Introduction

Under a good approximation, the fluid-filled catheter can be characterized as a second-order linear system.5,6 Gauge pressure measurements using fluid-filled catheters are widely used in medical practice for blood pressure measurements and are known to distort pressure signals depending on the catheter system transfer function.5,6 The output signal can be corrected using an inversed transfer function,5 or by harmonic analysis.7 The transfer function is affected by the measurement system components: the catheter diameter, length, material, the fluid viscosity, presence of air bubbles inside the extension tubing, and the pressure transducer characteristics.3,6 Other applications require simultaneous pressure measurements from two locations, or a single differential measurement, such as in aortic stenosis catheterization2,4 or in measurement of pressure drop along arterial flow. For example, in clinical practice, pressure drop is used as a surrogate for flow1 (e.g., across a coronary stenosis), since direct and accurate flow measurement inside arteries is very difficult. In comparison to gauge pressure measurement using a single fluid-filled catheter and a gauge pressure sensor, measurement and restoration of differential pressure using a fluid-filled double-lumen catheter and a differential sensor might be more challenging since the two pressure lines are connected via a common sensor, and therefore dependent.

We have conducted many tests in our lab of pressure gradient measurements along a fluid flow in a cylindrical tube using a differential pressure transducer and a fluid-filled double-lumen catheter, and found that the most significant effect on the distortion of the output signal is due to dynamic common mode pressure (CMP) effect. The effect of CMP distortion on the differential pressure measurement will be demonstrated in “Validation of Pressure Drop Restoration” section. CMP is the line pressure that is common to both ports. CMP is not directly measured by differential pressure sensors, however in dynamic systems with fluctuating pressure a CMP distortion effect can superimpose any dynamics on the differential measurement, especially for very low differential pressure readings (e.g., range 0–2.5 kPa). In general, piezoresistive differential pressure sensors are subjected to CMP artifacts due to slight difference in gains with forward vs. reverse pressurization. The CMP effect increases due to physical differences (e.g., small kink in the wall or micro-bubble of air in one of the channels) between the negative and positive channels, when small diameter extension tubing is connected to the differential sensor ports. The physical differences, which are relatively more significant in small diameter catheters, results in a different time delay between the channels and therefore increase the CMP effect. It is almost impossible to avoid the CMP effect when using small diameter fluid-filled extension tubing (e.g., medical catheters) since it is significantly affected by the presence of air bubbles inside the tubing. Therefore, all the air bubbles inside the working fluid should be removed even if it were depressurized initially. The resulting CMP effect can be an order of magnitude higher than the true differential signal. An example can be seen in Fig. 1 where the CMP effect has distorted the differential measurement which resulted in a signal that is unsynchronized with the flow wave. The flow and pressure conditions in this example are depicted in Fig. 1, and the experimental setup is fully described in the “Materials and Methods” section. To our knowledge, the subject of restoration of differential pressure signals affected by CMP artifact in double-lumen fluid-filled catheters has not been discussed in the literature to date.

Figure 1
figure 1

An example of pressure drop measurement using a double-lumen fluid-filled catheter where the delay mismatch between the positive and negative pressure channels caused the pressure drop output signal to be unsynchronized with the gauge pressure and flow signals

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The objective of the present study was to restore the original differential pressure signal from the distorted differential pressure measurement that was recorded from a fluid-filled double-lumen catheter. We developed a method to facilitate the restoration of the pressure gradient signal, and proposed a restoration function, demonstrated its efficacy against three extreme representative cases of CMP effects, and compared the restored signals to undistorted reference signals. We also tested the repeatability and efficacy of the function over 1 h.

Materials and Methods

Restoration Function

We have noticed a mismatch in the transfer functions between the positive and negative pressure channels, which is also expressed as differences in time delays. The sensing element inside the differential transducer measures gauge pressure changes first in the channel with the smaller delay, and a fraction of time later in the channel with the larger delay. The distorted output signal is amplified into positive or negative values, depending on which of the pressure channels has the smaller delay. The restoration function is based on CMP being mainly affected by the above mismatch.

