MBI Publications

MBI Publications for Mathematical Modeling (8)

  • P. Grajdeanu and L. Moore
    Effect of tubular inhomogeneities on filter properties of thick ascending limb of Henle's loop
    Mathematical BiosciencesVol. 209 No. 2 (2007) pp. 564-592

    Abstract

    We used a simple mathematical model of rat thick ascending limb (TAL) of the loop of Henle to predict the impact of spatially inhomogeneous NaCl permeability, spatially inhomogeneous NaCl active transport, and spatially inhomogeneous tubular radius on luminal NaCl concentration when sustained, sinusoidal perturbations were superimposed on steady-state TAL flow. A mathematical model previously devised by us that used homogeneous TAL transport and fixed TAL radius predicted that such perturbations result in TAL luminal fluid NaCl concentration profiles that are standing waves. That study also predicted that nodes in NaCl concentration occur at the end of the TAL when the tubular fluid transit time equals the period of a periodic perturbation, and that, for non-nodal periods, sinusoidal perturbations generate non-sinusoidal oscillations (and thus a series of harmonics) in NaCl concentration at the TAL end. In the present study we find that the inhomogeneities transform the standing waves and their associated nodes into approximate standing waves and approximate nodes. The impact of inhomogeneous NaCl permeability is small. However, for inhomogeneous active transport or inhomogeneous radius, the oscillations for non-nodal periods tend to be less sinusoidal and more distorted than in the homogeneous case and to thus have stronger harmonics. Both the homogeneous and non-homogeneous cases predict that the TAL, in its transduction of flow oscillations into concentration oscillations, acts as a low-pass filter, but the inhomogeneities result in a less effective filter that has accentuated non-linearities.
  • P. Grajdeanu, R. Schugart, A. Friedman, C. Valentine, A. Agarwal and B. Rovin
    A mathematical model of venous neointimal hyperplasia formation
    Theoretical Biology and Medical ModellingVol. 5 No. 2 (2008)

    Abstract

    In hemodialysis patients, the most common cause of vascular access failure is neointimal hyperplasia of vascular smooth muscle cells at the venous anastomosis of arteriovenous fistulas and grafts. The release of growth factors due to surgical injury, oxidative stress and turbulent flow has been suggested as a possible mechanism for neointimal hyperplasia.
    In this work, we construct a mathematical model which analyzes the role that growth factors might play in the stenosis at the venous anastomosis. The model consists of a system of partial differential equations describing the influence of oxidative stress and turbulent flow on growth factors, the interaction among growth factors, smooth muscle cells, and extracellular matrix, and the subsequent effect on the stenosis at the venous anastomosis, which, in turn, affects the level of oxidative stress and degree of turbulent flow. Computer simulations suggest that our model can be used to predict access stenosis as a function of the initial concentration of the growth factors inside the intimal-luminal space.
    The proposed model describes the formation of venous neointimal hyperplasia, based on pathogenic mechanisms. The results suggest that interventions aimed at specific growth factors may be successful in prolonging the life of the vascular access, while reducing the costs of vascular access maintenance. The model may also provide indication of when invasive access surveillance to repair stenosis should be undertaken.
  • X. Liu, P. Srinivasan, E. Collard, P. Grajdeanu, J. Zweier and A. Friedman
    Nitric Oxide Diffusion Rate is Reduced in the Aortic Wall
    Biophysical JournalVol. 94 No. 5 (2008) pp. 1880-1889q

    Abstract

    Endogenous nitric oxide (NO) plays important physiological roles in the body. As a small diatomic molecule, NO has been assumed to freely diffuse in tissues with a diffusion rate similar to that in water. However, this assumption has not been tested experimentally. In this study, a modified Clark-type NO electrode attached with a customized aorta holder was used to directly measure the flux of NO diffusion across the aortic wall at 37°C. Experiments were carefully designed for accurate measurements of the apparent NO diffusion coefficient D and the partition coefficient α in the aortic wall. A mathematical model was presented for analyzing experimental data. It was determined that α = 1.15 ± 0.11 and D = 848 ± 45 μm2/s (n = 12). The NO diffusion coefficient in the aortic wall is nearly fourfold smaller than the reported diffusion coefficient in solution at 37°C, indicating that NO diffusion in the vascular wall is no longer free, but markedly dependent on the environment in the tissue where these NO molecules are. These results imply that the NO diffusion rate in the vascular wall may be upregulated and downregulated by certain physiological and/or pathophysiological processes affecting the composition of tissues.
  • R. Bertram, P. Grajdeanu and S. Jafri
    Using phase relations to indtify potential mechanisms for metabolic oscillations in isolated beta-cell mitochondria
    IsletsVol. 1 No. 2 (2009) pp. 87-94

