Nonlinear Assessment of Cerebral Autoregulation from Spontaneous Blood Pressure and Cerebral Blood Flow Fluctuations

Abstract

Cerebral autoregulation (CA) is an most important mechanism responsible for the relatively constant blood flow supply to brain when cerebral perfusion pressure varies. Its assessment in nonacute cases has been relied on the quantification of the relationship between noninvasive beat-to-beat blood pressure (BP) and blood flow velocity (BFV). To overcome the nonstationary nature of physiological signals such as BP and BFV, a computational method called multimodal pressure-flow (MMPF) analysis was recently developed to study the nonlinear BP–BFV relationship during the Valsalva maneuver (VM). The present study aimed to determine (i) whether this method can estimate autoregulation from spontaneous BP and BFV fluctuations during baseline rest conditions; (ii) whether there is any difference between the MMPF measures of autoregulation based on intra-arterial BP (ABP) and based on cerebral perfusion pressure (CPP); and (iii) whether the MMPF method provides reproducible and reliable measure for noninvasive assessment of autoregulation. To achieve these aims, we analyzed data from existing databases including: (i) ABP and BFV of 12 healthy control, 10 hypertensive, and 10 stroke subjects during baseline resting conditions and during the Valsalva maneuver, and (ii) ABP, CPP, and BFV of 30 patients with traumatic brain injury (TBI) who were being paralyzed, sedated, and ventilated. We showed that autoregulation in healthy control subjects can be characterized by specific phase shifts between BP and BFV oscillations during the Valsalva maneuver, and the BP–BFV phase shifts were reduced in hypertensive and stroke subjects (P < 0.01), indicating impaired autoregulation. Similar results were found during baseline condition from spontaneous BP and BFV oscillations. The BP–BFV phase shifts obtained during baseline and during VM were highly correlated (R > 0.8, P < 0.0001), showing no statistical difference (paired-t test P > 0.47). In TBI patients there were strong correlations between phases of ABP and CPP oscillations (R = 0.99, P < 0.0001) and, thus, between ABP–BFV and CPP–BFV phase shifts (P < 0.0001, R = 0.76). By repeating the MMPF 4 times on data of TBI subjects, each time on a selected cycle of spontaneous BP and BFV oscillations, we showed that MMPF had better reproducibility than traditional autoregulation index. These results indicate that the MMPF method, based on instantaneous phase relationships between cerebral blood flow velocity and peripheral blood pressure, has better performance than the traditional standard method, and can reliably assess cerebral autoregulation dynamics from ambulatory blood pressure and cerebral blood flow during supine rest conditions.

Appendix

Signal Decomposition

The main concept of the MMPF method is to probe nonlinear BP–BFV relationship by concentrating on intrinsic components of BP and BFV signals that have simplified temporal structures but still can reflect nonlinear interactions between two physiological variables. The first step of the MMPF is to decompose each signal into multiple intrinsic mode functions (IMFs), each mode representing the frequency-amplitude modulation at a specific time scale corresponding to different physiologic influences. To achieve this, the original MMPF used the empirical mode decomposition (EMD) method (Huang et al. 1998a). For a time series x(t) with at least 2 extremes, the EMD applies a sifting procedure to extract IMFs one by one from a smallest time scale to the largest time scale

$$ \begin{aligned} {{x(t)} \hfill} & {{ = s_{1} (t) + r_{1} (t)}\hfill} \\ {{} \hfill} & {{ = s_{1} (t) + s_{2} (t) + r_{2} (t)}\hfill} \\ {{} \hfill} & { \vdots \hfill} \\ {{} \hfill} & {{ =s_{1} (t) + s_{2} (t) + \cdots + s_{n} (t)} \hfill}\\\end{aligned} $$

(1)

where s _k (t) is the kth IMF and $r_{k} (t) = x(t) - {\sum\limits_{i = 1}^k {s_{i} (t)} }$ is the residual after extracting the first k IMF. There are six steps in the extraction of the kth IMF:

• (i) Initialize $h_{0} (t) = h_{{i - 1}} (t) = r_{{k - 1}} (t)$ (if k = 1, h₀(t) = x(t)), where i = 1;

• (ii) Extract local minima/maxima of h _i−1(t) (if the total number of minima and maxima is less than 2, $s_{k} (t) = h_{{i - 1}} (t)$ and stop the whole EMD process);

• (iii) Obtain upper envelope (from maxima) and lower envelope (from minima) functions p(t) and v(t) using cubic spline fittings to interpolate local minima and maxima of h _i−1(t), respectively;

• (iv) Calculate the $h_{i} (t) = h_{{i - 1}} (t) - (p(t) + v(t))/2;$

• (v) Calculate the standard deviation (SD) of (p(t) + v(t))/2;

• (vi) If SD is small enough (less than a chosen threshold SD_max, typically between 0.2 and 0.3) (Huang et al. 1998b), the kth IMF is assigned as $s_{k} (t) = h_{i} (t)$ and $r_{k} (t) = r_{{k - 1}} (t) - s_{k} (t);$Otherwise repeat steps (ii) to (v) for i + 1 until SD < SD_max

The above procedure is repeated to for k + 1 to obtain different IMFs at different scales until there are less than 2 minima or maxima in a residual r _k (t) which will be assigned as the last IMF (see the step ii above).

The EMD can extract the true oscillation components embedded in the original signal without presuming oscillation frequency. However, for nonstationary signals with intermittent oscillations, a limitation of EMD can be caused by the “mode mixing” problem, i.e., a mode obtained from EMD could comprise of oscillations with different wavelengths (corresponding to different physiological functions) at various temporal locations or oscillations corresponding to a physiological function appear in different modes at different temporal locations (Huang et al. 1998a). In order to reliably extract the spontaneous oscillations in BP and BFV during baseline conditions, the improved MMPF method uses a noise-assisted EMD algorithm, namely the Ensemble Empirical Mode Decomposition (EEMD) (Wu and Huang 2005). The EEMD consists of an ensemble of the EMD decompositions of data with added white noise and treats the resultant means of the corresponding intrinsic mode functions from different decompositions as the final result. Shortly, for a time series x(t), the EEMD includes the following steps:

• (i) Generate a new signal y(t) from the original time series x(t) by superposing to x(t) a white noise with amplitude equal to 10% of the standard deviation of x(t) (applying noise with larger amplitude requires more realizations of decompositions);

• (ii) Perform the EMD on y(t) to obtain intrinsic mode functions;

• (iii) Repeat steps (i)–(ii) m times with different white noise to obtain an ensemble of intrinsic mode functions (IMFs) $\{ s^{1}_{k} (t),k = 1,2...,n\} ,\;\{ s^{2}_{k} (t),k = 1,2...,n\} , \ldots ,\{ s^{m}_{k} (t),k = 1,2...,n\} ;$

• (iv) Calculate the average of intrinsic mode functions ${\left\{ {s_{k} (t),k = 1,2...,n} \right\}},$ where

  $$ s^{{}}_{k} (t) = \frac{1} {m}{\sum\limits_{i = 1}^m {s^{i}_{k} (t)} }. $$

The last two steps are applied to reduce noise level and to ensure that the obtained IMFs reflect the true oscillations in the original time series x(t). In this study, we repeated decomposition m = 100 times so that the final noise level is approximately less than 1% (=amplitude of white noise/${\sqrt m }$).

The EEMD approach overcomes the mode-mixing problem and ensures the decompositions to compass the range of possible solutions in the sifting process and to collate the signals of different scale in the proper IMF naturally.