We correlated the CMP artifact with the gauge pressure and its derivatives, which are measured near the catheter tip:

$$ P_{\text{cmp}} = m_{0} + m_{1} P_{\text{g}} + m_{2} \frac{{dP_{\text{g}} }}{dt} + m_{3} \frac{{d^{2} P_{\text{g}} }}{{dt^{2} }} + \cdots + m_{n} \frac{{d^{n - 1} P_{\text{g}} }}{{dt^{n - 1} }} $$
(1)

where P cmp (mmHg) is the CMP artifact, P g (mmHg) is the gauge pressure, and m 0 to m n are coefficients. The series described in Eq. (1) was based on observations from our experiments, in which we have identified a similarity in signal morphology between the measured P cmp and the first derivative of the gauge pressure (\( {{dP_{\text{g}} } \mathord{\left/ {\vphantom {{dP_{\text{g}} } {dt}}} \right. \kern-0pt} {dt}} \)). We then hypothesized that P g itself as well as higher order derivatives of P g also correlate with P cmp.

The measured pressure drop (P dm) is assumed to be composed of the true pressure drop (P d) and the CMP (P cmp):

$$ P_{\text{dm}} = P_{\text{d}} + P_{\text{cmp}} $$
(2)

The criterion under which the assumption is valid is that the time response of the differential measurement system (differential sensor and extension tubing or catheters) is sufficient enough for the application; otherwise, P d cannot be related as ‘true’ pressure drop.

For two identical channels which are exposed to the same pressure source P g, the differential measurement should be zero (P d = 0), therefore:

$$ P_{\text{dm}} = P_{\text{cmp}} $$
(3)

Using Eq. (3) we can measure P cmp directly and fit it according to Eq. (1) to find the coefficients m 0 to m n . Plugging the coefficients into Eq. (2) facilitate the calculation of P d:

$$ P_{\text{d}} = P_{\text{dm}} - (m_{0} + m_{1} P_{\text{g}} + m_{2} \frac{{dP_{\text{g}} }}{dt} + m_{3} \frac{{d^{2} P_{\text{g}} }}{{dt^{2} }} + \cdots + m_{n} \frac{{d^{n - 1} P_{\text{g}} }}{{dt^{n - 1} }}) $$
(4)

The correlation of Eq. (1) is optimized by adjusting the time phase between the measured P cmp and the gauge pressure derivatives \( ({{d^{n - 1} P_{\text{g}} } \mathord{\left/ {\vphantom {{d^{n - 1} P_{\text{g}} } {dt^{n - 1} }}} \right. \kern-0pt} {dt^{n - 1} }}) \). While performing the experiments, a time phase was noticed between the aforementioned signals. This phase is expressed as ‘hysteresis’ like graph when plotting the measured P cmp with the gauge pressure derivatives (Fig. 2). The graph can be rotated to achieve better linear fitting by adjusting this phase. The constants m 0 to m n for the time phase with the highest fitting R 2 value are retained and used for the signal restoration in Eq. (4). The percent weight of each term of Eq. (1) is calculated according to Eqs. (5) and (6):

$$ W_{0} (t) = \frac{{\left| {m_{0} } \right|}}{{\left| {m_{0} } \right| + \sum\limits_{i = 1}^{n} {\left| {m_{i} \frac{{d^{i - 1} P_{\text{g}} (t)}}{{dt^{i - 1} }}} \right|} }} \times 100 $$
(5)
$$ W_{n} (t) = \frac{{\left| {m_{n} \frac{{d^{n - 1} P_{\text{g}} (t)}}{{dt^{n - 1} }}} \right|}}{{\left| {m_{0} } \right| + \sum\limits_{i = 1}^{n} {\left| {m_{i} \frac{{d^{i - 1} P_{g} (t)}}{{dt^{i - 1} }}} \right|} }} \times 100 $$
(6)

where W n (t) (%) is the percent weight of the term n from the total sum of weights at time t. According to the weights we can determine what is the maximal term n to be used in Eq. (4).

Figure 2
figure 2

Example of a typical P cmp(dP g /dt) curve with ‘hysteresis’ like shape. The P cmp(dP g /dt) curve can be rotated to fit a linear shape in a good approximation by adjusting the phase between the P cmp and dP g /dt signals

