygen shunting, as the increase in acidity is higher in the veins compared to that in the arteries. We conclude that while substantial arterial-to-venous oxygen shunting might take place in the postglomerular vasculature, the net amount of oxygen shunted at the preglomerular vasculature appears to be marginal.

Computational Model for Tumor Oxygenation Applied to Clinical Data on Breast Tumor Hemoglobin Concentrations Suggests Vascular Dilatation and Compression

We present a computational model for trans-vascular oxygen transport in synthetic tumor and host tissue blood vessel networks, aiming at qualitatively explaining published data of optical mammography, which were obtained from 87 breast cancer patients. The data generally show average hemoglobin concentration to be higher in tumors versus host tissue whereas average oxy-to total hemoglobin concentration (vascular segment RBC-volume-weighted blood oxygenation) can be above or below normal. Starting from a synthetic arterio-venous initial network the tumor vasculature was generated by processes involving cooption, angiogenesis, and vessel regression. Calculations of spatially resolved blood flow, hematocrit, oxy- and total hemoglobin concentrations, blood and tissue oxygenation were carried out for ninety tumor and associated normal vessel networks starting from various assumed geometries of feeding arteries and draining veins. Spatial heterogeneity in the extra-vascular partial oxygen pressure distribution can be related to various tumor compartments characterized by varying capillary densities and blood flow characteristics. The reported higher average hemoglobin concentration of tumors is explained by growth and dilatation of tumor blood vessels. Even assuming sixfold metabolic rate of oxygen consumption in tumorous versus host tissue, the predicted oxygen hemoglobin concentrations are above normal. Such tumors are likely associated with high tumor blood flow caused by high-caliber blood vessels crossing the tumor volume and hence oxygen supply exceeding oxygen demand. Tumor oxy- to total hemoglobin concentration below normal could only be achieved by reducing tumor vessel radii during growth by a randomly selected factor, simulating compression caused by intra-tumoral solid stress due to proliferation of cells and extracellular matrix. Since compression of blood vessels will impede chemotherapy we conclude that tumors with oxy- to total hemoglobin concentration below normal are less likely to respond to chemotherapy. Such behavior was recently reported for neo-adjuvant chemotherapy of locally advanced breast tumors.

Computer Simulations of the Tumor Vasculature: Applications to Interstitial Fluid Flow, Drug Delivery, and Oxygen Supply

Tumor vasculature, the blood vessel network supplying a growing tumor with nutrients such as oxygen or glucose, is in many respects different from the hierarchically organized arterio-venous blood vessel network in normal tissues. Angiogenesis (the formation of new blood vessels), vessel cooption (the integration of existing blood vessels into the tumor vasculature), and vessel regression remodel the healthy vascular network into a tumor-specific vasculature. Integrative models, based on detailed experimental data and physical laws, implement, in silico, the complex interplay of molecular pathways, cell proliferation, migration, and death, tissue microenvironment, mechanical and hydrodynamic forces, and the fine structure of the host tissue vasculature. With the help of computer simulations high-precision information about blood flow patterns, interstitial fluid flow, drug distribution, oxygen and nutrient distribution can be obtained and a plethora of therapeutic protocols can be tested before clinical trials. This chapter provides an overview over the current status of computer simulations of vascular remodeling during tumor growth including interstitial fluid flow, drug delivery, and oxygen supply within the tumor. The model predictions are compared with experimental and clinical data and a number of longstanding physiological paradigms about tumor vasculature and intratumoral solute transport are critically scrutinized.

Integrative models of vascular remodeling during tumor growth

Malignant solid tumors recruit the blood vessel network of the host tissue for nutrient supply, continuous growth, and gain of metastatic potential. Angiogenesis (the formation of new blood vessels), vessel cooption (the integration of existing blood vessels into the tumor vasculature), and vessel regression remodel the healthy vascular network into a tumor-specific vasculature that is in many respects different from the hierarchically organized arterio-venous blood vessel network of the host tissues. Integrative models based on detailed experimental data and physical laws implement in silico the complex interplay of molecular pathways, cell proliferation, migration, and death, tissue microenvironment, mechanical and hydrodynamic forces, and the fine structure of the host tissue vasculature. With the help of computer simulations high-precision information about blood flow patterns, interstitial fluid flow, drug distribution, oxygen and nutrient distribution can be obtained and a plethora of therapeutic protocols can be tested before clinical trials. In this review, we give an overview over the current status of integrative models describing tumor growth, vascular remodeling, blood and interstitial fluid flow, drug delivery, and concomitant transformations of the microenvironment. WIREs Syst Biol Med 2015, 7:113-129. doi: 10.1002/wsbm.1295 For further resources related to this article, please visit the WIREs website.

CONFLICT OF INTEREST

The authors have declared no conflicts of interest for this article.

