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Edited by: Timothy W. Secomb, University of Arizona, United States

Reviewed by: Bart Spronck, Yale University, United States; Jung Hee Seo, Johns Hopkins University, United States

This article was submitted to Computational Physiology and Medicine, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In this paper, tapered vs. uniform tube-load models are comparatively investigated as mathematical representation for blood pressure (BP) wave propagation in human aorta. The relationship between the aortic inlet and outlet BP waves was formulated based on the exponentially tapered and uniform tube-load models. Then, the validity of the two tube-load models was comparatively investigated by fitting them to the experimental aortic and femoral BP waveform signals collected from 13 coronary artery bypass graft surgery patients. The two tube-load models showed comparable goodness of fit: (i) the root-mean-squared error (RMSE) was 3.3+/−1.1 mmHg in the tapered tube-load model and 3.4+/−1.1 mmHg in the uniform tube-load model; and (ii) the correlation was

Cardiovascular disease (CVD) is the leading cause of mortality and morbidity that imposes profound impact on health and economy in the United States as well as globally (Benjamin et al.,

To date, non-invasive brachial arterial blood pressure (BP) measured by the auscultation technique remains the mainstay of CV health and disease assessment (Black et al.,

To exploit the superior clinical value of the central aortic BP while still leveraging the convenience of distal (e.g., brachial) BP measurement, many attempts have been made to derive central aortic BP from distal BP measurement(s). Currently prevalent approach is a population-based, frequency-domain transformation known as the Generalized Transfer Function (GTF), which converts a distal [e.g., brachial (Sharman et al.,

Tube-load (TL) model has the potential to serve as an alternative to the frequency response model used in the GTF technique as well as to offer new opportunities toward patient-specific assessment of CV health by virtue of its two unique strengths: (i) it is characterized by a small number of parameters as opposed to the frequency response model, thus facilitating individualization with small amount of data; and (ii) all its parameters are equipped with physiological implications, thus facilitating the assessment of patient-specific CV health based on the individualized TL model parameters in conjunction with the BP waveform data. In fact, when combined with techniques for estimating subject-specific TL model parameters, the TL model has been shown to be very useful in estimating and monitoring arterial hemodynamic indices (Zhang et al.,

Despite its success thus far, the simplicity of the uniform lossless TL model motivates investigations for its potential improvement by incorporating more realistic components. In particular, arteries exhibit tapering, bifurcations, and BP loss. In a series of our prior work, we investigated the TL models equipped with bifurcations and pressure loss to demonstrate that such extension of the uniform lossless TL model may lead to small but statistically significant improvement in its goodness of fit. As a follow-up work, the goal of this study was to investigate if there is any benefit in incorporating the geometric tapering into the TL model. To achieve the goal, this study comparatively investigated the tapered vs. uniform TL models as mathematical representation for BP wave propagation in human aorta. The relationship between the aortic inlet and outlet BP waves was formulated based on the exponentially tapered and uniform TL models. Then, the validity of the two TL models was comparatively investigated by fitting them to the experimental aortic and femoral BP waveform signals collected from 13 coronary artery bypass graft surgery patients.

This paper is organized as follows. Experimental data, the TL models, and the data analysis details are given in section Methods. Section Results presents key results, which are interpreted and discussed in section Discussion. Section Conclusion provides conclusions derived from the study.

The experimental data collected in our prior work (Rashedi et al.,

Data used in this work were collected right before or after the cardiopulmonary bypass. Following the induction of anesthesia and before the cardiopulmonary bypass, a catheter was inserted into the femoral artery. Then, a cannula was inserted into the ascending aorta by a surgeon immediately before or after the cardiopulmonary bypass. Then, ascending aortic and femoral arterial BP waveforms were recorded at a sampling rate of 1 kHz for up to 2 min.

In this study, a variant of the exponentially tapered TL model of the aorta developed by Fogliardi et al. (

where θ_{1} = _{0} and _{0} denote the inertance and compliance per unit length at the tube inlet; and (v) _{p} and _{c0} denote the terminal load resistance and tube characteristic impedance at the tube inlet, respectively. The uniform TL model is derived as a simplified case of the tapered TL model when _{1} = 0 and

It is noted that θ_{1} implies the absolute extent of tapering between the tube inlet and outlet cross-sectional radii:

Exponentially tapered _{1} = _{0}, and _{0} denote the inertance and compliance per unit length at the tube inlet; and _{p} and _{c0} denote the terminal load resistance and tube characteristic impedance at the tube inlet, respectively. Uniform TL model is derived as a special case of exponentially tapered TL model when _{2} and θ_{3}.

The validity of the tapered and uniform TL models was investigated and compared by fitting the models to the ascending aortic and femoral arterial BP waveforms associated with each subject on an individual basis. Details follow.

