## Abstract

We examined the validity and usefulness of a low-pass filter (LP_{FILTER}) to reduce point-to-point variability and enhance parameter estimation of the kinetics of blood flow (BF). Computer simulations were used to determine the power spectrum of simulated responses. Moreover, we studied the leg BF response to a single transition in four subjects during supine knee-extension exercise using three methods of data processing [beat-by-beat, average of 3 cardiac cycles (AVG_{3 BEATS}), and LP_{FILTER}]. The power spectrum of BF containing the kinetics information (≤0.2 Hz) did not overlap with the oscillations due to muscle contraction and cardiac cycle (simulations and Doppler measurements). There were no significant differences between the parameter estimates for a two-exponential model using Beat-by-Beat, AVG_{3 BEATS}, and LP_{FILTER} (*P* > 0.05; *n* = 4). However, LP_{FILTER} (cutoff = 0.2 Hz) resulted in a significantly lower standard error of the estimate for all parameters (*P* < 0.05). The means ± SD for the standard error of the estimate for Beat-by-Beat, AVG_{3 BEATS}, and LP_{FILTER} were, respectively, time constant-*phase 1* = 5.0 ± 1.1 s, 4.5 ± 2.1 s, and 0.3 ± 0.2 s; time delay-*phase 2* = 17.8 ± 7.9 s, 12.8 ± 7.5 s, and 1.4 ± 1.4 s; time constant-*phase 2* = 15.8 ± 4.6 s, 9.9 ± 2.9 s, and 1.1 ± 0.5 s. In conclusion, LP_{FILTER} appeared to be a valid procedure providing a high signal-to-noise ratio and data density and thus LP_{FILTER} resulted in the smallest confidence interval for parameter estimates of BF kinetics.

- low-pass filter
- Doppler ultrasound
- signal processing
- knee-extension exercise

the redundancy of processes associated with the muscle blood flow response to exercise (2, 10) has led to a growing interest in the kinetics across the rest-to-exercise transition. In this context, Doppler ultrasound has been used to measure the dynamics of blood flow response in humans (19) and animals (1). Despite good signal acquisition (i.e., little measurement error), these measurements are characteristically “noisy” during exercise because of cardiac cycle and contraction-induced oscillations in blood flow (11) (see Fig. 1). This is relevant for data interpretation because, to make physiological inferences from kinetic parameters and to compare the effects of interventions on control processes, it is paramount that the confidence interval (CI) for parameter estimation be as narrow as possible. In general, the signal-to-noise ratio and density of data across the transition phase (see discussion) are the main factors affecting the confidence of parameter estimation of kinetics responses (9).

The kinetics of blood flow are characterized by a fast increase (*phase 1*) followed by a slower response, beginning ∼10–20 s after the onset of exercise (*phase 2*) (for review, see Ref. 20). This biphasic characteristic and rapid early response increases the sensitivity of parameter estimation (and the CI) to the method used for kinetics analysis. Several methods have been used to preprocess the raw data [e.g., beat-by-beat (5) and contraction-relaxation cycle (18)]. Beat-by-beat data have a temporal resolution that seems adequate for determining blood flow kinetics; however, the low signal-to-noise ratio leads to unacceptably high CIs for the resulting parameter estimates. Conversely, averaging the blood flow response over a contraction-relaxation cycle will improve the signal-to-noise ratio, but with the undesirable effect of reducing the density of data across the transition phase, which might prevent adequate determination of the fast responding processes (*phase 1*). Ensemble averaging of multiple exercise replicates has also been used to improve the signal-to-noise ratio and accentuate the kinetic features (e.g., Refs. 7, 12, 18). However, to achieve acceptable CIs for noisy responses such as blood flow, it may be necessary to ensemble average an unreasonable number of transitions (see discussion) (8). This would severely limit the utility of this approach in scenarios such as patient evaluation in which only a limited number of transitions are feasible.

