Numerical study of oxygen diffusion from capillary to tissues during hypoxia with external force effects

A mathematical model to simulate oxygen delivery through a capillary to tissues under the influence of an external force field is presented. The multi-term general fractional diffusion equation containing force terms and a time dependent absorbent term is taken into account. Fractional calculus is applied to describe the phenomenon of sub-diffusion of oxygen in both transverse and longitudinal directions. A new computational algorithm, i.e., the new iterative method (NIM) is employed to solve the spatio-temporal fractional partial differential equation subject to appropriate physical boundary conditions. Validation of NIM solutions is achieved with a modified Adomian decomposition method (MADM). A parametric study is conducted for three loading scenarios on the capillary-radial force alone, axial force alone and the combined case of both forces. The results demonstrate that the force terms markedly influence the oxygen diffusion process. For example, the radial force exerts a more profound effect than axial force on sub-diffusion of oxygen indicating that careful manipulation of these forces on capillary tissues may assist in the effective reduction of hypoxia or other oxygen depletion phenomena

A Review of Mathematical Models for the Formation of\ud Vascular Networks

Mainly two mechanisms are involved in the formation of blood vasculature: vasculogenesis and angiogenesis. The former consists of the formation of a capillary-like network from either a dispersed or a monolayered population of endothelial cells, reproducible also in vitro by specific experimental assays. The latter consists of the sprouting of new vessels from an existing capillary or post-capillary venule. Similar phenomena are also involved in the formation of the lymphatic system through a process generally called lymphangiogenesis.\ud \ud A number of mathematical approaches have analysed these phenomena. This paper reviews the different modelling procedures, with a special emphasis on their ability to reproduce the biological system and to predict measured quantities which describe the overall processes. A comparison between the different methods is also made, highlighting their specific features

Regulation of blood flow and volume exchange across the microcirculation

Oxygen delivery to cells is the basic prerequisite of life. Within the human body, an ingenious oxygen delivery system, comprising steps of convection and diffusion from the upper airways via the lungs and the cardiovascular system to the microvascular area, bridges the gap between oxygen in the outside airspace and the interstitial space around the cells. However, the complexity of this evolutionary development makes us prone to pathophysiological problems. While those problems related to respiration and macrohemodynamics have already been successfully addressed by modern medicine, the pathophysiology of the microcirculation is still often a closed book in daily practice. Nevertheless, here as well, profound physiological understanding is the only key to rational therapeutic decisions. The prime guarantor of tissue oxygenation is tissue blood flow. Therefore, on the premise of intact macrohemodynamics, the microcirculation has three major responsibilities: 1) providing access for oxygenated blood to the tissues and appropriate return of volume;2) maintaining global tissue flood flow, even in the face of changes in central blood pressure;and 3) linking local blood flow to local metabolic needs. It is an intriguing concept of nature to do this mainly by local regulatory mechanisms, impacting primarily on flow resistance, be this via endothelial or direct smooth muscle actions. The final goal of microvascular blood flow per unit of time is to ensure the needed exchange of substances between tissue and blood compartments. The two principle means of accomplishing this are diffusion and filtration. While simple diffusion is the quantitatively most important form of capillary exchange activity for the respiratory gases, water flux across the blood-brain barrier is facilitated via preformed specialized channels, the aquaporines. Beyond that, the vascular barrier is practically nowhere completely tight for water, with paracellular filtration giving rise to generally low but permanent fluid flux outwards into the interstitial space at the microvascular high pressure segment. At the more leaky venular aspect, both filtration and diffusion allow for bidirectional passage of water, nutrients, and waste products. We are just beginning to appreciate that a major factor for maintaining tissue fluid homeostasis appears to be the integrity of the endothelial glycocalyx

A novel in vivo tumor oxygen profiling assay: Combining functional and molecular imaging with multivariate mathematical modeling

