Modelling relation between Pulse Transit Time (PTT) and blood pressure (BP) is a critical step in BP estimation for wearable technology. Recognizing the limitation of assuming constant vessel and blood conditions, we developed a simplified pulsatile flow model to analyze how various factors affect PTT values. Our research focuses on the impact of mechanical characteristics, such as vessel diameter, wall thickness, blood viscosity, and pressure, on PTT measurements and subsequent BP estimation. Measurements were conducted using accelerometer sensors within a custom-designed mock circulatory loop. This setup allowed for the testing of a wide range of pressure values and pulsation rates, as well as the modification of viscosity in blood-mimicking liquids across different vessel models. We employed the Moens-Korteweg conversion model for pressure estimation, initially trained on PTT data from a specific setup parameter combination, and subsequently tested with data from varied setup parameters. We observed high correlation levels (r = 0.93 ± 0.09) paired with high error (RMSE = 163 ± 100 mHg), suggesting potential inaccuracies in pressure estimation. We present the recorded signals and discuss how alterations in physical conditions influence PTT values and the precision of BP estimation.
1 Introduction
The demand for continuous and non-intrusive health tracking has driven advancements in the field of blood pressure (BP) monitoring. While the conventional cuff-based approach remains the current standard for BP measurement, recommended by hypertension experts for clinical evaluation [1], its inherent limitations, including intermittent measurements, discomfort, and the need for user compliance have fueled a search for more innovative and user-friendly alternatives. Therefore, various approaches have been investigated over the last few decades, aiming to develop wearable and cuffless method of BP measurement. It is included in the latest generation of smart watches [2, 3], wristbands [4], armbands [5], or rings [6, 7]; there are proposals to measure BP using smartphones [8] or diverse types of skin patches [9‐11]. Despite of the fact that some of the devices are already available on the market, accuracy of the measurement is often questionable; consequently, global BP guidelines do not endorse the utilization of wearable devices for diagnostic and treatment decisions. [1, 12, 13]. Moreover, validation against cuff-based measured BP is being criticized as a reliable reference for this purpose, and thus changes in requirements are called for [14, 15]. Consequently, research on cuffless non-invasive BP measurement techniques is still ongoing.
One of the methods utilized in some of the aforementioned solutions is indirect estimation of BP based on measurement of Pulse Transit Time (PTT). PTT is the time between two pulse waves propagating on the same cardiac cycle from two separate arterial sites. It is assumed that PTT is inversely related to BP, since with increasing BP, increasing distending pressure and decreasing arterial compliance, pulse wave velocity (PWV) rises and thus PTT shortens [4]. The PTT-based methodology has garnered significant attention within the research community due to its potential for wearable applications and its apparent simplicity. From a hardware perspective, the non-invasive detection of pulse can be achieved relatively easily using various modalities and sensor types. However, the relationship between BP and PTT is intricate and difficult to model. Both values are influenced by interconnected factors, such as blood viscosity, vessel stiffness and diameter, wave reflections along the arterial tree, varying shear rates in blood flow, cardiac output (determined by heart rate and stroke volume), sympathetic system activity, and the overall condition of the vasculature. Consequently, the measured values of pulse delays can be difficult to interpret, showing little to no value for beat-to-beat BP estimation [16]. Thus, an understanding of the cardiac flow properties is crucial in development of new methods and tools for cardiovascular health monitoring and improvement of the PTT-BP conversion models.
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1.1 Mechanical Properties of Blood Vessels, Flow, and Pressure
The characterization of blood flow presents inherent challenges due to the irregular structure of blood and the disruptive influence of red blood cells on viscosity. Additionally, vessel thickness varies throughout the arterial tree, introducing further complexity to the modeling process. Blood vessels endure forces from blood flow and surrounding tissues. The blood viscosity causes different levels of shear stress occurring tangentially to the vessel lumen, influencing the ease with which blood flows through the vessels and the resulting impact on vascular dynamics. Depending on factors such as the hematocrit levels, plasma viscosity, and the properties of red blood cells, blood viscosity is difficult to define with single number. Common agreement is that the normal range is between 3.5 and 5.5 cP, however the values can be different in the large arteries, the veins, and the microcirculation [17, 18]. Although 70% of blood vessel walls consist of water, the rest is a complex mix of collagen, elastic fibers, proteoglycans, and vascular cells organized in layers. These layers differ in thickness and composition across vessel types and diameters. Large arteries have a thick media layer with more elastin, while small arteries have more smooth muscle cells; veins have a thinner media layer and less elastic tissue [19]. The vessel wall is structured to withstand and transmit forces from blood flow, pressure, and surrounding tissues. The composite characteristics of the vascular wall result in distinctive mechanical properties when responding to physiological forces. In the example of artery, the response depends on the pressure inside the artery: low pressure states (<80 mmHg) engage the response from soft elastin fibers, while higher pressures cause stiff collagen fibers to dominate the response in order to avoid vessel damage [19, 20].
