A multiformalism and multiresolution modelling environment: application to the cardiovascular system and its regulation

Abstract

The role of modelling and simulation in the systemic analysis of living systems is now clearly established. Emerging disciplines, such as systems biology, and worldwide research actions, such as the Physiome Project or the Virtual Physiological Human, are based on an intensive use of modelling and simulation methodologies and tools. One of the key aspects in this context is to perform an efficient integration of various models representing different biological or physiological functions, at different resolutions, spanning through different scales. This paper presents a multiformalism modelling and simulation environment (M2SL) that has been conceived to ease model integration. A given model is represented as a set of coupled and atomic model components that may be based on different mathematical formalisms with heterogeneous structural and dynamical properties. A co-simulation approach is used to solve these hybrid systems. The pioneering model of the overall regulation of the cardiovascular system proposed by Guyton and co-workers in 1972 has been implemented under M2SL and a pulsatile ventricular model based on a time-varying elastance has been integrated in a multi-resolution approach. Simulations reproducing physiological conditions and using different coupling methods show the benefits of the proposed environment.

Figure 1

On the left side: Functional diagram representing the hierarchy of a coupled model M^c, composed of a set of atomic and coupled sub-models (solid arrows represent the relation ’is a component of’). On the right side: Dynamically-created simulator hierarchy, defined at runtime by the Root Coordinator (S). Each model is associated with the corresponding atomic simulator or coordinator (dotted arows). Grey-levels represent different mathematical formalisms used to define each model and its corresponding simulator.

Figure 2

Functional diagram of an example coupled model $M_1^c$ composed of n atomic models and its corresponding simulator hierarchy.

Figure 3

Graphical representation of the time synchronization schemes implemented in the library, based on the example of coupled models on figure 2. a) Fixed-step method; b) Adaptive synchronisation and simulation, with the smallest atomic timestep and c) Synchronisation at a fixed time-step (δt_c) and atomic simulation with independent, adaptive time-steps δt_a;i.

Figure 4

Simplified class diagram of the M2SL implementation for the G72 model. The class ’Guyton72’ is the coupled model that links all other atomic models as components. The description formalism F of each component is also displayed: algebraic equations (AE) and ordinary differential equations (ODE).

Figure 5

Comparison of M2SL simulations (black curves) with the original Guyton model (dotted curves) during BM1 (sudden severe muscle exercise). Total experiment time was 9 min. VUD (urinary output in ml/min), PVO (muscle venous oxygen pressure in mm Hg), PMO (muscle cell oxygen pressure in mm Hg), PA (mean arterial pressure in mm Hg), AUP (sympathetic stimulation in ratio to normal), QLO (cardiac output in l/min), BFM (muscle blood flow, in l/min), and MMO (rate of oxygen usage by muscle cells in ml O2/minK1)

Figure 6

Comparison of M2SL simulations (black curves) with the original Guyton model (dotted curves) during BM2 (atrioventricular fistula). Total experiment time was 9 days (216 hours). VEC (extracellular fluid volume in litres), VB (blood volume in litres), AU (sympathetic stimulation ratio to normal), QLO (cardiac output in l/min), RTP (total peripheral resistance in mm Hg/l/min), PA (mean arterial pressure in mm Hg), HR (heart rate in beats/min), ANC (angiotensin concentration ratio to normal), VUD (urinary output in ml/min)

Figure 7

Evolution of δt_a;i(t) (in min·10⁻³) for the main atomic models of the M2SL G72 implementation during the simulation of BM1, using time-synchronisation strategy ST2.

Figure 8

Evolution of δt_a;i (in min·10⁻³) for the main atomic models of the M2SL G72 implementation during the simulation of BM1, using time-synchronisation strategy ST3.

Figure 9

a) Implementation of the pulsatile left ventricle and valves. PLA(left atrial pressure), PA(arterial pressure), QLAO (ventricular inflow), QLO (ventricular outflow) b) Simulated pulsatile left ventricular pressure (black curves) and arterial pressure (grey curves) for one beat c) Simulated pulsatile right ventricular pressure (black curves) and pulmonary arterial pressure (grey curves) for one beat.

Figure 10

Comparison of M2SL simulations of Guyton model coupled with pulsatile ventricles (black curves) with the original Guyton model (dotted curves) during BM1 (sudden severe muscle exercise). Total experiment time was 9 min. VUD (urinary output in ml/min), PVO (muscle venous oxygen pressure in mm Hg), PMO (muscle cell oxygen pressure in mm Hg), PA (mean arterial pressure in mm Hg), AUP (sympathetic stimulation in ratio to normal), QLO (cardiac output in l/min), BFM (muscle blood flow, in l/min), and MMO (rate of oxygen usage by muscle cells in ml O2/minK1)