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Experimental Setup

A closed flow loop in vitro system was driven by a piston pump (Enabler™, HemoDynamics™ Systems Ltd., Yokneam, IL), a compliance chamber, a custom made piston converter for low flow rates, and custom made valves (Fig. 3). A rigid tube model (5 mm inner diameter, 15 cm long), which was connected to the flow loop, was placed inside a water bath. A water reservoir supported a constant water head for a baseline hydrostatic pressure of 80 mmHg (10.66 kPa) inside the tube. The gauge pressure was further controlled by resistance adjustment using a tap upstream of the reservoir. The water in the bath and in the reservoir was heated to 37 ± 0.2 °C (310.15 ± 0.2 K) using temperature controllers. Temperatures were continuously monitored directly from the flow loop downstream to the tube using a thermocouple (5SC-TT-T, Omega Engineering Inc.) and a digital temperature controller (PXR4, Fuji Electric Systems Co., Ltd.). Gauge pressure was measured 5 cm upstream of the tube using a gauge pressure transducer (PX242A-005G, Omega Engineering Inc.), a T-connector, and a fluid-filled extension tubing (5 mm inner diameter, 20 cm long PVC tubing). Flow rate was measured 20 cm upstream of the model using an ultrasonic flow-meter probe (Model 6NRB490, Transonic Systems Inc. Ithaca, NY). Pressure drop measurements inside the flow were taken from the tube lumen by a wet/wet differential pressure transducer via a direct connection to the tube, or through a 5 French (1.7 mm) 18/18 gauge (0.9/0.9 mm) double-lumen fluid-filled catheter (Morpheus®, AngioDynamics Inc., Latham, NY). The differential pressure transducer has a range of 0–18.7 mmHg and was calibrated to an accuracy of 0.03% full scale (PX419, Omega Engineering Inc.). The double lumen catheter was cut to a length of 45 cm, and two side-holes were punctured near the catheter tip, one in each of the catheter lumens, 3 cm apart. The lumens were then sealed with epoxy glue in the region between the side-holes and the catheter tip to avoid the remains of air pockets in that region. The proximal end of the lumen with the proximal side-hole was connected to the positive port of the differential pressure transducer, and the proximal end of the lumen with the distal side-hole was connected to the negative port of the transducer.

Figure 3
figure 3

Schematic illustration of the experimental setup

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Data acquisition and analysis of the gauge pressure, flow rate, and pressure drop were performed using a PC with an I/O board, and dedicated software (LabView, National Instruments Corporation, Austin Texas). Pressure and flow signals were sampled at a rate of 1 kHz. Pulsatile flow cycles were adjusted to approximately 0–300 mL/min (0–5 × 10−6 m3/s) at a rate of 1 Hz. The resistance tap and compliance chamber were manipulated to achieve pulsatile gauge pressures of approximately 80–120 mmHg. The acquired signals were filtered by the software using a 7 order IIR Butterworth low-pass filter with a cutoff frequency of 7 Hz. All the variables listed in Eqs. (1)–(6) were automatically monitored, acquired and calculated online by our custom made software.

The working fluid was distilled water treated to remove all small air bubbles by heating and vacuuming. The water in the flow loop was heated to approximately 50 °C to reduce the solubility of air. The heating was then turned off and the water reservoir was connected to a vacuum pump for approximately 1 h to remove soluble air bubbles. The heating was then turned on again to keep the flow loop temperature steady at 37 °C. Both ports of the double-lumen catheter were carefully flushed with distilled water for the removal of air bubbles from the catheter lumens. The catheter was then left with its tip inside the flow loop for approximately 1 h to allow the system to settle.

Common Mode Pressure Estimation

Pulsatile flow was applied through the flow loop, and the double-lumen catheter was positioned as depicted in Fig. 4 for the CMP estimation. The catheter was positioned with its two side-holes in a no-flow region exposed to the pulsatile gauge pressure. It was verified that there were no flow or recirculation regions near the catheter tip and side-holes by flow visualization of dye injection. The catheter was particularly positioned to equalize P d to zero. In this position, the differential pressure transducer reading equals the CMP distortion, as explained above. Ten consecutive gauge pressure (P g) and pressure drop (P dm) cycles were recorded simultaneously by our own software. The gauge pressure derivatives were calculated in real-time from the gauge pressure signal using a 5 points centered discretization scheme. The software was then used to optimize for the best fit between P dm and the gauge pressure derivatives (see Eq. (1)) as it scanned different time delays (1 ms steps) between P dm and\( ({{d^{n - 1} P_{\text{g}} } \mathord{\left/ {\vphantom {{d^{n - 1} P_{\text{g}} } {dt^{n - 1} }}} \right. \kern-0pt} {dt^{n - 1} }}) \). In every time delay checked the Least Square fitting method was adopted, and the best fit was determined according to the SVD algorithm. The coefficients m 0 –m 3 of the best fit were saved and used later on to restore P d using Eq. (4), and to calculate the percent weights of the derivative terms (W n (t)) using Eq. (5) and Eq. (6). The CMP effect was calculated as the peak-to-peak value of P cmp (PCM p2p), and the CMP estimation error (Err cmp) was calculated as the peak-to-peak of P dm–P cmp, which ideally should be perfectly zero.

