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Division of Medical Physics, University of Leicester, Leicester Royal Infirmary, Leicester LE1 5WW; and Division of Medicine for the Elderly, University of Leicester, Glenfield Hospital, Leicester LE3 9QP, United Kingdom
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX
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The linear dynamic relationship between
systemic arterial blood pressure (ABP) and cerebral blood flow velocity
(CBFV) was studied by time- and frequency-domain analysis methods. A
nonlinear moving-average approach was also implemented using
Volterra-Wiener kernels. In 47 normal subjects, ABP was measured with
Finapres and CBFV was recorded with Doppler ultrasound in both middle
cerebral arteries at rest in the supine position and also during ABP
drops induced by the sudden deflation of thigh cuffs. Impulse response functions estimated by Fourier transfer function analysis, a
second-order mathematical model proposed by Tiecks, and the linear
kernel of the Volterra-Wiener moving-average representation provided
reconstructed velocity model responses, for the same segment of data,
with significant correlations to CBFV recordings corresponding to
r = 0.52 ± 0.19, 0.53 ± 0.16, and 0.67 ± 0.12 (mean ± SD), respectively. The correlation coefficient for the linear plus quadratic kernels was 0.82 ± 0.08, significantly superior to that for the linear models
(P < 10
6). The supine linear
impulse responses were also used to predict the velocity transient of a
different baseline segment of data and of the thigh cuff velocity
response with significant correlations. In both cases, the three linear
methods provided equivalent model performances, but the correlation
coefficient for the nonlinear model dropped to 0.26 ± 0.25 for the
baseline test set of data and to 0.21 ± 0.42 for the thigh cuff
data. Whereas it is possible to model dynamic cerebral autoregulation
in humans with different linear methods, in the supine position a
second-order nonlinear component contributes significantly to improve
model accuracy for the same segment of data used to estimate model
parameters, but it cannot be automatically extended to represent the
nonlinear component of velocity responses of different segments of data or transient changes induced by the thigh cuff test.
cerebral blood flow; mathematical model; thigh cuff test; Volterra-Wiener kernels; cerebral hemodynamics
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX
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THE TEMPORAL RELATIONSHIP between beat-to-beat changes in mean arterial blood pressure (MABP) and mean cerebral blood flow (MCBF) is modulated by the mechanism of cerebral autoregulation, which tends to maintain MCBF relatively constant despite changes in systemic MABP. In animals, sudden changes in ABP are transmitted to the cerebral circulation, inducing similar changes in CBF, but under normal conditions the CBF tends to return to its original value with a time constant of a few seconds (13, 26, 28, 32). A similar transient response of CBF to ABP disturbances has been observed in humans, and it is usually referred to as dynamic cerebral autoregulation (1, 12, 19). The recent advent of transcranial Doppler ultrasound (TCD) has been instrumental in the study of the dynamic relation between changes in systemic ABP and CBF because it has the temporal resolution required to detect fast changes in CBF, unlike more traditional methods of measuring CBF. Although the Doppler technique measures blood velocity rather than absolute flow, these two quantities will change in parallel unless the diameter of the insonated vessel changes significantly, which has not been shown to occur (11, 18).
Studies of dynamic cerebral autoregulation have used different techniques to induce rapid changes in ABP. These include the sudden deflation of thigh cuffs (1, 29), Valsalva maneuvers (12, 30), forced breathing (8), periodic squatting (4), or tilting (2). Other investigators have relied on spontaneous fluctuations in ABP to observe the corresponding transient changes in CBF velocity (CBFV) (5, 9, 10, 14, 17, 21-24, 33).
Dynamic cerebral autoregulation has been shown to be disturbed by a number of pathophysiological conditions (1, 2, 4, 8, 10, 21, 23, 24, 29) and to correlate well with assessments of static autoregulation (21, 29). Both physiological and clinical studies can benefit from the dynamic approach compared with the more classic view of cerebral autoregulation as a static phenomenon (25, 29). The time course of the CBF transient response can shed light on the structure and efficiency of the feedback mechanisms responsible for CBF control (9, 22, 23, 33). With the use of noninvasive methods to estimate ABP and CBF, it is possible to conduct repeated tests, thus facilitating studies of the influence of other variables or physiological interventions (e.g., posture, blood gases, drugs) on CBF regulation. Modeling the dynamic pressure-flow (or velocity) relationship represents an important element of this research as a conceptual and quantitative tool to refine our understanding and assessment of dynamic cerebral autoregulation for physiological studies and clinical applications.
