On the mathematical modelling of cerebral autoregulation

Abstract

Cerebral autoregulation is the process by which blood flow to the brain is maintained despite changes in arterial blood pressure. Experiments using transcranial Doppler ultrasonography allow rapid measurements of blood flow velocity in the middle cerebral artery. Measurements of this velocity and a subject's arterial blood pressure are used in the assessment of the dynamic cerebral autoregulatory response. Two mathematical models representing the dynamic cerebral autoregulation response as a feedback mechanism, dependent on pressure and flow respectively, are derived. For each model two parameters are introduced, a rate of restoration and a time delay. Solutions for both flow between fixed plates and flow in a rigid pipe are obtained using Laplace transform methods. In both cases solutions for the velocity are found for a general arterial blood pressure, allowing the model to be applied to any experiment that uses changes in arterial blood pressure to assess dynamic cerebral autoregulation. Velocity profiles are determined for the thigh cuff and vacuum box experiments, modelled as a step change and sinusoidal variation in pressure gradient in the middle cerebral artery respectively. The influence of the underlying heart and breathing cycles on measurements obtained from the vacuum box experiments is assessed, before results derived using the mathematical model with a flow dependent feedback mechanism are compared with data from the two experiments. The comparisons yield similar estimates for the rate of restoration and time delay suggesting that these parameters could be independent of the pressure change stimulus and depend only on the main features of the dynamic cerebral autoregulation process. The modelling also indicates that for imposed oscillatory variations in arterial blood pressure a small phase difference between the pressure and velocity waveforms does not necessarily imply impaired autoregulation. The ratio between the percentage variation in maximum velocity and pressure can be used, along with the phase difference, to indicate more accurately the nature of the autoregulatory response. Finally, the relationship between arterial blood pressure and pressure gradient in the middle cerebral artery is modelled using electrical analogue theory. The influence of this relationship on the autoregulation model for flow in a rigid pipe is investigated.

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