Excess pressure as an analogue of blood flow velocity

Abstract

Introduction

Derivation of blood flow velocity from a blood pressure waveform is a novel technique which could have potential clinical importance. Excess pressure, calculated from the blood pressure waveform via the reservoir-excess pressure model, is purported to be an analogue of blood flow velocity, but this has never been examined in detail, which was the aim of this study.

Methods

Intra-arterial blood pressure was measured sequentially at the brachial and radial arteries via fluid filled catheter simultaneously with blood flow velocity waveforms recorded via Doppler ultrasound on the contralateral arm (n=98, aged 61±10, 72% male). Excess pressure was derived from intra-arterial blood pressure waveforms using pressure-only reservoir-excess pressure analysis.

Results

Brachial and radial blood flow velocity waveform morphology were closely approximated by excess pressure derived from their respective sites of measurement (median cross-correlation coefficient r=0.96 and r=0.95 for brachial and radial comparisons respectively). In frequency analyses, coherence between blood flow velocity and excess pressure was similar for brachial and radial artery comparisons (brachial and radial median coherence=0.93 and 0.92 respectively). Brachial and radial blood flow velocity pulse heights were correlated with their respective excess pressure pulse heights (r = 0.53, p <0.001 and r = 0.43, p <0.001 respectively).

Conclusion

Excess pressure is an analogue of blood flow velocity, thus affording the opportunity to derive potentially important information related to arterial blood flow using only the blood pressure waveform.

Introduction

Continuous non-invasive recording of blood pressure (BP) and flow is valuable in the settings of anaesthesiology, cardiology and emergency care for the hemodynamic assessment and management of the critically ill. Several methods exist for recording continuous non-invasive BP, many of which are straightforward to apply and nondemanding for the operator [1]. Continuous non-invasive measures of blood flow velocity and volumetric flow are also possible via Doppler ultrasound. However, Doppler-capable devices can be prohibitively expensive and require the presence of a skilled operator to hold the transducer at a fixed angle over the artery. Given the interdependence of BP and blood flow, methods have been proposed whereby volumetric blood flow may be estimated via analysis of the BP waveform (e.g. pulse contour analysis), thus circumventing the challenges posed by conventional blood flow assessment [2–5]. Yet, methods utilising pulse contour analysis have been shown to be inaccurate during hemodynamic instability [6].

The reservoir-excess pressure model is a heuristic model of arterial hemodynamics that separates the measured BP waveform into reservoir pressure and excess pressure components. [7,8] These components can be derived from peripheral arterial BP waveforms recorded invasively or non-invasively [9,10]. In their original study outlining the reservoir-excess pressure model, Wang et al. [7] demonstrated striking similarities between the shape of the excess pressure and volumetric flow waveforms in the dog aorta. More recently, excess pressure derived from non-invasively acquired carotid artery waveforms closely approximated aortic volumetric flow in humans. [11] Thus, excess-pressure represents a potential opportunity to measure clinically relevant information related to both BP and flow from only the arterial BP waveform and without the requirement for specialised flow-monitoring equipment. However, the equivalency of excess pressure to blood flow velocity has never been simultaneously compared using invasive BP and direct measurement of blood flow velocity in humans, which was the aim of this study.

Methods

Participants

A total of 146 individuals were approached for inclusion in the study at the Royal Hobart Hospital (Hobart, Australia) prior to elective coronary angiography. Study exclusion criteria included inter-arm cuff systolic and/or diastolic BP difference >5 mmHg (n=5), the presence of aortic stenosis or arrhythmias (n=8), arterial access only available via the femoral artery (n=7) and technical or medical issues arising that prevented the measurement of study variables (n=13). Additionally, data capture was unsuccessful at the brachial and radial arteries in 5 and 9 individuals respectively, and 6 individuals declined study participation. Reservoir-excess pressure model analysis failed to meet pre-specified quality control (P∞ > 0 and <diastolic BP) in 14 individuals (brachial n=5, radial n=9), so complete data were available for 97 brachial and 89 radial comparisons of flow velocity with excess pressure. Participants’ clinical history (hypertension, smoking and hyperlipidaemia status) and anthropometric measurements were obtained from coronary angiography pre-assessment documentation. Clinical information was collected from the hospital digital health records. All participants gave written informed consent and ethical approval was granted by the University of Tasmania Human Research Ethics Committee.