To date, there is no method that achieves an optimal balance between signal-to-noise ratio and data density to accentuate the underlying kinetics of blood flow and, consequently, provide narrow CIs of parameter estimates. In this context, filtering the blood flow data in the frequency domain might be an alternate option to time-domain filtering (beat-by-beat and contraction-relaxation cycle); however, this approach requires minimal overlap between the frequency spectrum of the underlying kinetics response and the main sources of “noise” in blood flow. To the best of our knowledge, the frequency-domain characteristics of the oscillatory components of blood flow and the underlying kinetic response have not been determined.

The purpose of the present study was to examine the validity and usefulness of filtering blood flow data in the frequency-domain before characterizing the kinetic response. We hypothesized that the main sources of noise in blood flow during exercise would be muscle contraction and cardiac cycle (11, 16, 25) (for example see Fig. 1), and that the frequency content of these oscillations would not overlap with the frequency range of the slower processes determining the underlying kinetics response because, in general, contraction frequency and heart rate are high-frequency processes. This would permit the application of a low-pass filter to the raw Doppler blood flow data in which the higher frequencies associated with muscle contraction and cardiac cycle would be eliminated. The advantage of this filtering procedure would be a lower standard error of the estimate for the parameters of frequency-domain filtered data compared with those obtained with time-domain methods, representing a substantial improvement in the CI for each kinetic parameter.

## METHODS

#### Computer simulations.

The dynamic (and biphasic) increase in blood flow after the onset of exercise was simulated with exponential equations. The default parameter values for simulations were baseline = 0.5 arbitrary units (AU), *A*_{1} = 2 AU, *A*_{2} = 3 AU, τ_{1} = 2 s, TD_{2} = 15 s, and τ_{2} = 15 s, where *A* is amplitude, τ is time constant, TD is time delay, and 1 and 2 are phases of the blood flow response. (see *Eq. 1*), similar to a previous computer modeling study (4). For simulations of responses without noise the physiologically relevant parameter values are *A*_{1} as percentage of total amplitude (*A*_{tot} = *A*_{1} + *A*_{2}), τ_{1}, TD_{2}, and τ_{2}. For these parameters our default values are in agreement with results observed in healthy subjects (12, 16, 18).

The oscillations in leg blood flow (LBF) superimposed on the underlying kinetics response were simulated by a sinusoidal function [*y*(*t*) = *A*·sin(360 *t*/*T*)], where *A* is mean-to-peak amplitude and *T* is period of oscillations (i.e., *T* = 1/*f* where *f* is frequency of oscillations). The known sources of oscillations at rest and exercise are cardiac frequencies (16), very-low-frequency oscillations (26), and muscle contraction (11, 16, 25). For both heart rate (100 beats/min, *f* = 1.67 Hz) and muscle contraction (40 min^{−1}, *f* = 0.67 Hz), we simulated *A* = 2 AU, which represented oscillations equal to 80% of the simulated increase in blood flow from rest to exercise. The mean-to-peak amplitude of the very-low-frequency oscillations (6 min^{−1}, *f* = 0.1 Hz) was arbitrarily set as 0.5 AU. In addition, we simulated the existence of random noise (range −0.75 to 0.75 AU) on blood flow measurements. The oscillatory components and random noise were added to the underlying simulated biphasic increase in “mean” blood flow to qualitatively approximate the raw data of Doppler ultrasound measurements during rest and exercise. It is important to note that a constant heart rate was simulated for the rest-to-exercise transition. Clearly, these simulations are an oversimplification of the real blood flow response; however, they permit examination of the effects of known sources of noise on the time- and frequency-domain characteristics of LBF response to exercise.

#### Human studies.

The experiments were conducted in four healthy individuals (2 men and 2 women, age 28 ± 9.0 yr and body mass 65.8 ± 13.8 kg). The protocol was explained in detail to each subject, who signed a consent form after reading a description of the study. The subjects were familiarized with the exercise protocol from participation in previous studies in the laboratory. The study was approved by the Institutional Review Board for Research Involving Humans Subjects at Kansas State University.

