Integration of detailed modules in a core model of body fluid homeostasis and blood pressure regulation

Abstract

This paper presents a contribution to the definition of the interfaces required to perform heterogeneous model integration in the context of integrative physiology. A formalization of the model integration problem is proposed and a coupling method is presented. The extension of the classic Guyton model, a multi-organ, integrated systems model of blood pressure regulation, is used as an example of the application of the proposed method. To this end, the Guyton model has been restructured, extensive sensitivity analyses have been performed, and appropriate transformations have been applied to replace a subset of its constituting modules by integrating a pulsatile heart and an updated representation of the renin–angiotensin system. Simulation results of the extended integrated model are presented and the impacts of their integration within the original model are evaluated.

Introduction

The central role of modeling and simulation in the analysis of biological and physiological systems is now established, and numerous mathematical models of physiological systems can be found in the literature. This is particularly important in the domain of cardiovascular and renal (CVR) pathophysiology. A number of models focusing on structural (or vertical) integration have been proposed, for example, for the multi-scale analysis of the electrical activity of the heart (Clayton et al., 2011, Clayton and Panfilov, 2008), or for cardiac electromechanics (Kerckhoffs et al., 2006, Nordsletten et al., 2011). These structurally-integrated models have proven useful for understanding various pathological conditions. However, they are often complex in terms of the number of state variables or structural elements represented, and they may lack an appropriate physiological description of boundary conditions. Such models are typically computationally intensive and difficult to analyze, to identify, and thus to exploit in a practical context.

Models aiming at functional (or horizontal) integration, representing the interaction between different organs or physiological subsystems, are particularly suited for the analysis of multifactorial pathologies. These models are easier to manage numerically and mathematically, since they are usually based on a lumped-parameter representation. However, their clinical applicability may be limited by the fact that their constitutive elements generally lack the level of detail required to address certain pathophysiological functions.

The work presented here is focused on the analysis of the dynamic and integrated behavior of the cardiovascular and renal systems (CVR), which are involved in major public health pathologies, such as heart failure and hypertension. These CVR pathologies are complex and multifactorial, strongly drawing their clinical features and consequences from intertwined and dynamic interactions between genotype, phenotype, and environment (McMurray, 2010). This very complexity (number of elements, multiscale interactions, adaptations, nonlinearity…) makes a complete horizontal and vertical integration impossible.

One way to overcome these limitations is to represent the various physiological components of interest by separate specific models, developed at distinct levels of structural complexity, as a function of the targeted clinical application. However, such different models are often developed under a variety of mathematical formalisms, use distinct structural resolutions, or present significant differences in their intrinsic dynamics. Coupling these heterogeneous models into a multi-resolution approach presents a number of methodological and technical difficulties, particularly:

• the creation of an appropriate environment based on a modular, horizontally integrated ‘core model’ and on specific tools for modeling and simulating a set of coupled heterogeneous models; and

• the definition of an interfacing method for coupling these formally heterogeneous models, while preserving the stability and the essential characteristics of each integrated model.

Concerning the first point, the classic Guyton model (Guyton et al., 1972), a multi-organ, integrated systems model of blood pressure regulation, was implemented within the SAPHIR project (funded by the French ANR BioSYS program and selected as an Exemplar Project of the VPH NoE¹), as an example of system-level horizontal integration that can be useful for the definition of an extensible core model (Thomas et al., 2008). This implementation was based on an object-oriented multi-resolution modeling tool (M2SL) that allowed us to create the corresponding modules of the Guyton model as individual physiological and functional components. These components were coupled through specific input/output interfaces, without the need to explicitly specify integration step-sizes for each module, despite the wide range of time-scales covered (Hernández et al., 2009).

This paper focuses on the second point. We present a modeling approach in which a system-level ‘core model’, devoted to functional integration, is selectively improved by interfacing more detailed models of specific functions, defined at different levels of structural integration. This is demonstrated with concrete application examples. Section 2 formalizes the problem and presents a general method for coupling heterogeneous models. Section 3 presents results from an extensive sensitivity analysis of the original Guyton model, as well as two examples of the application of the proposed method for the replacement of some modules of the Guyton model by updated models: a pulsatile model of cardiac activity and an updated representation of the renin–angiotensin system. Finally, section 4 places the present work within the context of integrative physiology and outlines some perspectives.