## Abstract

The coherence function has been used in transfer function analysis of dynamic cerebral autoregulation to assess the statistical significance of spectral estimates of gain and phase frequency response. Interpretation of the coherence function and choice of confidence limits has not taken into account the intrinsic nonlinearity represented by changes in cerebrovascular resistance due to vasomotor activity. For small spontaneous changes in arterial blood pressure (ABP), the relationship between ABP and cerebral blood flow velocity (CBFV) can be linearized, showing that corresponding changes in cerebrovascular resistance should be included as a second input variable. In this case, the standard univariate coherence function needs to be replaced by the multiple coherence, which takes into account the contribution of both inputs to explain CBFV variability. With the use of two different indicators of cerebrovascular resistance index [CVRI = ABP/CBFV and the resistance-area product (RAP)], multiple coherences were calculated for 42 healthy control subjects, aged 20 to 40 yr (28 ± 4.6 yr, mean ± SD), at rest in the supine position. CBFV was measured in both middle cerebral arteries, and ABP was recorded noninvasively by finger photoplethysmography. Results for the ABP + RAP inputs show that the multiple coherence of CBFV for frequencies <0.05 Hz is significantly higher than the corresponding values obtained for univariate coherence (*P* < 10^{−5}). Corresponding results for the ABP + CVRI inputs confirm the principle of multiple coherence but are less useful due to the interdependence between CVRI, ABP, and CBFV. The main conclusion is that values of univariate coherence between ABP and CBFV should not be used to reject spectral estimates of gain and phase, derived from small fluctuations in ABP, because the true explained power of CBFV in healthy subjects is much higher than what has been usually predicted by the univariate coherence functions.

- cerebral autoregulation
- coherence function
- transfer function analysis
- multivariate modeling
- transcranial Doppler ultrasound

transfer function analysis (TFA) is a useful technique to study physiological systems whose properties are characterized by linear or quasi-linear dynamic relationships between two or more physiological time series (6, 36). One area that has benefited from this approach is the study of dynamic cerebral blood flow (CBF) autoregulation in humans (27, 28, 40). Aaslid et al. (2) have extended the classical concept of “static” autoregulation (20) by showing that a sudden change in arterial blood pressure (ABP) leads to a relatively fast recovery of CBF to its original level. This transient response of the autoregulatory mechanisms is what is now termed “dynamic” cerebral autoregulation (CA) (2). TFA of dynamic CA was initially proposed by Giller (13), who modeled CA as an input-output relationship between beat-to-beat values of ABP and CBF, as estimated by transcranial Doppler ultrasound (TCD) recordings of CBF velocity (CBFV). The many contributions that followed have shown a very good agreement between different research groups for estimates of gain (or amplitude frequency response) and phase in healthy human subjects (4, 7, 10, 19, 27–29, 37, 40). One of the main features of these frequency domain functions is the positive values of phase for frequencies below 0.1 Hz. In this frequency range, the gain is relatively low but tends to rise continuously at higher frequencies, suggesting that dynamic CA can be regarded as a kind of high-pass filter (7).

One intrinsic problem with TFA in physiology is the inherent assumption of linearity and the difficulty of performing measurements that are free of noise or without the interference of covariates. The univariate coherence function between the input and output has been used to address these problems by measuring the fraction of output power that can be linearly explained by the input variable at each frequency (3). Estimates of coherence vary between 0 and 1.0. A value of 1.0 implies perfect linear coupling between input and output, whereas a value of 0 indicates an output that is linearly independent from the input. In applications of TFA to human studies of dynamic CA, the coherence function has served two main purposes. First, it has been used to select estimates of gain and phase between CBFV and ABP for further analyses based on the statistical significance of their coupling. Second, it has been used in clinical studies as an indicator of the integrity of dynamic CA in its own right. The rationale behind the latter, as proposed by Giller (13), is that a working dynamic CA should lead to uncoupling between CBFV and ABP and hence to low values of coherence. On the other hand, in the absence of autoregulation, changes in ABP will be closely followed by changes in CBF and coherence will tend toward 1.0. These two perspectives on the interpretation and use of the coherence function are contradictory and can lead to considerable bias in physiological and clinical studies. With active autoregulation, values of coherence will be low, leading to the rejection of estimates of gain and phase in the region of interest where dynamic CA is most efficient, that is, below 0.05 Hz (4, 7, 10, 19, 29, 37). To avoid this problem, several investigators have performed their analyses at frequencies above 0.05 Hz, where dynamic CA is less efficient and values of gain and phase might have reduced sensitivity to detect deterioration of dynamic CA in pathological conditions, such as carotid artery disease (15, 17, 25, 35). Others (21) have argued that the low values of coherence simply invalidate any attempts to describe dynamic CA as a time-invariant linear system.