The protocol consisted of supine single-leg knee-extension exercise (2 min rest + 6 min exercise) performed at a light-to-moderate work rate so as to achieve a steady state of cardiovascular variables. The knee-extension exercise was performed at a contraction rate of 40 min^{−1}. To maintain the desired cadence, the subjects were aided by the signal from a metronome, which was started at the beginning of the resting period to avoid anticipatory responses. Subjects performed one rest-to-exercise transition. One subject returned to the laboratory on 2 different days (in addition to the initial visit) to complete several transitions to the same work rate as performed on the first day to establish reproducibility. In two visits the subject performed two transitions, with 25–30 min of recovery between each transition, yielding a total of five exercise bouts (3 days) at the same work rate.

Femoral artery blood velocity (BV) was measured by pulsed-wave Doppler ultrasound (model 500V, Multigon Industries) using a 4-MHz probe with fixed angle of insonation of 45° and full gate width. The Doppler probe was placed above the common femoral artery of the right (exercising) leg ∼2–3 cm proximal to the femoral artery bifurcation. The probe position was determined by echo-Doppler ultrasound. The audio signal was processed and converted to velocity as described in detail by Lutjemeier et al. (11). The data were collected at 200 Hz and stored for offline analysis. To calculate beat-by-beat BV (and flow), the 200-Hz data were integrated over each R-R interval of the ECG determined by a modified lead I configuration. Femoral artery diameter was measured at rest by echo-Doppler ultrasound (Vivid 3-Pro, GE) with a 7.5-MHz linear probe. Previous studies have shown that femoral artery diameter did not change significantly during exercise (3, 12, 15). Thus resting diameter was used to calculate LBF, where LBF = BV·π*r*^{2} (BV in cm/s and *r* is femoral artery radius in cm) and was multiplied by 0.06 to convert from milliliters per second to liters per minute.

#### Spectral analysis and filtering.

A fast Fourier transform [Cooley-Tukey's algorithm, Hanning window (27)] was used to examine the frequency domain (spectral) characteristics of the simulated and directly measured blood flow response to exercise. Spectral analyses were done on data sets with 15 s of rest and 180 s of exercise (rest-to-exercise transition). This time window included ∼90–120 s of “steady state.” The Fourier transform assumes stationary data, and inclusion of the dynamic increase in blood flow violates this assumption. However, this analysis was important to establish the range of spectral frequencies containing information on the biphasic kinetics of blood flow (see results) and to determine the cutoff frequency for filtering the data.

The spectral analysis of blood flow responses (computer simulations and in vivo measurements) demonstrated that the frequency range containing the kinetics information did not overlap with higher frequency noise such as those induced by heart rate and muscle contraction (see results). Thus a frequency-domain filtering procedure was used to eliminate the higher frequency noise (i.e., low-pass filter). The filtering was done using a default low-pass function of SigmaPlot (lowpass.xfm, SigmaPlot 2001, Systat Software). The cutoff for filtering was titrated to produce the lowest noise while retaining data with frequencies fundamental for describing the biphasic kinetics of blood flow. To examine the effects of filtering on the underlying kinetics response, we applied the low-pass filter (LP_{FILTER}) on computer-simulated data, filtering out frequencies ≥0.1 and 0.2 Hz (i.e., eliminating 99.9 and 99.8% of high frequencies, respectively, for 200-Hz data). The cutoff frequency determined by this analysis was then used for filtering the raw Doppler blood flow data for kinetics analysis.