Blood flow velocity and intra-arterial blood pressure acquisition

Methods relating to the recording of intra-arterial BP have previously been published. [12] Intra-arterial BP was recorded using a fluid filled catheter with intra-arterial access via the right radial artery. Blood flow velocity was recorded using two-dimensional pulsed Doppler ultrasound with 12 MHz linear-array transducer (Vivid i, GE Healthcare, Chicago, IL, USA; Figure 1). Pulsed wave Doppler flow velocities were recorded at a transmission frequency of 12 MHz, with a fixed angle of insonation of 60 degrees and sample volume encompassing the lumen cross-section. The envelope of peak instantaneous blood flow velocity was derived offline using EchoPAC software (GE Healthcare, Chicago, IL, USA; Figure 1) and converted to text format for analysis using automated line tracing software. Immediately following completion of the coronary angiography procedure, the catheter was positioned in the right mid-brachial artery and continuous intra-arterial BP was recorded. Simultaneously with intra-arterial brachial BP recordings, blood flow velocity was recorded from the mid-brachial artery on the contralateral arm. Following successful data capture at the brachial artery, the intra-arterial catheter was pulled back to the radial artery and intra-arterial radial BP was recorded simultaneously with radial blood flow velocity recorded on the contralateral arm. Intra-arterial BP was recorded at a sampling frequency of 1000 Hz. Continuous intra-arterial BP and blood flow velocity waveforms were then ensemble averaged using up to 7 cardiac cycles (no less than 4) and the ensembled waveforms were cropped to the shortest cardiac cycle. The ensemble averaged waveforms were used for analysis.

Figure 1.

Example of reservoir-excess pressure parameters derived from an ensemble averaged brachial blood pressure waveform (left panel). Measurement of brachial artery Doppler ultrasound flow velocity (right panel).

Derivation of excess pressure waveform

Analysis of the ensemble averaged BP waveforms was performed using custom-written scripts in MATLAB (The MathWorks Inc, USA). Reservoir pressure (P_res) was estimated from:

$$\frac{d{P}_{res}}{dt}=ks(P-P_{res})-kd(P_{res}-P_{\infty })$$ (1)

where P is the total measured pressure, ks is the systolic rate constant, kd is the diastolic rate constant and P _∞ is the arterial asymptotic pressure. This first-order linear differential equation was solved as:

$$P_{res}=e^{-(ks+kd)t}{\displaystyle \underset{0}{\overset{t}{\int }}P(t^{\prime })e^{(ks+kd)t^{\prime }}d{t}^{\prime }}+\frac{kd}{ks+kd}\times (1-e^{-(ks+kd)t})P_{\infty }$$ (2)

The diastolic parameters, kd and were P _∞ estimated by fitting an exponential curve to P during diastole, and ks was estimated by minimizing the sum of squares of error between P and obtained over diastole. To calculate the excess pressure, P_res was subtracted from P. Derivation of P_res, excess pressure, ks and kd from the BP waveform can be seen in Figure 1.

Additional processing

Excess pressure and flow velocity waveforms were zero normalised and temporally aligned by cross-correlation. Prior to normalising, the pulse height of blood flow velocity and excess pressure waveform was calculated by subtracting the minimum value from the maximum value (e.g. maximum flow velocity – minimum flow velocity).

Statistical analysis

Unless stated otherwise, data are expressed as mean ± SD or n (%). All statistical analyses were performed in R, version 3.5.3 for Windows (R Foundation for Statistical Computing, Vienna, Austria). Cross-correlation (via the ccf function) and magnitude-squared coherence (via the spec.pgram function) were used to compare agreement between excess pressure waveforms and blood flow velocity waveforms in the time and frequency domain respectively. In time domain analyses, values of coherence were quantified between 0 and 1 and interpreted in a similar manner to the Pearson’s R coefficient. Specifically, a coherence value of 0 indicates no causal relationship between excess pressure and flow, whereas a coherence value of 1 indicates a linear frequency response between excess pressure and flow. To reduce the influence of noise (from high frequencies), the mean of coherence values up to 10Hz were calculated. Zou’s confidence interval was used to determine differences in cross-correlation coefficients between groups [13,14]. Linear regression and Pearson’s r were used to examine associations of continuous variables after the assumption of linearity was confirmed by examination of residuals. Read re-read reliability of flow velocity and BP measures was determined by two-way mixed model analysis and summarized by the intraclass correlation.