Figure 4
figure 4

Schematic illustration of the catheter position for the CMP estimation. The double lumen catheter is positioned such that the two side-holes are located in a no-flow region (P d = 0). This position of the catheter is used for finding the restoration function constants m 0 –m 3 . The ± signs near the side-holes represent the pressure channel sign. The filled blue arrows represent the flow direction

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Validation

The double-lumen fluid-filled catheter was positioned inside the tube lumen to position its two side-holes adjacent to the model side-holes. Figure 5 shows the position of the double lumen catheter and the tube model for validation of the restored pressure drop (P d). The tube was comprised of two perpendicular side connections, 3 cm apart, for direct measurement with the differential pressure transducer. These measurements could be considered undistorted since the water-line from the tube lumen to the transducer sensing element was wide, short, and rigid (5 mm inner diameter, 10 cm long, PVC tubing). Ten consecutive cycles of the restored pressure drop were recorded successively using the double-lumen catheter and then other, similar cycles, directly using the tube side-holes. The catheter remained in place during both measurements in order to eliminate differences in pressure drops due to the catheter presence. Comparison between the catheter measurement and the direct measurement was then performed offline qualitatively, and quantitatively as well by root mean square error (RMSE):

$$ RMSE = \sqrt {\frac{{\sum\limits_{i = 1}^{n} {\left( {\left( {P_{\text{d}} {direct}} \right)_{i} - (P_{\text{d}} {estimated})_{i} } \right)^{2} } }}{n}} $$
(7)
Figure 5
figure 5

Schematic illustration of the catheter position for validating the pressure drop signal restoration. The pressure drop is measured once through the double-lumen catheter (P dm), and then through direct connection to the circular tube (P d direct), using the same differential sensor. In both cases the catheter position remains the same. The filled blue arrows represent the flow direction

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Experimental Plan

The validation of the restoration function was tested for variable CMP cases. Three extreme representative cases were presented in this manuscript: case 1—low CMP, and cases 2–3—high CMP. In cases 2–3 we intentionally inserted a small air bubble into the positive port of the differential pressure transducer to increase the CMP effect. The CMP in case 3 was slightly lower than in case 2, and with different morphology. In each of the three cases we first identified and fitted the CMP distortion when the catheter was placed in a no-flow region. Then, we placed the catheter tip inside the rigid tube to validate the restored pressure drop in a flow region.

Another test was performed for evaluating the repeatability and accuracy of the CMP fitting over 1 h period. In this case we placed the catheter in the no-flow region and calculated the coefficients of the restoration function as was described in “Common Mode Pressure Estimation” section, and then left the catheter in place for 1 h while the flow loop system was still running. After 1 h, we recorded the Err cmp value in comparison to its value at time t = 0. In addition, we calculated the new coefficients for t = 1 h, and compared them to the coefficients found at time t = 0. For statistical reasons, calculation of the coefficients at time t = 1 h was repeated three times.

Results

Estimation of Common Mode Pressure

Figure 6 provides representative distributions of flow rate and gauge pressure that were used in the rigid tube during the experiments. We have conducted many experiments, but for clarity we selected three representative cases. CMP estimates for cases 1–3 are shown qualitatively in Fig. 7 and quantitatively in Table 1. It can be seen that the estimated CMP is in very good agreement with the measured signal, both for case 1, where the CMP is relatively small (0.17 mmHg), and for case 2 and 3, where the CMP is significantly higher (1.49 and 1.28 mmHg, respectively) and with different signal morphology. The fitting R 2 values of Eq. (1) were better than 0.99 for all the cases. The error of the CMP estimation was calculated to be 0.014 mmHg for case 1, 0.070 mmHg for case 2, and 0.048 mmHg for case 3. Therefore, the CMP of the restored pressure drop signal was reduced by a factor of 12.1, 21.2, and 26.7 for cases 1, 2, and 3 respectively.

Figure 6
figure 6

Representative flow rate and gauge pressure curves used during the experiments

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Figure 7
figure 7

Measured and estimated CMP in cases 1–3. The estimation of CMP is based on the fitting according to Eq. (1)

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Table 1 Summary of the CMP fitting parameters
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The percent weights of the restoration function terms were calculated according to Eqs. (5) and (6). In all three cases the percent weights of terms with P g derivative order higher than 2 were smaller than 0.2%. These terms were therefore discarded from the restoration equations, meaning that n = 3 is the maximal index used.

