Modelling blood partial pressures of the human cardiovascular/respiratory system

doi:10.1016/j.amc.2006.09.016

Applied Mathematics and Computation

Volume 187, Issue 2, 15 April 2007, Pages 1100-1108

https://doi.org/10.1016/j.amc.2006.09.016 Get rights and content

Abstract

This paper aims to design a mathematical model for determining blood pressures response to cardiac and respiratory parameters. For this purpose a three-compartmental model is investigated to provide a two-nonlinear coupled ordinary differential equations. Stability conditions are established and inverse technics are proposed for identifying model parameters. The validation of the model is achieved throughout a comparative study with an existing model.

Introduction

One of an important topic to human health is the control of the cardiovascular and respiratory system. The knowledge of this control mechanism will be of very helpful to improving diagnostics and treatments of cardio-respiratory diseases. However, it should be noticed that there are more and more situations where the cardio-respiratory system is very solicited. This is, for example, the case of high-level sportsmen. The cardiovascular and the respiratory systems are connected so that the autoregulation mechanism are not well known yet. The question that often arises is of determining heart rate and alveolar ventilation for controlling systemics arterial and venous blood pressures to prevent cardiovascular accidents. For a healthy subject, it is well known that the heart rate and the alveolar ventilation depend on wether he is trained or untrained.

A number of dynamic models of the human cardio-respiratory system have been proposed since the 1950s. Most of them arise from the compartmental theory [4], [5], [7], [8]. These models consist in solving control optimal problems of nonlinear differential equations with cumbersome terms, leading to unstable solutions. An interesting global model is proposed in [7] but it possesses unstable equilibrium states. This model requires that we must first search stable equilibrium states before computing the solution on an interval (0, T) with small value of T for the initial state which is very closed to the equilibrium state. Such a model does not allow to understand a long-term cardio-respiratory system in the case of aerobic physical activities. In the present paper, we design a new model that provides global solutions by the mean of inverse processes using existing global models of cardiovascular and respiratory system. The results obtained make it possible to find approximately an optimal way to maintain blood partial pressures to its desired values that are clinical worthy of being accepted.

The outline of this paper is as follows. In Section 2, we build the mathematical model as well as the inverse technique for computing unknown constants and functions of the model. Model parameters are computed in Section 3. In Section 4, we present numerical results for a healthy subject.

Section snippets

Outline of the model

In this section, we would like to design a mathematical model for determining blood partial pressures with respect to heart rate and alveolar ventilation. Based on physiology properties of the human cardiovascular and respiratory systems we propose a three-compartmental model composed of the systemic arterial compartment (SAC), the systemic venous compartment (SVC) and the alveolar compartment (AC).

These three compartments include two circuits (systemic and pulmonary) which are arranged in

Computing parameters of the model

Let T be a positive time parameter. We now describe the process for identifying constants α, β and functions f and g. Let us fix a positive integer parameter N and consider $\begin{matrix} {\underset{̲}{P}}_{vs}^{δ} = (P_{vs}^{δ} (t_{1}), \dots, P_{vs}^{δ} (t_{N}))^{T}, \\ {\underset{̲}{P}}_{as}^{δ} = (P_{as}^{δ} (t_{1}), \dots, P_{as}^{δ} (t_{N}))^{T}, \end{matrix}$ where $P_{vs}^{δ} (t_{k})$ , $P_{as}^{δ} (t_{k})$ are measured data at the time $t_{k} = \frac{kT}{N}$ representing idealized values P_vs(t_k) and P_as(t_k); δ is the perturbation parameter due to some imprecisions on measured data.

Mathematically we express the identification problem as the following inverse

Test results

To test our models we consider the acute respiratory response to graded dynamic exercise in a 30 year old trained women whose mean values are given in Table 1. The case of aerobic exercise is considered. These mean values are for healthy subjects. The autoregulation process states that the cardiovascular and respiratory systems evolves in the optimal way toward these values. This suggests us to solving the following optimal control problem: $\min_{H, {\dot{V}}_{A}} \{‖ P_{vs} - P_{vs}^{e} ‖^{2} + ‖ P_{as} - P_{as}^{e} ‖^{2} + ‖ H - H_{e} ‖^{2} + ‖ {\dot{V}}_{A} - {\dot{V}}_{A_{e}} ‖^{2}\}$

Concluding remarks

In this work, we have investigated a mathematical model that describes blood partial pressures responses to cardiac and respiratory parameters (heart and ventilation rate). The cardiovascular and respiratory system is comprised of a multitude of elements. The increasing necessity to interpret the meaning of measurable variables such as heart rate, ventilation capacity, and blood pressure under both physiological and pathological conditions has imposed the need for relatively simple models that

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Applied Mathematics and Computation

Modelling blood partial pressures of the human cardiovascular/respiratory system

Abstract

Introduction

Section snippets

Outline of the model

Computing parameters of the model

Test results

Concluding remarks

Regularization of Inverse Problem

Matlab Guide

A mathematical model for fundamental regulation processes in the cardiovascular model

J. Math. Biol.