Results

Clinical characteristics

Clinical characteristics of study participants are presented in Table 1. Participants were middle to older age and consisting of mostly males and often with coronary artery disease defined by mild to severe narrowing in at least one coronary vessel. Hemodynamic variables of study participants are presented in Table 2.

Table 1.

Clinical characteristics of study participants

Variable	Mean ± SD or n (%)
Age (years)	61.1 ± 10.4
Sex (male)	70 (72)
Height (cm)	170.1 ± 10.8
Weight (kg)	87.4 ± 16.5
BMI (kg/m2)	30.2 ± 4.5
Family history of CVD	55 (60)
Hypertension	37 (41)
Current smoker	18 (19)
Hyperlipidaemia	65 (70)
Type 2 diabetes mellitus	28 (30)
Coronary artery disease	71 (76)

n = 97. CVD, cardiovascular disease; SD, standard deviation.

Table 2.

Hemodynamic variables of study participants

Variable	Mean ± SD
Brachial
Invasive SBP (mmHg)	135.7 ± 23.5
Invasive DBP (mmHg)	66.5 ± 10.5
Invasive mean arterial pressure (mmHg)	94.3 ± 13.0
Peak flow velocity (cm/s)	68.1 ± 18.6
Mean flow velocity (cm/s)	9.1 ± 4.3
Heart rate (bpm)	62.8 ± 10.9
Peak excess pressure (mmHg)	41.4 ± 13.5
Excess pressure integral (mmHg)
Radial
Invasive SBP (mmHg)	141.8 ± 24.7
Invasive DBP (mmHg)	66.8 ± 10.6
Invasive mean arterial pressure (mmHg)	94.2 ± 13.2
Peak flow velocity (cm/s)	50.6 ± 16.3
Mean flow velocity (cm/s)	8.5 ± 5.4
Heart rate (bpm)	61.9 ± 10.7
Peak excess pressure (mmHg)	48.5 ± 14.7
Excess pressure integral (mmHg)	8.05 ± 3.5

n = 97. BP, blood pressure; SD, standard deviation.

Comparison of excess pressure with blood flow velocity

Comparisons of blood flow velocity and excess pressure waveforms can be seen in Figure 2. Brachial artery blood flow velocity was highly cross-correlated with excess pressure derived from brachial BP waveforms (Table 3, cross-correlation coefficient range = 0.72 to 0.99). Similarly, radial artery blood flow velocity was highly cross-correlated with excess pressure derived from radial artery BP waveforms (Table 3, cross-correlation coefficient range = 0.81 to 0.99). In frequency analyses, coherence between blood flow velocity and excess pressure was similar for brachial and radial artery comparisons (brachial median coherence = 0.93 and radial median coherence = 0.92). 32% of brachial artery excess pressure and flow comparisons had coherence values ≥0.95 and 73% had coherence ≥0.90. 27% of radial artery comparisons had coherence values ≥0.95 and 70% had coherence values ≥0.90. Mean square error was 0.09 ± 0.07 for brachial waveform comparisons and 0.11 ± 0.09 for radial waveform comparisons. Root mean square difference for blood flow velocity and excess pressure were 0.29 ± 0.11 and 0.32 ± 13 for brachial and radial artery comparisons respectively. Brachial blood flow velocity pulse height was linearly associated with excess pressure pulse height (r = 0.52, p <0.001). Similarly, radial blood flow velocity pulse height was associated with excess pressure pulse height (r = 0.43, p <0.001). Comparisons of wave intensity patterns using measured flow velocity and excess pressure are provided in the supplemental material.

Figure 2.

Comparisons of flow velocity (solid line) with excess pressure (dashed line) and respective blood flow velocity-excess pressure loop. Xcor is the cross-correlation coefficient. Figures represent typical examples of excess pressure and flow velocity comparisons in 75^th percentile, median and 25^th percentile of cross-correlation value. Coherence was 0.98 (minimum = 97) for the 75^th percentile example, 0.94 (minimum = 89) for the median example, and 0.86 (minimum = 71) for the 25^th percentile example.

Table 3.

Cross-correlation coefficients (r value) of flow velocity with excess pressure derived from brachial and radial arteries

Arterial site	25th percentile	Median	75th percentile
Brachial	0.94	0.96	0.97
Radial	0.92	0.95	0.96

Differences in cross-correlation coefficient between groups

There was no difference in brachial cross-correlation coefficient between individuals stratified by sex (95% confidence interval[CI] = -0.047, 0.062), hypertension status (95%CI = -0.035, 0.078), presence of coronary artery disease (95%CI = -0.053, 0.049), hyperlipidaemia (95%CI = -0.026, 0.086), smoking status (95%CI = -0.050, 0.058) or type 2 diabetes status (95%CI = -0.0650, 0.0158). Similarly, there were no differences in radial cross-correlation coefficients between groups.