Rejections of estimates of dynamic CA gain and/or phase due to low values of coherence are usually based on the assumption of a low signal-to-noise ratio and tend to ignore the fact that coherence can also be low due to the contribution of multiple inputs or the presence of nonlinearities (3). In this study, we tested the hypothesis that, if the inherent nonlinearity of dynamic CA that is due to changes in cerebrovascular resistance is taken into account, then the values of coherence are much higher than previously estimated in the frequency range below 0.05 Hz.

## METHODS

#### Subjects.

Data from healthy subjects aged ≤40 yr were obtained from several previous studies of CA where recordings were performed under standardized conditions (5, 29, 31, 34). None of the subjects had a history of cardiovascular, hypertensive, autonomic nervous system, or neurological disease. All subjects gave fully informed, written consent, and all studies were approved by the local Leicestershire Research Ethics Committee.

#### Measurements.

Subjects avoided caffeine, nicotine, and alcohol for 12 h before the CA recordings were performed and were studied at least 2 h postprandial. Studies were conducted in a dedicated research room kept at a constant temperature (20–24°C) with external stimuli minimized. Subjects were in the supine position with the head elevated at 15°.

The middle cerebral arteries (MCA) were insonated bilaterally as described by Aaslid et al. (1), and CBFV was measured by using TCD (SciMed QVL 842X; SciMed, Bristol, UK). A three-lead surface ECG was fitted, and beat-to-beat ABP was measured noninvasively by a servo-controlled plethysmograph (Finapres 2300; Ohmeda, Englewood, CO) on the middle finger of the left hand, supported at midaxillary level throughout the studies. In most subjects, arterial Pco_{2} (Pa_{CO2}) was monitored transcutaneously (TINA, Radiometer, Copenhagen, Denmark). In a subgroup of subjects, Pa_{CO2} was monitored via a closely fitting face mask and an infrared capnograph (Capnogard, Novametrix Medical Systems, Wallingford, CT).

A 10-min recording was made with the patient breathing spontaneously. The TCD, Finapres BP, ECG, and CO_{2} output signals were continuously recorded on digital tape (DAT, Sony PC-108M).

#### Data analysis.

Data recorded on tape were downloaded onto a microcomputer in real time. A fast Fourier transform (FFT) was used to extract the maximum frequency velocity envelope with temporal resolution of 5 ms. The ABP, ECG, and CO_{2} signals were sampled at a rate of 200 samples/s, and ABP was calibrated at the start of each recording. All signals were visually inspected to identify artifacts or noise, and narrow spikes were removed by linear interpolation. The two CBFV signals were subjected to a median filter with a window width of five samples, and all signals were low-pass filtered by a zero-phase Butterworth filter with a cutoff frequency of 20 Hz.

The beginning and end of each cardiac cycle were detected on the ECG, and mean beat-to-beat values were calculated for the two CBFV channels, ABP, and heart rate. An index of cerebrovascular resistance (CVRI) was estimated by the ratio of mean ABP to mean CBFV for each heartbeat for both MCAs (10). A second estimate of dynamic changes in CVRI that is independent of mean values of ABP and CBFV was obtained from the instantaneous relationship between ABP and CBFV from the slope of the linear regression, CBFV = *a*·ABP + *b*, as described previously (34). For each cardiac cycle, the inverse of the linear regression slope is referred to as the “resistance-area product” (RAP = 1/*a*) to differentiate it from CVRI (12). All beat-to-beat estimates were interpolated by using a third-order polynomial and resampled at 0.2-s intervals to generate a time series with a uniform time base (29).

