. Author manuscript; available in PMC: 2021 Jul 1.

Published in final edited form as: Biomed Signal Process Control. 2020 Apr 17;60:101947. doi: 10.1016/j.bspc.2020.101947

A robust Fourier-based method to measure pulse pressure variability

Sebastian Acosta ^a, Mubbasheer Ahmed ^b, Suellen M Yin ^c, Ken M Brady ^d, Daniel J Penny ^a, Craig G Rusin ^a,^*

PMCID: PMC7384614 NIHMSID: NIHMS1587835 PMID: 32719724

Abstract

Objective

To propose a new method to estimate pulse pressure variability (PPV) in the arterial blood pressure waveform.

Methods

Traditional techniques of calculating PPV using peak finding have a fundamental flaw that prevents them from accurately resolving PPV for small tidal volumes, limiting the use of PPV to only mechanical ventilated patients. The improved method described here addresses this limitation using Fourier analysis of an oscillatory signal that exhibits a time-varying modulation of its amplitude. The analysis reveals a constraint on the spectral representation that must be satisfied for any oscillatory signal that exhibits a time-varying modulation of its amplitude. This intrinsic mathematical structure is taken advantage of in order to improve the robustness of the algorithm.

Results

The applicability of the method is tested using synthetic data and 100 h of physiologic data collected from patients admitted to Texas Children’s Hospital.

Significance and conclusion

The proposed method accurately recovers values of PPV at signal-to-noise ratios six times smaller than the traditional method. This is a significant advance for the potential use of PPV to recognize fluid responsiveness during low tidal volume ventilation or spontaneous breathing for which the signal-to-noise ratio is expected to be small.

Keywords: Arterial pulse pressure variation, Fluid responsiveness, Fourier analysis

1. Introduction

Critically ill patients are vulnerable to organ injury/failure due to cellular hypoxia. Thus, the preservation of adequate oxygen delivery across the body is at the core of critical care medicine. Fluid administration is a routinely employed intervention in states of shock and hypovolemia when augmentation of the cardiac preload state is presumed to lead to improvement in cardiac output. When the administration of fluid resuscitation leads to an increase in cardiac output, the patient is “fluid responsive” i.e. the patient currently resides on the ascending segment of the Frank-Starling curve. Increased cardiac output leads to reduction tissue hypoxia which helps prevent organ injury. However fluid administration without an associated increase in cardiac output can lead to further organ injury and morbidity, particularly in patients with heart failure [1]. In studies designed to examine fluid responsiveness, only 40–70 % of adults with circulatory failure demonstrated an increase in cardiac output with fluid administration [2]. As a result, determining if a patient is fluid responsive or not is an important clinical question.

A clinician’s ability to gauge intravascular volume status of a patient using bedside exams (skin turgor, urine output) and variables such as heart rate, central venous pressure, pulmonary artery occlusion pressure, or blood pressure have consistently shown to be poor [3]. More reliable measures for predicting fluid responsiveness include pulse pressure variation (PPV) and stroke volume variation (SVV) [4–6]. The physical principles responsible for PPV and SVV are illustrated in Fig. 1. A change in intrathoracic pressure is felt simultaneously on the ascending aorta and on the vena cava. On one hand, the pressure variation on the aorta induces an instantaneous additive variation in the mean arterial pressure. On the other hand, a reduction in vena cava blood flow is induced during the early inspiration phase of positive-pressure ventilation [4,7]. This diminished right-ventricle preload causes, after a pulmonary transit time, a reduction in left-ventricle preload which is manifested as a momentary decline in cardiac output and pulse pressure [8–10]. With spontaneous breathing or negative pressure ventilation, an inverse effect would be anticipated [11]. This pulmonary transit time has been estimated to be approximately two to three cardiac cycles [8].

Fig. 1. — Diagram illustrating the propagation of pressure variations triggered by the oscillations in airway pressure. MAP: mean arterial pressure. PP: pulse pressure. RV: right ventricle. LV: left ventricle.

Beginning with Michard et al. in 2000, several studies have defined the PPV threshold for fluid responsiveness under positive pressure ventilation to be >14 % of the pulse amplitude [6,12]. A meta-analysis including 22 studies with 807 adult patients has found a pooled sensitivity of 88 % and specificity of 89 %, when utilizing a PPV threshold of 13 % in predicting fluid responsiveness [6]. A meta-analysis of 12 studies and 438 pediatric patients found dynamic variables, including pulse pressure variation and systolic pressure variation, were not able to predict fluid responsiveness [13]. There are only four studies limited to pediatric patients with atrial septal defects, ventricular septal defects, and/or tetralogy of Fallot that have found pulse pressure variation or stroke volume variation to predict fluid responsive in the post-operative period [14–16]. All studies were done on positive pressure ventilated patients.

It is expected that tidal volumes for spontaneously breathing patients will be lower than those on mechanical ventilation leading to less robust estimates of PPV. Soubrier et al. evaluate 32 adult patients who received volume expansion and found PPV (using a 12 % threshold) to have a high specificity (92 %) and low sensitivity (63 %) in predicting fluid responsiveness [11]. The authors concluded spontaneous breathing to be less reliable than positive pressure ventilation in predicting fluid responsiveness due to low sensitivity. It should be noted that in both positive and negative pressure ventilation, the magnitude of cardio-pulmonary interactions is dependent on tidal volume among other parameters [17–24]. Lower tidal volume will necessarily induce a weaker cardiopulmonary interaction. In such cases, variations in cardiac filling may not be large enough to induce a measurable PPV. Furthermore, tidal volume will not be constant in patients who are breathing spontaneously, complicating the estimation of PPV.

The present paper is focused on improving the accuracy and robustness of measuring PPV. We will first reveal existing limitations in the traditional mathematical algorithms employed to estimate PPV from blood pressure measurements. We will then describe a new algorithm capable of greater precision and robustness, especially in the small PPV regime where noise can easily overwhelm the physiologic signals. The performance of both the traditional and new algorithms are characterized using synthetic data as well as physiologic recordings from patients.