Purpose: The objective of this study is to develop and test a novel high spatio-temporal in vivo assay to quantify tumor oxygenation and hypoxia. The assay implements a biophysical model of oxygen transport to fuse parameters acquired from in vivo functional and molecular imaging modalities. ^ Introduction: Tumor hypoxia plays an important role in carcinogenesis. It triggers pathological angiogenesis to supply more oxygen to the tumor cells and promotes cancer cell metastasis. Preclinical and clinical evidence show that anti-angiogenic treatment is capable of normalizing the tumor vasculature both structurally and functionally. The resulting normalized vasculature provides a more efficient and uniform microcirculation that enhances oxygen and drug delivery to the tumor cells and improves second-line treatments such as traditional radiation or chemotherapy. Early studies using the overall or average tumor hypoxia as a prognostic biomarker of anti-angiogenic therapy efficacy was ambivalent; however, recent studies have discovered that the etiology of hypoxia and its heterogeneity could be used as reliable prognostic biomarkers. The capability to longitudinally map tumor hypoxia with high spatial and temporal resolution has the potential to enhance fundamental cancer research and ultimately cancer patient care. ^ Method: A novel methodology to identify and characterize tumor hypoxia by fusing the physiological hemodynamic parametric maps obtained from functional and molecular imaging modalities and technique using a modified Krogh model of oxygen transport (MPO2) was developed. First, simulations studies were performed to validate this technique. Microscopy data of tumor and brain tissue (control) provided both the vasculature and rheology data. A Green\u27s function algorithm was used to solve the ordinary differential equation and calculate the oxygen profile at a microscopic scale (15 μm) (GPO2), which was used as a reference. From this data, simulated physiological maps (perfusion, fractional plasma volume, fractional interstitial volume) and hemoglobin status (oxygen saturation, hemoglobin concentration) was used as input to MPO2 and used to calculate pO2 levels as a function of scanner spatial resolution and noise. Second, MPO2 was compared to pO2 measurements in xenograft breast tumors using OxyLite oxygen sensor as a Gold Standard, where DCE-CT and PCT-S images were acquired to obtain hemodynamic images. Finally, the vascular physiology measurements obtained from an anti-angiogenic therapeutic study in pancreatic tumors was applied to MPO2 and compared to therapeutic response. ^ Results: The simulation results using Green\u27s function pO2 as standard showed that the MPO2 model performance was dependent on the spatial resolution (voxel size) of the images. Sensitivity and error analysis of this model were also investigated in this study. These oxygen transport simulations results suggest the oxygen saturation and hemoglobin concentration were two key factors in tissue oxygenation, and concomitant with blood perfusion and tumor metabolic rate. Comparisons of the pO2 profile obtained from MPO2 and OxyLite probe in MCF7 tumor model demonstrated a significant correlation and approached a slope of one (after accounting for a few outliers). Simulation studies implementing the physiological data obtained from the anti-angiogenic therapeutic study in pancreatic tumors using the MPO2 model agreed with the experimental findings that blood perfusion is a valuable prognostic biomarker in therapeutic efficacy. This model also predicted the oxygenation improvement difference from two vascular renormalization modes (topological normalization and geometrical normalization). ^ Conclusion: The results from the simulation and in vivo studies demonstrated the feasibility of this novel hypoxia assay. Simulation results of the pancreatic tumors provide an example of the impact the MPO2 model in conjunction with imaging can provide when evaluating the therapeutic significance of various normalization modes in anti-angiogenic therapy, and suggests potential approaches to further improve anti-angiogenic therapy efficacy