For step changes in ABP, as obtained by the sudden deflation of thigh cuffs, the CBFV response has been modelled by a second-order differential equation with a set of parameters that can be used to grade the performance of autoregulation (29). The typical CBFV transient that follows the negative step change in ABP and the phase changes observed during forced breathing have also prompted analogies between the dynamic autoregulatory response and the behavior of a simple high-pass filter (8, 9, 33). Empirical modeling has also been performed by means of frequency-domain analysis through calculation of the coherence, transfer function, and phase-frequency response between CBFV and ABP (5, 9, 10, 14, 17, 21-23, 33). Some studies have also obtained estimates of the CBFV impulse response, which reflects the dynamic CBFV response to an impulselike disturbance in ABP (22, 23, 33). This approach is quite attractive because the time integral of the impulse response represents the CBFV step response, which is approximately the same transient generated by the sudden deflation of thigh cuffs (1, 29). We have recently demonstrated that the model proposed by Tiecks et al. (29) can also be used to estimate the impulse response of dynamic autoregulation during baseline spontaneous fluctuations in ABP (24).
All of these models rely on the assumption that the dynamic autoregulatory mechanism can be approximated by a linear system. This assumption can be questioned on theoretical grounds, because changes in pressure lead to changes in cerebrovascular resistance. Because resistance is in the denominator of Pouseuille's law, and because it depends on the fourth power of arteriolar diameter, which may be influenced by myogenic, neurogenic, endothelial, or metabolic mechanisms (25), it becomes clear that there are several nonlinearities that cannot be ignored (31). On the other hand, the assumption of linear systems has been justified for the case of spontaneous fluctuations in ABP, for example, because of the relatively small changes in ABP compared with the larger pressure drops produced by the thigh cuff technique. The objective of the present study is to examine the limitations of the linear assumption and to compare the relative performance of different modeling options. For this purpose we have adopted the approach pioneered by Marmarelis and co-workers (6, 7) in studies of the dynamic renal autoregulation of rats, showing that the renal regulatory mechanism cannot be represented by a linear system. Their analysis was based on the Volterra-Wiener representation of dynamic nonlinear systems (15, 16), which we have also used to obtain linear and quadratic models for the dynamic relationship between beat-to-beat changes of ABP and CBFV in a large group of normal subjects.
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METHODS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX
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Subjects and measurements. Subjects recruited were free from cardiovascular disease as determined from history, full physical examination, and a 12-lead surface electrocardiogram (ECG). None was taking any medication known to affect the autonomic nervous or cardiovascular system. All studies were conducted in the morning, at least 2 h after a light breakfast, and the subjects were asked to refrain from alcohol-, nicotine-, or caffeine-containing products for a minimum of 12 h. Measurements were performed in a quiet, dimly lit room at constant ambient temperature (23°C) with subjects lying supine on a couch with the head supported by two pillows and the right arm supported at atrial height. Arterial blood pressure was measured with a noninvasive finger blood pressure monitor (Finapres 2300, Ohmeda) that has been extensively validated against intravascular measurements (20, 27) and has also been used in previous studies of dynamic autoregulation (5, 14, 24, 33). A finger cuff of the appropriate size was attached to the middle finger of the right hand. During the recording the servo-adjust mechanism of the Finapres was disabled, and an ABP calibration signal was recorded before each measurement. An electrocardiogram (ECG) was obtained using three standard surface chest leads, and a transcutaneous gas monitor (TINA, Radiometer) was attached to record carbon dioxide partial pressure. A dual-channel TCD (SciMed QVL 120) was used to measure CBFV in the right and left middle cerebral arteries (MCA) using 2-MHz probes held immobile by a purpose-built head frame.
The ABP, dual-channel CBFV Doppler-shift signals, and ECG were continuously recorded on digital tape for posterior analysis (DAT, Sony PC 108M). After a 30-min rest period, signals were recorded for two 5-min periods with subjects breathing normally at rest. To perform the thigh cuff test, wide blood pressure cuffs were attached to both legs and inflated 40 mmHg above systolic pressure for 45 s. The cuffs were rapidly deflated, and signals were recorded for another 2 min. Only one thigh cuff test was performed on each subject. Tests were rejected if the ABP drop was <10 mmHg or if it was not accompanied by a simultaneous drop in CBFV in both channels.Data analysis. The DAT recording was downloaded to a microcomputer in real time. A fast Fourier transform (FFT) was used to extract the maximum frequency velocity envelope, using a time window of 6.25 ms. The ABP and ECG signals were sampled at 200 samples/s per channel.