2. Traditional methods (peak-finding)

Traditional algorithms for estimating PPV are based on finding the systolic peaks and diastolic troughs of the arterial pressure waveform. The pulse pressure (PP) can be estimated on a beat-by-beat basis as the difference between adjacent peaks and troughs. An illustration of the maximum PP and minimum PP within a respiratory cycle is shown in Fig. 2 which displays an arterial blood pressure waveform over a few seconds. Within a respiratory cycle, the maximum PP and minimum PP are found, and the PPV can then be calculated using the equation:

P P V_{old} = 2 \frac{\max (P P) - \min (P P)}{\max (P P) + \min (P P)} .

(1)

Fig. 2. — A patient’s arterial blood pressure waveform. Max(PP): Maximum pulse pressure over a respiratory cycle. Min(PP): Minimum pulse pressure over a respiratory cycle. Peaks and troughs in the waveform are used to estimate pulse pressure on a beat-by-beat basis by finding its maximum and minimum values over each respiratory cycle.

Thus, an estimation of PPV is obtained at every respiratory cycle. An average of such estimations can be computed over many respiratory cycles in order to reduce the effects of noise. A brief description of this algorithm is found in [5,25]. Improvements have been proposed in [26–28]. Sun et al. [28] investigated the performance of existing algorithms, as well as their own proposed algorithm, to estimate PPV during surgery. The first step in all these methods is to find peaks and troughs in the pressure waveform, followed by interpolation and/or Kernel smoothing to define a continuous-in-time pulse pressure signal from which PPV can be estimated.

A synthetic model was constructed to better understand the theoretical performance of PPV estimation algorithms based on peak finding. The blood pressure waveform, u (t), contains a cardiac component u_c (t) oscillating at the heart rate (HR). Periodic changes in intrathoracic pressure due to breathing induce changes in cardiac pumping, modulating the pulse pressure (amplitude of the cardiac component). This effect can be expressed as the product A(t)u_c (t) where the amplitude modulation A(t) oscillates at the respiratory rate (RR). Using these observations, the arterial blood pressure waveform was modeled as follows:

u (t) = u_{m} + (1 + α u_{r} (t + t_{d})) u_{c} (t) + u_{r} (t) + η (t),

(2)

where u_m is the mean value of blood pressure, u_r (t) is the respiratory component of the blood pressure, u_c (t) is the cardiac component, and η(t) represents random noise. The factor A (t) = (1 + α u_r (t + t_d)) represents the amplitude modulation of the cardiac oscillations (pulse pressure) which oscillates at the frequency of the respiratory component u_r. The components u_m, u_r, and u_c don’t have to be sinusoidal necessarily. However, they need to have spectral power in disjoint frequency bands. In practice, this requirement allows us to distinguish one component from the others. The factor α, known as the transmission coefficient, quantifies the portion of the respiratory component that affects the pulse pressure modulation. We allow for the presence of a time delay t_d between the respiratory component u_r (t) and the amplitude modulation (1 + αu_r(t + td)).The fluctuations in right ventricle preload lead to changes in left ventricle filling after a short delay due to the blood flow pulmonary transit time [3,7,18,29–32]. This lag is accounted for by the time delay t_d ≤ 0. See Fig. 1 for details and the Appendix A for a mathematical description of this process. Under the assumed form (2), the pulse pressure variability is given by:

P P V = 2 \frac{\max_{t} (1 + α u_{r} (t + t_{d})) - \min_{t} (1 + α u_{r} (t + t_{d}))}{\max_{t} (1 + α u_{r} (t + t_{d})) + \min_{t} (1 + α u_{r} (t + t_{d}))} .

Assuming that the respiratory component u_r oscillates with equal magnitude above and below zero, that is, min(u_r) = − max(u_r) then the formula for PPV simplifies to

P P V_{e x a c t} = 2 α \max (u_{r}) .

(3)

Using Eqs. (2) and (3), we can now derive the error of the PPV estimate associated with the approach in Eq. (1). The error can be estimated by calculating the difference between the imposed value of PPV (by selecting the parameter α) and the PPV value recovered by the algorithm. In the case of zero PPV (equivalent to setting α = 0) the presence of u_r (t) in (2) affects the estimation of the height of peaks and depth of troughs in the cardiac oscillations in u_c (t). In the best case scenario, when the largest slope of u_r (t) coincides with a peak or trough of u_c (t), then we have that

\max (P P) = 2 A_{c} + | u_{r} (t + d t) - u_{r} (t) | \approx 2 A_{c} + 2 π f_{r} A_{r} d t \cos 2 π f_{r}

\min (P P) = 2 A_{c} - | u_{r} (t + d t) - u_{r} (t) | \approx 2 A_{c} - 2 π f_{r} A_{r} d t \cos 2 π f_{r}

where dt = 1 /(2f_c) is half of a cardiac period, A_r is the amplitude of the respiratory component, and A_c is the amplitude of the cardiac component. Plugging these estimations into (1), we find that the traditional method overestimates PPV for small transmission coefficient as follows

\lim_{α \to 0} P P V_{t r a d i t i o n a l} \geq π \frac{A_{r} f_{r}}{A_{c} f_{c}} \cos (π \frac{f_{r}}{f_{c}}) .

(4)

This analysis suggests that there is a fundamental lower bound to the value of PPV that can be resolved by traditional peak-finding methods. This behavior is illustrated in Fig. 3. Sun et al. [28,33] have recognized a similar limitation from peak-finding-based algorithms where baseline variations in the pressure waveform may be strong enough to alter peaks and troughs differently, thereby compromising the naive calculation of PPV. Sun et al. proposed to filter out the baseline (low frequency variations) from the interpolated pulse pressure waveform in order to eliminate this undesired effect. We believe that (4) is a potential explanation for the limited utility of using PPV in situations with low tidal volume, as the naive peak-finding methods are mathematically unable to accurately estimate PPV in these cases.