#### Ordinary and multiple coherences.

For spontaneous fluctuations in ABP and CBFV, the respective frequency domain transforms *P*(*f*) and *V*(*f*) were computed with the FFT algorithm. The auto- and cross-spectra were calculated as (1) and (2) where the expected value of the complex product *E* [] was obtained by smoothing the spectra with a triangular window and averaging eight segments of data with 512 samples (102 s) each.

In conventional TFA, the ordinary (or univariate) coherence between ABP and CBFV is calculated as (3) (3) where *G*_{VV}(*f*) is the autospectra of CBFV. In this case, the corresponding transfer function between CBFV and ABP is given by (4) To take into account the nonlinear contribution of resistance, we express CBFV as (5) where *R* can represent either the CVRI or the RAP parameters described above.

Because of the spontaneous fluctuations in ABP, CBFV and *R* will also fluctuate around mean values *CBFV*_{0} and *R*_{0}, leading to (6) where ABP_{0} is the mean value of ABP and Δ*v*, Δ*p*, and Δ*r* represent small changes in CBFV, ABP, and *R*, respectively. Assuming that Δ*r* ≪ *R*_{0}, (7) Substituting in *Eq. 6*: (8) Neglecting second-order products (Δr·Δp ≈ 0) and using *Eq. 5*: (9) *Equation 9* shows that for small changes in resistance, as observed during spontaneous fluctuations of ABP and CBFV, the intrinsic nonlinearity resulting from a time-varying resistance can be approximated as a second, separate contribution to changes in velocity. *Equation 9* can be written in the frequency domain as (3) (10) where *P*(*f*) and *R*(*f*) are the FFT transforms of Δ*p*(*t*) and Δ*r*(*t*), respectively, and *H*_{PV}(*f*) and *H*_{RV}(*f*) are their partial transfer functions as shown in Fig. 1. From *Eq. 9*, (11) (12) The appendix shows that, for a system with multiple inputs, the total, or multiple coherence, depends not only on the partial transfer functions but also on the coupling between ABP and resistance, as reflected by a third transfer function, *H*_{PR}(*f*) (Fig. 1), which can be defined as (13) With the use of the two partial transfer functions and the corresponding cross-spectra, it is then possible to obtain an expression for the multiple coherence of CBFV, γ^{2}_{M}(*f*), which takes into account the contributions of both ABP and resistance (3). Estimates of univariate and multiple coherences as well as the three transfer functions [*H*_{PV}(*f*), *H*_{RV}(*f*), and *H*_{PR}(*f*)] were obtained by using either CVRI or RAP as the resistance parameter and ABP as the driving pressure. For simplicity, rather than looking separately at the gain and phase frequency responses, we expressed the transfer functions by the corresponding time domain step responses of which the temporal patterns reflect the contributions of both amplitude and phase (3). Rather than adopting the theoretical values expected for the partial transfer functions (*Eqs. 11* and *12*), these functions were estimated as described in the appendix, and the corresponding step responses were normalized by the expected theoretical values.

To test the consistency and quality of the estimated transfer functions, the classical univariate transfer function between ABP and CBFV was recalculated according to the cascade represented in Fig. 1 as *H*_{PV}(*f*) + *H*_{PR}(*f*)·*H*_{RV}(*f*), and the corresponding step response was obtained with the inverse FFT transform.

#### Statistics.

Results are presented as means ± SD, except in ⇓Figs. 3–7 where error bars are plotted as ± SE for the sake of clarity. The variability of ABP was assessed by the coefficient of variation given by percent standard deviation divided by the mean for all values of mean ABP in each recording. A surrogate data analysis procedure was adopted to estimate the 95% confidence limits of null univariate and multiple coherences. For each data file, the sample values of the input signal(s) were shuffled randomly, thus destroying its coherence with the output. This procedure was repeated 250 times for each subject, generating >10,000 coherence estimates. From its probability distribution, the 95% confidence limit was obtained by numerical methods. This method has the advantage of maintaining the statistical distribution of the original data. The surrogate data shuffling procedure was also used to obtain multiple coherences by using the original ABP signals, but shuffled RAP inputs, to test the effect of simply adding a second independent input. Individual estimates of univariate and multiple coherence and the step responses corresponding to the transfer functions *H*_{PV}(*f*), *H*_{RV}(*f*) and *H*_{PR}(*f*) were averaged for the entire population, and the standard deviation was also calculated for each frequency bin or time interval. Differences between estimates obtained for different selections of input and output variables were assessed with Student’s *t*-test. A value of *P* < 0.05 was considered significant.

