Multivariate System Identification for Cerebral Autoregulation

Abstract

The effect of spontaneous beat-to-beat mean arterial blood pressure (ABP) fluctuations and breath-to-breath end-tidal carbon dioxide $(P_{ETCO_2})$ and end-tidal oxygen $(P_{ETO_2})$ fluctuations on beat-to-beat cerebral blood flow velocity (CBFV) variations is studied using a multiple coherence function. Multiple coherence is a measure of the extent to which the output, CBFV, can be represented as a linear time invariant system of multiple input signals. Analysis of experimental measurements from 13 different healthy subjects reveal that, with additional inputs, $P_{ETCO_2}$ and $P_{ETO_2},$ the multiple coherence for frequencies  < 0.05 Hz is significantly higher than the corresponding values obtained for univariate coherence with a single input of ABP. The result illustrates that the low value of univariate coherence at small frequencies may be due to the effects of $P_{ETCO_2}$ and $P_{ETO_2}$ fluctuations on CBFV variability. Moreover, it is also found that the transfer function between ABP and CBFV time series identified from previous univariate techniques at low frequencies can be modified by CO ₂ and O ₂ reactivity and no longer represents pressure autoregulation only. Multivariate system identification provides a technique of incorporating additional variability and recovering from this artifact. Finally, a physiologically based model and its linear transfer function are used as a simulation tool to investigate possible causes of low univariate coherence.

Appendix

The schematic of the model used here is shown in Fig. 9. The full description of the system can be found in Payne17. All differential equations used in the model are summarized below:

Figure 9

Schematic of model. P _a , systemic arterial pressure; R _la , resistance of nonregulating arterial compartment; P ₁, R _sa , and C _a , pressure, resistance, and compliance of regulating arterial compartment; R _sv , resistance of capillary compartment and small veins; C _v , venous compliance; P ₂, venous pressure; P _v and R _lv , venous pressure and resistance of large veins, respectivly; P _ic , intracranial pressure; C _ic , intracranial compliance

In arterial compartment ³:

$$ \frac{V_{sa}}{{\bar{V}}_{sa}} = \sqrt{\frac{\bar{R}_{sa}}{R_{sa}}}, $$

(A.1)

$$ \frac{\hbox{d}V_{sa}}{\hbox{d}t}= \frac{\hbox{d}(C_a(P_1-P_{ic}))}{\hbox{d}t}, $$

(A.2)

$$ \frac{\hbox{d}V_{sa}}{\hbox{d}t}= \frac{P_a-P_1}{R_{la}+R_{sa}/2}-\frac{P_1-P_2}{R_{sa}/2+R_{sv}}. $$

(A.3)

In capillary and venous compartments:

$$ \frac{\hbox{d}V_v}{\hbox{d}t} =C_v\frac{\hbox{d}(P_2-P_{ic})}{\hbox{d}t}, $$

(A.4)

$$ C_v=\frac{1}{k_{ven}(P_2-{\bar{P}_{ic}}-P_{v1})}, $$

(A.5)

$$ \frac{\hbox{d}V_v}{\hbox{d}t} =\frac{P_1-P_2}{R_{sa}/2+R_{sv}}-\frac{P_2-P_v}{R_{lv}}, $$

(A.6)

where k _ven and P _v1 are constants.

The feedback system:

$$ \tau_q\frac{\hbox{d}x_q}{\hbox{d}t}+x_q = G_q\left(\frac{q-{\bar{q}}}{\bar{q}}\right), $$

(A.7)

$$ \tau_{CO_{2}}\frac{\hbox{d}x_{CO_{2}}}{\hbox{d}t}+x_{CO_{2}} =fn\left(\frac{P_{a_{CO_{2}}}}{{\bar{P}}_{a_{CO_{2}}}}\right), $$

(A.8)

$$ fn\left(\frac{P_{a_{CO_{2}}}}{{\bar{P}}_{a_{CO_{2}}}}\right) = 0.3+3\,\hbox{tanh}\left(\frac{P_{a_{CO_{2}}}}{{\bar{P}}_{a_{CO_{2}}}}-1.1\right), $$

(A.9)

where q is the microvascular CBF:

$$ q=\frac{P_1-P_2}{R_{sa}/2+R_{sv}}. $$

(A.10)

It is then assumed that the two feedback mechanisms act in a linearly additive manner:

$$ x=x_{CO_2}-x_q, $$

(A.11)

and that x then modifies C _a by means of the following sigmoidal relationship from:26

$$ C_{a} = {\bar{C}}_{a}+ \frac{1}{2} \left\{\begin{array}{c} \Updelta{{C}_{a}^{+}} \hbox{tanh} (\frac{2x}{\Updelta{C_a^+}})\quad \hbox{ if }x > 0\\ \Updelta{{C}_{a}^{-}}\hbox{tanh}(\frac{2x}{\Updelta{C_a^-}})\quad \hbox{ if }x < 0 \end{array}\right\}. $$

(A.12)

