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

Using a continuum model to predict closure time of gaps in intestinal epithelial cell layers

A two-dimensional continuum model of collective cell migration is used to predict the closure of gaps in intestinal epithelial cell layers. The model assumes that cell migration is governed by lamellipodia formation, cell-cell adhesion, and cell-substrate adhesion. Model predictions of the gap edge position and complete gap closure time are compared with experimental measures from cell layer scratch assays (also called scratch wound assays). The goal of the study is to combine experimental observations with mathematical descriptions of cell motion to identify effects of gap shape and area on closure time and to propose a method that uses a simple measure (e.g., area) to predict overall gap closure time early in the closure process. Gap closure time is shown to increase linearly with increasing gap area; however, gaps of equal areas but different aspect ratios differ greatly in healing time. Previous methods that calculate overall healing time according to the absolute or percent change in gap area assume that the gap area changes at a constant rate and typically underestimate gap closure time. In this study, data from scratch assays suggest that the rate of change of area is proportional to the first power or square root power of area

Combining Theoretical and Experimental Techniques to Study Murine Heart Transplant Rejection

The quality of life of organ transplant recipients is compromised by complications associated with life-long immunosuppression, such as hypertension, diabetes, opportunistic infections, and cancer. Moreover, the absence of established tolerance to the transplanted tissues causes limited long-term graft survival rates. Thus, there is a great medical need to understand the complex immune system interactions that lead to transplant rejection so that novel and effective strategies of intervention that redirect the system toward transplant acceptance (while preserving overall immune competence) can be identified. This study implements a systems biology approach in which an experimentally based mathematical model is used to predict how alterations in the immune response influence the rejection of mouse heart transplants. Five stages of conventional mouse heart transplantation are modeled using a system of 13 ordinary differential equations that tracks populations of both innate and adaptive immunity as well as proxies for pro- and anti-inflammatory factors within the graft and a representative draining lymph node. The model correctly reproduces known experimental outcomes, such as indefinite survival of the graft in the absence of CD4(+) T cells and quick rejection in the absence of CD8(+) T cells. The model predicts that decreasing the translocation rate of effector cells from the lymph node to the graft delays transplant rejection. Increasing the starting number of quiescent regulatory T cells in the model yields a significant but somewhat limited protective effect on graft survival. Surprisingly, the model shows that a delayed appearance of alloreactive T cells has an impact on graft survival that does not correlate linearly with the time delay. This computational model represents one of the first comprehensive approaches toward simulating the many interacting components of the immune system. Despite some limitations, the model provides important suggestions of experimental investigations that could improve the understanding of rejection. Overall, the systems biology approach used here is a first step in predicting treatments and interventions that can induce transplant tolerance while preserving the capacity of the immune system to protect against legitimate pathogens

Bifurcation study of blood flow control in the kidney.

Renal blood flow is maintained within a narrow window by a set of intrinsic autoregulatory mechanisms. Here, a mathematical model of renal hemodynamics control in the rat kidney is used to understand the interactions between two major renal autoregulatory mechanisms: the myogenic response and tubuloglomerular feedback. A bifurcation analysis of the model equations is performed to assess the effects of the delay and sensitivity of the feedback system and the time constants governing the response of vessel diameter and smooth muscle tone. The results of the bifurcation analysis are verified using numerical simulations of the full nonlinear model. Both the analytical and numerical results predict the generation of limit cycle oscillations under certain physiologically relevant conditions, as observed in vivo

Predicting retinal tissue oxygenation using an image-based theoretical model

Impaired oxygen delivery and tissue perfusion have been identified as significant factors that contribute to the loss of retinal ganglion cells in glaucoma patients. This study predicts retinal blood and tissue oxygenation using a theoretical model of the retinal vasculature based on confocal microscopy images of the mouse retina. These images reveal a complex and heterogeneous geometry of vessels that are distributed non-uniformly into multiple distinct retinal layers at varying depths. Predicting oxygen delivery and distribution in this irregular arrangement of retinal microvessels requires the use of an efficient theoretical model. The model employed in this work utilizes numerical methods based on a Green's function approach to simulate the spatial distribution of oxygen levels in a network of retinal blood vessels and the tissue surrounding them. Model simulations also predict the blood flow rates and pressures in each of the microvessels throughout the entire network. As expected, the model predicts that average vessel PO2 decreases as oxygen demand is increased. However, the standard deviation of PO2 in the vessels nearly doubles as oxygen demand is increased from 1 to 8 cm3 O2/100 cm3/min, indicating a very wide spread in the predicted PO2 levels, suggesting that average PO2 is not a sufficient indicator of oxygenation in a heterogeneous vascular network. Ultimately, the development of this mathematical model will help to elucidate the important factors associated with blood flow and metabolism that contribute to the vision loss characteristic of glaucoma

