Multiscale bio-chemo-mechanical model of intimal hyperplasia

We consider a computational multiscale framework of a bio-chemo-mechanical model for intimal hyperplasia. With respect to existing models, we investigate the interactions between hemodynamics, cellular dynamics and biochemistry on the development of the pathology. Within the arterial wall, we propose a mathematical model consisting of kinetic differential equations for key vascular cell types, collagen and growth factors. The luminal hemodynamics is modeled with the Navier–Stokes equations. Coupling hypothesis among time and space scales are proposed to build a tractable modeling of such a complex multifactorial and multiscale pathology. A one-dimensional numerical test-case is presented for validation by comparing the results of the framework with experiments at short and long timescales. Our model permits to capture many cellular phenomena which have a central role in the physiopathology of intimal hyperplasia. Results are quantitatively and qualitatively consistent with experimental findings at both short and long timescales.

     

A statistical model of the VA/Q distribution

A "blocks model" is proposed to model the distribution of the ventilation-perfusion ratio (VA/Q distribution). This model is developed from statistical principles and enables the estimated VA/Q distribution to be interpreted in a straightforward and intuitive manner. Estimation of parameters of the blocks model uses a constrained weighted least-squares procedure. Although developed initially to estimate VA/Q distributions from data generated by the multiple inert gas elimination technique (P. D. Wagner, H. A. Saltzman, and J. B. West, J. Appl. Physiol. 36: 588-599, 1974), the blocks method is applicable to any problem in which the unknown distribution is related to the data through an ill-posed integral equation and is particularly suited for problems in which the data are scarce. The method is illustrated with several examples--hypothetical data representing a wide range of VA/Q distributions as well as some real data.     

An experimental approach to the fundamental principles of hemodynamics

An experimental model has been developed to give students hands-on experience with the fundamental laws of hemodynamics. The proposed experimental setup is of simple construction but permits the precise measurements of physical variables involved in the experience. The model consists in a series of experiments where different basic phenomena are quantitatively investigated, such as the pressure drop in a long straight vessel and in an obstructed vessel, the transition from laminar to turbulent flow, the association of vessels in vascular networks, or the generation of a critical stenosis. Through these experiments, students acquire a direct appreciation of the importance of the parameters involved in the relationship between pressure and flow rate, thus facilitating the comprehension of more complex problems in hemodynamics.     

Prediction of circulatory equilibrium in response to changes in stressed blood volume

Accurate prediction of cardiac output (CO), left atrial pressure (PLA), and right atrial pressure (PRA) is a prerequisite for management of patients with compromised hemodynamics. In our previous study (Uemura et al. Am J Physiol Heart Circ Physiol 286: H2376-H2385, 2004), we demonstrated a circulatory equilibrium framework, which permits the prediction of CO, PLA, and PRA once the venous return surface and integrated CO curve are known. Inasmuch as we also showed that the surface can be estimated from single-point CO, PLA, and PRA measurements, we hypothesized that a similar single-point estimation of the CO curve would enable us to predict hemodynamics. In seven dogs, we measured the PLA-CO and PRA-CO relations and derived a standardized CO curve using the logarithmic function CO = SL[ln(PLA - 2.03) + 0.80] for the left heart and CO = SR[ln(PRA - 2.13) + 1.90] for the right heart, where SL and SR represent the preload sensitivity of CO, i.e., pumping ability, of the left and right heart, respectively. To estimate the integrated CO curve in each animal, we calculated SL and SR from single-point CO, PLA, and PRA measurements. Estimated and measured CO agreed reasonably well. In another eight dogs, we altered stressed blood volume (-8 to +8 ml/kg of reference volume) under normal and heart failure conditions and predicted the hemodynamics by intersecting the surface and the CO curve thus estimated. We could predict CO [y = 0.93x + 6.5, r2 = 0.96, standard error of estimate (SEE) = 7.5 ml.min(-1).kg(-1)], PLA (y = 0.90x + 0.5, r2= 0.93, SEE = 1.4 mmHg), and PRA (y = 0.87x + 0.4, r2= 0.91, SEE = 0.4 mmHg) reasonably well. In conclusion, single-point estimation of the integrated CO curve enables accurate prediction of hemodynamics in response to extensive changes in stressed blood volume.     