Fig. 3. — Comparison between the proposed (new) and the traditional (old) methods for estimating PPV. The range of PPV was realized in the model (2) by varying the transmission coefficient α from 10⁻⁴ /mmHg to 10⁻¹ /mmHg. In the absence of noise (A), the old method fails to estimate the PPV accurately for small values of PPV. The new method is extremely precise for small and large values of PPV. In the presence of 5% pink noise (B), the average of 250 realizations is shown (circles) along with the 5th and 95th percentile curves (solid lines). All of the other parameters are fixed as follows: A_m = 100mmHg, A_c = 20 mmHg, A_r = 6mmHg, f_c = 100cpm, f_r = 20cpm.

3. Proposed method (Fourier)

The proposed method relies on the observation that respiratory and cardiac components appear as a product in the PPV term in Eq. (2). The product of functions in the time domain translates into convolution of factors in the frequency domain. Fig. 4 illustrates the Fourier spectrum of the patient’s blood pressure waveform shown in Fig. 2. The presence of peaks at the respiratory rate (RR) and at the heart rate (HR) can easily be seen. There are also complex- conjugate peaks at the frequencies (−RR) and (−HR). The cardiac peak at the heart rate (HR), is convolved with the respiratory peaks at (RR) and (−RR), giving raise to energy supported in the vicinity of (HR−RR) and (HR + RR), respectively. In other words, the variation in pulse pressure due to respiration induces the appearance of peaks at the frequencies (HR−RR) and (HR + RR) as seen in Fig. 4. The method proposed below is based on the detection of convolved Fourier components in the vicinity of the frequencies (HR−RR) and (HR + RR). We highlight the difference with other Fourier-based approaches from [34,35] that seek to determine the height of the spectral peak at RR (respiratory component as shown in Fig. 4). This peak quantifies the oscillation in mean pressure rather than the oscillation in the pulse pressure. See Fig. 1.

Fig. 4. — The Fourier transform of a patient’s blood pressure waveform. RR: Respiratory rate. HR: Heart rate. The proposed method to measure PPV is based on the detection of the convolved components supported in the vicinity of the frequencies HR−RR and HR + RR to quantify amplitude modulation of the pulse pressure.

In order to use Eq. (3) to calculate PPV, estimates of the transmission coefficient α and the amplitude of the respiratory component, max(u_r), are needed. We accomplish this task using the discrete fast Fourier transform, $F$ , with frequencies ranging from −f_s/2 to +f_s/2 where f_s is the frequency at which the signal u (t) is sampled. The amplitude of the respiratory component can be estimated as follows:

\max (u_{r}) \approx \sqrt{2} ‖ {F (u) |}_{f_{r}} ‖

(5)

where f_r is the frequency band of the respiratory component. In order to estimate the transmission coefficient α, we use the Fourier-Convolution and the Time-Shift theorems in the model (2) to obtain

F (u) = F (u_{m}) + F (u_{c}) + α [e^{i ω t_{d}} F (u_{r})] * F (u_{c}) + F (u_{r}) + F (η) .

(6)

Since the frequency bands of the respiratory f_r and cardiac f_c components are known and do not overlap, we can simultaneously extract $F (u_{r}) = {F (u) |}_{f_{r}}$ and $F (u_{c}) = {F (u) |}_{f_{c}}$ from the measured signal $F (u)$ , up to the presence of noise $F (η)$ . The convolution $e^{i ω t_{d}} F (u_{r}) * F (u_{c})$ is supported on the convolved frequencies which we denote by f_c * f_r. Therefore, we estimate the transmission coefficient α and the time-shift t_s as the optimizers of the following problem,

(α_{o p t}, t_{d, o p t}) = a r g m i n ‖ {F (u) |}_{f_{c} * f_{r}} - {{α e^{i ω t_{d}} F (u) |}_{f_{r}} ⇑ F (u) |}_{f_{c}} ‖ .

(7)

As a result, the unknown parameters α and t_d are fitted to the measured data ${F (u) |}_{f_{c} * f_{r}}$ , ${F (u) |}_{f_{r}}$ and ${F (u) |}_{f_{c}}$ . This fitting process leads to a robust method with respect to uncorrelated noise. Once the parameters α and t_d are fitted to the data, the optimal α_opt is used with (5) and (3) to obtain the proposed estimation of the pulse pressure variability as

P P V_{n e w} = 2 \sqrt{2} α_{o p t} ‖ {F (u) |}_{f_{r}} ‖ .

(8)

4. Results

In this section we compare the results from the proposed Fourier-based algorithm to estimate PPV and the traditional algorithm based on peak finding. Both methods are applied to synthetic and patient data.

4.1. Synthetic data

Synthetic data is used to analyze the behavior of the traditional (old) algorithm and of the proposed (new) algorithm under controlled settings. Let u_m =A_m, $u_{c} (t) = A_{c} \sin (2 π f_{c} t)$ , $u_{r} (t) = A_{r} \sin (2 π f_{r} t)$ , and let η(t) be pink noise. We choose the amplitudes and frequencies as follows: A_m = 100mmHg, A_c = 20mmHg, A_r = 6mmHg,f_c = 100cycles/min,f_r = 20 cycles/min. Pink noise is defined such that the standard deviation of η is a chosen percentage of the cardiac amplitude A_c. The transmission coefficient α varies from from 10⁻⁴ /mmHg to 10⁻¹ /mmHg. As indicated by Eq. (4), the old method significantly overestimates PPV when PPV values are small. See Fig. 3. The new method can recover the model value of PPV from measurements over a much larger range of values. Even in the presence of 5% noise, the average estimation of the new method demonstrates accurate parameter recovery down to values of PPV below 0.01.

