Cardiac Modelling

Confronting a model with heart rate and blood pressure data

The cardiovascular system may be investigated by observing fluctuations in the heart rate, blood pressure and rate of respiration. Its time evolution is governed by the baroreflex control mechanism, where the sympathetic and vagal nerves compete to increase and decrease the heart rate respectively. A nonlinear delay-differential equation model is constructed to describe this control mechanism and to explore the interactions between the heart rate and blood pressure. In this model, a time delay gives rise to the oscillations in the blood pressure known as Mayer waves. The model maintains an intrinsically stable heart rate in the absence of nervous control, and features baroreflex influence on both heart rate and peripheral resistance. The effect of respiratory sinus arrhythmia (RSA) is introduced using a sinusoidal driving component. Clinical recordings obtained by carefully controlling the rate and depth of respiration are used to test the suitability of the model for representing the complicated physiology of the cardiovascular system. The model is shown to be able to reproduce many of the empirical characteristics observed in these biomedical signals, including RSA, Mayer waves and synchronization. Key physiological parameters in the model, including the time delay and levels of sympathetic and vagal activity, could provide useful diagnostic information about the state of the cardiovascular system [1].

Baroreflex schematic The baroreflex system (see figure to the left) acts by detecting arterial pressure and sending signals to the brainstem or medulla, which responds via either parasympathetic (fast) and sympathetic (slow) signals that change the heart rate and peripheral resistance of the arterioles and capillaries. In the model, the fast-acting parasympathetic system is assumed to be instantaneous and the slow-acting sympathetic system is modelled as depending on the blood pressure with a delay of $&tau$ = 3 seconds. The model also includes the intrinsic controlled behaviour that would be present with no central nervous system. Indeed the resulting natural frequency affects the response dynamics of the cardiac system to baroreflex feedback.

In the absence of nervous control the sino-atrial node will pulse at approximately $h$ ₀ = 100 beats per minute and the mean arterial blood pressure is approximately $p$ ₀ = 100 mmHg. Our model for heart rate and mean blood pressure, which extends previous research k [2,3], is given by a dimensionless system of delay differential equations for the heart rate, $h* = h/h$ ₀, and mean arterial blood pressure, $p* = p/p$ ₀. The expression for heart rate includes $&beta$ -sympathetic, vagal, and intrinsic (sinoatrial) response and the expression for pressure includes mechanical inflation by the heart pump action and the $&alpha$ -sympathetic response. Using $'$ to denote differentiation with respect to dimensionless time, $t* = t/&tau$ , the delay differential equations are

&epsilon_h h*'	=	&beta g₁ /(1 + &gamma g₂) - &nu g₂ + &delta (1 - h*),
&epsilon_p p*'	=	&mu h* - p* / (1 + &alpha g₁),

where the Hill function,

g(p) = 1/(1 +p

ⁿ), is employed to describe the pressure-dependent baroreceptor control,

g₁	=	g(p₁* + r₁),
g₂	=	1 - g(p* + r₂),

and the notation

p

₁* denotes the delayed function

p*(t*-1)

. The forcing effect of respiration with frequency

f

_r enters via

r₁	=	A₁ sin [2&pi f_r &tau(t*-1) - &phi],
r₂	=	A₂ sin [2&pi f_r &tau t* - &phi].

Further details about the model and the parameter values are provided in a [1].

Matlab files for the model in [1] are available for download:

cardiacdgp.m	Matlab file for running the data generating process of the cardiac delay differential equation model
cardiacf.m	Matlab file containing the cardiac delay differential equation model
cardiachistory.m	Matlab file containing the cardiac history required for running the model

Further information is available from Patrick McSharry. Suggestions and comments are welcome.

[1] P.E. McSharry, M.J. McGuinness and A.C. Fowler (2005). Confronting a cardiovascular system model with heart rate and blood pressure data. Computers in Cardiology 32: 587-590

[2] A.C. Fowler and M.J. McGuinness (2004). A delay recruitment model of the cardiovascular control system. J Math Biol 51(5):508-26

[3] J.T. Ottensen (1997). Modelling of the baroreflex-feedback mechanism with time-delay. J Math Biol 36:41–63