A Physical Model-Based Approach to One-Point Calibration of Pulse Transit Time to Blood Pressure

Abstract

Objective:

To develop a novel physical model-based approach to enable 1-point calibration of pulse transit time (PTT) to blood pressure (BP).

Methods:

The proposed PTT-BP calibration model is derived by combining the Bramwell-Hill equation and a phenomenological model of the arterial compliance (AC) curve. By imposing a physiologically plausible constraint on the skewness of AC at positive and negative transmural pressures, the number of tunable parameters in the PTT-BP calibration model reduces to 1. Hence, as opposed to most existing PTT-BP calibration models requiring multiple (≥2) PTT-BP measurements to personalize, the PTT-BP calibration model can be personalized to an individual subject using a single PTT-BP measurement pair. Equipped with the physically relevant PTT-AC and AC-BP relationships, the proposed approach may serve as a universal means to calibrate PTT to BP over a wide BP range. The validity and proof-of-concept of the proposed approach were evaluated using PTT and BP measurements collected from 22 healthy young volunteers undergoing large BP changes.

Results:

The proposed approach modestly yet significantly outperformed an empiric linear PTT-BP calibration with a group-average slope and subject-specific intercept in terms of bias (5.5 mmHg vs 6.4 mmHg), precision (8.4 mmHg vs 9.4 mmHg), mean absolute error (7.8 mmHg vs 8.8 mmHg), and root-mean-squared error (8.7 mmHg vs 10.3 mmHg, all in the case of diastolic BP).

Conclusion:

We demonstrated the preliminary proof-of-concept of an innovative physical model-based approach to one-point PTT-BP calibration.

Significance:

The proposed physical model-based approach has the potential to enable more accurate and convenient calibration of PTT to BP.

I. Introduction

Cuffless blood pressure (BP) monitoring is an actively pursued technology that has the potential to advance hypertension management [1], [2]. One popular approach to cuffless BP monitoring is pulse transit time (PTT) [3]. PTT, the time required for an arterial pulse to travel from a proximal site to a distal site in the arterial tree, is inversely proportional to BP and thus has been extensively studied as a surrogate measure of BP. Prior work has developed a number of innovative PTT measurement approaches based on various physiological signals (especially those compatible with wearables), including photoplethysmogram (PPG) [4], electrical bio-impedance [5], [6], ballistocardiogram (BCG) [7], and seismocardiogram (SCG) [8] signals, to list a few.

For PTT to be practically useful for BP monitoring, PTT in millisecond (ms) units must be correctly calibrated to BP in millimeters mercury (mmHg) units. A myriad of mathematical models (called PTT-BP calibration models) have been developed to facilitate the calibration of PTT to BP, either empirically or inspired by the physics underlying the PTT-BP relationship [9]. A long-standing challenge in PTT-BP calibration lies in how to best reconcile accuracy and convenience. Most PTT-BP calibration models have multiple tunable parameters (e.g., even the popular empiric linear PTT-BP calibration model (see Table II in [3] and many references therein), which relates PTT and BP using a straight line, includes two tunable parameters: slope and intercept) [3]. These parameters must be personalized to enable accurate PTT-BP calibration. Personalization of multiple parameters in the calibration model to an individual requires multiple concurrent PTT and BP measurements from the individual over a range of BP values. But, acquiring multiple PTT-BP measurements at various BP values is inconvenient in that it requires artificial BP-perturbing interventions and simultaneous recording of PTT and BP [3]. To reconcile accuracy (by maximally personalizing the calibration model) and convenience (by minimizing the interventions and measurements required to personalize the calibration model), most existing work employs a one-point calibration scheme where all the tunable parameters but one are fixed at an appropriate values (e.g., group-average values) while the remaining parameter is personalized using a PTT-BP measurement [8], [10]–[12]. In this regard, the use of an empiric two-parameter (e.g., linear) PTT-calibration model equipped with one parameter pre-determined, e.g., as a group-average value and another parameter personalized based on a PTT-BP measurement pair appears to be the common state-of-the-art. However, prior work focused on innovating one-point PTT-BP calibration methods is very rare, and there is no well-established method for one-point PTT-BP calibration to the best of our knowledge.