The ABP signal was calibrated at the start of each recording, and all signals were visually inspected for artifacts or noise. Narrow spikes on the CBFV signals were removed by linear interpolation, and the four signals were low-pass filtered with a zero-phase, eighth-order, Butterworth digital filter with a cutoff frequency of 20 Hz. The beginning and end of each cardiac cycle were detected from the upstroke of the arterial pulse-pressure wave, and the MABP and mean CBFV (MCBFV) were calculated for each cycle. Ectopics occurring during the thigh cuff test led to the rejection of the data as did more than one ectopic per 30 s during the baseline recording. Occasional ectopics could be marked and removed by linear interpolation. Spline interpolation was used to resample the data at 0.2 s to create a uniform time base. For baseline recordings, the MABP and MCBFV time series were normalized by the mean value during the 5-min period and subtracted by 1. The resulting zero-mean signals, reflecting relative changes in ABP and CBFV, are represented by P(t) and V(t), respectively, where t is time. A similar procedure was adopted for the thigh cuff, but in this case the reference values were the mean values of MABP and MCBFV preceding the cuff deflation. Four linear models and one nonlinear (quadratic) model were analyzed by assuming P(t) as the input and V(t) as the output. The mathematical formulation for the different models is given in the APPENDIX. The simplest linear model, labeled "zero-order model," corresponds to the approximation V(t) = P(t). Because both signals are normalized in relative units, it is assumed that all velocity changes are purely reflecting the pressure transients in the absence of autoregulation. The corresponding impulse response for this model is iZ(t) = 1 for t = 0 and iZ(t) = 0 for all other values of t. Although the zero-order model can be expected to perform poorly, it fits the purpose of providing a "background" condition against which all other models can be compared. The model proposed by Tiecks et al. (29) was implemented using exactly the same set of equations and parameters proposed by the authors (see APPENDIX). Different combinations of the model parameters (T, D, and K, defined in APPENDIX) were selected corresponding to 10 grades of autoregulation expressed by a dynamic autoregulation index (ARI) ranging from 0 (absence of autoregulation) to 9 (best autoregulation). The Tiecks model was fitted to baseline data by selecting the value of ARI leading to the minimum quadratic error between the model-generated velocity [VT(t)] and V(t). The corresponding impulse response [iT(t)] was calculated as the first derivative of VT(t). Another impulse response estimate was obtained using the FFT approach as described previously (23). The P(t) and V(t) time series were divided into 4 segments with 256 samples (51.2 s) each, multiplied by a cosine-tapered window, and transformed with the FFT algorithm using 30% superposition of segments (Welch method). The corresponding frequency resolution is 0.019 Hz per harmonic. The auto- and cross-spectra, the coherence function, and the transfer function (see APPENDIX) were estimated using the average of the 4 segments of data and smoothed with a 3-point triangular window, leading to spectral estimates with 18 degrees of freedom per harmonic (3). The phase-frequency response was estimated without unwrapping (23). The inverse FFT was applied to obtain the time-domain impulse response [iF(t)] after low-pass filtering the transfer function with a cutoff frequency of 1 Hz. The Volterra-Wiener approach to estimate the linear and nonlinear representation of the dynamic P(t)-V(t) relationship was implemented as proposed by Marmarelis (15) with the use of Fortran software specially written for this purpose. We have taken advantage of the orthogonal property of Laguerre polynomials to expand the first- and second-order (quadratic) kernels, thus obtaining a more efficient and accurate estimation of these models (15). This method will be referred to as the Wiener-Laguerre representation for nonlinear systems. The software Lysis 6.2, distributed by the Biomedical Simulations Resource, University of Southern California, was used as a benchmark to test and validate our own software. As described in the APPENDIX, the Volterra series contains a term for each order considered. Because both P(t) and V(t) have zero mean, the constant term was not estimated. Only the second term, k1(t), was used to obtain the impulse response corresponding to the linear assumption. To obtain a nonlinear, quadratic model, both the first- and second-order terms were considered (see APPENDIX). The Wiener formulation assumes that the input signal is random, but, as demonstrated by Marmarelis (15, 16), this condition can be relaxed when Laguerre polynomials are used to expand the different kernels. To increase the whiteness of P(t), we decimated both input and output signals by a factor of five, taking every fifth sample to increase the sampling interval to 1 s. Decimation reduces the frequency bandwidth of the signals to 0.5 Hz (3), which is appropriate to describe the fastest components of cerebral autoregulation (5, 14, 23, 33). To estimate the linear and quadratic kernels, we used segments of data with 160-s duration, which is approximately the same signal duration used for obtaining the FFT impulse response, iF(t). The Laguerre expansion used 10 terms with a value of
= 0.2 (see
APPENDIX). The number of lags used
for both the linear and nonlinear kernels was 32.
All of the analyses were conducted separately for the CBFV signals from
the right and left MCA. Separate average curves of iT(t),
iF(t),
and
k1(t)
were also obtained for each MCA by averaging the individual impulse
responses of all subjects studied. Similarly, the average second-order
kernel,
k2(t, t),
was obtained from the mean of all subjects.
The performance of each linear model was assessed by comparing the
predicted model response
[VM(t)]
with the real data,
V(t). A standard procedure was adopted for all models using the convolution operation to calculate the model response from the impulse response and
the input signal, P(t), as
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(1) |
Statistical analysis. Model-predicted velocity responses (Eq. 1) were compared with velocity data by means of Pearson's correlation coefficient (r) and the mean square error (MSE). The latter represents a more rigorous test because it takes into account the amplitude similarity between model and data in addition to the temporal pattern that is reflected by the correlation coefficient. The whiteness of model residuals was assessed with the Durbin-Watson test. For each model, means ± SD of r and MSE were calculated for all subjects for the right and left MCA responses. Results for the two MCAs were grouped if no significant differences were observed between the right and left sides. Comparisons between models were performed for each subject by testing the change in r with the z test and the MSE with the F test. The sign test was then used to assess the significance of the differences between models for all subjects. A value of P < 0.05 was considered significant.