Validation of Pressure Drop Restoration

Following the determination of CMP fitting coefficients, we were able to process pressure drop measurements from a flow region and restore the true pressure drop. Comparison of pressure drop measurements from cases 1–3 can be seen in Fig. 8. It can be seen that the estimated pressure drops, which were calculated using Eq. (4) on the catheter measurements (P dm), were in very good agreement with the undistorted pressure drops measured, taken directly from the model (P d direct). The RMSE values for case 1 were 0.076 mmHg for P dm and 0.033 mmHg for P d estimated. In case 2 the RMSE values were 0.378 and 0.058 mmHg for P dm and P d estimated respectively. The RMSE values for case 3 were 0.364 and 0.015 mmHg for P dm and P d estimated respectively. Therefore an improvement in the RMSE can be seen in all three cases.

Figure 8
figure 8

Five consecutive pressure drop restoration cycles of cases 1–3. P dm is the raw measurement from the catheter, P d estimated is the restored pressure drop from the catheter measurement, and P d direct is the pressure drop measured directly from the model

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Repeatability

We have checked the repeatability of the CMP fitting coefficients over 1 h period, and summarized it in Table 2. The maximal percent change for the fitting coefficients was 3.12 ± 0.48%, which accounted for increased CMP fitting error of 0.002 mmHg, as the Err cmp at time t = 0 h was 0.060 mmHg, and at t = 1 h was increased to 0.062 mmHg. The maximal P g derivative order used in this case for Eq. (1) is the first order derivative, as the maximal percent weight of the second derivative term (W 3 (t)) was 0.1%.

Table 2 Summary of the CMP fitting coefficients change over 1 h test
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Discussion

In clinical practice, measurement of intravascular pressure drop requires small diameter catheters with high fidelity sensing elements. To the best of our knowledge, there are no commercial catheters that can measure intravascular pressure drops in a single cavity over short distances of 1–3 cm, nor could we find published reports. The main reason is that such measurements require extremely high accuracy of the measurement system (on the scale of 10−2 mmHg), and most of the catheter systems use gauge pressure transducers with an accuracy that does not exceed 0.5 mmHg. Differential pressure transducers with the required accuracy exist and can be connected to fluid-filled catheters. Practically, however, CMP distortions are superimposed on the differential measurement and the smaller and longer the catheter is the smaller the signal-to-noise ratio, therefore significantly reducing the differential measurement accuracy.

Our newly developed method for the reduction of CMP distortion, allows reliable and accurate differential pressure measurement inside rigid tubes. The method incorporates two steps: (1) Placement of the catheter tip in a no-flow region, where it is still exposed to the fluctuating gauge pressure. CMP distortion is measured using the formula that we have developed above (Eq. (1)); (2) Placement of the catheter tip in a flow region, where we wish to perform pressure drop measurements. In this stage the original pressure drop (Eq. (4)) is restored based on the CMP fitting from stage 1. The restoration function, although appears similar to a form of the Taylor series, differs significantly: In the Taylor series, the derivatives are of the same function that we want to approximate, whereas in our case the derivatives are of a different function (P g). This difference makes it possible to continue P cmp approximation even when the catheter is no longer in the no-flow region. As stated above, a number of varying CMP cases were tested for validation of the restoration function, from which three extreme representative cases are presented.

In step 1 we placed the catheter tip in a no-flow region and assumed that the measured signal is actually the CMP. This allowed us to isolate it and find a fitting formula that best describes it. This assumption was later supported by the successful validation of the pressure drop measurements.

The CMP fitting using Eq. (1) was excellent, with R 2 value better than 0.99. This was shown for three representative cases of low (case 1) and high (cases 2–3) CMP distortions, and for different CMP distortion morphology (cases 2 and 3). In case 2 of higher CMP distortion, the distortion was an order of magnitude larger than the pressure drop itself (Fig. 8).

Validation of the restored pressure drop signal was also proved to be very successful in all the cases tested. The restored signal was closer to the reference signal even when the CMP distortion was very low (case 1). Case 1 was practically difficult to simulate as it required tedious manipulations with the working fluid for the reduction of soluble air bubbles. While case 1 could be obtained in vitro, cases 2–3 may represent better measurement conditions for clinical use where manipulation of the catheter fluids is minimal.