This paper concerns the development of a novel physical model-based approach to one-point PTT-BP calibration. Our PTT-BP calibration model is derived by combining the Bramwell-Hill equation and a phenomenological model of arterial compliance (AC) curve. Its novelty is that it eliminates simplifying approximations used in previously reported calibration models, e.g., neglecting arterial cross-sectional area changes or limiting to high BP regime [9]. We made our calibration model compatible with one-point calibration by imposing a physiologically plausible constraint pertaining to the skewness of AC at positive and negative transmural pressures: that the skewness of the AC curve in positive and negative transmural pressure regimes may be formalized into a function of age and gender groups [13], [14]. Equipped with the physically relevant PTT-AC and AC-BP relationships, the proposed approach may serve as a universal means to enable one-point calibration of PTT to BP over a wide BP range across diverse demographics. Using PTT and BP measurements collected from 22 healthy young volunteers experiencing large BP changes, the validity and proof-of-concept of the proposed physical model-based one-point PTT-BP calibration approach were evaluated and compared with an empiric linear one-point PTT-BP calibration with pre-determined group-average slope and subject-specific intercept.

This paper is organized as follows. Section II presents the proposed physical model-based one-point PTT-BP calibration approach. Section III describes the experimental data employed in this work with emphasis on PTT as well as the data analysis methods to evaluate the accuracy of the proposed approach. Section IV summarizes the main results, which are discussed in Section V. Section VI concludes this paper with future directions.

II. Physical Model-Based PTT-BP Calibration

The physical model-based PTT-BP calibration model is derived by combining the Bramwell-Hill equation and a popular phenomenological model of the AC curve. According to the Bramwell-Hill equation, PTT is related to BP as follows:

$$\tau =\eta \sqrt{\frac{C\left(p\right)}{A\left(p\right)}}$$ (1)

where $\tau$ is PTT, $C\left(p\right)$ and $A\left(p\right)$ are AC and arterial cross-sectional area as functions of BP $p$, and $\eta =l\sqrt{\rho }$ with the length between the PTT measurement sites $l$ and the blood density $\rho$. In this work, we adopted the exponential AC curve commonly used in prior work [15], [16]:

$$C(p)=A_C\left[e^{\frac{p-p_0}{{\sigma }_N}}u\left(-\left(p-p_0\right)\right)+e^{-\frac{p-p_0}{{\sigma }_P}}u\left(p-p_0\right)\right]$$ (2)

where $A_C$ is the maximum AC value at zero transmural pressure (i.e., at $p=p_0),p_0$ is the BP at which AC attains its maximum (which is assumed to be zero in this work), ${\sigma }_P$ and ${\sigma }_N$ are parameters characterizing the width of the AC curve in positive and negative transmural pressure regimes, respectively, and $u(\cdot )$ is the step function ($u\left(x\right)=1$ if $x>0$, $u\left(x\right)=0.5$ if $x=0$, and $u\left(x\right)=0$ if $x<0$). This AC curve has two key desired properties: (i) it fits the experimental data reasonably well and (ii) it yields a closed-form expression for arterial cross-sectional area. Indeed, integrating (2) yields the following closed-form expression for the arterial cross-sectional area:

$$A(p)=A_C\left[{\sigma }_N{e}^{\frac{p-p_0}{{\sigma }_N}}u\left(-\left(p-p_0\right)\right)+\left\{{\sigma }_N+{\sigma }_P(1-e^{-\frac{p-p_0}{{\sigma }_P}})\right\}u\left(p-p_0\right)\right]$$ (3)

Substituting (2) and (3) into (1) yields the following physical model-based PTT-BP calibration model:

$$\tau =\eta \sqrt{\frac{\left[e^{\frac{p-p_0}{{\sigma }_N}}u\left(-\left(p-p_0\right)\right)+e^{-\frac{p-p_0}{{\sigma }_P}}u\left(p-p_0\right)\right]}{\left[{\sigma }_N{e}^{\frac{p-p_0}{{\sigma }_N}}u\left(-\left(p-p_0\right)\right)+\left\{{\sigma }_N+{\sigma }_P\left(1-e^{-\frac{p-p_0}{{\sigma }_P}}\right)\right\}u\left(p-p_0\right)\right]}}$$ (4)