Read re-read analysis

Intraclass correlation coefficient for peak blood flow velocity was 0.99 (95%CI = 0.99 to 0.99). Intraclass correlation coefficient for peak excess pressure was 0.99 (95%CI = 0.99 to 1).

Discussion

The aim of the present study was to determine the relationship of excess pressure to blood flow velocity derived from brachial and radial artery waveforms. Our main finding was that the envelope of excess pressure derived from either brachial or radial artery waveforms corresponded closely to the measured blood flow velocity envelope at each arterial site. These findings highlight that important information about the blood flow waveform may be derived from assessment of the BP waveform alone, which may be adapted for use in the clinical setting. We envision our findings may provide useful information for continuous hemodynamic monitoring and may facilitate more detailed BP waveform analysis which require measured flow, such as wave intensity and wave separation analyses. Nevertheless, future studies are needed to determine the usefulness of our findings for these purposes.

A continuous, non-invasive and operator independent method for accurate assessment of stroke volume is highly sought after for improving clinical decisions in the critical care setting. Pulse contour analysis is a method whereby stroke volume is estimated from the BP waveform and has been the focus of numerous investigations [15]. Several techniques employing pulse contour analysis have attempted to exploit the relationship between the windkessel-related BP and blood flow to estimate stroke volume. [15,16] However, these methods require individual patient calibration with a reference standard to achieve accurate absolute values of stroke volume. In this regard, calibration is often performed via transpulmonary thermodilution which necessitates intra-arterial access to the pulmonary artery. Additionally, calibration methods assume fixed arterial properties, but these change over time, ultimately resulting in inaccurate estimates and a requirement for frequent re-calibration. [17,18] Interestingly, Kamoi et al. [19] used a porcine model to show that the estimation of stroke volume may be optimised via the application of the reservoir-excess pressure model by reducing the number of fixed assumptions in the derivation of the windkessel pressure. They went on to show that the excess-pressure model in combination with pulse-wave velocity measures may facilitate more precise estimates of vessel dimensions and further improve estimates of stroke volume from the BP waveform alone. [20] Furthermore, a quantitative estimate of flow velocity may be achieved by scaling the peak of the excess pressure waveform to 1m/s, which, based on recent large population studies in Norway and Korea, seems a reasonable estimate for an assumed peak velocity. [21–23] This may facilitate the estimation of parameters such as stroke distance (analogous to stroke index) and minute distance (analogous to cardiac index). Yet, this method provides only an approximate estimation of a quantitative flow velocity waveform and as a result, its clinical value will be greatly restricted.

As a result of the inverse relationship between the absolute magnitude of flow and characteristic impedance, surrogate flow waveforms for use in wave separation analysis do not require calibration. [3,24] Thus, previous investigators have employed a triangular flow approximation method for pressure only wave separation analysis, where flow in the aorta is assumed to be triangular in shape. [2,3] However, a more physiologically representative blood flow waveform, such as we have examined in this current study for excess pressure, may provide better results for the purpose of wave separation analysis. [24–26] Furthermore, in a recent study it was shown that aortic wave intensity analysis performed using excess pressure as a surrogate flow velocity waveform provides reasonable estimates of wave intensity parameters. The concordance between excess pressure and flow velocity observed in the present study indicate that excess pressure derived from peripheral artery BP waveforms may also prove useful for wave intensity analysis. Yet, among some individuals the concordance between excess pressure and flow velocity was poor and the implications of this for wave separation and wave intensity analyses need to be determined. Future work should aim to identify appropriate cut-off values for what constitutes good agreement between excess pressure and flow velocity. As it stands, excess pressure may provide a reasonable surrogate waveform for wave separation and intensity analysis, in most, but not all, individuals. [21,24]