#### Kinetics analysis.

The raw Doppler data from five transitions of one subject were time aligned to the onset of exercise and ensemble averaged to produce a single data set. For these transitions, a low-pass filter (LP_{FILTER}; cutoff *f* = 0.2 Hz) was then applied to the 200-Hz ensemble-averaged data. Finally, the data were resampled at 10 Hz for nonlinear regression analysis, i.e., from the 200-Hz data set we retrieved every 20th datum. The resampling was necessary because using a personal computer for curve fitting of a single response with 36,000 data points (180 s at 200 Hz) with a two-exponential model would usually take 30–45 min and frequently lock up the computer before achieving the convergence criteria of the nonlinear regression. Importantly, for data sets in which the curve fitting could be successfully completed at 200 Hz, we observed that the parameter estimates of blood flow kinetics (see *Eq. 1* below) for 10 Hz (resampled) data were, within round-off error, almost identical to the 200-Hz data. The Beat-by-Beat data were interpolated (zero-order interpolation) at 10 Hz and ensemble averaged for analysis of blood flow kinetics.

The kinetics of blood flow for a single transition of each subject (*visit 1*) were determined for the Beat-by-Beat, average of three consecutive beats (AVG_{3 BEATS}), and LP_{FILTER} of raw Doppler data. For the latter, the Doppler data (200 Hz) were filtered and resampled at 10 Hz as described above. The time course of LBF was determined by nonlinear regression using a least squares technique (SigmaPlot 2001, Systat Software). A two-component model was used to describe the response (7, 12, 16, 23) (1) where BSL is baseline, *A* is amplitude, TD is time delay, and τ is time constant of each phase of the LBF response. The nonlinear regression results of interest for this study were the parameter estimate describing the LBF kinetics and the standard error of the estimate (SEE) for each parameter, which describes the confidence of parameter estimation. A reduction in the SEE represents a decrease, and therefore improvement in the CI for estimation of kinetic parameters.

#### Statistical analysis.

The parameter estimate and SEE for blood flow kinetics were determined from nonlinear regression analysis (Marquadt-Levenberg algorithm, SigmaPlot 7.01, Systat Software) of data preprocessed with the three different methods (Beat-by-Beat, AVG_{3 BEATS}, and LP_{FILTER}). The SEE for Beat-by-Beat, AVG_{3 BEATS}, and LP_{FILTER} were compared by one-way ANOVA. The Fisher least significant difference test was used for post hoc analysis. Significance was declared when *P* < 0.05. Data are presented as means ± SD.

## RESULTS

The time- and frequency-domain characteristics of the simulated blood flow response are shown in Fig. 2. The spectral analysis of simulated blood flow + noise identified the three simulated oscillatory components (as expected) and a lower frequency component (in the range 0 to 0.1 Hz), which appeared to contain the information about the underlying kinetics response. However, filtering the data by removing frequencies greater than 0.1 Hz distorted the simulated blood flow (Fig. 3). As shown in Table 1, we observed that, within the range of kinetic parameters previously described for blood flow response in healthy subjects (12, 16, 18), a cutoff frequency of 0.2 Hz retained sufficient frequency information so as to accurately characterize the biphasic kinetics of blood flow, including the transitional changes at exercise onset (Fig. 3, *bottom inset*) and the Phase I–II transition (Fig. 3, *top inset*).

To examine further the validity of our simulations and establish reproducibility, we determined the frequency characteristics of the raw Doppler data from a single transition and from five transitions ensemble averaged (Fig. 4). The power spectrum of these responses qualitatively resembled the computer simulations. Moreover, the overall frequency-domain characteristics of each transition were grossly similar (Fig. 5). The primary effect of ensemble averaging several transitions was a reduction in the power of components with frequencies >0.1 Hz (Fig. 4*B*), which is in effect qualitatively similar to a low-pass filter with cutoff of ∼0.1 to 0.2 Hz.

The effects of LP_{FILTER} on the raw Doppler data ensemble averaged are shown in Fig. 6. As predicted by the computer simulations, the LP_{FILTER} eliminated the primary sources of noise and revealed the underlying biphasic increase in LBF. This noise-reducing effect was greater than that achieved by ensemble averaging five transitions (after 10-Hz interpolation) of Beat-by-Beat data (Fig. 6*C*). The LP_{FILTER} was applied to each of the five transitions and their kinetics were determined (Table 2). For each of the five transitions the overall kinetics (mean response time) of LBF after LP_{FILTER} were virtually identical to the corresponding raw (unfiltered) Doppler data when both were analyzed by a monoexponential function (Table 2). These results demonstrate that LP_{FILTER} did not slow the overall kinetics of blood flow, which is a concern with any filtering procedure. The coefficient of variation (CV) for baseline and steady-state blood flow was <10%, which is similar to a previous study in our laboratory (11). However, the CV for each kinetic (i.e., time-dependent) parameter ranged from 12 to 67%, suggesting substantial day-to-day variability.

