Potential measurement error from vessel reflex and multiple light paths in dual-wavelength retinal oximetry.

PURPOSE: This study aims to characterize the dependence of measured retinal arterial and venous saturation on vessel diameter and central reflex in retinal oximetry, with an ultimate goal of identifying potential causes and suggesting approaches to improve measurement accuracy. METHODS: In 10 subjects, oxygen saturation, vessel diameter and optical density are obtained using Oxymap Analyzer software without diameter correction. Diameter dependence of saturation is characterized using linear regression between measured values of saturation and diameter. Occurrences of negative values of vessel optical densities (ODs) associated with central vessel reflex are acquired from Oxymap Analyzer. A conceptual model is used to calculate the ratio of optical densities (ODRs) according to retinal reflectance properties and single and double-pass light transmission across fixed path lengths. Model-predicted values are compared with measured oximetry values at different vessel diameters. RESULTS: Venous saturation shows an inverse relationship with vessel diameter (D) across subjects, with a mean slope of -0.180 (SE = 0.022) %/µm (20 < D < 180 µm) and a more rapid saturation increase at small vessel diameters reaching to over 80%. Arterial saturation yields smaller positive and negative slopes in individual subjects, with an average of -0.007 (SE = 0.021) %/µm (20 < D < 200 µm) across all subjects. Measurements where vessel brightness exceeds that of the retinal background result in negative values of optical density, causing an artifactual increase in saturation. Optimization of model reflectance values produces a good fit of the conceptual model to measured ODRs. CONCLUSION: Measurement artefacts in retinal oximetry are caused by strong central vessel reflections, and apparent diameter sensitivity may result from single and double-pass transmission in vessels. Improvement in correction for vessel diameter is indicated for arteries however further study is necessary for venous corrections.

Modeling the Potential of Treg-Based Therapies for Transplant Rejection: Effect of Dose, Timing, and Accumulation Site.

Introduction: The adoptive transfer of regulatory T cells (Tregs) has emerged as a method to promote graft tolerance. Clinical trials have demonstrated the safety of adoptive transfer and are now assessing their therapeutic efficacy. Strategies that generate large numbers of antigen specific Tregs are even more efficacious. However, the combinations of factors that influence the outcome of adoptive transfer are too numerous to be tested experimentally. Here, mathematical modeling is used to predict the most impactful treatment scenarios. Methods: We adapted our mathematical model of murine heart transplant rejection to simulate Treg adoptive transfer and to correlate therapeutic efficacy with Treg dose and timing, frequency of administration, and distribution of injected cells. Results: The model predicts that Tregs directly accumulating to the graft are more protective than Tregs localizing to draining lymph nodes. Inhibiting antigen-presenting cell maturation and effector functions at the graft site was more effective at modulating rejection than inhibition of T cell activation in lymphoid tissues. These complex dynamics define non-intuitive relationships between graft survival and timing and frequency of adoptive transfer. Conclusion: This work provides the framework for better understanding the impact of Treg adoptive transfer and will guide experimental design to improve interventions.

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.

Ocular hemodynamics and glaucoma: the role of mathematical modeling.

PURPOSE: To discuss the role of mathematical modeling in studying ocular hemodynamics, with a focus on glaucoma. METHODS: We reviewed recent literature on glaucoma, ocular blood flow, autoregulation, the optic nerve head, and the use of mathematical modeling in ocular circulation. RESULTS: Many studies suggest that alterations in ocular hemodynamics play a significant role in the development, progression, and incidence of glaucoma. Although there is currently a limited number of studies involving mathematical modeling of ocular blood flow, regulation, and diseases (such as glaucoma), preliminary modeling work shows the potential of mathematical models to elucidate the mechanisms that contribute most significantly to glaucoma progression. CONCLUSION: Mathematical modeling is a useful tool when used synergistically with clinical and laboratory data in the study of ocular blood flow and glaucoma. The development of models to investigate the relationship between ocular hemodynamic alterations and glaucoma progression will provide a unique and useful method for studying the pathophysiology of 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.

Mathematical modeling approaches in the study of glaucoma disparities among people of African and European descents.

Open angle glaucoma (OAG) is a severe ocular disease characterized by progressive and irreversible vision loss. While elevated intraocular pressure (IOP) is a well-established risk factor for OAG, the progression of OAG in many cases, despite IOP treatment, suggests that other risk factors must play significant roles in the development of the disease. For example, various structural properties of the eye, ocular blood flow properties, and systemic conditions have been identified as risk factors for OAG. Ethnicity has also been indicated as a relevant factor that affects the incidence and prevalence of OAG; in fact, OAG is the leading cause of blindness among people of African descent. Numerous clinical studies have been designed to examine the possible correlation and causation between OAG and these factors; however, these studies are met with the challenge of isolating the individual role of multiple interconnected factors. Over the last decade, various mathematical modeling approaches have been implemented in combination with clinical studies in order to provide a mechanical and hemodynamical description of the eye in relation to the entire human body and to assess the contribution of single risk factors to the development of OAG. This review provides a summary of the clinical evidence of ocular structural differences, ocular vascular differences and systemic vascular differences among people of African and European descent, describes the mathematical approaches that have been proposed to study ocular mechanics and hemodynamics while discussing how they could be used to investigate the relevance to OAG of racial disparities, and outlines possible new directions of research.

Continuum model of collective cell migration in wound healing and colony expansion.

Collective cell migration plays an important role during wound healing and embryo development. Although the exact mechanisms that coordinate such migration are still unknown, experimental studies of moving cell layers have shown that the primary interactions governing the motion of the layer are the force of lamellipodia, the adhesion of cells to the substrate, and the adhesion of cells to each other. Here, we derive a two-dimensional continuum mechanical model of cell-layer migration that is based on a novel assumption of elastic deformation of the layer and incorporates basic mechanical interactions of cells as well as cell proliferation and apoptosis. The evolution equations are solved numerically using a level set method. The model successfully reproduces data from two types of experiments: 1), the contraction of an enterocyte cell layer during wound healing; and 2), the expansion of a radially symmetric colony of MDCK cells, both in the edge migration velocity and in cell-layer density. In accord with experimental observations, and in contrast to reaction-diffusion models, this model predicts a partial wound closure if lamellipod formation is inhibited at the wound edge and gives implications of the effect of spatially restricted proliferation.

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.

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.

Theoretical model of metabolic blood flow regulation: roles of ATP release by red blood cells and conducted responses.

A proposed mechanism for metabolic flow regulation involves the saturation-dependent release of ATP by red blood cells, which triggers an upstream conducted response signal and arteriolar vasodilation. To analyze this mechanism, a theoretical model is used to simulate the variation of oxygen and ATP levels along a flow pathway of seven representative segments, including two vasoactive arteriolar segments. The conducted response signal is defined by integrating the ATP concentration along the vascular pathway, assuming exponential decay of the signal in the upstream direction with a length constant of approximately 1 cm. Arteriolar tone depends on the conducted metabolic signal and on local wall shear stress and wall tension. Arteriolar diameters are calculated based on vascular smooth muscle mechanics. The model predicts that conducted responses stimulated by ATP release in venules and propagated to arterioles can account for increases in perfusion in response to increased oxygen demand that are consistent with experimental findings at low to moderate oxygen consumption rates. Myogenic and shear-dependent responses are found to act in opposition to this mechanism of metabolic flow regulation.