This physical model-based PTT-BP calibration model involves two tunable parameters: ${\sigma }_N$ and ${\sigma }_P$, which specify the width of the AC curve in negative and positive transmural pressure regimes, respectively. Prior reports have shown that the shape of the AC curve depends largely on age and gender [13], [14]. Hence, we hypothesize that the shape of the AC curve, and its skewness in positive and negative transmural pressure regimes in particular, can be treated as consistent within appropriate age and sex range:

$$\frac{A\left(p_0+ϵ\right)-A\left(p_0\right)}{A\left(p_0\right)-A\left(p_0-ϵ\right)}=\alpha$$ (5)

where $\alpha$ is a constant defining the skewness of the AC curve at a pressure level $ϵ\mathrm{m}\mathrm{m}\mathrm{H}\mathrm{g}$ away from $p_0:\alpha >1$ if the AC curve is right-skewed and $\alpha <1$ if the AC curve is left-skewed. Note that the value of $\alpha$ must be specified in advance, e.g., based on available data. For a pre-specified $\alpha$, (5) imposes a constraint on the relationship between ${\sigma }_P$. Hence, a PTT-BP measurement in conjunction with (5) armed with the value of $\alpha$ can personalize (4), which enables its one-point calibration to a subject associated with the PTT-BP measurement.

III. Methods

A. Experimental Data and PTTs

To evaluate the validity and proof-of-concept of the proposed physical model-based one-point PTT-BP calibration approach described in Section II, we used experimental data collected from 22 healthy volunteers (age 22±4 years; gender 19 males and 3 females; height 177±11 cm; weight 75±1 kg) in our prior work [7]. All the subjects underwent three BP-perturbing intervention periods with rest periods in between them: initial rest (R1)-mental arithmetic (MA)-rest (R2)-cold pressor (CP)-rest (R3)-physical exercise (PE). Throughout the experiments, we measured reference cuff BP using the volume clamp method (ccNexfin, Edwards Lifesciences, Irvine, CA, USA) and two PTTs using a weighing scale-like platform (9260AA6, Kystler Group, Winterthur, Switzerland): (i) a PTT measured as the time interval between the I wave in the BCG signal and the foot of the PPG signal at the instep of a foot (called hereafter the BCG-PPG PTT) and (ii) another PTT measured as the time interval between the I wave and the J wave in the BCG signal (called hereafter the BCG I-J interval) [7].

Consistently to our prior work [7], we extracted six PTT-BP data pairs from each subject. In each intervention or rest period in each subject, we averaged reference BP and PTTs over non-overlapping five-beat intervals. Then, we identified the five-beat intervals in each period where diastolic BP (DP) and systolic BP (SP) attained extremum values: noting that all the BP-perturbing interventions employed in the protocol increase BP, we chose the intervals associated with minimum DP and SP for rest periods and maximum DP and SP for intervention periods. In this way, we created 132 DP-PTT measurement pairs and 132 SP-PTT measurement pairs pertaining to the 22 subjects. Each pair included reference BP and two PTTs (i.e., BCG-PPG PTT and BCG I-J interval). The data were used to evaluate the proposed physical model-based one-point PTT-BP calibration approach as described in detail below.

B. Data Analysis

We analyzed the 132 BP-PTT measurement pairs created in Section III.A to quantitatively evaluate the proposed physical model-based PTT-BP calibration approach and compare it to an empiric linear PTT-BP calibration with a group-average slope and subject-specific intercept. Details follow.

1). Determination of Configurable Parameters:

To personalize the physical model-based PTT-BP calibration model (4) using a PTT-BP measurement pair, the values of $\alpha$ and $ϵ$ in (5) must be given a priori. To likewise personalize the empiric linear PTT-BP calibration model using a PTT-BP measurement pair, the group-average slope must be given a priori. Hence, we pre-determined the values of $\alpha$ and $ϵ$ as well as the group-average slope using the available data (i.e., 132 DP-PTT and 132 SP-PTT pairs).