Numerical Simulation of Physiological Blood Flow in 2-way Coronary Artery Bypass Grafts

The Coronary Artery Bypass Graft (CABG) yields excellent results and remains the modern standard of care for treatment of occlusive disease in the cardiovascular system. However, the development of anastomotic Intimal Hyperplasia (IH) and restenosis can compromise the medium-and-long term effects of the CABG. This problem can be correlated with the geometric configuration and hemodynamics of the bypass graft. A novel geometric configuration was proposed for the CABG with two symmetrically implanted grafts for the purpose of improving the hemodynamics. Physiological blood flows in two models of bypass grafts were simulated using numerical methods. One model was for the conventional bypass configuration with a single graft (1-way model); the other model was for the proposed bypass configuration with two grafts (2-way model). The temporal and spatial distributions of hemodynamics, such as flow patterns and Wall Shear Stress (WSS) in the vicinity of the distal anastomoses, were analyzed and compared. Calculation results showed that the 2-way model possessed favorable hemodynamics with uniform longitudinal flow patterns and WSS distributions, which could decrease the probability of restenosis and improve the effect of the surgical treatment. Concerning the limitations of the 2-way bypass grafts, it is necessary to perform animal experiments to verify the viability of this novel idea for the CABG.     

Multistable Dynamics Mediated by Tubuloglomerular Feedback in a Model of Coupled Nephrons

To help elucidate the causes of irregular tubular flow oscillations found in the nephrons of spontaneously hypertensive rats (SHR), we have conducted a bifurcation analysis of a mathematical model of two nephrons that are coupled through their tubuloglomerular feedback (TGF) systems. This analysis was motivated by a previous modeling study which predicts that NaCl backleak from a nephron’s thick ascending limb permits multiple stable oscillatory states that are mediated by TGF (Layton et al. in Am. J. Physiol. Renal Physiol. 291:F79–F97, 2006); that prediction served as the basis for a comprehensive, multifaceted hypothesis for the emergence of irregular flow oscillations in SHR. However, in that study, we used a characteristic equation obtained via linearization from a single-nephron model, in conjunction with numerical solutions of the full, nonlinear model equations for two and three coupled nephrons. In the present study, we have derived a characteristic equation for a model of any finite number of mutually coupled nephrons having NaCl backleak. Analysis of that characteristic equation for the case of two coupled nephrons has revealed a number of parameter regions having the potential for differing stable dynamic states. Numerical solutions of the full equations for two model nephrons exhibit a variety of behaviors in these regions. Some behaviors exhibit a degree of complexity that is consistent with our hypothesis for the emergence of irregular oscillations in SHR.     

A new integrated method for analyzing heart mechanics using a cell-hemodynamics-autonomic nerve control coupled model of the cardiovascular system

A model of the cardiovascular system coupling cell, hemodynamics, and autonomic nerve control function is proposed for analyzing heart mechanics. We developed a comprehensive cardiovascular model with multi-physics and multi-scale characteristics that simulates the physiological events from membrane excitation of a cardiac cell to contraction of the human heart and systemic blood circulation and ultimately to autonomic nerve control. A lumped parameter model is used to compute the systemic and pulmonary circulations interacting with the cardiac cell mechanism. For autonomic control of the cardiovascular system, we used the approach suggested by Heldt et al. [2002. Computational modeling of cardiovascular response to orthostatic stress. J. Appl. Physiol. 92, 1239–1254] (Heldt model), including baroreflex and cardiopulmonary reflexes. We assumed sympathetic and parasympathetic pathways for the nerve control system. The cardiac muscle response to these reflex control systems was implemented using the activation-level changes in the L-type calcium channel and sarcoplasmic/endoplasmic reticulum calcium ATPase function based on experimental observations. Using this model, we delineated the cellular mechanism of heart contractility mediated by nerve control function. To verify the integrated method, we simulated a 10% hemorrhage, which involves cardiac cell mechanics, circulatory hemodynamics, and nerve control function. The computed and experimental results were compared. Using this methodology, the state of cardiac contractility, influenced by diverse properties such as the afterload and nerve control systems, is easily assessed in an integrated manner.     