## RESULTS

A total of 46 data files were available for analysis, but four subjects were removed due to poor quality signals in at least one CBFV channel. The 42 remaining subjects (20 males) were aged between 20 and 40 yr (27.8 ± 4.6 yr, mean ± SD), with clinical ABP of 121 ± 15/73 ± 11 mmHg. Their baseline values for noninvasive ABP and the other values are given in Table 1. None of the variables considered was significantly different for the right- and left-side recordings. Both end-tidal and transcutaneous estimates of Pa_{CO2} remained stable in all subjects. The coefficient of variation for mean ABP ranged from 2.8% to 9.1%, with 14 subjects showing values above 5%. Very similar results were obtained for the right and left MCA for all analyses performed.

The 95% confidence limits for null univariate and multiple coherences, estimated by randomizing the input signal(s), were 0.44 and 0.60, respectively. These limits are shown in Fig. 2, with the population mean univariate coherence between ABP and CBFV and the multiple coherences for the ABP + RAP and ABP + CVRI inputs. Also represented is the multiple coherence for the ABP input combined with the randomized RAP input. The mean univariate coherence for ABP and CBFV is below the 95% confidence limit for frequencies <0.06 Hz, and the mean multiple coherence with randomized RAP input is also below the corresponding 95% confidence limit in the same frequency range (Fig. 2). On the other hand, the mean multiple coherences for ABP + RAP and ABP + CVRI inputs are both above the 95% confidence limit. The reasons for the latter approaching 1.0 are discussed in *Coherence functions*. In the frequency region <0.05 Hz, each coherence function in Fig. 2 is significantly different (*P* < 0.0001) from the next function above or below it.

The classical CBFV step response to a single ABP input showed an initial sudden rise in CBFV followed by a rapid return to lower values characteristic of a working CA (Fig. 3). The step responses corresponding to the partial transfer function *H*_{PV}(*f*) are significantly different (*P* < 10^{−5}). For the combination of ABP + CVRI inputs, the CBFV step response approximates a perfect step, showing very good agreement with the theoretical value predicted by *Eq. 11* (Fig. 3). The corresponding step response for the RAP input in combination with ABP is not as similar to a perfect step (Fig. 3), but it is significantly greater than the univariate step response for *t* > 3 s (*P* < 10^{−5}).

Step responses for the *H*_{RV}(*f*) transfer function are shown in Fig. 4. Again, the best approximation to the expected theoretical value given by *Eq. 12* was obtained with the ABP + CVRI inputs. The ABP + RAP combination led to a significantly lower step response.

The gain, phase, and step responses corresponding to the *H*_{PR}(*f*) transfer function are represented in Figs. 5 and 6. This function describes the coupling between ABP and RAP or CVRI (Fig. 1). For both the RAP and CVRI outputs, the gain is fairly constant for frequencies <0.25 Hz (Fig. 5*A*), whereas the phases are consistently negative (Fig. 5*B*). For the CVRI step response to an ABP input, the curve shows a gradual rise during ∼3 s, to reach a plateau, in good agreement with the value predicted by *Eq. 9* for a working autoregulation (Fig. 6). The corresponding curve for the RAP response to the ABP input rises slower but stabilizes at approximately the same value as the CVRI response (Fig. 6).

The reconstruction of the univariate CBFV step response to the ABP input based on the partial transfer functions and the *H*_{PR}(*f*) transfer function, according to the cascade represented in Fig. 1, led to very good approximations as depicted in Fig. 7 for both the ABP + CVRI and ABP + RAP inputs.