First, we determined two $\alpha$ values for physical model-based one-point PTT-BP calibration pertaining to DP and SP. Using our data, we formulated and solved a two-loop optimization problem to determine $\alpha$ values that minimize the BP estimation errors associated with DP and SP, respectively. Note that the optimization problem was solved twice, once for DP and once for SP, respectively. We set $ϵ$ to 20 mmHg in order to account for large change in arterial cross-sectional area while avoiding the plateau regimes in (3). In the outer loop, $\alpha$ was optimized to minimize the BP estimation errors aggregated over all the 22 subjects. We used the physical model-based PTT-BP calibration model, which was personalized with the PTT-BP measurement pair at R1 in the inner loop, to calculate the 110 BP estimation errors (pertaining to R2, R3, MA, CP, and PE of 22 subjects). Then, we summarized them into the root-mean-squared error (RMSE). Then, $\alpha$ was optimized to minimize the RMSE. In the inner loop, ${\sigma }_N$ and ${\sigma }_P$ were optimized on a subject-by-subject basis, based on the value of $\alpha$ specified in the outer loop and the prespecified value of $ϵ$. We optimized ${\sigma }_N$ and ${\sigma }_P$ to personalize the physical model-based PTT-BP model (4) so that the subject-specific PTT-BP calibration model (4) (i) passes through the PTT-BP measurement pair at the subject’s R1 and (ii) satisfies the constraint (5). In this way, two $\alpha$ values pertaining to DP and SP compatible with the $ϵ$ value of 20 mmHg were obtained.

Second, we also determined two group-average slope values for empiric linear one-point PTT-BP calibration pertaining to DP and SP, which minimize the BP estimation errors associated with DP (i.e., 132 DP-PTT pairs) and SP (132 SP-PTT pairs). The linear PTT-BP calibration model is given by:

$$p={\theta }_1\tau +{\theta }_2$$ (6)

where ${\theta }_1$ is slope and ${\theta }_2$ is intercept. Our approach to personalize the intercept while fixing the slope at a group-average value is motivated by the previous observation that personalizing intercept led to superior BP estimation accuracy to personalizing slope [8]. We determined the group-average slopes pertaining to DP and SP using the 110 PTT-BP measurement pairs in all subjects except those at R1 (which were excluded since they were used to personalize the intercept in Section III.B.3 below) using the least-squares method.

2). Quantitative Evaluation of Physical Model-Based PTT-BP Calibration:

For each subject and each BP level (i.e., DP and SP), we used PTT-BP measurement at R1 to personalize the physical model-based PTT-BP calibration model (4) to the subject by finding the values of ${\sigma }_N$ and ${\sigma }_P$ so that the subject-specific PTT-BP calibration model (4) (i) passes through the PTT-BP measurement pair at R1 and (ii) satisfies the constraint (5) equipped with $\alpha$ and $ϵ$ pre-determined in Section III.B.1. Then, we used the remaining five PTT-BP measurements (at R2, R3, MA, CP, and PE) to calculate five BP estimation errors corresponding to the five measurements, as the difference between the reference BP and BP estimated by the personalized PTT-BP calibration model. Then, we expressed the ultimate performance of the proposed physical model-based one-point PTT-BP calibration approach on a subject-by-subject basis in terms of bias, precision, mean absolute error (MAE), and RMSE. We evaluated these performance measures for both DP and SP. Then, we summarized these metrics across all subjects in terms of median and interquartile range (IQR).

3). Comparison with Empiric Linear Calibration:

To compare the physical model-based PTT-BP calibration approach with a popular empiric linear PTT-BP calibration, we evaluated the performance of the linear PTT-BP calibration as follows. For each subject and each BP level, we used the group-average slope pre-determined in Section III.B.1 as ${\theta }_1$ in (6). Then, we used PTT-BP measurement at R1 to personalize the empiric linear PTT-BP calibration model (6) to the subject by selecting ${\theta }_2$ so that the subject-specific empiric linear PTT-BP model (6) passes through the PTT-BP measurement pair at R1. Then, we used the remaining five PTT-BP measurements (at R2, R3, MA, CP, and PE) to calculate five BP estimation errors corresponding to the five measurements in the same way as in Section III.B.2. Then, we expressed the ultimate performance of the empiric linear one-point PTT-BP calibration approach on a subject-by-subject basis in terms of bias, precision, MAE, and RMSE. We evaluated these performance measures for both DP and SP. Then, we summarized these metrics across all subjects in terms of median and interquartile range (IQR).

We determined the significance in the difference between the proposed physical model-based versus empiric linear one-point PTT-BP calibration approaches by applying the Wilcoxon signed-rank test to all four performance measures obtained on a subject-by-subject basis with a significance level of p=0.05.