The mean work rate for a single transition performed by four subjects was 10.3 ± 2.6 W, which elicited a mean heart rate equal to 89.0 ± 9.4 beats/min and LBF of 1.32 ± 0.23 l/min at the steady state of exercise. A representative response showing the three methods of data processing used for kinetics analysis (Beat-by-Beat, AVG_{3 BEATS}, and LP_{FILTER}) is depicted in Fig. 7. The parameter estimates describing the blood flow response for each method were not significantly different (Table 3). However, the SEE for each parameter was significantly lower for LP_{FILTER} compared with Beat-by-Beat and AVG_{3 BEATS} (Fig. 8). The LP_{FILTER} method decreased the SEE by 85–95% compared with Beat-by-Beat and AVG_{3 BEATS}.

## DISCUSSION

In the present study we examined, for the first time, the time- and frequency-domain characteristics of arterial blood flow responses after the onset of exercise. The principal and novel findings were that the low-frequency range (<0.5 Hz) of the power spectrum contained the information necessary to describe the biphasic kinetics of blood flow adjustment, and that the primary sources of noise in Doppler measurements (muscle contraction and cardiac cycle) occurred at higher frequencies and thus did not overlap with the frequency spectrum of the blood flow kinetics. These observations provided the bases for applying a low-pass filter (cutoff = 0.2 Hz) to the Doppler blood flow data. For single rest-exercise transitions, the LP_{FILTER} method produced the lowest noise with highest temporal resolution compared with processing the data as Beat-by-Beat and AVG_{3 BEATS}. The implications for kinetics analysis were a significant reduction in the SEE and, therefore, improvement in the 95% CI for each kinetic parameter.

The ensemble averaging of blood flow from several similar transitions to reduce the sample-to-sample variation and improve kinetic parameter estimation is made under the premise that the sample-to-sample variation has a normal distribution (i.e., Gaussian or “white” noise); however, the frequency analysis of raw Doppler data suggests that the variability in blood flow is non-white noise. Even though we have not performed a thorough analysis of time-domain noise distribution as previously done for pulmonary O_{2} uptake and phosphocreatine (9, 17), a fundamental characteristic of white (Gaussian) noise is the uniform distribution of power in the frequency spectrum. Therefore, the non-Gaussian properties of blood flow noise were evidenced by distinct peaks encountered in the power spectrum associated with muscle contraction and cardiac cycle (Fig. 4), which agreed nicely with the results from computer simulations (Fig. 2). For Doppler measurements, the noise related to muscle contraction showed a narrow frequency range, suggesting small variations in contraction frequency, and the greatest power spectral density. In contrast, the spectral characteristics of oscillations induced by cardiac cycle demonstrated a wide frequency range, which reflected the dynamic increase in heart rate after the onset of exercise (rest 60 ± 3.1 beats/min, exercise 89 ± 9.4 beats/min). The most important observation, however, was that the contraction and cardiac cycle noise did not overlap with the power spectrum describing the underlying dynamic increase in blood flow. This is a prerequisite to apply the LP_{FILTER} because, in the presence of overlap, the cutoff frequency necessary to eliminate the primary sources of noise would also filter out frequency components essential to describe the underlying kinetics and, therefore, distort the response (e.g., Fig. 3). On the basis of comparison of kinetics of simulated and filtered responses (Table 1 and Fig. 3), we chose a cutoff frequency = 0.2 Hz for LP_{FILTER} because a lower cutoff (e.g., 0.1 Hz) distorted the *phase 1* of blood flow, leading to results that differed by more than 10% from the simulated value. However, we acknowledge that the “optimal” cutoff frequency for LP_{FILTER} will vary according to the underlying kinetics of blood flow (Table 1); thus the cutoff frequency must be tailored to the characteristics of each type of response.