Therefore, the placement of the sensors in PTT measurement will have an impact on PTT-BP way beyond just a distance. Commonly, ECG R-peak is used as a proximal point, paired with the slope of PPG pulse measured periphery, e.g. on the finger. There is no agreement on which endpoints are the best choice. On one hand, PTT estimation along central arteries seems to be promising, because of central arterial wall properties and little interference caused by vasomotion and wave reflection [21]. On the other hand, distal waveforms are measured often with satisfactory results, with measurement from heart to toes and fingers showing better correlation with cuff-based BP than from heart to earlobes [22, 23].
1.2 Mathematical Models to Estimate BP Based on PTT
Modeling of PTT-BP relation is based on the works by Moens and Korteweg on the flow in tubes. The velocity of the fluid wave was be determined as a function of vessel and fluid characteristics [24, 25]:
where PWV – pulse wave velocity, d is distance between sensors, ρ is the blood density, r is the inner radius of the vessel, h is the vessel wall thickness, and E stands for Young’s modulus describing the elasticity of the arterial wall. Young’s modulus E is not a constant, but it varies nonlinearly with pressure [26]:
$$ E\left( P \right) = E_{0} \,e^{\alpha P} $$
(2)
where E0 is the zero-pressure modulus, α is a constant that depends on the vessel and P is pressure. Deriving a formula directly from Eqs. (1) and (2), we get logarithmic relation between PTT and BP [27]:
Alternative formulas have been proposed, defining the relation e.g. as linear [28] or polynomial [29]. However, there are inaccuracies in all models deriving from Moens-Korteweg equation, as they make multiple presumptions to simplify the complicated relationship. The primary presumptions are that smooth muscle contraction and viscous effects are negligible, arterial elasticity is not significantly modified by aging or disease during measurement, and the timing points of the distal pulse are free of wave reflections. Additionally, both approaches assume that the thickness and diameter of the vessel remain constant for changing BP level, and that the vessel wall is thin and can be modeled as an unchanging thin shell. However, the thickness-to-radius ratio of human vessel is beyond the limit for a thin shell, and the change of the radius of a human artery can reach ∼30% due to BP changes [30].
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In this study, we aim to assess the impact of variations in blood viscosity, vessel diameter, and vessel wall thickness on the error in BP prediction. We compare how the model trained on the same environmental parameters will respond to change in one of said parameters. Measurements of PTT were conducted within a simplified mock circulatory loop. This setup allows testing across a wide range of pressure values and pulsation rates, along with precise control over the viscosity of the blood-mimicking liquid and the parameters of the flow architecture elements. The controllable environment facilitates the analysis of components influencing the recorded signal characteristics, the derived PTT, and subsequent pressure estimation.
2 Methodology
2.1 Pulsatile Flow Simulation
Haemodynamic simulations were performed using a mock circulatory loop consisting of pulsatile flow generator, an artificial blood vessel, pressure control, pulse sensors and blood-mimicking liquid, see Fig. 1. The system employs a dosing pump (Injecta, Athena AT.MT4) with a pumping rate set to 1 Hz. The pump induces a circulation of fluid inside of the tube system, with a detectable pulsation corresponding to the pumping rate. For blood vessel emulation we used tubes made of latex, with the manufacturer determined hardness K = 40 (Shore A). System tubing was made with tubes of inner diameter 9 mm and wall thickness 2 mm. To study the influence of vessel dimensions and elastic properties on measured signals, system architecture included a replaceable test tube (see Fig. 1).
Fig. 1.
Mock circulatory loop emulating pulsatile flow with known pulsation rate and variable pressure levels. Water-glycerol solutions of different concentrations were serving as a blood mimicking liquid. Latex test tubes of different dimensions were used for emulating the vessels.
Table 1.
Parameters of elastic tubes used in the experiment.
