Combined Transfer Function Analysis and Modelling of Cerebral Autoregulation

Abstract

The clinical importance of cerebral autoregulation has resulted in a significant body of literature that attempts both to model the underlying physiological processes and to estimate the mathematical relationships between clinically measurable variables, the most common of which are Arterial Blood Pressure and Cerebral Blood Flow Velocity. These approaches have, however, rarely been used together to interpret clinical data. A simple model of cerebral autoregulation is thus proposed here, based on a flow dependent feedback mechanism with gain and time constant that adjusts arterial compliance. Analysis of this model shows that it closely approximates a second order system for typical values of physiological parameters. The model parameters can be optimally estimated from available experimental data for the Impulse Response (IR), yielding physiologically reasonable values, although there is one free parameter that must be fixed. The effects of changes in feedback gain and time constant are found to be significant on the predicted IR and can thus be estimated robustly from experimental data. The effects of elevated baseline Intracranial Pressure (ICP) are found to be exactly equivalent to a reduced feedback gain, although the solution is much less sensitive to the former effect. A transfer function approach can be used to estimate autoregulation status clinically using a physiologically-based model, thus providing greater insight into the processes that govern cerebral autoregulation.

APPENDIX

To derive the linear transfer function, small changes about the basal conditions are assumed using a Taylor series expansion. Since the resulting equations will all be linear, the Laplace transform is used to convert the differential equations into a transfer function.

Equation (2) becomes:

$$\Delta R_{{\rm sa}} = - 2\frac{{\Delta V_{{\rm sa}} }}{{V_{{\rm sa}} }}R_{{\rm sa}} .$$

(A1)

Equation (3) becomes:

$$s\Delta V_{{\rm sa}} = sC_a \Delta p_1 + s\Delta C_a \left( {p_1 - p_{{\rm ic}} } \right).$$

(A2)

Equations (4) and (7) combine to give:

$$sC_v \Delta p_2 = \frac{{\Delta p_1 - \Delta p_2 }}{{\left( {{{R_{{\rm sa}} } \mathord{\left/{\vphantom {{R_{{\rm sa}} } 2}} \right.\kern-\nulldelimiterspace} 2} + R_{{\rm sv}} } \right)}} - q\frac{{{{\Delta R_{{\rm sa}} } \mathord{\left/{\vphantom {{\Delta R_{{\rm sa}} } 2}} \right.\kern-\nulldelimiterspace} 2}}}{{\left( {{{R_{{\rm sa}} } \mathord{\left/{\vphantom {{R_{{\rm sa}} } 2}} \right.\kern-\nulldelimiterspace} 2} + R_{{\rm sv}} } \right)}} - \frac{{\Delta p_2 }}{{R_{{\rm lv}} }}.$$

(A3)

Equation (6) becomes:

$$s\Delta V_{{\rm sa}} = \frac{{\Delta p_a - \Delta p_1 }}{{\left( {{{R_{{\rm la}} + R_{{\rm sa}} } \mathord{\left/{\vphantom {{R_{{\rm la}} + R_{{\rm sa}} } 2}} \right.\kern-\nulldelimiterspace} 2}} \right)}} - q\frac{{{{\Delta R_{{\rm sa}} } \mathord{\left/{\vphantom {{\Delta R_{{\rm sa}} } 2}} \right.\kern-\nulldelimiterspace} 2}}}{{\left( {{{R_{{\rm la}} + R_{{\rm sa}} } \mathord{\left/{\vphantom {{R_{{\rm la}} + R_{{\rm sa}} } 2}} \right.\kern-\nulldelimiterspace} 2}} \right)}} - \Delta q.$$

(A4)

Equations (8) and (10) combine to give:

$$\Delta C_a = - C_a G\frac{{\Delta q}}{{q\left( {1 + s\tau } \right)}}.$$

(A5)

Equation (9) becomes:

$$\Delta q = \frac{{\Delta p_1 - \Delta p_2 }}{{\left( {{{R_{{\rm sa}} } \mathord{\left/{\vphantom {{R_{{\rm sa}} } 2}} \right.\kern-\nulldelimiterspace} 2} + R_{{\rm sv}} } \right)}} - q\frac{{{{\Delta R_{{\rm sa}} } \mathord{\left/{\vphantom {{\Delta R_{{\rm sa}} } 2}} \right.\kern-\nulldelimiterspace} 2}}}{{\left( {{{R_{{\rm sa}} } \mathord{\left/{\vphantom {{R_{{\rm sa}} } 2}} \right.\kern-\nulldelimiterspace} 2} + R_{{\rm sv}} } \right)}}.$$

(A6)

The six Eqs. (A1)/(A6), can be used to eliminate the unwanted variables $\left( {\Delta V_{{\rm sa}}, \,\Delta R_{{\rm sa}}, \,\Delta C_a, \,\Delta p_1, \,\Delta p_2 } \right)$ to derive the expression in Eq. (11). The transfer function for V _MCA can be found from the flow rate:

$$V_{{\rm MCA}} = \frac{1}{A}\frac{{p_a - p_1 }}{{\left( {R_{{\rm la}} + {{R_{{\rm sa}} } \mathord{\left/{\vphantom {{R_{{\rm sa}} } 2}} \right.\kern-\nulldelimiterspace} 2}} \right)}}.$$

(A7)

Hence for small changes:

$$\frac{{\Delta V_{{\rm MCA}} }}{{V_{{\rm MCA}} }} = \frac{{\Delta p_a - \Delta p_1 }}{{\left( {p_a - p_1 } \right)}} - \frac{{{{\Delta R_{{\rm sa}} } \mathord{\left/{\vphantom {{\Delta R_{{\rm sa}} } 2}} \right.\kern-\nulldelimiterspace} 2}}}{{\left( {R_{{\rm la}} + {{R_{{\rm sa}} } \mathord{\left/{\vphantom {{R_{{\rm sa}} } 2}} \right.\kern-\nulldelimiterspace} 2}} \right)}},$$

(A8)

since area is assumed constant. This can then be calculated from the expressions in Eqs. (27)–(32).