## DISCUSSION

Several previous studies have shown that dynamic CA is not influenced by aging (5, 22, 29, 39). However, preliminary analysis of our database indicated that RAP and CVRI showed an increase for subjects above 50 yr old, and for this reason we limited the analysis to healthy subjects that were 40 yr old or less. Although these data were acquired as part of different studies in the period 2000–2005, the environmental conditions, equipment, and measurement procedures were exactly the same for all subjects.

#### Coherence functions.

From the results presented, it is clear that the main hypothesis of the study has been confirmed. In the frequency range <0.06 Hz, the standard univariate coherence was below the 95% confidence limit for null coherence, suggesting lack of linear coupling between input and output as claimed by several previous studies. However, when the nonlinear contribution of cerebrovascular (or RAP) resistance is taken into account, for small spontaneous fluctuations, CBFV is determined by two separate inputs, and in this case, the multiple coherence shows much higher values, being well above the 95% confidence limit for frequencies up to 0.25 Hz (Fig. 2). It is well known that multiple coherence is always greater than the univariate coherence. In fact, it can be demonstrated that for two independent inputs, the multiple coherence is the sum of the individual univariate coherences (3). For this reason, we have randomized the RAP input signal, making it independent of ABP and CBFV, to test the extent to which the increase in multiple coherence was due to the simple addition of a second input. The results shown in Fig. 2 confirm that the multiple coherence does increase; however, in this case, the result is nonsignificant, because most values lie below the 95% confidence limit for frequencies below 0.06 Hz (Fig. 2). Taken together, these results confirm that the ABP + RAP inputs lead to a multiple coherence function that is statistically significant and also significantly greater than the univariate coherence. Nevertheless, some of the results shown need to be properly interpreted to avoid misunderstandings. This applies mainly to the multiple coherence values approaching 1.0 for the ABP + CVRI inputs. Because CVRI is defined by the ratio between ABP and CBFV, this parameter cannot be seen as independent from the mean beat-to-beat values of ABP and CBFV, and hence it is not surprising that the corresponding multiple coherence function is very close to 1.0. Although this multiple coherence does not contribute to a better understanding of the dynamic determinants of CBFV, it provides reassurance that the computational methods employed lead to a very good approximation of the expected theoretical value. On the other hand, the result that really confirms our main hypothesis is the highly significant increase in multiple coherence for the ABP + RAP inputs in the frequency region below 0.05 Hz, in comparison to the standard univariate coherence between ABP and CBFV (Fig. 2). This conclusion follows from the complete independence of beat-to-beat estimates of RAP from the corresponding same beat estimates of mean ABP and CBFV. As described in methods, RAP is derived as the linear regression slope of the instantaneous pressure-velocity relationship for each cardiac cycle. For a given cardiac cycle, the mean [ABP, CBFV] pair represents a single point on the regression line, and this point can be seen as the fulcrum around which it is possible to rotate the regression line to obtain an infinite number of slopes and hence RAP values. In summary, the independence of fluctuations in RAP from corresponding changes in ABP and CBFV is demonstrated by the possibility of an infinite number of RAP values for each mean [ABP, CBFV] pair. For the ABP + RAP inputs, the population mean multiple coherence is significantly less than unity but reaches values >0.7 for frequencies <0.05 Hz, and this is much higher than the typical values of ∼0.4, or less, that we and others have found for the standard coherence between ABP and CBFV (10, 14, 18, 29, 35, 37, 40, 41).