Using a Mathematical Model to Analyze the Role of Probiotics and Inflammation in Necrotizing Enterocolitis

Background: Necrotizing enterocolitis (NEC) is a severe disease of the gastrointestinal tract of pre-term babies and is thought to be related to the physiological immaturity of the intestine and altered levels of normal flora in the gut. Understanding the factors that contribute to the pathology of NEC may lead to the development of treatment strategies aimed at re-establishing the integrity of the epithelial wall and preventing the propagation of inflammation in NEC. Several studies have shown a reduced incidence and severity of NEC in neonates treated with probiotics (beneficial bacteria species). Methodology/Principal Findings: The objective of this study is to use a mathematical model to predict the conditions under which probiotics may be successful in promoting the health of infants suffering from NEC. An ordinary differential equation model is developed that tracks the populations of pathogenic and probiotic bacteria in the intestinal lumen and in the blood/tissue region. The permeability of the intestinal epithelial layer is treated as a variable, and the role of the inflammatory response is included. The model predicts that in the presence of probiotics health is restored in many cases that would have been otherwise pathogenic. The timing of probiotic administration is also shown to determine whether or not health is restored. Finally, the model predicts that probiotics may be harmful to the NEC patient under very specific conditions, perhaps explaining the detrimental effects of probiotics observed in some clinical studies. Conclusions/Significance: The reduced, experimentally motivated mathematical model that we have developed suggests how a certain general set of characteristics of probiotics can lead to beneficial or detrimental outcomes for infants suffering from NEC, depending on the influences of probiotics on defined features of the inflammatory response. © 2010 Arciero et al

Modeling the Afferent Dynamics of the Baroreflex Control System

In this study we develop a modeling framework for predicting baroreceptor firing rate as a function of blood pressure. We test models within this framework both quantitatively and qualitatively using data from rats. The models describe three components: arterial wall deformation, stimulation of mechanoreceptors located in the BR nerve-endings, and modulation of the action potential frequency. The three sub-systems are modeled individually following well-established biological principles. The first submodel, predicting arterial wall deformation, uses blood pressure as an input and outputs circumferential strain. The mechanoreceptor stimulation model, uses circumferential strain as an input, predicting receptor deformation as an output. Finally, the neural model takes receptor deformation as an input predicting the BR firing rate as an output. Our results show that nonlinear dependence of firing rate on pressure can be accounted for by taking into account the nonlinear elastic properties of the artery wall. This was observed when testing the models using multiple experiments with a single set of parameters. We find that to model the response to a square pressure stimulus, giving rise to post-excitatory depression, it is necessary to include an integrate-and-fire model, which allows the firing rate to cease when the stimulus falls below a given threshold. We show that our modeling framework in combination with sensitivity analysis and parameter estimation can be used to test and compare models. Finally, we demonstrate that our preferred model can exhibit all known dynamics and that it is advantageous to combine qualitative and quantitative analysis methods

Theoretical model of blood flow autoregulation: roles of myogenic, shear-dependent, and metabolic responses

The autoregulation of blood flow, the maintenance of almost constant blood flow in the face of variations in arterial pressure, is characteristic of many tissue types. Here, contributions to the autoregulation of pressure-dependent, shear stress-dependent, and metabolic vasoactive responses are analyzed using a theoretical model. Seven segments, connected in series, represent classes of vessels: arteries, large arterioles, small arterioles, capillaries, small venules, large venules, and veins. The large and small arterioles respond actively to local changes in pressure and wall shear stress and to the downstream metabolic state communicated via conducted responses. All other segments are considered fixed resistances. The myogenic, shear-dependent, and metabolic responses of the arteriolar segments are represented by a theoretical model based on experimental data from isolated vessels. To assess autoregulation, the predicted flow at an arterial pressure of 130 mmHg is compared with that at 80 mmHg. If the degree of vascular smooth muscle activation is held constant at 0.5, there is a fivefold increase in blood flow. When myogenic variation of tone is included, flow increases by a factor of 1.66 over the same pressure range, indicating weak autoregulation. The inclusion of both myogenic and shear-dependent responses results in an increase in flow by a factor of 2.43. A further addition of the metabolic response produces strong autoregulation with flow increasing by a factor of 1.18 and gives results consistent with experimental observation. The model results indicate that the combined effects of myogenic and metabolic regulation overcome the vasodilatory effect of the shear response and lead to the autoregulation of blood flow