On the basis of the non-Gaussian distribution of blood flow noise, it is possible that ensemble averaging might actually enhance the underlying oscillatory characteristic induced by muscle contraction and cardiac cycle instead of eliminating these oscillations and improve the signal-to-noise ratio for kinetics analysis. However, we observed that ensemble averaging five transitions substantially reduced the noise in the raw Doppler data (Fig. 4*A*). This might be explained by the fact that, after time alignment of each transition for the onset of exercise, the occurrence of muscle contraction and heartbeat are not superimposable from test to test because of the within-subject variability of heart rate (baseline, steady state, and dynamics) and timing of muscle contraction. In the frequency domain, averaging several like transitions reduced the power spectrum of high-frequency components. This was evidenced by a 75% reduction in the area under 0.6–0.7 Hz and a 60% reduction for 1.10–1.50 Hz, but a similar area under the frequency spectrum for 0–0.2 Hz (Fig. 4*B*). Thus ensemble averaging blood flow data worked in a manner similar to, but less effective than, a low-pass filter in diminishing the sample-to-sample variability.

To eliminate or reduce the noise of blood flow measurements so as to improve the precision of estimating the kinetic parameters of blood flow, investigators have used different approaches for data processing [beat-by-beat (5, 7, 12), 1-s time bins (19, 23), and averaging over a contraction-relaxation cycle (16, 18, 22)]. These strategies are based on the observations of Lamarra et al. (9), who defined the CI for τ or TD (*K*_{1}) as (2) where S_{0} is the standard deviation of the fluctuations around the mean (noise), ΔY_{SS} is the amplitude of increase in the investigated variable from baseline to steady-state exercise (“signal”), and *L̂* is determined by the time constant and data density. Therefore, decreasing the standard deviation by averaging data over longer time bins (13, 24) will improve (i.e., decrease) the CI of parameter estimation. However, these averaging procedures have the limitation of decreasing the temporal resolution for kinetics analysis (i.e., increase in *L̂*), which is frequently overlooked in the context of kinetics analysis.

The kinetics of blood flow after the onset of exercise are biphasic (*Eq. 1*) with an initial fast component (τ_{1} ≈ 1–5 s) followed by a slower response (τ_{2} ≈ 10–40 s) that starts or emerges 10–30 s after the start of exercise (for review, see Ref. 20). The fast *phase 1* of blood flow appears to be crucial for maintaining adequate O_{2} delivery after the onset of exercise (4); therefore, these kinetics need to be adequately characterized. The theoretical predictions of Lamarra et al. (9) indicated that, for a fixed noise and sampling frequency, the CI of τ improves as the kinetics become slower because of the increase in density of data across the transition phase. This has direct implications for investigation of blood flow kinetics because the rapidity of the early response (τ_{1}) and *phase 1*-*phase 2* transition (TD_{2} in *Eq. 1*), and the relatively low amplitude of *phase 1* (∼50% of total for moderate exercise) requires high sampling frequencies to determine the biphasic kinetics with acceptable degree of confidence. In this setting, the 95% CI of the kinetic parameters will be very sensitive to the averaging procedure. This was demonstrated in the present study by comparing three methods (Beat-by-Beat, AVG_{3 BEATS}, and LP_{FILTER}) to process the blood flow data before kinetics analysis.