Inner diameter (mm)
Wall thickness (mm)
Outer diameter (mm)
Thickness-to-radius ratio h0
Tube A
9
2
13
0.44
Tube B
9
2.5
14
0.56
Tube C
10
2
14
0.4
Tube D
10
3
16
0.6
×
Four tubes with different parameters were used in the experiments, with their dimensions shown in Table 1. Pressure of the fluid in the system is controlled using disc pump (TTP Ventus). Two pressure sensors (Honeywell sensing and solutions, SSCDRRN005PDAA5) with measurement accuracy of ±2.5% (full scale span) were placed at both ends of the test tube. In order to determine propagation velocity of the pressure waves travelling along the tubes, we used accelerometers (ACM) (model LIS344ALH from STMicroelectronics), which allow detection of the exact moment of the pulse appearance in corresponding set locations. The distance between ACMs was 45 cm. ACMs are sensing the tube displacement caused by travelling pulse wave that can be easily distinguished as sharp acceleration responses enabling exact time determination. All sensors used in the setup are connected to NI USB – 6289 multifunction I/O device with data acquisition at 5 kHz sampling rate. The pump and pressure are controlled via custom-made LabView software.
Water-glycerol mixture of different proportions served to mimic the blood on different viscosity levels. The temperature of the glycerol mixture influences its viscosity and therefore it requires temperature control. If we assume the viscosity of human blood is approximately 4 cP [18], it is almost equivalent to the viscosity of a 40% glycerin solution. A temperature sensor was placed inside the liquid tank. Viscosity of used mixtures were calculated using the standard formula [31, 32]. Tested concentrations together with their density and viscosity levels in measured temperature are shown in Table 2:
Table 2.
Glycerol concentrations used in the study. The temperature of the liquid was in the range of 22 ℃.
Glycerol concentration
Density [kg/m3]
Dynamic viscosity [Ns/m2]
Dynamic viscosity [cP]
0%
997.61
0.0010
0.96
20%
1056.2
0.0019
1.88
30%
1085.2
0.0028
2.81
40%
1113.5
0.0044
4.48
45%
1127.3
0.0058
5.81
50%
1140.9
0.0077
7.71
2.2 Data Acquisition
Twenty-four scenarios were tested using four different tube diameters, each measured with six concentrations of aqueous-glycerol mixtures. Pressure levels in the system were continuously increasing in the range from 0 to 220 mmHg. Measurements for each combination were repeated three times in order to ensure results repeatability, with PTT value being an averaged value of three repetitions. An example of the signal recorded with ACM with the peaks detected for each pulse is shown on Fig. 2. Signal quality was good with clear separation between pulse complexes from consecutive pressure cycles. For this reason, raw signal was used directly. The offset value has been removed, so that the signal was oscillating around zero.
Fig. 2.
Part of the signal recorded using ACM with detected pulses marked with red asterisks. Acceleration waveforms are sufficiently similar to typical waveforms when measuring heartbeat.
×
2.3 Statistical Analysis
Data are expressed as means ± standard deviations or percentages. Root means square error (RMSE), Pearson’s correlation coefficient (R) and Bland–Altman analysis were used for the evaluation of agreement between the two methods of pressure estimation. Bland-Altman rates included RPC, reproducibility coefficient (±1.96 * SD values) and CV, coefficient of variation (SD of mean values in %).
3 Results
3.1 System Stability Testing
The stability test was conducted using a 50% glycerol liquid. Twelve different pressure levels were applied, and the dosing pump was operated at a frequency of 1 Hz. The experiment was repeated five times. For the analysis, three random pressure levels (40 mmHg, 80 mmHg, and 180 mmHg) were chosen, and individual random acceleration pulses and pressure pulses were examined. The Pressure 1 signal was utilized in determining the pressure levels. The analysis using longer signal periods was hindered due to the pump driving clock pulse exhibiting jitter up to 0.04 s, causing the pulses to be out of phase. Stability was assessed by identifying the maximum PTT errors (the differences between the highest and lowest PTT values) in each situation and calculating the mean and standard deviation of the errors. The pump driving clock pulses were precisely set to the same phase. Figure 3 shows the pressure and acceleration pulses in different pressure levels. The average maximum PTT error was 9 ms and the standard deviation 3.9 ms. The number of used acceleration pulses was 45.
Fig. 3.
Changes in pulse signal shapes when detected by pressure sensors (A) and acceleration sensors (B) for pressure levels of 40 mmHg, 80 mmHg and 180 mmHg and 50% water-glycerol solution.
×
The second part of the stability test was measurement of the single pulse signal shape, when liquids of different glycerol concentrations are used in the similar size of the test tube. Observed signals are displayed on Fig. 4:
Fig. 4.