Analysis of ligand-receptor binding by the difference method

A new method has been proposed for analysis of experimental data on ligand-receptor binding at equilibrium. This method makes it possible to detect heterogeneity of a receptor system in cases where the contribution of the high-affinity site to total binding is rather small and the problem of graphic discrimination of a model cannot be solved unambiguously by other methods. The difference method permits us to exclude experiments on measuring nonspecific binding. A computer program for analysis of ligand-receptor binding has been worked out in which the difference method and traditional methods of binding isotherm analysis are realized. Numerical modeling has shown that the best strategy in experimental data processing is the treatment of total binding isotherms by both the difference method and regression analysis, including the nonspecific binding constant as one of the regression parameters.     

A simple mathematical model of the interaction between intracranial pressure and cerebral hemodynamics

Ursino, Mauro, and Carlo Alberto Lodi. A simplemathematical model of the interaction between intracranial pressure andcerebral hemodynamics. J. Appl.Physiol. 82(4): 1256-1269, 1997.A simplemathematical model of intracranial pressure (ICP) dynamics oriented toclinical practice is presented. It includes the hemodynamics of thearterial-arteriolar cerebrovascular bed, cerebrospinal fluid (CSF)production and reabsorption processes, the nonlinear pressure-volumerelationship of the craniospinal compartment, and a Starling resistormechanism for the cerebral veins. Moreover, arterioles are controlledby cerebral autoregulation mechanisms, which are simulated by means ofa time constant and a sigmoidal static characteristic. The model isused to simulate interactions between ICP, cerebral blood volume, andautoregulation. Three different related phenomena are analyzed: thegeneration of plateau waves, the effect of acute arterial hypotensionon ICP, and the role of cerebral hemodynamics during pressure-volume index (PVI) tests. Simulation results suggest the following:1) ICP dynamics may become unstablein patients with elevated CSF outflow resistance and decreasedintracranial compliance, provided cerebral autoregulation is efficient.Instability manifests itself with the occurrence of self-sustainedplateau waves. 2) Moderate acutearterial hypotension may have completely different effects on ICP,depending on the value of model parameters. If physiological compensatory mechanisms (CSF circulation and intracranial storage capacity) are efficient, acute hypotension has only negligible effectson ICP and cerebral blood flow (CBF). If these compensatory mechanismsare poor, even modest hypotension may induce a large transient increasein ICP and a significant transient reduction in CBF, with risks ofsecondary brain damage. 3) The ICPresponse to a bolus injection (PVI test) is sharply affected, viacerebral blood volume changes, by cerebral hemodynamics andautoregulation. We suggest that PVI tests may be used to extractinformation not only on intracranial compliance and CSF circulation,but also on the status of mechanisms controlling CBF.

     

Partial hepatectomy hemodynamics changes: Experimental data explained by closed-loop lumped modeling

The liver function may be degraded after partial liver ablation surgery. Adverse liver hemodynamics have been shown to be associated to liver failure. The link between these hemodynamics changes and ablation size is however poorly understood. This article proposes to explain with a closed-loop lumped model the hemodynamics changes observed during twelve surgeries in pigs. The portal venous tree is modeled with a pressure-dependent variable resistor. The variables measured, before liver ablation, are used to tune the model parameters. Then, the liver partial ablation is simulated with the model and the simulated pressures and flows are compared with post-operative measurements. Fluid infusion and blood losses occur during the surgery. The closed-loop model presented accounts for these blood volume changes. Moreover, the impact of blood volume changes and the liver lobe mass estimations on the simulated variables is studied. The typical increase of portal pressure, increase of liver pressure loss, slight decrease of portal flow and major decrease in arterial flow are quantitatively captured by the model for a 75% hepatectomy. It appears that the 75% decrease in hepatic arterial flow can be explained by the resistance increase induced by the surgery, and that no hepatic arterial buffer response (HABR) mechanism is needed to account for this change. The different post-operative states, observed in experiments, are reproduced with the proposed model. Thus, an explanation for inter-subjects post-operative variability is proposed. The presented framework can easily be adapted to other species circulations and to different pathologies for clinical hepatic applications.     