The simplest and most logical averaging of blood flow data is over a cardiac cycle (Beat-by-Beat); however, the resulting profile has a low signal-to-noise ratio (CV = 33%, Fig. 7*A*) that results in large SEE for kinetic parameters (Fig. 8) and in some occasions has masked the biphasic nature of the blood flow response to exercise (5). In the present study, the Beat-by-Beat method demonstrated a waxing and waning of the noise in blood flow (Fig. 7*A*), likely because the heartbeat occurred during muscle relaxation at some points, generating large oscillations in flow, but at other periods the cardiac cycle coincided with muscle contraction, generating small oscillations (11, 13, 16, 25) (e.g., Fig. 1). The procedure of averaging over longer time bins is of diminishing return because the decrease in the S_{0} is counterbalanced by a reduction in data density (leading to an increase in *L̂*; see *Eq. 2*). Accordingly, AVG_{3 BEATS} showed a small decrease in the SEE that was not significantly different from Beat-by-Beat for most kinetic parameters (Fig. 8); however, with inclusion of more subjects, these differences would probably reach significance. In contrast to time-domain filtering, LP_{FILTER} provided an excellent combination of a decrease in blood flow noise while maintaining high temporal resolution for kinetics description (which would tend to minimize *L̂* in *Eq. 2*). These effects were confirmed by the very low SEE of each kinetic parameter (Fig. 8). For all parameters the SEE of LP_{FILTER} was significantly lower than Beat-by-Beat and AVG_{3 BEATS}, demonstrating a substantial improvement in the 95% CI.

The improvement in the 95% CI is relevant for comparison of kinetic parameters between separate transitions and becomes more critical when investigators are limited to single transitions, such as in pharmacological and clinical studies of blood flow kinetics (e.g., Refs. 21, 22). However, a shortcoming of single transitions is the potential for within-day and day-to-day variability of the “true” underlying kinetic parameters for a given subject. Even with good compromise between signal-to-noise ratio and data density obtained by LP_{FILTER}, we observed a large within-subject variability of kinetic parameters determined for five transitions collected over three days (Table 1). This suggests that ensemble averaging of multiple transitions may still be necessary to minimize variability and accentuate the underlying physiological response. For the subject who repeated five transitions, we also compared the effects of LP_{FILTER} to beat-by-beat data (10-Hz interpolated) ensemble averaged. As previously shown (9, 17), the effects of ensemble averaging data with low signal-to-noise ratio on the 95% CI is limited because the latter will be improved by a factor of , where *n* is the number of repetitions (9, 17). For example, for the blood flow response shown in Fig. 6, to obtain a 95% CI for beat-by-beat data (10-Hz interpolated) similar to that achieved with LP_{FILTER} for a given parameter (τ or TD), it would be necessary to average ∼225 more transitions (estimated from Ref. 9). This further emphasizes the advantage of using LP_{FILTER} to process blood flow data for kinetics analysis, irrespective of using single or multiple transitions ensemble averaged (Figs. 6–8).

The present analysis was conducted with supine one-leg knee-extension exercise of moderate intensity at 40 contractions/min; thus the possibility to extrapolate our results to different exercise modes, intensities, and contraction frequencies must be considered. Compared with our results for supine knee-extension exercise, other exercise modes may result in differences in amplitude and kinetics of the blood flow response. From the perspective of frequency analysis, the total amplitude of blood flow response is a “dc shift” located at 0 Hz on the power spectrum and thus would not be affected by LP_{FILTER}. Hence, the major concern for extrapolating our results to other exercise modalities, intensities, or contraction frequencies is the kinetics (i.e., time-dependent characteristics) of the adjustment in blood flow. Specifically, situations in which blood flow kinetics are substantially faster than examined herein will require higher cutoff frequencies to avoid distortion of the kinetic response (Table 1 and Fig. 3). With regard to contraction frequency, the applicability of our results will depend on the effects of contraction frequency on blood flow kinetics. If we assume, for example, that blood flow kinetics are independent of contraction frequency, our data would suggest that ∼15 contractions/min (or 0.25 Hz) is the lower limit to be able to apply the LP_{FILTER} with a cutoff = 0.2 Hz (i.e., eliminating the frequency components ≥12 min^{−1}). However, there is no upper limit for contraction frequency because LP_{FILTER} will eliminate all high-frequency components. Likewise, heavy exercise is characterized by the existence of a slow increase in blood flow after *phase 2* (7, 14), which will not limit the use of LP_{FILTER} because the frequency components describing this process are lower than 0.2 Hz (i.e., time constant longer than the kinetics of *phase 2*) and, thus, retained after applying the LP_{FILTER}. Therefore, the filtering procedure used in the present study should be valid for different exercise modes and intensities. However, for low-contraction frequencies, the most appropriate cutoff frequency will depend on the effects of contraction frequency on blood flow kinetics.