Attenuation of the acceleration pulse amplitude, as the effect of viscosity change. Tested concentrations: 0%, 30%, 40%, 50%.
×
3.2 Influence of Liquid Viscosity and Tube Dimensions on PTT Levels
Figure 5A shows the variations in PTT levels across different viscosities of the blood mimicking liquid, all measured within the same tube. Numerical values for all tubes presented as a mean PTT change in comparison to reference concentration of 40% are shown in Table 3. Conversely, Fig. 5B displays the PTT variations observed across different tubes while maintaining a consistent viscosity of the blood-mimicking liquid. Changes of PTT values as compared to Tube A serving as a reference are shown in Table 4.
Fig. 5.
Changes in PTT values for: A. different viscosities of glycerol solution in Tube A. B. different tube parameters, 40% glycerol solution.
Table 3.
Variations in PTT levels (ms) caused by changes in liquid viscosity in reference to 40%.
0%
20%
30%
40%
45%
50%
Tube A
−4.2 ± 0.5
−2.5 ± 0.6
−1.3 ± 0.5
0
2.7 ± 0.7
3.9 ± 0.9
Tube B
−5.8 ± 0.2
−3.7 ± 0.3
−2.4 ± 0.9
0
2.5 ± 0.1
4.0 ± 0.3
Tube C
−3.6 ± 1.0
−1.9 ± 0.9
−1.3 ± 0.9
0
1.6 ± 0.2
2.5 ± 0.8
Tube D
−7.2 ± 1.2
−0.9 ± 0.6
−2.3 ± 0.5
0
1.6 ± 2.1
2.0 ± 0.5
Table 4.
Variations in PTT levels (ms) caused by changes in liquid viscosity in reference to Tube A.
Concentration
Tube A
Tube B
Tube C
Tube D
0%
0
0.5 ± 0.7
5.5 ± 0.2
0.8 ± 0.2
20%
0
0.9 ± 0.9
5.5 ± 0.9
5.6 ± 0.9
30%
0
1.0 ± 0.7
4.9 ± 0.6
3.1 ± 0.4
40%
0
2.1 ± 1.0
4.9 ± 1.3
4.0 ± 1.6
45%
0
1.9 ± 0.6
3.8 ± 0.4
2.9 ± 2.6
50%
0
2.2 ± 0.4
3.4 ± 0.4
2.1 ± 0.4
×
3.3 Regression Model Based on Moens-Korteweg Equation
A regression model was created based on PTT and BP levels measured for a reference viscosity 40% and Tube B, following the formula (3). Correlation plot and Bland-Altman plot are shown on Fig. 6A–B. Model parameters are shown on Fig. 6C.
Fig. 6.
A: Correlation plot between estimated and measured pressures, r = 0.96. B: Bland-Altman plot: mean difference = 0, RPC = 11.96, C: model parameters.
×
Model parameters (Fig. 6C) were used to estimate pressure values based on the PTT obtained using data from different liquid viscosities and tube dimensions. Relative errors resulting from these estimations are shown in Table 5. Correlation coefficients are shown on Fig. 7A, RMSE values are shown on Fig. 7B.
Table 5.
Relative error between estimated and measured values.
Concentration
Tube A
Tube B
Tube C
Tube D
0%
3.2 ± 1.6
2.8 ± 1.2
0.5 ± 0.5
2.5 ± 1.2
20%
2.4 ± 1.3
2.2 ± 1.8
−0.5 ± 0.5
−0.5 ± 0.4
30%
1.8 ± 1.0
1.4 ± 1.1
−0.8 ± 0.7
0.1 ± 0.1
40%
1.2 ± 0.9
−0.01 ± 0.1
−1.2 ± 0.5
−1.2 ± 1.0
45%
−0.1 ± 0.3
−1.3 ± 0.7
−2.1 ± 0.8
−2.2 ± 3.1
50%
−0.9 ± 0.5
−2.2 ± 1.3
−2.6 ± 1.3
−1.9 ± 0.8
Fig. 7.
A. Correlation coefficients B. RMSE between estimated and measured pressure values for all viscosities and tubes combinations.
×
4 Discussion
In this study, we used a system with simulated pulsatile flow in order to study PTT, vessel dimensions and viscosity relationship in a controlled environment, and without wave reflections. Four tubes made of the same material but with different thickness-to-radius ratio were used in the experiments, and six concentrations of water-glycerol solutions were mimicking blood of different viscosity levels. Pressure level in the system was gradually increased from 20 to 220 mmHg. Pulsations were detected using ACMs placed on two ends of the test tube. Calculated PTT was then compared with simultaneously measured pressure.