It is known that shallow breathing (low tidal volume) may result in changes in pulse pressure that are too small to be measured properly using the traditional method [5,11,24,36–42]. We synthetically model this scenario by selecting a fixed transmission coefficient and recovering PPV over a range of respiratory amplitudes using (2). The parameters for this analysis are u_m = A_m, $u_{c} (t) = A_{c} \sin (2 π f_{c} t)$ , $u_{r} (t) = A_{r} \sin (2 π f_{r} t)$ , and η(t) be pink noise, with the amplitudes and frequencies being A_m = 100 mmHg, A_c = 20 mmHg, α = 0.01/mmHg, f_c = 100cycles/min, f_r = 20 cycles/min. For this analysis, the respiratory amplitude A_r varies from 0.1 mmHg to 40 mmHg. Fig. 5 displays the behavior of the new and old methods to estimate the PPV with 5 %, 10 % and 20 % noise. Results indicate that the new Fourier-based method is much more precise and less sensitive to noise than the old method. On average, the minimum recoverable value of PPV using the new method is approximately 6 times smaller than the minimum recoverable PPV value using the traditional method.

The robustness of these methods can also be characterized by estimating the signal-to-noise ratio at which each method begins to lose accuracy. The amplitude of the signal to be recovered is α A_rAc. The amplitude of noise is given as std(η) = σA_c where σ > 0 is a chosen constant. In Fig. 5, this constant σ is 0.05, 0.1, and 0.2 respectively. The signal-to-noise ratio is therefore defined as $SNR = \frac{α A_{r} A_{c}}{σ A_{c}} = \frac{α A_{r}}{σ}$ .

Based on the plots displayed in Fig. 5, the signal-to-noise ratio at which the old method begins to lose accuracy is approximately 1:3, while the new method is approximately 1:18, indicating a factor of 6 improvement in SNR.

As provided by (7), the new method is also able to estimate the time delay between the oscillations in mean blood pressure and the oscillations in pulse pressure. The synthetic model allows us to assess the performance of the method in the recovery of this time delay. Fig. 5 illustrates the performance of the new method to estimate the time delay td ranging from 0 s to 3 s and for 5 %, 10 % and 20 % pink noise. As indicated by the figure, the difference between the input time delay to the model and measured time delay were small, resulting in residuals that were close to zero over the range of time delay simulated.

4.2. Patient data

While synthetic data is helpful for theoretical studies, it is necessary to test both algorithms on data obtained from human subjects to measure their performance under real-world conditions. Retrospective data was obtained from patients admitted to the pediatric intensive care unit (PICU) of Texas Children’s Hospital. High-resolution physiologic data is continuously captured from all PICU patients by the Sickbay platform (Medical Informatics Corp, Houston, TX). The heart rate, respiratory rate, and the arterial blood pressure waveform were obtained in 60 min intervals around 100 administrations of sodium chloride boluses (dosage = 1000 mL or 20 cc/kg). The cohort of subjects consists of 57 patients (54 % male and 46 % female). The distribution of race is as follows: 41 Caucasian, 13 African American, 1 Asian and 2 unknowns. At time of bolus administration, the median age of the patients was 4.4 years with interquartile range of 8.8 years. Data analysis was completed under an approved protocol by the IRB of Baylor College of Medicine.

Before the data was analyzed, it was filtered to remove common artifacts that occur in the clinical environment. For example, arterial lines can be clogged or flushed preventing accurate measurements of the blood pressure waveform from being obtained. A quality measure was developed to filter out inadmissible epochs of data. Two thousand epochs of 1-min long blood pressure waveforms were randomly selected and manually labeled as either “clean” or “artifact”. After labeling the data, 6 metrics associated with the Fourier transform of the blood pressure waveform were identified as factors for a logistic regression to fit the labeling. These metrics are: the spectral power in the cardiac frequency band, the range of the cardiac frequency band, the first and second moments of $| F (u) |$ , the entropy of low frequency components of $| F (u) |$ , and the entropy of the high frequency components of $| F (u) |$ . Half of the data (1000 epochs randomly selected) was employed as a training set to fit the regression coefficients. The other half (1000 epochs) of the data was employed as a test set to quantify the performance of the logistic regression to discriminate between clean data and data with artifacts. The area under the ROC curve for recognition of artifact-free waveforms is 0.94.

Both methods of calculating PPV were applied to the recorded physiologic data. Estimations of PPV were compared using parity plots and Pearson’s correlation coefficient. A data quality threshold equal to 0.15 was chosen to rule out inadmissible data. Approximately 35 % of the data is excluded since it falls below this quality threshold. Fig. 6 displays the parity plots for the admissible data. The traditional method demonstrated good agreement with the new method for large values of PPV. However, as expected from the analysis of synthetic data, there is significant disagreement for small values of PPV due to the traditional method’s overestimation of PPV in that range. Fig. 6A illustrates that results from patient data follow the same behavior observed in the synthetic data. In this case, the correlation coefficient is 0.74. Fig. 6B displays the adjusted parity behavior after the model curve from synthetic data was set as the parity line to correct the limitations in the old method. This correlation coefficient for the adjusted data is 0.82. The new method also renders the transmission coefficient, α, and the time delay, td, for the patient data. These results are displayed in Fig. 7. The most common time delay found in the patient data set is between 1.5–2.0 s, which matches clinical and physiological expectations of 2–3 cardiac cycles for this population. To our knowledge this is the first direct measurement of pulmonary transit delay in spontaneously breathing subjects using only an arterial pressure waveform.

Fig. 6. — Parity plots between the proposed (new) method and the traditional (old) method for estimating PPV. Panel (A) shows how the measured patient data follows the same behavior observed in the synthetic data (solid curve). The correlation coefficient is 0.74 and $r \hat{2}$ value is 0.86. Panel (B) displays the adjusted parity behavior after the synthetic data line (solid curve from left panel) has been set as the parity line to correct for the behavior observed in the traditional method. This adjusted correlation coefficient is 0.82 and corresponding $r \hat{2}$ value is 0.90.