As for blood flow, there has been great interest in the kinetics of minute ventilation, pulmonary O_{2} uptake (3, 5, 7, 9, 17), and phosphocreatine (from ^{31}P-magnetic resonance spectroscopy) (e.g., Ref. 17). Is it possible to use the LP_{FILTER} to reduce the noise of pulmonary O_{2} uptake or phosphocreatine and thereby decrease the number of trials needed to improve the confidence of kinetics analysis? These variables have Gaussian distribution of noise (9, 17), thus we anticipate uniform distribution of noise on their frequency spectrum, suggesting that application of the LP_{FILTER} might be possible. However, it would require a different strategy of data processing because, for example, pulmonary O_{2} uptake data are a time series in which each breath is a discrete event of variable duration, in contrast to the analog blood flow using Doppler ultrasound, which can be evenly sampled in time.

#### Limitations.

In the present study we assumed constant femoral artery diameter from rest to exercise to calculate blood flow from Doppler measurements of BV. However, as mentioned above, femoral artery diameter did not change significantly during knee-extension exercise (3, 12, 15).

It was outside the scope of our study to establish a generic cutoff frequency for low-pass filtering of blood flow data for kinetics analysis. However, we observed that using a cutoff of 0.5 Hz dramatically reduced the noise observed for direct measurements of blood flow, whereas the fast-responding characteristics of simulated responses were preserved. In fact, a cutoff frequency of 0.5 Hz also resulted in very low SEE for the kinetic parameters of blood flow from five transitions ensemble averaged (1.0–4.7% for 0.5 Hz vs. 0.4–3.5% for 0.2 Hz; see Table 4), but investigators must keep in mind that this cutoff frequency would only be appropriate for exercise performed with a muscle contraction frequency greater than 30 min^{−1} (see above).

The large within-subject variability of blood flow kinetics (Table 2) poses a limitation for using a single transition to compare the effects of Beat-by-Beat, AVG_{3 BEATS}, and LP_{FILTER} on the SEE of kinetic parameters. Although the variability might be attributed to Doppler measurement errors (6), careful inspection of the velocity waveforms did not reveal any loss of signal or noise in the transition phase. Furthermore, the intrasubject variability for kinetics of multiple transitions will not affect our comparison of Beat-by-Beat, AVG_{3 BEATS}, and LP_{FILTER} for a single transition, because the same data set was used for each method.

In summary, we have demonstrated that the low-frequency range (≤0.5 Hz) of the power spectrum contained the information necessary to describe the kinetics of blood flow, that the main sources of noise in Doppler measurements of BV were muscle contraction and heart rate, and that these did not overlap with the frequency spectrum of the biphasic kinetics. Moreover, ensemble averaging several like transitions had an effect qualitatively similar to, but less effective than, a low-pass filter (i.e., reduce or eliminate high-frequency noise). These observations suggest that LP_{FILTER} was a valid procedure to pretreat blood flow data for kinetics analysis. In this setting, LP_{FILTER} yielded the highest signal-to-noise ratio and temporal resolution compared with beat-by-beat data and the average of three consecutive beats, which resulted in significantly lower SEE for all kinetic parameters describing the blood flow response. The direct consequence was a substantial improvement in the 95% CI of each kinetic parameter.

## GRANTS

This work was supported in part by American Heart Association, Grant-in-Aid no. 0151183Z to T. J. Barstow. L. F. Ferreira was supported by a Fellowship from the Ministry of Education/CAPES, Brazil.

## Footnotes

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- Copyright © 2006 the American Physiological Society