Original ResearchIntegration of detailed modules in a core model of body fluid homeostasis and blood pressure regulation
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:
- 1.
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
- 2.
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.
Section snippets
Core model
Two different versions of the classic Guyton models were re-implemented during the SAPHIR project: the initial version, published in 1972 (MG72) (Guyton et al., 1972) and a more complete version that has been used in other work by people from Guyton’s group (MG92) (Montani and Van Vliet, 2009).2 No complete formal
Whole-model sensitivity analysis
Using the Morris method described above, we carried out an extensive sensitivity analysis using the 1992 version of Guyton’s model, implemented by us to allow looping over the various model parameters. Here, p = 50, and the size of the perturbations, ∆, was taken to be 5/(p − 1); that is, for each parameter, the range of values between its minimum and its maximum values was split into 50 slices, and the size of the perturbation ∆ corresponded to one-tenth of this range. In our MG92, we have 296
Conclusions and perspectives
With the emergence of integrative physiology and international projects such as the IUPS Physiome and the European Virtual Physiological Human (VPH), an increasing interest exists today toward the integration of different physiological models, which may cover different functions and be developed at various scales, under distinct mathematical formalisms. This paper presents a contribution to the formalization of the seldom-covered problem of the appropriate definition of the interfaces required
Acknowledgments
We gratefully acknowledge the financial support of the French National Research Agency (ANR grants ANR-06-BYOS-0007-01 (SAPHIR), and ANR-08-SYSC-002 (BIMBO)), and the European Community’s Seventh Framework Program (FP7/2007-2013, grant agreement no. 223920, VPH-NoE). This work was also aided by discussions within the GdR STIC-Santé CNRS 2647 – INSERM.
Editor’s note: Please see also related communications in this issue by Joshi et al. (2011) and Aslanidi et al. (2011).
Abbreviation list
- AARK
- afferent glomerular arteriolar resistances
- ACE
- angiotensin-converting enzyme
- ADH
- antidiuretic hormone
- ADHMV
- effect of ADH on nonrenal vascular resistance
- AGT
- liver-derived angiotensinogen
- AHM
- vasopressin effect
- AM
- aldosterone effect
- AMM
- overall multiplier factor for muscle autoregulation
- ANCSNS
- sensitivity controller of ANM
- Ang I
- angiotensin I
- Ang II
- angiotensin II
- ANM
- general angiotensin multiplier effect
- ANS
- autonomic nervous system
- ANU
- angiotensin effect on arterial resist + venous volume
- ANUVN
- effect of angiotensin
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