The high values of coherence for frequencies below 0.05 Hz in healthy subjects do not invalidate the hypothesis of Giller (13) that the standard ABP-CBFV univariate coherence would rise in the very low frequency region in subjects with impaired CA. Under these circumstances, what will happen is a reduction in the contribution of RAP, which can be seen as a reduction of the nonlinear component of dynamic CA. As a consequence, a higher fraction of CBFV variability will be directly explained by the ABP input and the univariate coherence will rise accordingly. The main implication of our findings, though, is to send a strong message toward the way the coherence function is used in TFA studies of dynamic CA. The results in Fig. 2 clearly indicate that it is not possible to dismiss spectral estimates of amplitude and phase for frequencies where univariate coherence is less than a preestablished threshold on grounds of poor statistical significance. Such procedure should only be applied to the total, multiple coherence, e.g., for the ABP + RAP inputs. Hitherto, thresholds of univariate coherence of ∼0.4–0.5 have been used by most investigators to accept spectral estimates at selected frequency bands (4, 7, 10, 15, 19, 35, 40). Different methods have been used to estimate confidence limits for univariate and multiple coherence functions (23). In many cases, Monte Carlo simulations have been performed by using Gaussian noise to obtain the distribution of coherence under the null hypothesis. We preferred to use the random shuffling of the input signals to maintain the original signal record duration and statistical distribution. The resulting 95% confidence limits of 0.44 for univariate and 0.60 for the multiple coherence compare well with the theoretical 95% confidence limits derived by Miranda de Sá et al. (23) of 0.46 and 0.66, respectively, when using eight segments of data for each estimate.

#### Limitations of study.

Addressing this point first will help with the interpretation of some of the step responses shown in Figs. 3–6. The main limitations of this and similar studies of dynamic CA in humans are due to the noninvasive measurement techniques employed. Transcranial Doppler ultrasound can provide measurements of CBFV with high temporal resolution but not absolute blood flow. Changes in CBFV will only reflect changes in flow if the artery of interest, e.g., MCA, is properly insonated and if its diameter remains approximately constant. Significant changes in MCA cross-sectional area have not been observed during relatively large changes in mean ABP and Pa_{CO2} (38) and therefore are less likely to be manifested at rest when much smaller fluctuations of these variables are taking place. The problem with appropriate insonation, though, is more of a concern in our case. Continuous good quality recordings of CBFV are not universally feasible due to anatomical differences in skull thickness and position of the temporal acoustic window (1). Inadequate insonation of the MCA will translate as artifacts and noise in the CBFV signal, and this will tend to depress estimates of coherence for both the univariate and multiple cases. For this reason, we have visually inspected all recordings and have rejected four subjects with excessive noise in at least one of the CBFV channels. The univariate coherence between ABP and CBFV (Fig. 2*A*) is in excellent agreement with what has been reported by many investigators (10, 14, 18, 29, 35, 37, 40, 41), and there is no reason to suspect that its lower values, in comparison with the multiple coherence curves, might be due to excessive noise in the CBFV signals.

Similarly to most other studies of dynamic CA in humans, we have performed noninvasive measurements of ABP with arterial volume clamping in the finger. It is generally accepted that with careful positioning of the cuff transducer, the Finapres device can provide beat-to-beat estimates of mean ABP that closely follow changes in systemic ABP (16). In our case, though, the main problem lies with estimates of RAP, which, to a certain extent, depend on the ABP waveform. As the pulse wave propagates from the ascending aorta to the finger artery, the ABP waveforms changes its temporal pattern, and it is not clear how much this can affect estimates of RAP. Our approach to minimize this uncertainty has been through careful visual scrutiny of estimated values together with the use of large samples to control for variability. Accordingly, we have averaged eight segments of data in each individual spectral estimate and have also averaged results from a relatively large sample of 42 healthy subjects. Although our estimates of RAP seem to have led to consistent results in most cases, there are instances where distortions might have played a part, as, for example, in the low amplitude of the step response corresponding to the *H*_{RV}(*f*) transfer function in Fig. 4. The extensive use of the Finapres and related devices in studies of dynamic CA warrants more research into the reliability of estimates of RAP to allow more sophisticated models of CBF regulation (34).

The approximation involved in the derivation of *Eq. 9* is valid if spontaneous changes in mean ABP, CBFV, and resistance (CVRI or RAP) are relatively small. Changes in mean CBFV and resistance result from changes in mean ABP, and the distribution of its coefficient of variation confirmed that ABP variability remained within acceptable limits. The step response reconstruction shown in Fig. 7 also works as a kind of “check sum” of the validity of the approximations involved in *Eq. 9*.