IV. Results

Table I summarizes the changes in BP (DP and SP) observed in the subjects during the experiments. All three BP-perturbing interventions induced large changes in both DP and SP, with the overall DP and SP changes of 13–42 mmHg and 25–66 mmHg, respectively. Hence, the experimental data was ideally suited to the evaluation of the proposed physical model-based one-point PTT-BP calibration approach. Table II summarizes the performance of the physical model-based versus empiric linear approaches to one-point PTT-BP calibration pertaining to the BCG-PPG PTT. Fig. 1 shows representative examples of one-point PTT-BP calibration based on the physical model-based versus empiric linear approaches pertaining to the BCG-PPG PTT. Fig. 2 and Fig. 3 show the error distributions associated with the physical model-based versus empiric linear one-point PTT-BP calibration approaches pertaining to the BCG-PPG PTT in terms of the Bland-Altman plot and the correlation plot, respectively. The same results pertaining to the BCG I-J interval are presented in Supplemental Material (Table SI and Fig. S1–Fig. S3). For DP, both BCG-PPG PTT and BCG I-J interval achieved <6 mmHg in median bias and approximately 8–9 mmHg in median precision. For SP, both BCG-PPG PTT and BCG I-J interval achieved approximately <8 mmHg in median bias and 12–13 mmHg in median precision. In general, MA, CP, and PE were associated with relatively large errors, due presumably to the constraints that (i) the calibration model must pass through R1 and (ii) the ratio between ${\sigma }_N$ and ${\sigma }_P$ must be $\alpha$.

TABLE I.

Changes in blood pressure (BP) in response to BP-perturbing interventions (mean (SE)).

	R1	MA	R2	CP	R3	PE	Overall Change
DP [mmHg]	75.0 (2.1)	93.1 (2.4)	72.7 (2.0)	89.6 (2.2)	72.1 (1.9)	89.8 (2.3)	27.6 (1.6)
SP [mmHg]	112.7 (3.7)	138.1 (4.0)	110.6 (3.6)	133.7 (4.0)	108.2 (3.5)	139.3 (4.)	42.6 (2.4)

TABLE II.

Bias, precision, mean absolute error (MAE), and root-mean-squared error (RMSE) of PTT-BP calibration: BCG-PPG PTT (median (IQR)).

(a) Diastolic BP Errors
	Bias [mmHg]	Precision [mmHg]	MAE [mmHg]	RMSE [mmHg]
Physical Model-Based One-Point PTT-BP Calibration	5.5 (6.5)	8.4 (3.6)	7.8 (2.6)	8.7 (3.9)
Empiric Linear One-Point PTT-BP Calibration	6.4 (5.2)*	9.4 (3.7)*	8.8 (3.0)*	10.3 (4.7)*
(b) Systolic BP Errors
	Bias [mmHg]	Precision [mmHg]	MAE [mmHg]	RMSE [mmHg]
Physical Model-Based One-Point PTT-BP Calibration	7.7 (11.1)	11.8 (5.7)	11.3 (6.6)	14.3 (8.0)
Empiric Linear One-Point PTT-BP Calibration	8.9 (9.8)*	13.7 (5.5)*	14.0 (6.7)*	15.6 (9.3)*

^*: p<0.05 relative to physical model-based one-point PTT-BP calibration.

Fig. 1.

Representative example of physical model-based and empiric linear one-point PTT-BP calibration in a subject. (a) Diastolic BP via BCG-PPG PTT (physical model-based vs empiric linear calibration: bias 1.9 mmHg vs 2.0 mmHg; precision 3.8 mmHg vs 4.2 mmHg; MAE 2.8 mmHg vs 3.1 mmHg; RMSE 3.9 mmHg vs 4.2 mmHg). (b) Systolic BP via BCG-PPG PTT (physical model-based vs empiric linear calibration: bias 6.4 mmHg vs 6.8 mmHg; precision 8.7 mmHg vs 10.5 mmHg; MAE 7.3 mmHg vs 9.3 mmHg; RMSE 10.1 mmHg vs 11.6 mmHg).

Fig. 2.

Error distributions associated with proposed physical model-based versus empiric linear one-point PTT-BP calibration approaches in terms of Bland-Altman plot (all subjects). (a) BCG-PPG PTT-DP. (b) BCG-PPG PTT-SP.

Fig. 3.

Error distributions associated with proposed physical model-based versus empiric linear one-point PTT-BP calibration approaches in terms of correlation plot (all subjects). (a) BCG-PPG PTT-DP. (b) BCG-PPGPTT-SP.