Fig. 7. — (A) Distribution of the transmission coefficient obtained from measured patient data. This coefficient quantifies the transmission of oscillatory changes in mean pressure at the respiratory frequency into oscillatory changes in pulse pressure. (B) Distribution of the time delay between oscillatory changes in pulse pressure and mean pressure at the respiratory frequency obtained from measured patient data.

5. Conclusion and limitations

A new method for measuring pulse pressure variability over the respiratory cycle based on nonlinear Fourier analysis of the arterial pressure waveform has been described. This method is a significant advance over previous methods of calculating PPV based on peak finding, which have a fundamental mathematical limit on the measured value that can be recovered from an arterial pressure waveform recording. This restriction may potentially be the underlying factor which has limited the use of PPV for assessing fluid responsiveness to only mechanically ventilated patients with large tidal volumes, as this is the only condition under which the traditional PPV methods are accurate.

The new method accounts for the respiratory influence on the cardiac performance by using the mathematical structure of the oscillatory model (2). Specifically, the new method looks for convolved components of cardiac and respiratory oscillations in the proper frequency bands, consistent with the mathematical structure described in Eq. (2). Since many sources of noise do not conform to this constraint, such noise is rejected during the parameter recovery process. As a result, the new method performs robustly in the presence of high levels of noise. The new method remains robust at signal-to-noise ratios approximately 6 times smaller than a traditional method. This may allow for the use of PPV to recognize the fluid responsive state in patients who are spontaneous breathing or are on mechanical ventilation with low tidal volume, where the signal-to-noise ratio is expected to be small.

The proposed method is not only able to estimate the PPV, but also the transmission coefficient α, as well as the pulmonary transit time delay td for a patient. It is well-known that PPV is dependent on the strength of the ventilation or tidal volume [17–24,42]. The same patient, with the same cardiovascular state, can be induced to render different values of PPV by adjusting the tidal volume of mechanical ventilation. The transmission coefficient, α, provides a normalized PPV where the strength of the respiratory component is removed in order to reveal the intrinsic fluid responsiveness status of the patient. The time delay parameter, td, estimates the pulmonary transit time [3,7,18,29–32] which is associated with the lag between variations in the right ventricular preload pressure and the aortic flow. Therefore, this time delay parameter may potentially be employed as a non-invasive measure of pulmonary vascular resistance. However, testing of this hypothesis is needed. See the Appendix A for a mathematical description of this process based on transfer function theory. This approach neglects nonlinear and viscoelastic effects that may contribute to additional phase delays between pressure and flow. More refined mathematical analysis or numerical simulations may be needed to estimate the precise interplay between these hemodynamic factors. We also note that model (2) describes the pressure waveform in the aorta rather than the radial artery where pressure is commonly measured in the clinical setting. The pressure waveform is known to distort as it travels from the ascending aorta to the radial artery. Hence, model (2) may not fully account for dispersive, dissipative and reflective effects imposed by the vascular network. See Chapter 26 in [43] are references therein.

Since this proposed method relies on the application of the FFT, then it is required to have a sufficiently large sampling frequency to avoid aliasing and a sufficiently long window of time to avoid spectral leakage. The sampling frequency from a standard monitor is 120 Hz which is large enough for our purposes. The length of the time window determines an adequate frequency resolution to distinguish peaks and other components in the spectrum of the arterial pressure waveform. The resolution in the frequency-domain is given by Δf = 1 /T where T is the length of the time window. This frequency resolution should be much smaller (one tenth as a rule of thumb) than the minimum respiratory rate RR_min encountered in physiologic data. For instance, if RR_min = 10 cpm = 1/6 Hz then it would be needed to choose T ≈ 60 sec. Consequently, the proposed method is limited to render an estimate of PPV once a minute, approximately. By contrast, other methods may render a PPV estimation once per breath cycle. Another limitation of our method is that it requires the frequency bands around RR and HR−RR to be non-overlapping. This condition is generally satisfied. However, there may be some clinical scenarios when this requirement is not met. The method also requires having simultaneous measurements of the blood pressure waveform, the heart rate and respiratory rate.

The primary purpose of estimating PPV is to assess fluid responsiveness. Ongoing work is being carried out to validate the proposed method to predict change in stroke volume which is the gold standard for determining fluid responsiveness. Also, there is need to validate the proposed method in other clinical settings, including a larger cohort of patients stratified by age and cardiovascular conditions. As soon as meaningful results are obtained, they will be reported in a forthcoming publication.

Funding

This work was supported in part by grants from the National Institute of Health (1R01HL142994) as well as the American Heart Association and Children’s Heart Foundation (16BGIA27490024).

Appendix A

We describe a mathematical model for the transfer function from slow preload/afterload pressure oscillations (caused by breathing) to the oscillation of the cardiac output at the aorta. This model justifies the choice of a time-delayed pulse pressure modulation for the arterial pressure waveform (2). We consider the most proximal systemic arteries (sa), systemic veins (sv), pulmonary arteries (pa) and pulmonary veins (pv). We take the right and left ventricles to be elastic chambers whose elastances oscillate at the heart rate (HR). The atria can be effectively lumped with the venous vessels. Following Hoppensteadt and Peskin, Section 1.4 in [44], the blood flow rate Q through the left ventricle satisfies

Q = HR (\frac{P_{p v}}{E_{d i a}^{L}} - \frac{P_{s a}}{E_{s y s}^{L}}) = \frac{HR}{E_{d i a}^{L}} (P_{p v} - \frac{E_{d i a}^{L}}{E_{s y s}^{L}} P_{s a}) .