Another limitation that should be considered is the potential contribution of spontaneous variability of Pa_{CO2} to the multiple coherence of CBFV. Previous studies (9, 24, 30) have shown that short-term fluctuations in CBFV can be partially explained by breath-by-breath changes in end-tidal CO_{2}. From these, it could be expected that including Pa_{CO2} as a third input variable would lead to higher multiple coherences in relation to having only the ABP + RAP inputs. End-tidal CO_{2} was only measured in a subgroup of our population; in the remainder, Pa_{CO2} was monitored transcutaneously, and the poor temporal resolution of this technique precluded any attempts of dynamic modeling. However, even if we had end-tidal measurements in all our subjects, it would be a hard task to obtain frequency domain estimates of multiple coherence with three inputs due to the need to include another three transfer functions to take into account the coupling between Pa_{CO2}, CBFV, and the other input variables. Future work to increase the dimensionality of inputs is likely to be more successful if time domain multivariate modeling (9, 24, 30) is used instead of the frequency domain approach that we have adopted.

Finally, we used ABP as an approximation for cerebral perfusion pressure, which is acceptable for subjects with low values of intracranial pressure (ICP), as should be expected in this group of healthy young subjects. However, in situations where ICP might be elevated and highly variable, as is often observed in patients with severe head injury, ICP is likely to influence cerebral blood flow variability, and, therefore, it might have a strong effect on the multiple coherence of CBFV.

#### Step responses.

The classical CBFV step response for changes in ABP (Fig. 3) is in very good agreement with previous estimates (9, 11, 29–31), including studies where ABP was recorded with intravascular catheters (27, 32). One interesting consequence of expressing dynamic CA with the linearized model in Fig. 1 and *Eq. 9* is the different perspective it brings to the interaction of the main determinants of CBFV. Rigorously, the standard linear univariate transfer function between ABP and CBFV is only valid if the cerebrovascular resistance parameter in *Eq. 5* is constant. If we accept that cerebrovascular resistance is allowed to change due to vasomotor activity, then *Eq. 9* is the best linear approximation, although other more complex nonlinear models could also be considered (24, 29). What *Eq. 9* is telling us, though, is that the immediate dependence of CBFV on ABP (*H*_{PV}) and RAP or CVRI (*H*_{RV}) is entirely passive, all regulatory mechanisms being reflected by the relationship between ABP and RAP, i.e., *H*_{PR}(*f*). This line of reasoning sheds a different light on previous attempts to model the ABP-CVRI relationship as proposed by Edwards et al. (10, 11). Moreover, the passive transfer functions also give a much more consistent description of cerebrovascular physiology. Accordingly, the effect of a step change in ABP is simply to increase CBF, assuming that no compliance or inertial effects are included. After the ABP change, autoregulation will attempt to restore CBF to its original value, but this takes place due to the change in resistance predicted by transfer function *H*_{PR}(*f*) (Fig. 1). Again, if we only envisage the effect of resistance on CBF, i.e., *H*_{RV}(*f*), the expected effect is purely passive, with flow varying inversely with resistance as predicted by Poiseuille’s law. On purpose, we did not compute the multiple coherence based on the idealized partial transfer functions (*Eqs. 11* and *12*) but let their patterns and amplitudes be determined by the solution of *Eqs. A8* and *A9* to gain additional information about the validity of the approximations leading to *Eq. 9*. The resulting step responses in Figs. 3 and 4 are extremely encouraging. With the ABP + CVRI inputs, the step responses in Figs. 3 and 4 show a very good approximation to a perfect step with amplitude 1/*R*_{0} when normalized by the mean value of CVRI of the entire population. The results obtained for the RAP input were not as good (Fig. 3). In Fig. 3, the step response for the ABP + RAP inputs is significantly greater than the univariate step responses for *t* > 3 s, showing a tendency toward a flat response that does not reflect a marked reduction in CBFV due to autoregulation. In Fig. 4, the CBFV step response to changes in RAP is considerably lower than the expected theoretical value. On the other hand, the ABP-RAP step response (Fig. 6) shows a much better agreement in relation to the expected plateau value. Although these large differences in relation to expected values (*Eq. 9*) could be attributed to the aforementioned limitations of deriving RAP from Finapres measurements, we believe that they are more likely to arise from an oversimplification of the linearized model, whereby feedback effects of CBFV on cerebrovascular resistance are not included (Fig. 1). In other words, the model reflects myogenic regulatory mechanisms but not regulation of metabolic origin (26, 34). To take the possibility of myogenic regulation into account, it would be necessary to extend the model represented by Fig. 1 to include feedback loops allowing for the influence of CBFV on RAP (or CVRI) and possibly on critical closing pressure as well (26, 34).