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX
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We studied 47 subjects (27 male) with mean ± SD age of 66.9 ± 9.2 yr (range 44-80 yr), body mass index of 26.5 ± 3.4 kg/m2, heart rate of 75 ± 9 beats/min, clinical MABP of 101.6 ± 11.5 mmHg, systolic BP of 138.3 ± 15.2 mmHg, and diastolic BP of 83.3 ± 10.6 mmHg. Only 33 subjects had thigh cuff responses that were acceptable according to the criteria described previously. There were no major changes in transcutaneous CO2 during measurements or between the baseline recordings and the thigh cuff test.
The transfer function analysis results are presented in Fig.
1, and the corresponding impulse response
functions, estimated using the FFT method, are shown in Fig.
2A as mean
values for the population studied. At frequencies <0.1 Hz, the mean
coherence function was relatively low, suggesting uncoupling between
changes in CBFV and ABP. As the frequency increased to 
0.1 Hz, the
squared coherence reached a plateau around 0.6 for both right and left channels. The mean gain, or amplitude-frequency response (Fig. 1B), also increased with frequency,
in this case rising continuously up to a maximum at 0.2 Hz, leveling
off until reaching ~0.4 Hz, and then rising again to peak at 0.5 Hz.
The mean phase-frequency response (Fig.
1C) presented an inverse trend; it
was more positive below 0.1 Hz and then decreases more or less
continuously until crossing zero at ~0.35 Hz for both channels.
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Figure 2B presents the group-average Tiecks impulse response, which was almost identical for both channels, with the smallest SE for all linear models studied. The corresponding mean ± SD of the ARI for the combined right- and left-side responses was 4.5 ± 2.9.
The use of the Wiener-Laguerre method to treat the ABP-CBFV
relationship as linear led to the impulse response functions
k1(t) depicted in Fig. 2C, also showing
excellent agreement between the two different CBFV channels. The
nonlinear analysis, with the formulation described in the
APPENDIX, produced a first-order term
with a temporal pattern that resembles
k1(t)
in Fig. 2C and the mean second-order
kernel
k2(t, t),
shown in Fig. 3. The differences between
the linear and quadratic implementations of the Wiener-Laguerre method
are exemplified by a representative segment of data and the
corresponding model-predicted responses in Fig.
4. Figure 4 also shows the phase lead of
CBFV in relation to ABP during spontaneous oscillations in pressure.
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For the training set of data, the superiority of the nonlinear
representation, exemplified in Fig. 4, has been confirmed for the whole
group of subjects as indicated by the results in Table 1. As expected, the zero-order model gave
the worst results for both r and MSE.
The FFT and Tiecks methods yielded similar results, but for the linear
assumption the best results were obtained with the linear
Wiener-Laguerre approach. When applied to the change in correlation and
MSE, the sign test indicated that
1) the nonlinear method was
superior to the linear Wiener-Laguerre method
(P < 106);
2) the linear Wiener-Laguerre method
was superior to both the FFT and Tiecks methods
(P < 106);
3) the FFT and Tiecks methods were
not significantly different from each other; and
4) both the FFT and Tiecks methods
were superior to the zero-order model
(P < 106). Durbin-Watson tests
were inferior to the critical limit (d = 1.7, where d is the Durbin-Watson
statistic), indicating that residuals were serially
correlated in all models (Table 1). However, for the nonlinear
Wiener-Laguerre model there was a very significant (P < 108) change of this
statistic, showing an improvement in the direction of less correlated
residuals.
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Despite the fact that the nonlinear Wiener-Laguerre model was much
superior to all linear versions (Table 1), the estimated step responses
obtained with the quadratic and linear implementations of the
Wiener-Laguerre method show only a small difference in the temporal
pattern of the CBFV response to a step change in ABP, as indicated in
Fig. 5.
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Table 2 presents the model performance
results when impulse responses estimated from the training set were
used to predict the velocity responses of the test set of data. The
worst model in this case was the quadratic Wiener-Laguerre model
(P < 0.0002), followed by the
zero-order model (P < 106). The Tiecks, FFT,
and linear Wiener-Laguerre models had comparable performance, with
nonsignificant differences by the sign test between these three models,
except for the correlation coefficient, which was superior for the
linear Wiener-Laguerre model in relation to the FFT model
(P < 0.05). The Durbin-Watson test
indicated that residuals were serially correlated in every case,
although the quadratic Wiener-Laguerre model provided significantly
higher values in relation to the other models examined.
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The results of the estimation of CBFV responses during the thigh cuff
test are given in Table 3. In this case the
zero-order model performed even worse than in the baseline situations,
but neither the mean results for the FFT method nor those for the Tiecks method are very different from those in Table 1, although the SD
is larger. The linear Wiener-Laguerre method was only slightly superior
to the other linear methods, but the nonlinear model was clearly
inadequate, performing even worse than the zero-order model with
respect to MSE (Table 3). The sign test for these data indicated that
1) the nonlinear model was
significantly worse than the linear Wiener-Laguerre representation
(P = 0.0003);
2) the linear Wiener-Laguerre model
was not significantly different from the Tiecks method;
3) both the Tiecks and linear
Wiener-Laguerre methods were superior to the FFT method
(P = 0.008, P = 0.041); and
4) the FFT method was superior
to the zero-order model (P = 0.003). For all models the Durbin-Watson statistic indicated that
residuals were serially correlated.