(A1)

Here $E_{dia}^{L}$ and $E_{sys}^{L}$ are the left ventricular elastance at end-diastole and end-systole respectively. We assume that the oscillation in intrathoracic pressure translates into an additive component to the blood pressure in all vessels within the thoracic chamber. Hence, in the frequency domain, it is common to neglect the last term in the parenthesis in (A1) since $E_{dia}^{L} / E_{sys}^{L} ≪ 1$ and the oscillatory amplitudes, at the respiratory rate, for the preload P_pv(ω) and afterload P_sa(ω) are comparable. Similarly, the flow rate Q through the right ventricle is given by

Q = HR (\frac{P_{s v}}{E_{d i a}^{R}} - \frac{P_{p a}}{E_{s y s}^{R}}) = \frac{HR}{E_{d i a}^{R}} (P_{p v} - \frac{E_{d i a}^{R}}{E_{s y s}^{R}} P_{s a}) .

(A2)

The pulmonary vasculature can be modeled as a two-element windkessel [43,45] with pulmonary vascular resistance R_p and total compliance C_p. The effective governing equation in the frequency domain is

Q \frac{R_{p}}{1 + i ω τ} = P_{p a} - P_{p v} .

(A3)

The pulmonary time-constant τ = R_pC_p. Eqs. (A1)–(A3) and the law of impedances for elements connected in series yield an effective impedance

Z_{e f f} (ω) = \frac{E_{d i a}^{R} + E_{d i a}^{L}}{HR} + \frac{R_{p}}{1 + i ω τ} .

(A4)

Now, recall that the flow rate Q at the aorta can be expressed as Q = HR C_a PP where C_a is the compliance of the aorta and PP is the aortic pulse pressure. Therefore, according to (A4), the phase lag φ between the flow Q(ω) (or equivalently pulse pressure PP(ω)) and the preload pressure P_sv(ω) is given by.

\tan φ = \frac{Im Z_{e f f}}{Re Z_{e f f}} = \frac{- ω τ}{1 + (1 + {(ω τ)}^{2}) (\frac{E_{d i a}^{R} + E_{d i a}^{L}}{H R R_{p}})} .

(A5)

We take the angular frequency ω = 2π RR corresponding to the preload/afterload oscillating at the respiratory rate (RR). The associated time delay for these oscillations is t_d = φ/(2π RR) which is can determined uniquely modulo the respiratory period. This analysis justifies the structure (2) chosen to model the arterial pressure waveforms where the modulation of pulse pressure PP lags the oscillatory preload/afterload which is in phase with the intrathoracic pressure.

Footnotes

Declaration of Competing Interest

Dr. Rusin declares that he has a conflict with Medical Informatics Corp. Medical Informatics Corp did not financially support this work. All other authors have no conflicts of interest.

Appendix B. Supplementary data

Supplementary material related to this article can be found, in the online version, at doi: https://doi.org/10.1016/j.bspc.2020.101947.