    Abstract

    There is a great deal of evidence for the existence of metabolic oscillations in pancreatic β-cells. Mechanisms that have been proposed for these oscillations include glycolytic oscillations; oscillations due to the feedback of Ca2+ onto the mitochondrial inner membrane and on dehydrogenases; and oscillations intrinsic to the tricarboxylic (TCA) cycle or the downstream reactions of oxidative phosphorylation. MacDonald and co-workers (J. Biol. Chem., 278:51894-51900, 2003) showed examples of oscillations in TCA intermediates in isolated mitochondria from liver cells and pancreatic β-cells. These oscillations were clearly not due to oscillations in glycolysis or Ca2+ feedback. In this article we consider several potential mechanisms for these TCA oscillations, using mathematical modeling to determine the phase relations that would result between the citrate and NAD+ concentrations in each case. We demonstrate that negative feedback at only one feedback point, isocitrate dehydrogenase, produces the correct phase relation if oscillations are intrinsic to the TCA cycle. Alternatively, the correct phase relation results if oscillations are due to oscillations in oxidative phosphorylation feeding back onto the TCA cycle. This analysis shows that the observed phase relation between citrate and NAD(P) places strict limits on the potential mechanism for the metabolic oscillations in isolated mitochondria that were observed by MacDonald and co-workers.
  • X. Liu, P. Srinivasan, E. Collard, P. Grajdeanu, K. Lok, S. Boyle, A. Friedman and J. Zweier
    Oxygen regulates the effective diffusion distance of nitric oxide in the aortic wall
    Free Radic Biol MedVol. 48 No. 4 (2010) pp. 554-559

    Abstract

    Endothelium-derived nitric oxide (NO) is critical in maintaining vascular tone. Accumulating evidence shows that NO bioavailability is regulated by oxygen concentration. However, it is unclear to what extent the oxygen concentration regulates NO bioavailability in the vascular wall. In this study, a recently developed experimental setup was used to measure the NO diffusion flux across the aortic wall at various oxygen concentrations. It was observed that for a constant NO concentration at the endothelial surface, the measured NO diffusion flux out of the adventitial surface at [O2] = 0 μM is around fivefold greater than at [O2] = 150 μM, indicating that NO is consumed in the aortic wall in an oxygen-dependent manner. Analysis of experimental data shows that the rate of NO consumption in the aortic wall is first order with respect to [NO] and first order with respect to [O2], and the rate constant k1 was determined as (4.0 ± 0.3) — 103 Mˆ’1 sˆ’1. Computer simulations demonstrate that NO concentration distribution significantly changes with oxygen concentration and the effective NO diffusion distance at low oxygen level ([O2] ‰ 25 μM) is significantly longer than that at high oxygen level ([O2] = 200 μM). These results suggest that oxygen-dependent NO consumption may play an important role in dilating blood vessels during hypoxia by increasing the effective NO diffusion distance.
  • P. Grajdeanu, R. Schugart, A. Friedman, D. Birmingham and B. Rovin
    Mathematical framework for human SLE Nephritis: disease dynamics and urine biomarkers
    Theoretical Biology and Medical ModellingVol. 7 No. 14 (2010)

    Abstract

    Although the prognosis for Lupus Nephritis (LN) has dramatically improved with aggressive immunosuppressive therapies, these drugs carry significant side effects. To improve the effectiveness of these drugs, biomarkers of renal flare cycle could be used to detect the onset, severity, and responsiveness of kidney relapses, and to modify therapy accordingly. However, LN is a complex disease and individual biomarkers have so far not been sufficient to accurately describe disease activity. It has been postulated that biomarkers would be more informative if integrated into a pathogenic-based model of LN.
    This work is a first attempt to integrate human LN biomarkers data into a model of kidney inflammation. Our approach is based on a system of differential equations that capture, in a simplified way, the complexity of interactions underlying disease activity. Using this model, we have been able to fit clinical urine biomarkers data from individual patients and estimate patient-specific parameters to reproduce disease dynamics, and to better understand disease mechanisms. Furthermore, our simulations suggest that the model can be used to evaluate therapeutic strategies for individual patients, or a group of patients that share similar data patterns.
    We show that effective combination of clinical data and physiologically based mathematical modeling may provide a basis for more comprehensive modeling and improved clinical care for LN patients.
  • J. Day, L. Schlesinger and A. Friedman
    Tuberculosis research: Going forward with a powerful "Translation Systems Biology" approach
    Tuberculosis (Edinb)Vol. 90 No. 1 (2010) pp. 7-8

    Abstract

    Due to the complexity of the immune response to a Mycobacterium tuberculosis infection, identifying new, effective therapies and vaccines to combat it has been a problematic issue. Although many advances have been made in understanding particular mechanisms involved, they have, to date, proved insufficient to provide real breakthroughs in this area of tuberculosis research. The term €œTranslational Systems Biology€? has been formally proposed to describe the use of experimental findings combined with mathematical modeling and/or engineering principles to understand complex biological processes in an integrative fashion for the purpose of enhancing clinical practice. This opinion piece discusses the importance of using a translational systems biology approach for tuberculosis research as a means by which to go forward with the potential for significant breakthroughs to occur.
  • J. Day, A. Friedman and L. Schlesinger
    Modeling the host response to inhalation anthrax
    J Theor BiolVol. 276 No. 1 (2011) pp. 199-208

    Abstract

    Inhalation anthrax, an often fatal infection, is initiated by endospores of the bacterium Bacillus anthracis, which are introduced into the lung. To better understand the pathogenesis of an inhalation anthrax infection, we propose a two-compartment mathematical model that takes into account the documented early events of such an infection. Anthrax spores, once inhaled, are readily taken up by alveolar phagocytes, which then migrate rather quickly out of the lung and into the thoracic/mediastinal lymph nodes. En route, these spores germinate to become vegetative bacteria. In the lymph nodes, the bacteria kill the host cells and are released into the extracellular environment where they can be disseminated into the blood stream and grow to a very high level, often resulting in the death of the infected person. Using this framework as the basis of our model, we explore the probability of survival of an infected individual. This is dependent on several factors, such as the rate of migration and germination events and treatment with antibiotics.

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