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX
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Analyses of dynamic cerebral autoregulation based on spontaneous
fluctuations of ABP have often adopted transfer function analysis to
quantify the relationship between beat-to-beat changes in MCBFV and
MABP (5, 9, 10, 14, 17, 23, 33). Several investigators have reported
coherence-, gain-, and phase-response curves that allow cerebral
autoregulation to be interpreted as a frequency-dependent mechanism (5,
9, 14, 17, 22, 23, 33). The common characteristics of these findings
are a very significant phase lead of MCBFV in relation to ABP and a
coherence function that is relatively high for frequencies >0.2 Hz
[say,
2(f) > 0.6] but drops to values of 
0.50 for frequencies <0.1 Hz (14, 23, 33). This reduction in coherence has been interpreted as the
result of active autoregulation operating in this frequency range,
whereby changes in ABP are buffered and the corresponding fluctuations
in MCBFV are attenuated by feedback adjustments in cerebrovascular
resistance. As the frequency of the input ABP changes increases, the
autoregulatory mechanism cannot respond fast enough and both the
coherence and gain, or amplitude-frequency response, tend to increase.
These previous findings have been confirmed by the present study,
incorporating data from both MCA arteries of a large number of healthy
subjects. We have obtained frequency-domain functions of mean
coherence, gain, and phase that are very similar to those presented by
Zhang et al. (33). The relatively high values of squared coherence
(2) obtained by ourselves and
others (5, 14, 23, 33) indicate that a large fraction of the
beat-to-beat variability of Doppler velocity measurements in the MCA
can be explained by fluctuations in MABP. This is an important result
because of the ongoing discussion about the possibility of diameter
changes of the MCA affecting the flow-velocity relationship. Although
Newell et al. (18) have provided evidence that the relationship between
flow and velocity remains relatively constant during thigh cuff tests, we cannot rule out the possibility of small spontaneous changes in
diameter, leading to CBFV fluctuations of amplitude comparable to what
we have observed during baseline recordings. The high values of
coherence, however, indicate that if such fluctuations in diameter
occur, they have only a minor influence and cannot obscure the
relationship between MABP and CBFV. In fact, it is possible to assume
that these hypothetical changes in diameter are even less influential
because other factors, such as nonlinearity and noise, also have a
bearing on the variance unaccounted for (i.e., 1 
2). When our results on
transfer function analysis are compared with those reported by other
investigators, two aspects should be noted. First, we studied an older
population, ranging from 44 to 80 yr of age (mean 66.9 yr), but without
apparent qualitative differences in relation to studies involving
younger subjects. Second, we used dual-channel Doppler recordings,
obtaining an excellent agreement between the right- and left-side
measurements, not only for the transfer function analysis (Fig. 1) but
also for the other modeling approaches investigated (Fig. 2, Table 1).
These results are very encouraging regarding the confidence that can be
placed on the Doppler technique and the reproducibility of the analytic methods.
Within the linear assumption for the MABP-MCBFV relationship at rest, we have also studied the performance of the model of Tiecks et al. (29) and the Wiener-Laguerre representation proposed by Marmarelis et al. (6, 7, 15, 16). Contrary to our initial expectations, the Tiecks model gave results comparable to those of the transfer function approach, although it was originally proposed and validated for use with the thigh cuff test (29). These results should not be interpreted, however, as an indication that the Tiecks model is the best mathematical representation of the pressure-velocity relationship in the presence of an active autoregulation (31). As an empirical mathematical model, the formulation proposed by Tiecks et al. (29) was aimed solely at grading dynamic autoregulation for step changes in ABP and, consequently, does not provide any insights into the underlying physiology of cerebral autoregulation. On the other hand, the finding that the Tiecks model can provide a reasonable representation of dynamic autoregulation at rest has some relevant implications for the assessment of autoregulation for clinical purposes. Panerai et al. (24) have recently demonstrated that the Tiecks model used in conjunction with baseline measurements can detect the occurrence of carotid artery disease. Similar results could possibly be obtained with the transfer function method, but the Tiecks model has the potential advantage of better performance for short segments of data. As a drawback, this model cannot reflect the same temporal evolution of the negative phase of the impulse responses given in Fig. 2 as those of the transfer function and Wiener-Laguerre impulse responses.
For the linear methods studied, the best results were obtained with the
linear Wiener-Laguerre representation (Table 1). This approach has been
applied to study renal autoregulation in rats, but we are not aware of
other applications involving the human cerebral autoregulation. As
described in the APPENDIX, the method
requires the choice of three basic parameters, namely, the lag duration
L, the number of Laguerre functions
(J), and the value of for the
Laguerre functions. We have obtained satisfactory results with lags in
the range of 16-32 s and values of in the range of
0.05-0.2. We have adopted J = 10 but have not explored the sensitivity of the method to this parameter.