As stated above, in the linearized model, all information about the dynamic autoregulatory response is concentrated on the *H*_{PR}(*f*) transfer function (Fig. 1). The phase frequency responses for both indicators of resistance (RAP or CVRI) were consistently negative (Fig. 5*B*), confirming that changes in resistance are causally dependent on preceding changes in ABP. The corresponding step responses in Fig. 6 show that resistance responds relatively fast to an idealized step change in ABP. The CVRI response is in very good agreement with estimates previously reported by Edwards et al. (10, 11). Finally, the results of reconstructing the univariate ABP-CBFV step response from the other three transfer functions represented in Fig. 1 provides additional evidence of the consistency of the computational procedures, but it cannot be regarded as evidence of the accuracy of intermediate functions as exemplified by the distortions of the RAP partial transfer functions mentioned above.

In conclusion, we have shown that the standard univariate coherence function between ABP and CBFV fails to take into account the inherent nonlinearity of dynamic CA as reflected by changes in cerebrovascular resistance. For this reason, univariate coherence should be replaced by estimates of multiple coherence to assess the statistical significance of input-output relationships in the frequency domain. The choice of which parameter to adopt as an indicator of cerebral vasomotor activity is still overshadowed by limitations in continuous noninvasive measurements of ABP. CVRI is less prone to artifacts, but its dependence on ABP and CBFV does not allow informative estimates of multiple coherence, and for this reason RAP should be preferred for this purpose. On the other hand, CVRI provides more reliable estimates of the ABP-resistance transfer function, which, according to the linearized model, concentrates all the information about the regulatory mechanisms influencing CBFV. Future work along these lines should explore the potential contribution of other variables, such as ICP and breath-to-breath fluctuations in CO_{2}, to the multiple coherence of CBFV.

## APPENDIX

From Fig. 1, the measured output differs from the true CBFV, *v*(*t*), by the presence of noise, or (A1)

Assuming that the noise is uncorrelated to *v*(*t*), the autospectrum of *y*(*t*) can be calculated from *Eq. 1*, leading to (A2) The multiple coherence function is defined as the ratio of the ideal output spectrum due to the measured inputs, in the absence of noise, to the total output spectrum *G*_{yy}(*f*), which includes the noise term (3) (A3) From *Eq. 10*, *G*_{yy}(*f*) can be estimated as (A4) If the noise is uncorrelated with either input, these are the results: (A5) where the different auto- and cross-spectra were calculated as in *Eqs. 1* and *2*.

To estimate the partial transfer functions linking the inputs to *v*(*t*) (Fig. 1), we calculated the cross-spectra of the measured output *y*(*t*) with each input. Assuming the noise is uncorrelated to each input leads to (3) (A6) (A7) and the partial transfer functions for *P*(*f*) and *R*(*f*) can then be estimated as (A8) (A9) We found that *Eqs. A8* and *A9* provided more stable estimates than those of the corresponding formulation suggested by Bendat and Piersol (3) that depends on the estimation of γ^{2}_{PR}(*f*).

Once all the auto- and cross-spectra were calculated, *H*_{PV}(*f*) and *H*_{RV}(*f*) were obtained from *Eqs. A8* and *A9* and then substituted into *Eq. A5* to determine *G*_{yy}(*f*) and *G*_{VV}(*f*). From these, the multiple coherence was calculated with *Eq. A3.*

## Acknowledgments

We thank David H. Evans and Lingke Fan for development of the FFT Doppler analyzer and Michelle Moody for collection of a subset of data.

## Footnotes

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- Copyright © 2006 by the American Physiological Society