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. A detailed description of the linearization process can be found in Payne and Tarassenko.18 The differential equations governing the flow autoregulation above are thus linearized to the algebraic equations presented below:

$$ \Updelta{R_{sa}} = -2\frac{\Updelta{V_{sa}}}{V_{sa}}R_{sa}, $$

(A.13)

$$ s\Updelta{V_{sa}} = sC_a{\Updelta{P_1}}+s\Updelta{C_a}(P_1-P_{ic}), $$

(A.14)

$$ s\Updelta{V_{sa}} = \frac{\Updelta{P_a}-\Updelta{P_1}}{R_{la}+R_{sa}/2}-q\frac{\Updelta{R_{sa}}}{R_{la}+R_{sa}/2}-\Updelta{q}, $$

(A.15)

$$ \begin{aligned} s\Updelta{V_{v}} &= sC_v\Updelta{P_2}\\ &= \frac{\Updelta{P_1}-\Updelta{P_2}}{R_{sa}/2+R_{sv}}-q\frac{\Updelta{R_{sa}}/2} {R_{sa}/2+R_{sv}}-\frac{\Updelta{P_2}}{R_{lv}}, \end{aligned} $$

(A.16)

$$ C_a = \bar{C_a}-x, $$

(A.17)

$$ \Updelta{C_a} = -G_q\frac{\Updelta{q}}{q(1+s\tau_{q})}, $$

(A.18)

$$ \Updelta{q} = \frac{\Updelta{P_1}-\Updelta{P_2}}{R_{sa}/2+R_{sv}}-q\frac{\Updelta{R_{sa}}/2}{R_{sa}+R_{sv}}, $$

(A.19)

$$ V_{MCA} = \frac{1}{A}\frac{P_a-P_1}{R_{la}+R_{sa}/2}, $$

(A.20)

in which A is the cross-sectional area of the MCA, which is assumed to be invariant here. Therefore, beat-to-beat changes in mean velocity may represent predominantly beat-to-beat changes in cerebral blood flow. Hence for small pressure changes, velocity changes will be:

$$ \frac{\Updelta{V_{MCA}}}{V_{MCA}}= \frac{\Updelta{P_a}-\Updelta{P_1}}{P_a-P_1}-\frac{\Updelta{R_{sa}}/2} {R_{la}+R_{sa}/2}. $$

(A.21)

After simplification and approximation, the transfer functions for V _MCA is obtained as:

$$ H(s) = \frac{1}{\beta_1}\left\{ 1+\frac{(1-\beta_1)(1-\alpha_1)-\frac{2\alpha_2}{1+s\tau_q}} {(\alpha_1+2\beta_1-1)+(1-\beta_1)s\alpha_1\tau_a+(\frac{\alpha_2}{1+s\tau_{q}})(2-s\alpha_1\tau_a)}\right\}, $$

(A.22)

where the relevant nondimensional parameters and time constants are:

$$ \alpha_1 = \frac{{\bar{V}_{sa}}\big/{\bar{q}}}{{\bar{R}_{sa}}\bar{C_a}}, $$

(A.23)

$$ \alpha_2 = G_q\left[\left(\frac{{\bar{P}_a}-{\bar{P}_{ic}}}{{\bar{P}_a}-P_{vs}}\right)-\beta_1 \right]\big/{{\bar{C}_a}}, $$

(A.24)

$$ \beta_1 = \frac{R_{la}+{\bar{R}_{sa}}/2}{R_{total}}, $$

(A.25)

$$ \tau_a = (R_{la}+{\bar{R}}_{sa}/2){\bar{C}}_a. $$

(A.26)

The differential equations governing CO ₂ cerebrovascular reactivity are almost the same as those governing flow autoregulation. The difference exists only in two equations: Equation (19) becomes:

$$ s\Updelta{V_{sa}}=-\frac{\Updelta{P_1}}{R_{la}+R_{sa}/2}-q \frac{\Updelta{R_{sa}}}{R_{la}+R_{sa}/2}-\Updelta{q}. $$

(A.27)

Since there is no arterial pressure changes, the compliance equation [Eq. (28)] becomes:

$$ \Updelta{C_a}=-G_q\frac{\Updelta{q}}{q(1+s\tau_{q})}+G_{CO_2} \frac{\Updelta{Pa_{CO_2}}}{Pa_{CO_2}(1+s\tau_{CO_2})}. $$

(A.28)

After simplification, the transfer function V _MCA to CO ₂ changes is obtained as:

$$ \begin{aligned} J(s)&=\frac{\Updelta{V_{MCA}}/V_{MCA}}{\Updelta{Pa_{CO_2}/Pa_{CO_2}}}\\ &=\frac{1}{\beta_1}\left\{ 1+\frac{\alpha_2(2\beta_1+(1-\beta_1)s\alpha_1\tau_a)} {(\alpha_1+2\beta_1-1)+(1-\beta_1)s\alpha_1\tau_a+(\frac{\alpha_2}{1+s\tau_{q}})(2-s\alpha_1\tau_a)}\right\} \\ &\ast\frac{G_{CO_2}/G_q}{1+s\tau_{CO_2}}. \end{aligned} $$

(A.29)

where the definitions of [α₁, α₂, β₁, τ _q , τ _a ] are the same as before.