An Improved Mathematical Model for the Autonomic Regulation of Cardiovascular System

Abstract

The activity of the autonomic nervous system is hardly measurable. This study presents an improved mathematical model to estimate the baroreceptor nerve firing rate, the efferent parasympathetic and sympathetic response in a scenario of postural change from sitting to standing, based on observed blood pressure and heart rate changes. An optimization step is then applied to find the model parameters best fitting to observed cardiac data of healthy people and hypertensive patients using unbiased estimation and Nelder-Mead method. The experimental results on 59 subjects have shown that the improved model can describe autonomic regulation mechanism well and the estimated system parameters have clear clinical meaning.

Appendix

In this appendix we describe the part of chemical concentrations and heart rate modeling by the previous model and we use it.

Chemical concentrations: The concentration of neurotransmitters acetylcholine Cach and noradrenaline Cnor are defined using linear differential equations which depend on the sympathetic (Tsym) and parasympathetic (Tpar) responses.

$$ \begin{aligned} \frac{dCnor}{dt} & = \frac{ - Cnor + Tsym}{{\tau_{nor} }} \\ \frac{dCach}{dt} & = \frac{ - Cach + Tpar}{{\tau_{ach} }} \\ \end{aligned} $$

(8)

where $\tau_{nor}$ and $\tau_{ach}$ are two time parameters.

Heart rate: HR potential $\left( \psi \right)$ is linked with these chemical concentrations by an integrated model of the form.

$$ \frac{d\varphi }{dt} = H_{0} (1 + M_{S} C_{nor} - M_{P} C_{ach} ) $$

(9)

where

$$ \begin{aligned} M_{S} & = \frac{{\xi_{S}^{2} }}{{1 + \xi_{S}^{2} }} \\ M_{P} & = \frac{{\xi_{P}^{2} }}{{1 + \xi_{P}^{2} }} \\ \end{aligned} $$

(10)

where $H_{0}$ denotes intrinsic HR, $M_{s}$ and $M_{p}$ represent the strength of the response to changes in the concentrations. $\zeta_{s}$ and $\xi_{p}$ are the two parameters used to constrain $M_{s}$ and $M_{p}$ in the interval [0, 1], in order to bound HR within physiological values.

A heartbeat occurs when $\psi$ reaches 1, then is reset to 0. So the HR is defined as the following equation:

$$ HR = 1/(t_{\varphi = 1} - t_{\varphi = 0} ) $$

(11)