Fig. 4 shows the personalized AC curves associated with all the 22 subjects. All the AC curves derived from the proposed approach (i.e., (2) equipped with personalized ${\sigma }_N$ and ${\sigma }_P$ values subject to (5)) appeared physiologically plausible. However, they also exhibited substantial variability in shape despite the presence of the skewness constraint (5).

Fig. 4.

Personalized arterial compliance (AC) curves associated with all the 22 subjects. Noting that $A_C$ in (2) and (3) are cancelled, the presented curves correspond to $\frac{C\left(p\right)}{A_C}$ in mmHg⁻¹.

V. Discussion

Cuffless BP monitoring has the potential to advance our ability to prevent, detect, and control hypertension. Prior reports have developed a large number of innovative PTTs that show good correlation with BP based on various physiological signals. In contrast, progress related to accurate and convenient PTT-BP calibration (which must accompany the development of innovative PTTs to collectively advance cuffless BP monitoring as a whole) has been very slow. To the best of our knowledge, there is no established PTT-BP calibration method available in the literature. As a consequence, many existing reports employ ad-hoc and empiric PTT-BP calibration models and methods to achieve practical convenience and adequate ability to regress PTT-BP measurements [9]. The goal of this work is to develop and demonstrate the initial proof-of-concept of a novel physical model-based one-point PTT-BP calibration approach.

The proposed physical model-based one-point PTT-BP calibration approach exhibited reasonable performance (Table II as well as Fig. 1–Fig. 3). For DP, both BCG-PPG PTT and BCG I-J interval achieved bias (<6 mmHg) and precision (approximately 8–9 mmHg) levels with which hypertension may be screened at an accuracy comparable to conventional auscultation [17]. For SP, both BCG-PPG PTT and BCG I-J interval achieved precision levels (12–13 mmHg) with which hypertension may be screened at an accuracy comparable to conventional auscultation [17], although bias was quite larger than the conventional limit of 5 mmHg pertaining to oscillometry (<8 mmHg). However, note that these SP biases may not still be discouraging considering that the (unknown) bias between the gold standard auscultation BP and the non-invasive volume-clamping BP measurement used to collect BP in our experiments were not taken into account (which will relax the allowed bias limit to beyond 5 mmHg to likely cover the SP bias achieved in this work [17]).

Notably, the proposed physical model-based one-point PTT-BP calibration approach modestly yet significantly outperformed the empiric linear one-point PTT-BP calibration widely used in previous work (Table II and Fig. 2–Fig. 3 for BCG-PPG PTT as well as Table SI and Fig. S1–Fig. S3 for BCG I-J interval). For both DP and SP, the proposed physical model-based one-point PTT-BP calibration approach resulted in significantly smaller bias, precision, MAE, and RMSE than the empiric linear one-point PTT-BP calibration when used with the BCG-PPG PTT (Table II and Fig. 2–Fig. 3) as well as the BCG I-J interval (Table SI and Fig. S2–Fig. S3). In addition, the proposed physical model-based one-point PTT-BP calibration approach was more robust than the empiric linear one-point PTT-BP calibration in that it resulted in smaller number of subjects associated with large BP estimation errors. For DP, the proposed physical model-based one-point PTT-BP calibration approach resulted in 2 (BCG-PPG PTT) and 3 (BCG I-J interval) subjects with RMSE >15 mmHg, which was superior to the empiric linear one-point PTT-BP calibration which resulted in 3 (BCG-PPG PTT) and 5 (BCG I-J interval) subjects, respectively. For SP, the proposed physical model-based one-point PTT-BP calibration approach resulted in 5 (BCG-PPG PTT) and 7 (BCG I-J interval) subjects with RMSE >20 mmHg, which was superior to the empiric linear one-point PTT-BP calibration which resulted in 8 (BCG-PPG PTT) and 8 (BCG I-J interval) subjects, respectively. In all these large BP error cases, the empiric linear one-point PTT-BP calibration always resulted in large BP estimation error when the proposed physical model-based one-point PTT-BP calibration approach resulted in large BP estimation error. All in all, the results suggest that the proposed physical model-based one-point PTT-BP calibration approach may add unique value to the existing body of work on one-point PTT-BP calibration as a potentially universal means to conveniently calibrate aortic PTT to BP using a single PTT-BP measurement.