Optical Cerebral Blood Flow Monitoring of Mice to Men

This thesis describes cerebral hemodynamic monitoring with the optical techniques of diffuse optical spectroscopy (DOS) and diffuse correlation spectroscopy (DCS). DOS and DCS both employ near-infrared light to investigate tissue physiology millimeters to centimeters below the tissue surface. DOS is a static technique that analyzes multispectral tissue-scattered light intensity signals with a photon diffusion approach (Chapter 2) or a Modified Beer-Lambert law approach (Chapter 3) to derive tissue oxy- and deoxy-hemoglobin concentrations, which are in turn used to compute tissue oxygen saturation and blood volume (Section 2.13). DCS is a qualitatively different dynamic technique that analyzes rapid temporal fluctuations in tissue-scattered light with a correlation diffusion approach to derive tissue blood flow (Chapter 4). Further, in combination these measurements of blood flow and blood oxygenation provide access to tissue oxygen metabolism (Section 7.6). The new contributions of my thesis to the diffuse optics field are a novel analysis technique for the DCS signal (Chapter 5), and a novel approach for separating cerebral hemodynamic signals from extra-cerebral artifacts (Chapter 6). The DCS analysis technique extends the Modified Beer-Lambert approach for DOS to the DCS measurement. This new technique has some useful advantages compared to the correlation diffusion approach. It facilitates real-time flow monitoring in complex tissue geometries, provides a novel route for increasing DCS measurement speed, and can be used to probe tissues wherein light transport is non-diffusive (Chapter 5). It also can be used to filter signals from superficial tissues. For separation of cerebral hemodynamic signals from extra-cerebral artifacts, the Modified Beer-Lambert approach is employed in a pressure modulation scheme, which determines subject-specific contributions of extra-cerebral and cerebral tissues to the DCS/DOS signals by utilizing probe pressure modulation to induce variations in extra-cerebral hemodynamics while cerebral hemodynamics remain constant (Chapter 6). In another novel contribution, I used optical techniques to characterize neurovascular coupling at several levels of cerebral ischemia in a rat model (Chapter 7). Neurovascular coupling refers to the relationship between increased blood flow and oxygen metabolism and increased neuronal activity in the brain. In the rat, localized neuronal activity was increased from functional forepaw stimulation. Under normal flow levels, I (and others) observed that the increase in cerebral blood flow (surrogate for oxygen delivery) from forepaw stimulation exceeded the increase in cerebral oxygen metabolism by about a factor of 2. My measurements indicate that this mismatch between oxygen delivery and consumption are more balanced during ischemia (Chapter 7). In Chapters 2 and 3, I review the underlying theory for the photon diffusion model and the Modified Beer-Lambert law for DOS analysis. I also review the correlation diffusion approach for analyzing DCS signals in Chapter 4. My hope is that readers new to the field will find these background chapters helpful

Mathematical methods for modeling the microcirculation

The microcirculation plays a major role in maintaining homeostasis in the body. Alterations or dysfunctions of the microcirculation can lead to several types of serious diseases. It is not surprising, then, that the microcirculation has been an object of intense theoretical and experimental study over the past few decades. Mathematical approaches offer a valuable method for quantifying the relationships between various mechanical, hemodynamic, and regulatory factors of the microcirculation and the pathophysiology of numerous diseases. This work provides an overview of several mathematical models that describe and investigate the many different aspects of the microcirculation, including geometry of the vascular bed, blood flow in the vascular networks, solute transport and delivery to the surrounding tissue, and vessel wall mechanics under passive and active stimuli. Representing relevant phenomena across multiple spatial scales remains a major challenge in modeling the microcirculation. Nevertheless, the depth and breadth of mathematical modeling with applications in the microcirculation is demonstrated in this work. A special emphasis is placed on models of the retinal circulation, including models that predict the influence of ocular hemodynamic alterations with the progression of ocular diseases such as glaucoma

Development of a mathematical model to estimate intra-tumor oxygen concentrations through multi-parametric imaging

Background Tumor hypoxia is involved in every stage of solid tumor development: formation, progression, metastasis, and apoptosis. Two types of hypoxia exist in tumors—chronic hypoxia and acute hypoxia. Recent studies indicate that the regional hypoxia kinetics is closely linked to metastasis and therapeutic responses, but regional hypoxia kinetics is hard to measure. We propose a novel approach to determine the local pO2 by fusing the parameters obtained from in vivo functional imaging through the use of a modified multivariate Krogh model. Methods To test our idea and its potential to translate into an in vivo setting through the use of existing imaging techniques, simulation studies were performed comparing the local partial oxygen pressure (pO2) from the proposed multivariate image fusion model to the referenced pO2 derived by Green’s function, which considers the contribution from every vessel segment of an entire three-dimensional tumor vasculature to profile tumor oxygen with high spatial resolution. Results pO2 derived from our fusion approach were close to the referenced pO2 with regression slope near 1.0 and an r2 higher than 0.8 if the voxel size (or the spatial resolution set by functional imaging modality) was less than 200 μm. The simulation also showed that the metabolic rate, blood perfusion, and hemoglobin concentration were dominant factors in tissue oxygenation. The impact of the measurement error of functional imaging to the pO2 precision and accuracy was simulated. A Gaussian error function with FWHM equal to 20 % of blood perfusion or fractional vascular volume measurement contributed to average 7 % statistical error in pO2. Conclusion The simulation results indicate that the fusion of multiple parametric maps through the biophysically derived mathematical models can monitor the intra-tumor spatial variations of hypoxia in tumors with existing imaging methods, and the potential to further investigate different forms of hypoxia, such as chronic and acute hypoxia, in response to cancer therapies