Asymptotics of Conduction Velocity Restitution in Models of Electrical Excitation in the Heart

We extend a non-Tikhonov asymptotic embedding, proposed earlier, for calculation of conduction velocity restitution curves in ionic models of cardiac excitability. Conduction velocity restitution is the simplest non-trivial spatially extended problem in excitable media, and in the case of cardiac tissue it is an important tool for prediction of cardiac arrhythmias and fibrillation. An idealized conduction velocity restitution curve requires solving a non-linear eigenvalue problem with periodic boundary conditions, which in the cardiac case is very stiff and calls for the use of asymptotic methods. We compare asymptotics of restitution curves in four examples, two generic excitable media models, and two ionic cardiac models. The generic models include the classical FitzHugh–Nagumo model and its variation by Barkley. They are treated with standard singular perturbation techniques. The ionic models include a simplified “caricature” of Noble (J. Physiol. Lond. 160:317–352, 1962) model and Beeler and Reuter (J. Physiol. Lond. 268:177–210, 1977) model, which lead to non-Tikhonov problems where known asymptotic results do not apply. The Caricature Noble model is considered with particular care to demonstrate the well-posedness of the corresponding boundary-value problem. The developed method for calculation of conduction velocity restitution is then applied to the Beeler–Reuter model. We discuss new mathematical features appearing in cardiac ionic models and possible applications of the developed method.     

Mathematical modeling of metabolism and hemodynamics

We provide a mathematical study of a model of energy metabolism and hemodynamics of glioma allowing a better understanding of metabolic modifications leading to anaplastic transformation from low grade glioma. This mathematical analysis allows ultimately to unveil the solution to a viability problem which seems quite pertinent for applications to medecine.     

A quantitative description of membrane current and its application to conduction and excitation in nerve

This article concludes a series of papers concerned with the flow of electric current through the surface membrane of a giant nerve fibre (Hodgkinet al., 1952,J. Physiol. 116, 424–448; Hodgkin and Huxley, 1952,J. Physiol. 116, 449–566). Its general object is to discuss the results of the preceding papers (Section 1), to put them into mathematical form (Section 2) and to show that they will account for conduction and excitation in quantitative terms (Sections 3–6).     

On the computation of substrate levels in perfused tissues

The recent development of experimental techniques to study the isolated perfused heart and brain have renewed interest in the mathematical modeling of capillary-tissue structures. A new analytical representation is developed for the Krogh cylinder model for blood-tissue structures, and a scheme is presented for determining an asymptotic series approximation for this solution. This solution is explored for model parameters of experimental interest, and the contribution and effect of axial diffusion is demonstrated.     

Axonal excitability revisited

The original papers of Hodgkin and Huxley (J. Physiol. 116 (1952a) 449, J. Physiol. 116 (1952b) 473, J. Physiol. 116 (1952c) 497, J. Physiol. 117 (1952d) 500) have provided a benchmark in our understanding of cellular excitability. Not surprisingly, their model of the membrane action potential (AP) requires revisions even for the squid giant axon, the preparation for which it was originally formulated. The mechanisms they proposed for the voltage-gated potassium and sodium ion currents, IK, and INa, respectively, have been superceded by more recent formulations that more accurately describe voltage-clamp measurements of these components. Moreover, the current-voltage relation for IK has a non-linear dependence upon driving force that is well described by the Goldman-Hodgkin-Katz (GHK) relation, rather than the linear dependence on driving force found by Hodgkin and Huxley. Furthermore, accumulation of potassium ions in the extracellular space adjacent to the axolemma appears to be significant even during a single AP. This paper describes the influence of these various modifications in their model on the mathematically reconstructed AP. The GHK and K+ accumulation results alter the shape of the AP, whereas the modifications in IK and INa gating have surprisingly little effect. Perhaps the most significant change in their model concerns the amplitude of INa, which they appear to have overestimated by a factor of two. This modification together with the GHK and the K+ accumulation results largely remove the discrepancies between membrane excitability of the squid giant axon and the Hodgkin and Huxley (J. Physiol. 117 (1952d) 500) model previously described (Clay, J. Neurophysiol. 80 (1998) 903).     

Variable Selection for Clustering with Gaussian Mixture Models

Summary .  This article is concerned with variable selection for cluster analysis. The problem is regarded as a model selection problem in the model-based cluster analysis context. A model generalizing the model of Raftery and Dean (2006,  Journal of the American Statistical Association   101, 168–178) is proposed to specify the role of each variable. This model does not need any prior assumptions about the linear link between the selected and discarded variables. Models are compared with Bayesian information criterion. Variable role is obtained through an algorithm embedding two backward stepwise algorithms for variable selection for clustering and linear regression. The model identifiability is established and the consistency of the resulting criterion is proved under regularity conditions. Numerical experiments on simulated datasets and a genomic application highlight the interest of the procedure.     