Impulse responses obtained with the transfer function and Tiecks
methods indicate that the response is fairly completed for lag times of
8 s. This feature of dynamic autoregulation makes the
Wiener-Laguerre method a convenient approach because of the limited
time lag and, hence, computational burden required (15). Despite the
apparent practical advantage of the Tiecks model in providing a simple
index (ARI) for grading autoregulation, it should be possible to obtain
a similar measure for the linear Wiener-Laguerre approach, as indicated by the step response depicted in Fig. 5. Similarly to the Tiecks model, the linear Wiener-Laguerre can also provide robust estimates for
much shorter segments of data than we have adopted in our study.
Application of impulse responses derived with the three linear models in the training set to other two segments of data for each subject, the test set and the thigh cuff response, respectively, have shown similar performances for the three methods and also considerable robustness of the linear approach. Although extrapolation of dynamic responses to different segments of data can provide an indication of model robustness, in the case of cerebral autoregulation this test needs to be interpreted with caution because there is no evidence that the dynamic relationship between MABP and CBFV should be immutable. The opposite is more likely, in fact, because this relationship can be influenced by other variables, such as PCO2 and mental activation (1, 25), that may show small oscillations with time even during baseline recordings. The superiority of the linear Wiener-Laguerre model for the training set, which disappears for the other two segments of data, suggests that this method may be more sensitive to the time-varying influences of other variables that can affect the regulation of CBF.
We and others (5, 22, 23, 33) have used the small-signal argument to justify the utilization of linear models for baseline measurements. The finding that all three linear models produced satisfactory results when baseline impulse responses were applied to thigh cuff recordings suggests that these impulse responses can be regarded as a surrogate for situations with much larger changes in ABP such as the thigh cuff test, tilting, and Valsalva or lower body negative-pressure maneuvers. Zhang et al. (33) have obtained similar results showing that the linear impulse response derived for baseline measurements can predict the MCBFV for the thigh cuff test reasonably well, quoting a mean difference of 0.22 ± 1.45 cm/s between the model prediction and the data. However, according to their Fig. 5, it is likely that the error quoted includes the balance of positive and negative deviations, and therefore it is not comparable to our results using the mean quadratic error (Table 3). Interestingly enough, the Tiecks model presented a performance comparable to that of the linear Wiener-Laguerre representation and was significantly, albeit only slightly, superior to the transfer function approach (Table 3). Despite the fact that the thigh cuff test can provide an almost immediate visualization of the efficiency of the autoregulatory mechanism, our results indicate that similar information can be derived from measurements at rest. The feasibility of substituting the thigh cuff analysis with the modeling of baseline recordings has been recently demonstrated (24), but the present set of results provides more general and encouraging evidence that should facilitate the diffusion of cerebral autoregulation assessment as a clinical tool.
When the linear Wiener-Laguerre representation was expanded to include a quadratic term, excellent results were obtained in the training conditions (Table 1), but extension of the corresponding kernels to the test set (Table 2) and the thigh cuff data (Tables 3) led to disappointing results. It could be argued that the superior results obtained with the nonlinear model for the training set simply reflect the fact that the additional quadratic term can more easily fit the noise rather than reflect a true characteristic of the autoregulatory mechanism. Figure 4 shows that this is not the case and is representative of observations in the majority of subjects. The predicted velocity response for the nonlinear model does seem to represent important features of the MCBFV time series rather than simply following the noise. This conclusion is reinforced by the analysis of residuals as expressed by the Durbin-Watson test. Although none of the models provided uncorrelated residuals (Table 1), the nonlinear Wiener-Laguerre model led to a very significant change in the Durbin-Watson statistic, thus indicating that the quadratic term helps to explain nonrandom features of the CBFV time series. Nevertheless, the nonwhiteness of residuals for all the models studied is of concern and may be due to the fact that the output noise is not itself random because it may be the result of low-frequency oscillations induced by respiration, intracranial pressure, or small changes in MCA diameter, as mentioned previously. The two-dimensional mean kernel for the quadratic term k2(t, t), represented in Fig. 3, shows relatively large secondary peaks at lag values of several seconds. This feature is not observed in any of the linear impulse responses in Fig. 2. The presence of these peaks suggests that the second-order kernel may account for some delayed effects of pressure on MCBFV transients. This behavior is exemplified in Fig. 5, which shows that the main difference between the linear and nonlinear estimated responses occurs several seconds after the sudden ABP transition. Although the difference between the linear and nonlinear predicted responses is relatively small, this result cannot be generalized to other input functions because of the presence of the quadratic term.