References

[1].Arikan AA, Zappitelli M, Goldstein SL, Naipaul A, Jefferson LS, Loftis LL, Fluid overload is associated with impaired oxygenation and morbidity in critically ill children, Pediatr. Crit. Care Med 13 (3) (2012) 253–258, 10.1097/PCC.0b013e31822882a3. [DOI] [PubMed] [Google Scholar]
[2].Michard F,Teboul J-L, Predicting fluid responsiveness in ICU patients. A critical analysis of the evidence, Chest 121 (2002) 2000–2008, 10.1378/chest.121.6.2000. [DOI] [PubMed] [Google Scholar]
[3].Marik PE, Cavallazzi R, Vasu T, Hirani A, Dynamic changes in arterial waveform derived variables and fluid responsiveness in mechanically ventilated patients: a systematic review of the literature, Crit. Care Med. 37 (9) (2009) 2642–2647, 10.1097/CCM.0b013e3181a590da. [DOI] [PubMed] [Google Scholar]
[4].Michard F, Changes in arterial pressure during mechanical ventilation, Anesthesiology 103 (2005) 419–428. [DOI] [PubMed] [Google Scholar]
[5].Cannesson M, Aboy M, Hofer CK, Rehman M, Pulse pressure variation: where are we today? J. Clin. Monit. Comput 25 (1) (2011) 45–56, 10.1007/s10877-010-9229-1. [DOI] [PubMed] [Google Scholar]
[6].Monnet X, Marik PE,Teboul JL, Prediction of fluid responsiveness: an update, Ann. Intensive Care 6 (1) (2016) 111, 10.1186/s13613-016-0216-7. [DOI] [PMC free article] [PubMed] [Google Scholar]
[7].Marik PE, Lemson J, Fluid responsiveness: an evolution of our understanding, Br. J. Anaesth. 112 (4) (2014) 617–620, 10.1093/bja/aet590. [DOI] [PubMed] [Google Scholar]
[8].Morgan BC, Martin WE, Hornbein TF, Crawford EW, Guntheroth WG, Hemodynamic effects of intermittent positive pressure respiration, Anesthesiology 27 (5) (1966) 584–590. [DOI] [PubMed] [Google Scholar]
[9].Morgan BC, Crawford EW, Guntheroth WG, The hemodynamic effects of changes in blood volume during intermittent positive-pressure ventilation, Anesthesiology 30 (3) (1969) 297–305. [DOI] [PubMed] [Google Scholar]
[10].Denault AY, Gorcsan J, Pinsky MR, Dynamic effects of positive-pressure ventilation on canine left ventricular pressure-volume relations, J. Appl. Physiol. 91 (1) (2001) 298–308. [DOI] [PubMed] [Google Scholar]
[11].Soubrier S, et al. , Can dynamic indicators help the prediction of fluid responsiveness in spontaneously breathing critically ill patients? Intensive Care Med. 33 (7) (2007) 1117–1124, 10.1007/s00134-007-0644-9. [DOI] [PubMed] [Google Scholar]
[12].Michard F, et al. , Relation between respiratory changes in arterial pulse pressure and fluid responsiveness in septic patients with acute circulatory failure, Am. J. Respir. Crit. Care Med. 162 (2000) 134–138, 10.1164/ajrccm.162.1.9903035. [DOI] [PubMed] [Google Scholar]
[13].Gan H, Cannesson M, Chandler JR, Ansermino JM, Predicting fluid responsiveness in children: a systematic review, Anesth. Analg 117 (6) (2013) 1380–1392, 10.1213/ANE.0b013e3182a9557e. [DOI] [PubMed] [Google Scholar]
[14].Han D, et al. , Different predictivity of fluid responsiveness by pulse pressure variation in children after surgical repair of ventricular septal defect or tetralogy of Fallot, Paediatr. Anaesth 27 (10) (2017) 1056–1063, 10.1111/pan.13218. [DOI] [PubMed] [Google Scholar]
[15].Renner J, et al. , Prediction of fluid responsiveness in infants and neonates undergoing congenital heart surgery, Br. J. Anaesth. 108 (1) (2012) 108–115. [DOI] [PubMed] [Google Scholar]
[16].Lee JY, Kim JY, Choi CH, Kim HS, Lee KC, Kwak HJ, The ability of stroke volume variation measured by a noninvasive cardiac output monitor to predict fluid responsiveness in mechanically ventilated children, Pediatr. Cardiol. 35 (2) (2014) 289–294, 10.1007/s00246-013-0772-7. [DOI] [PubMed] [Google Scholar]
[17].Piccioni F, Bernasconi F, Tramontano GTA, Langer M, A systematic review of pulse pressure variation and stroke volume variation to predict fluid responsiveness during cardiac and thoracic surgery, J. Clin. Monit. Comput. 31 (4) (2017) 677–684, 10.1007/s10877-016-9898-5. [DOI] [PubMed] [Google Scholar]
[18].Scharf S, Brown R, Saunders N, Green L, Hemodynamic effects of positive-pressure inflation, J. Appl. Physiol. Resp. Environ. Exerc. Physiol 49 (1) (1980) 124–131. [DOI] [PubMed] [Google Scholar]
[19].Myatra SN, Prabu NR, DIvatia JV, Monnet X, Kulkarni AP, Teboul JL, The changes in pulse pressure variation or stroke volume variation after a ‘tidal volume challenge’ reliably predict fluid responsiveness during low tidal volume ventilation, Crit. Care Med. 45 (3) (2017) 415–421, 10.1097/CCM.0000000000002183. [DOI] [PubMed] [Google Scholar]
[20].De Backer D, Taccone FS, Holsten R, Ibrahimi F, Vincent JL, Influence of respiratory rate on stroke volume variation in mechanically ventilated patients, Anesthesiology 110 (5) (2009) 1092–1097, 10.1097/ALN.0b013e31819db2a1. [DOI] [PubMed] [Google Scholar]
[21].Reuter DA, et al. , Influence of tidal volume on left ventricular stroke volume variation measured by pulse contour analysis in mechanically ventilated patients, Intensive Care Med. 29 (3) (2003) 476–480, 10.1007/s00134-003-1649-7. [DOI] [PubMed] [Google Scholar]
[22].Szold A, Pizov R, Segal E, Perel A, The effect of tidal volume and intravascular volume state on systolic pressure variation in ventilated dogs, Intensive Care Med. 15 (1989) 368–371. [DOI] [PubMed] [Google Scholar]
[23].Mesquida J, Kim HK, Pinsky MG, Effect of tidal volume, intrathoracic pressure, and cardiac contractility on variations in pulse pressure, stroke volume, and intrathoracic blood volume, Intensive Care Med. 37 (2011) 1672–1679, 10.1007/s00134-011-2304-3. [DOI] [PMC free article] [PubMed] [Google Scholar]
[24].Zöllei É, et al. , Non-invasive detection of hypovolemia or fluid responsiveness in spontaneously breathing subjects, BMC Anesthesiol. 13 (2013) 40, 10.1186/1471-2253-13-40. [DOI] [PMC free article] [PubMed] [Google Scholar]
[25].Michard F, et al. , Clinical use of respiratory changes in arterial pulse pressure to monitor the hemodynamic effects of PEEP, Am. J. Respir. Crit. Care Med. 159 (3) (1999) 935–939, 10.1164/ajrccm.159.3.9805077. [DOI] [PubMed] [Google Scholar]
[26].Soltesz K, Robust computation of pulse pressure variations, Biomed. Signal Process. Control 39 (2018) 197–203, 10.1016/j.bspc.2017.07.021. [DOI] [Google Scholar]
[27].Aboy M, McNames J, Thong T, Phillips CR, Ellenby MS, Goldstein B, A novel algorithm to estimate the pulse pressure variation index DeltaPP, IEEE Trans. Biomed. Eng. 51 (12) (2004) 2198–2203, 10.1109/TBME.2004.834295. [DOI] [PubMed] [Google Scholar]
[28].Sun S, et al. , On algorithms for calculating arterial pulse pressure variation during major surgery, Physiol. Meas. 38 (12) (2017) 2101–2121, 10.1088/1361-6579/aa95a4. [DOI] [PubMed] [Google Scholar]
[29].Theres H, et al. , Phase-related changes in right ventricular cardiac output under volume-controlled mechanical ventilation with positive end-expiratory pressure, Crit. Care Med. 27 (5) (1999) 953–958, 10.1097/00003246-199905000-00033. [DOI] [PubMed] [Google Scholar]
[30].Jardin F, Farcot J-C, Gueret P, Prost J-F, Ozier Y, Bourdarias J-P, Cyclic changes in arterial pulse during respiratory support, Circulation 68 (2) (1983) 266–274. [DOI] [PubMed] [Google Scholar]
[31].Lansdorp B, et al. , Mechanical ventilation-induced intrathoracic pressure distribution and heart-lung interactions, Crit. Care Med 42 (9) (2014) 1983–1990, 10.1097/CCM.0000000000000345. [DOI] [PubMed] [Google Scholar]
[32].Michard F, Teboul J-L, Using heart-lung interactions to assess fluid responsiveness during mechanical ventilation, Crit. Care 4 (2000) 282–289. [DOI] [PMC free article] [PubMed] [Google Scholar]
[33].Sun S, et al. , Finger and forehead photoplethysmography-derived pulse-pressure variation and the benefits of baseline correction, J. Clin. Monit. Comput. 33 (2019) 65–75, 10.1007/s10877-018-0140-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
[34].Pybus DA, Real-time, spectral analysis of the arterial pressure waveform using a wirelessly-connected, tablet computer: a pilot study, J. Clin. Monit. Comput. 33 (1) (2019) 53–63, 10.1007/s10877-018-0145-0. [DOI] [PMC free article] [PubMed] [Google Scholar]
[35].Thiele RH, Colquhoun DA, Tucker-Schwartz JM, Gillies GT, Durieux ME, Radial-femoral concordance in time and frequency domain-based estimates of systemic arterial respiratory variation, J. Clin. Monit. Comput 26 (5) (2012) 393–400, 10.1007/s10877-012-9390-9. [DOI] [PubMed] [Google Scholar]
[36].De Backer D, Heenen S, Piagnerelli M, Koch M, Vincent JL, Pulse pressure variations to predict fluid responsiveness: influence of tidal volume, Intensive Care Med. 31 (4) (2005) 517–523, 10.1007/s00134-005-2586-4. [DOI] [PubMed] [Google Scholar]
[37].Heenen S, De Backer D, Vincent JL, How can the response to volume expansion in patients with spontaneous respiratory movements be predicted? Crit. Care 10 (4) (2006) R102, 10.1186/cc4970. [DOI] [PMC free article] [PubMed] [Google Scholar]
[38].Bronzwaer AS, Ouweneel DM, Stok WJ, Westerhof BE, Van Lieshout JJ, Arterial pressure variation as a biomarker of preload dependency in spontaneously breathing subjects – a proof of principle, PLoS One 10 (9) (2015), e0137364, 10.1371/journal.pone.0137364. [DOI] [PMC free article] [PubMed] [Google Scholar]
[39].De Backer D, Pinsky MR, Can one predict fluid responsiveness in spontaneously breathing patients? Intensive Care Med 33 (7) (2007) 1111–1113, PMID: 17508200. [DOI] [PubMed] [Google Scholar]
[40].Hong DM, Lee JM, Seo JH, Min JJ, Jeon Y, Bahk JH, Pulse pressure variation to predict fluid responsiveness in spontaneously breathing patients: tidal vs forced inspiratory breathing, Anaesthesia 69 (7) (2014) 717–722, 10.1111/anae.12678. [DOI] [PubMed] [Google Scholar]
[41].de RC Chaves F, et al. , Assessment of fluid responsiveness in spontaneously breathing patients: a systematic review of literature, Ann. Intensive Care 8 (1) (2018) 21, 10.1186/s13613-018-0365-y. [DOI] [PMC free article] [PubMed] [Google Scholar]
[42].Liu Y, et al. , Pulse pressure variation shows a direct linear correlation with tidal volume in anesthetized healthy patients, BMC Anesthesiol 16 (75) (2016) 1–7, 10.1186/s12871-016-0233-y. [DOI] [PMC free article] [PubMed] [Google Scholar]
[43].Westerhof N, Stergiopulos N, Noble MIM, Snapshots of Hemodynamics: an Aid for Clinical Research and Graduate Education, Springer, Boston, MA, 2005, 10.1007/b101305. [DOI] [Google Scholar]
[44].Hoppensteadt FC, Peskin CS, Modeling and Simulation in Medicine and the Life Sciences, 2nd edition, Springer-Verlag, New York, 2002, 10.1007/978-0-387-21571-6. [DOI] [Google Scholar]
[45].Westerhof N, Lankhaar J-W, Westerhof B, The arterial Windkessel, Med. Biol. Eng. Comput. 47 (February (2)) (2009) 131–141, 10.1007/s11517-008-0359-2. [DOI] [PubMed] [Google Scholar]