We have tested the stability of the ACM sensor response for system pressure of 40 mmHg, 80 mmHg and 180 mmHg. The higher the pressure, the higher observed PTT error, with the maximum value of 9 ms. Since the measured PTTs were staying in the range between 100–150 ms, this potential value variation can be a significant source of error. However, we have decided to average the value of measurements repeated with the same setup, in order to reduce the impact of random errors. We have also measured the influence of liquid viscosity on pulse signal attenuation- the observed effect was negligible (Fig. 4). This conclusion corresponds to the study by Ikenaga et al. [33], in which authors were comparing the attenuation of a pressure wave in a phantom of human circulation.
When observing the levels of PTT in the single tube, the results met the expectations- more viscous liquid resulted in longer PTT (Fig. 5A). It can be seen in Table 3, that although the effect was visible for all tested tubes, the observed changes were in the small range. However, when comparing results between different tubes, there was no straightforward pattern visible (Fig. 5B, Table 4). The potential source of this effect might be e.g. a small difference in distance between sensors, when the new test tube was attached to the system. It might be possible that tube radius non-uniformities were present, e.g. due to tube stretching or displacement.
We have calculated parameters of the transit time-pressure conversion model, based on Moens-Korteweg formula shown in (3). Even when tested on the original viscosity-tube combination, it resulted in deviation of ±11.96 mmHg, even though the correlation coefficient was very high (r = 0.96; Fig. 6). When applying similar model parameters to PTT values from other measurements, the correlation coefficients remained notably high (r = 0.93 ± 0.09, Fig. 7A); however, the RMSE values were found to be exceedingly elevated (RMSE = 163 ± 100 mHg, Fig. 7B). Analysis of relative error in Table 5 reveals positive error levels for concentrations lower than in the original model, and negative ones for the higher concentrations, when comparing the results for same tube as the original model (Tube B). It means, that for lower liquid viscosity the model was estimating too low pressure, whereas for higher viscosity the estimate was too high. However, further studies are needed to confirm such patterns and the severity of the effect.
Modeling blood flow is a complicated task. We have observed possible sources of errors in the presented setup. Pressure sensors in the presented setup were connected to the system with tubes of different diameter than the test tube, which might have resulted in wave reflections. The conversion of the pressure levels to mmHg was done based on single values of atmospheric pressure, which might have differed for measurements done in separate days. Every measurement was done with steadily increased pressure; however, the system dynamics might be different when pressure is dropping. It also might be beneficial to increase the number of measurement repetitions, in order to eliminate random errors. Our next step is to increase the complexity of the phantom setup, in order to study e.g. effect of bifurcation. Furthermore, testing the tubes of different materials and wall thickness can be used e.g. to study the effect of vessel stiffening, resulting from aging. Another possibility is to test different position of the phantom (now it was always horizontal), which would correspond to different body positions. In terms of analysis, other transit time-pressure conversion models need to be tested.
Since blood viscosity and vessel diameter can dynamically change [34], it is important to understand the effect of these parameters on PTT levels and improve PTT-BP calibration process accordingly. Alternatively, the methods such as single point measurement as could be tested [35], which could potentially enable avoiding the calibration error coming from viscosity and vessel changes.
5 Conclusion
This study investigated PTT in relation to blood viscosity, vessel dimensions, and pressure levels using a simulated pulsatile flow system. Our experiments utilized four tubes with varying thickness-to-radius ratios and six water-glycerol solutions to mimic different blood viscosities, all tested in gradually increasing system pressure. PTT was measured with ACMs and compared against simultaneous pressure readings. Our findings suggest that liquid viscosity and tube dimensions impact PTT and following pressure estimation. The Moens-Korteweg formula-based model showed high correlation but large RMSE when applied across different conditions, indicating the need for further refinement. These results highlight the complexities in accurately modeling blood flow and the influence of factors like viscosity and vessel dimensions on PTT. Future work will focus on enhancing our model’s accuracy and exploring effects like bifurcation and vessel stiffening. This study underscores the importance of considering dynamic changes in blood properties and vessel characteristics for effective PTT-based blood pressure monitoring.
Acknowledgments
AZ thanks Alfred Kordelin Foundation. This work was partly supported by the European Structural and Investment Funds - European Regional Development Fund (ERDF): EMUVALID, and the Academy of Finland Profi6 programme 6G-Enabling Sustainable Society (6GESS).
Disclosure of Interests
The authors have no competing interests to declare that are relevant to the content of this article.
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