In the aorta, wave intensity is dominated by a forward traveling compression wave in early systole followed by a forward travelling decompression wave immediately preceding diastole. Waves (forward and backward traveling) are present throughout the entire cardiac cycle but the intensity of backward traveling waves in the aorta during diastole is minimal. [27–29] Under these conditions, the excess pressure waveform is analogous to the blood flow velocity waveform being related to it through the characteristic impedance of the aorta. [30] However, in the peripheral arteries the contribution of backward traveling waves to the measured BP waveform is larger due to proximity to sites of impedance mismatch. [31,32] Indeed, reflected waves explain at least in part why the contribution of excess pressure to the BP waveform increases moving distally from the aorta. [10] In this regard, the concordance of excess pressure (wave related pressure) with directly measured flow velocity measured from peripheral artery waveforms, may deviate due to the contribution of backward traveling waves. [33] There was some evidence for this in our wave intensity analyses using measured flow velocity (supplemental material, Figure S1). In the 75^th percentile example (i.e. good concordance), there was minimal backward wave activity. Whereas, in the 25^th percentile example (i.e. poor concordance) there was noticeably greater backward wave activity. The overall importance of this in practice remains to be comprehensively determined, certainly we still observed strong relationships on average between excess pressure and blood flow velocity at peripheral arterial sites. In this regard, when using excess pressure as a flow surrogate for wave intensity analyses, we observed that the forward compression wave was somewhat comparable to that obtained by wave intensity analyses using the measured flow velocity. Nevertheless, we also observed that backward wave activity was not reproducible when using excess pressure as a flow surrogate, as evidenced by the median and 25^th percentile examples in Figure S1. Therefore, caution should be exercised before employing peripheral artery excess pressure for the purposes of wave intensity analysis and more detailed studies are needed to confirm the usefulness of our findings for these envisioned applications.

Previous studies have shown that excess pressure derived from peripheral artery BP waveforms is associated with cardiovascular events and impaired kidney function independent of conventional risk factors. [9,34,35] A numerical analysis of the reservoir-excess pressure model posits that excess pressure represents the additional work performed by the heart above the minimum required work. [36] This suggests that excess pressure may be a marker of circulatory inefficiency. This current study extends previous findings of the equivalency of aortic blood flow velocity with excess pressure to brachial and radial arteries. In this regard, excess pressure may facilitate more detailed BP waveform analyses, such as wave intensity analysis, which could provide useful clinical information for deeper BP phenotyping and risk stratification beyond excess pressure alone. [37] Altogether, when derived from peripheral artery BP waveforms, excess pressure may be considered a cardiovascular risk marker encompassing information on cardiovascular efficiency and local blood flow dynamics.

Limitations

Due to the nature of the clinical setting and procedure from which the data were collected, it was not possible to acquire simultaneous intra-arterial BP and blood flow velocity in the same arm. Thus, it is assumed that hemodynamics between arms are comparable, an assumption that may not be true for all individuals. However, inter-arm cuff systolic BP differences >5 mmHg was a study exclusion criterion and may have lessened the potential influence of inter-arm hemodynamic variability. Additionally, our study sample included mostly older people who were undergoing coronary angiography and is not representative of young, healthy individuals nor critical care patients for whom comprehensive hemodynamic monitoring is most valuable. Future studies should determine if the findings of the present study are generalisable to young, healthy individuals and patients in the surgical or intensive care settings. Finally, we provide some hypothesis generating discussion on the potential usefulness of our findings and though there was good agreement between excess pressure and flow velocity among many individuals, in others there are clear differences, which may have important implications for the envisioned applications of our findings. In future studies it would be valuable to identify potential predictors of poor concordance between excess pressure and flow to help refine this method.

Conclusion

Excess pressure derived via the pressure only reservoir-excess pressure model may represent a useful method for assessment of arterial hemodynamics and circulatory function. Previous studies have shown that aortic excess pressure is proportional to aortic volumetric flow. In the present study, excess pressure derived from peripheral artery BP waveforms corresponded closely to the measured flow velocity waveform. This is of potential clinical importance as continuous non-invasive recording of peripheral artery BP waveforms is easy to perform and thus, removes barriers associated with conventional methods of blood flow assessment. Indeed, many non-invasive continuous hemodynamic monitoring systems seek to estimate hemodynamic indices from the peripheral artery BP waveform, including finger, radial and brachial BP waveforms. [38] Therefore, our findings could have implications for the clinical assessment of hemodynamic parameters and may provide important information for improving continuous, non-invasive and operator independent hemodynamic monitoring.

Figure 3.

Radial blood flow velocity and excess pressure (A) with associated first derivatives (B) and frequency coherence (C). Complex blood flow velocity waveform morphology was well matched by excess pressure and maximum coherence occurred at a frequency of 0.7 Hz.