Unfortunately, the finding that the quadratic term produces significantly better models does not immediately shed much light on the mechanism of autoregulation. Moreover, it does not make sense to attempt to find a physiological interpretation for the second-order term. Whatever the nature of the nonlinearities involved (31), the quadratic term is only the first of their equivalent series expansion (see APPENDIX), and that it is how it should be interpreted. By applying the nonlinear Wiener-Laguerre method to study the renal autoregulation of rats, Chon et al. (6) have proposed that the third-order kernel is more relevant than the second-order term to represent the nonlinear behavior of the system. The same authors previously analyzed the second-order component in the frequency domain, using the two-dimensional Fourier transform, extracting information about possible interactions between the myogenic and tubuloglomerular feedback mechanisms for control of kidney blood flow (7). Similar investigations could be performed on the cerebral autoregulation, for example, to explore the possibility of interaction between the myogenic and metabolic control pathways (25, 31). One limitation for human studies, however, is that Chon et al. (6, 7) have imposed randomlike changes in ABP to increase whiteness and improve the convergence of estimation for higher order kernels. With spontaneous fluctuations we have no control of the nature and amplitude of the ABP transients, but we have been able to fit the second-order component using 160-s segments of data. However, for shorter segments of data the Wiener-Laguerre method failed because of poor conditioning of the single-value decomposition solution for the second-order kernel (see APPENDIX). It is an open question as to whether it would be possible to obtain the third-order component by using baseline data of similar duration in humans.
The most appropriate explanation for the poor results of the nonlinear
Wiener-Laguerre model, when extended to other segments of data, lies
with the temporal pattern of MABP fluctuations. Although the Wiener
method requires strict randomness of the input variable, Marmarelis
(15, 16) claims that this condition can be relaxed in the case of the
Laguerre transformation. Nevertheless, Chon et al. (6, 7) have imposed
disturbances in MABP to increase its randomness, and we are not aware
that a separate test set of data has been used to validate their
results. In our study, only spontaneous fluctuations in MABP were used,
and decimation to a sampling rate of 1 s was adopted to increase the
randomness of the MABP signal. Because of the larger number of
coefficients in the quadratic kernel compared with that in the linear
Wiener-Laguerre component (see
APPENDIX), we hypothesize that in
the absence of strict randomness the nonlinear Wiener-Laguerre model
becomes too sensitive to the particular MABP temporal pattern of the
training set. Consequently, when kernels are applied to different sets of data, nonrandom changes in MABP, such as the thigh cuff response, produce dominant quadratic components with aberrant responses, as shown
by the results in Tables 2 and 3. Further work is necessary to
determine whether increases in randomness of MABP will lead to parallel
increases in robustness of the nonlinear Wiener-Laguerre model. With
spontaneous fluctuations in ABP, it may be possible to increase
randomness by performing much longer baseline recordings (20
min) than we have adopted in the present study.
Notwithstanding these limitations, the very significant improvement in model performance obtained with the nonlinear Wiener-Laguerre method in the training segment of data deserves some attention because it may have important implications for clinical investigations and future research into dynamic autoregulation. First, investigators need to be aware of the general limitation of linear models when searching for mathematical models of cerebral dynamic autoregulation. This might be the case of attempts to improve empirical models (29) or obtain models that can provide greater insight into the physiology of the cerebral circulation. Both the FFT and linear Wiener-Laguerre techniques (Fig. 2, A and C) can be regarded as general linear models whose flexibility of coefficients (or impulse response) allows the representation of relatively complex mathematical models formed by a large set of linear differential equations. The differences in performance between the FFT and linear Wiener-Laguerre approaches and the Tiecks model (Table 1) provide an indication of how much can be gained by sticking to the linear formulation. On the other hand, nonlinear models, as proposed by Ursino and Lodi (31), stand a much better chance of explaining the dynamic relationship between MABP and CBFV beat-to-beat changes. Second, the clinical relevance of nonlinear models can only be established by future research. Linear models are much simpler to use in routine applications and have been shown to possess diagnostic value in certain conditions (23, 24, 29). What remains to be established, however, is whether nonlinear models can lead to significant improvements in diagnostic accuracy in different clinical conditions.
| |
APPENDIX |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX
|
|---|
Tiecks model. For an ABP drop P(t), normalized by the baseline values preceding the deflation of thigh cuffs, the relative pressure change dP(t) is given by (29)
|
(A1) |
|
(A2) |
|
(A3) |
|
(A4) |
Transfer function analysis. From the temporal sequences P(t) and V(t), the frequency-domain transforms P(f) and V(f) are computed with an FFT algorithm. The power spectrum of P(t) [GPP(f)] is calculated as
![G<SUB>PP</SUB>( <IT>f</IT> ) = E[P( <IT>f</IT> )*P( <IT>f</IT> )]](/content/vol277/issue3/fulltext/H1089/img006.gif)
|
(A5) |
![G<SUB>PV</SUB>( <IT>f</IT> ) = E[P( <IT>f</IT> )*<IT>V</IT>( <IT>f</IT> )]](/content/vol277/issue3/fulltext/H1089/img007.gif)
|
(A6) |
2(f)]
was estimated by (3)
|
(A7) |
|
(A8) |
(f) obtained
from the real part
HR(f)
and imaginary part
HI(f)
of the complex transfer function, this becomes
![‖H( <IT>f</IT> )‖ = [H<SUB>R</SUB>( <IT>f</IT> )<SUP>2</SUP> + H<SUB>I</SUB>( <IT>f</IT> )<SUP>2</SUP>]<SUP>½</SUP>](/content/vol277/issue3/fulltext/H1089/img010.gif)
|
(A9) |
|
(A10) |
![<IT>i</IT><SUB>F</SUB>(<IT>t</IT>) = ϝ<SUP>−1</SUP>[H( <IT>f</IT> )]](/content/vol277/issue3/fulltext/H1089/img012.gif)
|
(A11) |
1 represents the
inverse Fourier transform (3).