A Mathematical and Numerical Investigation of the Hemodynamical Origins of Oscillations in Microvascular Networks

Evidence is presented to show that self-sustained oscillations of purely hemodynamical origin are possible in some arcade-type microvascular networks supplied with steady boundary conditions, but that in others the oscillations disappear with sufficient reduction of the time step Δt, showing them to be numerical artefacts. In an attempt to elucidate the mechanisms involved in the onset of fluctuations, we proceed to perform a linear stability analysis for the convective model of Kiani et al. (Microvasc. Res. 45:219–232, 1993; Am. J. Physiol. 266(35):H1822–H1828, 1994), and show that this leads via a system of delay differential equations to a nonlinear eigenvalue problem. This result generalises the characteristic equation obtained by Carr et al. (Ann. Biomed. Eng. 33:764–771, 2005) and Geddes et al. (SIAM J. Appl. Dyn. Syst. 6(4):694–727, 2007) who solved a special case in a two node network. An implicit numerical method is proposed for the computation of blood flows in networks using the convective model. In a moderate size subnetwork of one of the networks chosen by Kiani et al. (Am. J. Physiol. 266(35):H1822–H1828, 1994), the topology, vessel lengths, and diameters of which were based on microvascular networks in the rat mesentery, we compare results generated using the original explicit numerical method of Kiani et al. (Am. J. Physiol. 266(35):H1822–H1828, 1994) with those from our implicit scheme. From the linear stability theory, a critical value D _RBC,crit of a red blood cell diameter parameter D _RBC in the plasma skimming model of Fenton et al. (Pflügers Arch. 403:396–401, 1985b) is identified for the onset of oscillations about steady state and both the explicit and implicit methods are used to calculate the inflow hematocrit solutions in all vessels of the subnetwork at the critical parameter value, subject to perturbed initial conditions. The results of the implicit method are demonstrated to be in excellent and superior agreement with the predictions of the linear analysis in this case. For values of D _RBC slightly larger than D _RBC,crit the bifurcating periodic solutions calculated using either the explicit or implicit schemes are characteristic of those of a supercritical Hopf bifurcation and the graphs of D _RBC vs. oscillation amplitude would seem to converge as Δt→0.     

A 1-D model to explore the effects of tissue loading and tissue concentration gradients in the revised Starling principle

The recent experiments in Hu et al. (Am J Physiol Heart Circ Physiol 279: H1724-H1736, 2000) and Adamson et al. (J Physiol 557: 889-907, 2004) in frog and rat mesentery microvessels have provided strong evidence supporting the Michel-Weinbaum hypothesis for a revised asymmetric Starling principle in which the Starling force balance is applied locally across the endothelial glycocalyx layer rather than between lumen and tissue. These experiments were interpreted by a three-dimensional (3-D) mathematical model (Hu et al.; Microvasc Res 58: 281-304, 1999) to describe the coupled water and albumin fluxes in the glycocalyx layer, the cleft with its tight junction strand, and the surrounding tissue. This numerical 3-D model converges if the tissue is at uniform concentration or has significant tissue gradients due to tissue loading. However, for most physiological conditions, tissue gradients are two to three orders of magnitude smaller than the albumin gradients in the cleft, and the numerical model does not converge. A simpler multilayer one-dimensional (1-D) analytical model has been developed to describe these conditions. This model is used to extend Michel and Phillips's original 1-D analysis of the matrix layer (J Physiol 388: 421-435, 1987) to include a cleft with a tight junction strand, to explain the observation of Levick (Exp Physiol 76: 825-857, 1991) that most tissues have an equilibrium tissue concentration that is close to 0.4 lumen concentration, and to explore the role of vesicular transport in achieving this equilibrium. The model predicts the surprising finding that one can have steady-state reabsorption at low pressures, in contrast to the experiments in Michel and Phillips, if a backward-standing gradient is established in the cleft that prevents the concentration from rising behind the glycocalyx.