Minute differences between the restored pressure drop (P d estimated) and the reference pressure drop (P d direct), were still noticed. These differences are attributed to these measurements being taken successively rather than simultaneously. Therefore the comparison was actually performed on similar but not identical cycles.

CMP fitting coefficients over 1 h showed very good repeatability. It demonstrated that the method was reliable for short time periods and kept its accuracy, as the CMP fitting error increased only by 0.002 mmHg, when the catheter remained static for 1 h in the no-flow region. For longer time scales, the manipulation with the method is quite easy, and takes approximately 2 min to be performed. It should be noted that non-gentle manipulation of the catheter by the user may change the CMP artifact, Therefore in order to keep the accuracy of the restored differential signal it is advisable to repeat the CMP fitting from time to time.

Accurate differential blood pressure measurements are essential in the clinical practice for applications such as assessment of aortic stenosis, coronary stenosis, and aortic valves. Enhancing the differential measurement accuracy to the scale of 10−2 mmHg can set the ground for future clinical applications. An example for such an application is currently under investigation by the authors of this manuscript: The extremely high accuracy provided by the presented restoration method allows the intravascular measurement of local blood pressure drops inside arteries, even in the absence of a stenosis. We have found that the measured pressure drops, which functionally indicate the flow rate, are also highly affected by local changes in the arterial compliance. A possible future application for this study is a coronary catheter that can both functionally diagnose lesions, and determine their compliance based on pressure drop measurements alone. This will provide more information to the interventionalists and allow better decision making before stenting.

To the best of the authors’ knowledge, there are no reports in the literature on CMP distortion removal from differential measurements using two-lumen fluid-filled catheters (or two single-lumen catheters) and a differential sensor (or two gauge sensors). Most prior-art cases deal with gauge pressure measurements using a measurement system that is comprised of a gauge pressure sensor and a single lumen catheter. In these studies the restoration methods rely on the assumption that the fluid-filled catheter measurement system is considered as a dynamic system fully characterized by its natural frequency and damping ratio,5 or on harmonic analysis.7 Differential pressure sensor connected to two extensions tubing (in our case double-lumen catheter) has practically two different transfer functions, which are dependent due to a common sensor, therefore making state-of-the-art restoration methods impertinent in this case. Theoretically it is possible to obtain a differential measurement using two separate gauge pressure catheter measurement systems, with separate restoration functions for each of them (based on natural frequency and damping ratio), however in such case the accuracy of the differential measurement will hardly be close to that of the differential sensor measurement system.

There are several limitations for in vivo applications: First, placing the catheter side-holes in a no-flow region, which is essential in order to get a fitting of the CMP distortion, is not a straightforward procedure. We recommend for the in vivo setting to incorporate an introducer sheet, or a retractable sleeve, as compartments for the catheter that can keep the catheter side-holes in a no-flow region. Following the fitting procedure, the sleeve can be retracted to expose the side-holes to the flow. Second, low range differential pressure sensors (e.g., 0–18 mmHg) are more sensitive to changes in the fluid properties (viscosity, density, temperature) than gauge pressure sensors for the physiological blood pressure range (0–300 mmHg), especially when connected to small bore and long extension tubing (e.g., medical catheters). For this limitation, the CMP fitting should be repeated from time to time even in the same system. It should be noted though that restoration methods that characterize the measurement system by its natural frequency and damping ratio5 or on harmonic analysis7 are also affected by the fluid properties6,7 and therefore it is reasonable to assume that these methods also prone to reduced accuracy when the characterizing parameters are found in one system and then used in other system. Another limitation is the difference between rigid tubes and compliant tubes. While proving our method to be valid in rigid tubes, arteries are compliant vessels with viscoelastic wall properties, which are subjected to pulse wave velocity (PWV) phenomena. PWV in compliant tubes is much smaller than in rigid tubes, and heterogeneity in the arterial wall properties (e.g., due to calcifications, or bifurcations) is responsible for pulse wave phenomena (e.g., reflecting waves) which can be negligible in rigid tubes due to the much higher PWV. Future work should focus on the validity of the method in compliant vessels, toward in vivo application in arteries.

The data presented in this study demonstrate the efficacy of our method that restores the original differential pressure signal from the distorted signal acquired by a fluid-filled double-lumen catheter. It was obtained by studying the CMP effects and comparing the restored signals to undistorted reference signals. Altogether, our study highlights the importance of the CMP and the restoration formula for small pressure gradients over a wide spectrum of applications.