Marmarelis method. The Volterra series expansion for a nonlinear dynamic system can be written as
|
|
(A12) |
|
|
(A13) |
j (t)
are obtained from the convolution integral
|
(A14) |
![<IT>L</IT><SUB><IT>j</IT></SUB>(<IT>t</IT>) = [&agr;<SUP><IT>t</IT>−<IT>j</IT></SUP>(1 − &agr;)]<SUP>½</SUP> <LIM><OP>∑</OP><LL><IT>k</IT>=0</LL><UL><IT>j</IT></UL></LIM> (−1)<SUP><IT>k</IT></SUP><FENCE><AR><R><C><SUP><IT>t</IT></SUP></C></R><R><C>·</C></R><R><C><SUB><IT>k</IT></SUB></C></R></AR></FENCE> <FENCE><AR><R><C><SUP><IT>j</IT></SUP></C></R><R><C>·</C></R><R><C><SUB><IT>k</IT></SUB></C></R></AR></FENCE> &agr;<SUP> <IT>j−k</IT></SUP>(1 − &agr;)<SUP><IT>k</IT></SUP>](/content/vol277/issue3/fulltext/H1089/img019.gif)
|
(A15) |
controls the amount of damping of the Laguerre function
and needs to be carefully selected for different applications. One
advantage of using the Laguerre series is the fact that, in practice,
Eq. A14 does not need to be computed
because the orthogonal set
j(t)
can be obtained with the recursive relation
|
(A16) |
|
(A17) |
j (t)
are calculated with Eqs. A16 and A17. We assume
c0 = k0 = 0, because
P(t) and V(t)
are zero mean. From Eq. A13, the set
of coefficients
c1( j) and
c2( j,
j) can be calculated by single-value decomposition (16). Finally, the first- and second-order kernels can then be obtained
from
|
(A18) |
|
(A19) |
| |
ACKNOWLEDGEMENTS |
|---|
S. Dawson was funded by The Stroke Association of the United Kingdom. The support of the Engineering and Physical Sciences Research Council (UK) is also gratefully acknowledged.
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FOOTNOTES |
|---|
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: R. B. Panerai, Dept. of Medical Physics, Leicester Royal Infirmary, Leicester LE1 5WW, United Kingdom (E-mail: rp9{at}le.ac.uk).
Received 29 September 1998; accepted in final form 19 April 1999.
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M. R. Edwards, J. K. Shoemaker, and R. L. Hughson Dynamic modulation of cerebrovascular resistance as an index of autoregulation under tilt and controlled PETCO2 Am J Physiol Regulatory Integrative Comp Physiol, September 1, 2002; 283(3): R653 - R662. [Abstract] [Full Text] [PDF]
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R. Schondorf, R. Stein, R. Roberts, J. Benoit, and W. Cupples Dynamic cerebral autoregulation is preserved in neurally mediated syncope J Appl Physiol, December 1, 2001; 91(6): 2493 - 2502. [Abstract] [Full Text] [PDF]
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K. Narayanan, J. J. Collins, J. Hamner, S. Mukai, and L. A. Lipsitz Predicting cerebral blood flow response to orthostatic stress from resting dynamics: effects of healthy aging Am J Physiol Regulatory Integrative Comp Physiol, September 1, 2001; 281(3): R716 - R722. [Abstract] [Full Text] [PDF]
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B. J. Carey, B. N. Manktelow, R. B. Panerai, and J. F. Potter Cerebral Autoregulatory Responses to Head-Up Tilt in Normal Subjects and Patients With Recurrent Vasovagal Syncope Circulation, August 21, 2001; 104(8): 898 - 902. [Abstract] [Full Text] [PDF]
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J. Sato, M. Tachibana, T. Numata, T. Nishino, and A. Konno Differences in the dynamic cerebrovascular response between stepwise up tilt and down tilt in humans Am J Physiol Heart Circ Physiol, August 1, 2001; 281(2): H774 - H783. [Abstract] [Full Text] [PDF]
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R. B. Panerai, S. L. Dawson, P. J. Eames, and J. F. Potter Cerebral blood flow velocity response to induced and spontaneous sudden changes in arterial blood pressure Am J Physiol Heart Circ Physiol, May 1, 2001; 280(5): H2162 - H2174. [Abstract] [Full Text] [PDF]
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