The notable variability associated with the subject-specific AC curves in Fig. 4 strongly suggest that fixing both ${\sigma }_N$ and ${\sigma }_P$ at group-average values (which leads essentially to the use of a group-average PTT-BP calibration model in all subjects) may not achieve accurate PTT-BP calibration. Hence, it supports the relevance of the one-point calibration strategy (i.e., personalizing a parameter in the PTT-BP calibration model to make it subject-specific) in the proposed physical model-based PTT-BP calibration approach.

Despite the promising initial proof-of-concept, this work has limitations. First, the physiologically plausible constraint on AC curve (5) (or equivalently, our hypothesis that the skewness of the AC curve in positive and negative transmural pressure regimes may be treated as consistent within appropriate age and sex range) was not directly validated. As a proof-of-concept study, the proposed physical model-based one-point PTT-BP calibration approach was tested only in heathy young subjects, which was not sufficient to establish the validity of (5). The optimal values of $\alpha$ in (5) derived from our work were: 1.03 (DP) and 1.05 (SP) for BCG-PPG PTT and 1.0 (DP) and 1.1 (SP) for BCG I-J interval, respectively. However, these values were derived using all the available data and lacks independent assessment (note, however, that the comparison between the proposed physical model-based versus empiric linear one-point PTT-BP calibration approaches may still be valid because both approaches used the values of $\alpha$ and ${\theta }_1$ derived with the same data). Although our hypothesis appeared to be effective in personalizing the physical model-based PTT-BP calibration model (4) based on a PTT-BP pair, its legitimacy must be investigated using, e.g., in vivo BP-aortic cross-sectional area measurements in a future work. An additional limitation related to the constraint and the hypothesis is associated with vascular aging. It is conceivable that the proposed physical model-based one-point PTT-BP calibration approach may use non-optimal $\alpha$ values in subjects whose vascular aging is not consistent with chronological aging when $\alpha$ is specified as a function of chronological aging. In this regard, it may be a rewarding exercise to specify $\alpha$ directly as a function of vascular aging, e.g., by evaluating PWV of each subject relative to established normal and reference values [19]. Second, DP and SP were estimated using the same PTT. Our PTTs represent DP since they are measured based on diastolic timing markers [7], [18]. Hence, it is not strictly correct to use the proposed physical model-based one-point PTT-BP calibration approach to estimate SP. Future work must develop novel ideas to calibrate DP and SP independently.

It is also worthwhile to investigate the effectiveness of the proposed physical model-based one-point PTT-BP calibration approach when combined with non-aortic PTTs. The PTTs used in this work are ideally suited to the evaluation of the proposed physical model-based one-point PTT-BP calibration approach relative to most existing PTTs (including pulse arrival time (PAT) measured as the time interval between the R wave in the ECG signal and the foot of the PPG signal at a finger [3]) in that they may represent PTT through the main aorta. First, the BCG-PPG PTT includes PTT pertaining to the aorta and PTT pertaining to the femoral/tibial arteries as the primary and secondary contributors, respectively. Second, the BCG I-J interval may correspond to PTT between aortic inlet (i.e., aortic valve) and outlet (presumably abdominal aorta or femoral artery) [18]. In our prior work, we demonstrated that both these PTTs outperformed PAT in their ability to estimate BP [7]. Considering that the Bramwell-Hill equation may be more compatible with the main aorta (whose wall is predominantly elastic with minimal presence of smooth muscles) than distal branch arteries (e.g., radial artery whose AC can be largely altered by smooth muscle contraction and relaxation), the two PTTs used in this work are well suited for evaluating the physical model-based PTT-BP calibration approach. But, the compatibility of the proposed approach to non-aortic PTTs is unknown. Considering that PTT may be acquired much more conveniently through non-aortic arteries (e.g., brachial and radial arteries via smartwatches) than through aortic arteries (e.g., femoral artery), elucidating the compatibility of the proposed physical model-based one-point PTT-BP calibration approach to non-aortic PTTs may extend its utility and impact.

VI. Conclusion

We demonstrated proof-of-concept of an innovative physical model-based approach to one-point PTT-BP calibration. Its initial promise warrants extensive follow-up investigations, including the refinement and optimization of the approach by exploiting various physics-based PTT-BP models and AC curves (e.g., linear-exponential [16] and Fisk [20] curves) as well as PTTs. It must also be evaluated in subjects in diverse age and gender groups in order to demonstrate its potential for universal effectiveness. With such endeavors, the proposed physical model-based one-point PTT-BP calibration approach may enable more accurate and convenient calibration of PTT to BP.