PERMALINK

A robust Fourier-based method to measure pulse pressure variability

Sebastian Acosta

Mubbasheer Ahmed

Suellen M Yin

Ken M Brady

Daniel J Penny

Craig G Rusin

Abstract

Objective

Methods

Results

Significance and conclusion

1. Introduction

Fig. 1.

2. Traditional methods (peak-finding)

Fig. 2.

Fig. 3.

3. Proposed method (Fourier)

Fig. 4.

4. Results

4.1. Synthetic data

Fig. 5.

4.2. Patient data

Fig. 6.

Fig. 7.

5. Conclusion and limitations

Funding

Appendix A

Footnotes

References

ACTIONS

PERMALINK

RESOURCES

Cite

Add to Collections

PERMALINK

A robust Fourier-based method to measure pulse pressure variability

Sebastian Acosta

Mubbasheer Ahmed

Suellen M Yin

Ken M Brady

Daniel J Penny

Craig G Rusin

Abstract

Objective

Methods

Results

Significance and conclusion

1. Introduction

Fig. 1.

2. Traditional methods (peak-finding)

Fig. 2.

Fig. 3.

3. Proposed method (Fourier)

Fig. 4.

4. Results

4.1. Synthetic data

Fig. 5.

4.2. Patient data

Fig. 6.

Fig. 7.

5. Conclusion and limitations

Funding

Appendix A

Footnotes

References

ACTIONS

PERMALINK

RESOURCES

Similar articles

Cited by other articles

Links to NCBI Databases