Renal oxygenation: preglomerular vasculature is an unlikely contributor to renal oxygen shunting

The primary aim of this study was to assess the plausibility of preglomerular arterial-to-venous oxygen shunting in the kidney. To this end, we have developed a segment-wise three-dimensional computational model that takes into account transport processes in arteries, veins, cortical tissue, and capillaries. Our model suggests that the amount of preglomerular oxygen shunting is negligible. Consequently, it is improbable that preglomerular shunting contributes to the hypothesized regulation of renal oxygenation. Cortical tissue oxygenation is more likely determined by the interplay between oxygen supply, either from the preglomerular vasculature or from capillaries, and oxygen consumption. We show that reported differences in permeability to oxygen between perfused and unperfused tissue may be explained by what we refer to as advection-facilitated diffusion. We further show that the preglomerular vasculature is the primary source of oxygen for the tissue when cortical consumption is high or renal arterial blood is highly oxygenated, i.e., under hyperoxemic conditions. Conversely, when oxygen demand in the tissue is decreased, or under hypoxemic conditions, oxygen is supplied predominantly by capillaries.

A MATHEMATICAL COUPLED MODEL OF OXYGEN TRANSPORT IN THE MICROCIRCULATION: THE EFFECT OF CONVECTION–DIFFUSION ON OXYGEN TRANSPORT

This paper is aimed at examining the effect of convection–diffusion on oxygen transport at the micro-level. A coupled model of the convection–diffusion and molecular diffusion of oxygen is developed, and the solid deformation resulting from capillary fluctuations and the seepage of tissue fluid are incorporated into this model. The results indicate that (1) the oxygen concentration calculated from this coupled model is higher than that given by molecular diffusion models, both within the capillaries and tissue (maximum difference of 16%); (2) convection–diffusion has the greatest effect in tissue surrounding the middle of the capillary, and enhances the amount of oxygen transported to cells far from the oxygen source; (3) larger permeability coefficients or smaller diffusion coefficients produce a more obvious convection–diffusion effect; (4) a counter-current flow occurs near the inlet and outlet ends of the capillary. This model also provides a foundation for the study of how oxygen affects tumor growth.

A computational model of a microfluidic device to measure the dynamics of oxygen-dependent ATP release from erythrocytes

Erythrocytes are proposed to be involved in blood flow regulation through both shear- and oxygen-dependent mechanisms for the release of adenosine triphosphate (ATP), a potent vasodilator. In a recent study, the dynamics of shear-dependent ATP release from erythrocytes was measured using a microfluidic device with a constriction in the channel to increase shear stress. The brief period of increased shear stress resulted in ATP release within 25 to 75 milliseconds downstream of the constriction. The long-term goal of our research is to apply a similar approach to determine the dynamics of oxygen-dependent ATP release. In the place of the constriction, an oxygen permeable membrane would be used to decrease the hemoglobin oxygen saturation of erythrocytes flowing through the channel. This paper describes the first stage in achieving that goal, the development of a computational model of the proposed experimental system to determine the feasibility of altering oxygen saturation rapidly enough to measure ATP release dynamics. The computational model was constructed based on hemodynamics, molecular transport of oxygen and ATP, kinetics of luciferin/luciferase reaction for reporting ATP concentrations, light absorption by hemoglobin, and sensor characteristics. A linear model of oxygen saturation-dependent ATP release with variable time delay was used in this study. The computational results demonstrate that a microfluidic device with a 100 µm deep channel will cause a rapid decrease in oxygen saturation over the oxygen permeable membrane that yields a measurable light intensity profile for a change in rate of ATP release from erythrocytes on a timescale as short as 25 milliseconds. The simulation also demonstrates that the complex dynamics of ATP release from erythrocytes combined with the consumption by luciferin/luciferase in a flowing system results in light intensity values that do not simply correlate with ATP concentrations. A computational model is required for proper interpretation of experimental data.

What is the efficiency of ATP signaling from erythrocytes to regulate distribution of O(2) supply within the microvasculature?

Erythrocytes appear to be ideal sensors for regulating microvascular O(2) supply as they release the potent vasodilator ATP in an O(2) saturation-dependent manner. Whether erythrocytes play a significant role in regulating O(2) supply in the complex environment of diffusional O(2) exchange among capillaries, arterioles, and venules, depends on the efficiency with which erythrocytes signal the vascular endothelium. If one assumes that the distribution of purinergic receptors is uniform throughout the microvasculature, then the most efficient site for signaling should occur in capillaries, where the erythrocyte membrane is in close proximity to the endothelium. ATP released from erythrocytes would diffuse a short distance to P(2y) receptors inducing an increase in blood flow, possibly the result of endothelial hyperpolarization. We hypothesize that this hyperpolarization varies across the capillary bed depending upon erythrocyte supply rate and the flux of O(2) from these erythrocytes to support O(2) metabolism. This would suggest that the capillary bed would be the most effective site for erythrocytes to communicate tissue oxygen needs. Electrically coupled endothelial cells conduct the integrated signal upstream where arterioles adjust vascular resistance, thus enabling ATP released from erythrocytes to regulate the magnitude and distribution of O(2) supply to individual capillary networks.

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.