In each subject, a 15 beat-long pair of ascending aortic and femoral arterial BP waveforms were extracted from the recorded data and then down-sampled at 100 Hz. The first 10 beat-long data (called the training data) were used for model fitting, while the remaining 5 beat-long data (called the testing data) were used for assessing the validity of the models thus fitted. In this way, the TL models could be tested in the same CV state as when they were trained using the data not presented in the training process. For the sake of model fitting, the following optimization problem was solved using MATLAB and its Optimization Toolbox in order to derive the optimal parameter estimates θ^{*} associated with each subject from the training data:

where _{1}, θ_{2}, θ_{3}}, when the aortic inlet BP (i.e., the ascending aortic BP) data were inputted. The domain Ω_{θ} was defined as Ω_{θ} = {θ|θ_{1} > 0, θ_{2} > 0, θ_{3} > 0} based on the physical meanings of the TL model parameters. The model-predicted aortic outlet BP

Finally,

It is well-known that PTT is the most critical high-sensitivity parameter in the uniform TL model (Sugimachi et al., _{2} over a physiologically plausible range while (ii) θ_{1} and θ_{3} were determined for each value of θ_{2}. In deriving the optimal θ_{1} and θ_{3} associated with each θ_{2}, multiple (85) initial guesses were employed to ensure that the solution obtained from the optimization problem corresponds to (or at least is very close to) global minimum. For each θ associated with each of the θ_{2} values examined, the cost function in Equation (3) was evaluated. Then, θ associated with the minimum cost function value was determined as θ^{*}. In this way, the integrity and accuracy of the estimated TL model parameters was maximized.

The validity of the TL models was then assessed using both testing and training data. The testing data were employed to assess (i) the goodness of fit including the root-mean-squared error (RMSE) and correlation coefficient (

where

_{1} = _{1} = _{1} = _{1} =

The validity metrics of the tapered vs. uniform TL models, including the root-mean-squared error (RMSE), correlation coefficient (

Tapered TL | 3.3+/−1.1 | 2.5+/−1.1 | 0.98+/−0.02 | 0.98+/−0.01 | 6 |

Tapered TL (1.7 ≤ q ≤ 3) | 3.9+/−1.1^{*} |
2.8+/−1.1^{*} |
0.97+/−0.02^{*}^{†} |
0.98+/−0.01^{*} |
0 |

Uniform TL | 3.4+/−1.1 | 2.8+/−1.0^{*} |
0.98+/−0.01 | 0.98+/−0.01^{*} |
7 |

Representative femoral blood pressure (BP) waveforms derived from tapered vs. uniform tube-load (TL) models when ascending aortic BP waveform was inputted.

Representative frequency responses of the two tube-load (TL) models in comparison with the non-parametric frequency response derived directly from the aortic inlet and outlet blood pressure (BP) waveforms.

Parameter values estimated for the two TL models.

Tapered TL | 78+/−16 | 0.55+/−0.19 | 0.6+/−0.7 |

Tapered TL (1.7 ≤ q ≤ 3) | 84+/−02^{†} |
0.73+/−0.09^{*}^{†} |
1.7+/−0.1^{*}^{†} |

Uniform TL | 70+/−13^{*} |
0.43+/−0.15^{*} |
0^{*} |

Comparison of individual-specific pulse transit time (PTT), reflection constant (Γ), and radius ratio (qL) values associated with tapered vs. uniform tube-load (TL) models.

Individual-specific pulse transit time (PTT) values associated with the two tube-load (TL) models in comparison to the PTT values derived directly from the aortic inlet and outlet blood pressure (BP) waveforms.

The uniform TL model has the potential to enable patient-specific assessment of CV health with its minimal number of physiologically interpretable model parameters that may be individualized using small amount of data. Despite its demonstrated success in CV health and disease monitoring applications, opportunities exist for its potential improvement by incorporating realistic components. In this study, the effect of adding an exponential tapering to the TL model as an approximation for aortic geometric tapering on its predictive performance and physiological relevance was investigated.

The exponentially tapered and uniform TL models exhibited comparable goodness of fit for the aortic outlet BP whose differences were not statistically significant, both in terms of RMSE and correlation coefficient (

The comparable goodness of fit between the exponentially tapered and uniform TL models was supported by the values of the tapering constant (θ_{1} = _{G}) and location (F_{G}) of its first peak, which may be the most practically critical peak considering the limited frequency contents of the aortic BP signals (Hahn et al., _{G} of the first peak in the frequency response while increasing its frequency coordinate F_{G} (_{G} (_{G} while decreasing F_{G} (_{G}) and location (F_{G}) of the first peak in the frequency response associated with the two TL model remain the same as dictated by the data to be fitted.

Parametric sensitivity of the frequency response associated with the tapered tube-load (TL) model. _{G} of the first peak in the frequency response while increasing its frequency coordinate F_{G}. Green dashed line: frequency response of uniform TL model. Blue dotted line: frequency response of tapered TL model (qL > 0) with pulse transit time (PTT) and reflection constant (Γ) identical to uniform TL model. _{G}. Red solid line: frequency response of tapered TL model in _{G} while decreasing F_{G}. Black solid line: frequency response of tapered TL model in

When the tapering constant was constrained in solving the model fitting problem in Equation (3) to restrict the aortic inlet-outlet radius ratio in the vicinity of its anatomically plausible value (1.7~3.0), the exponentially tapered TL model underperformed the uniform TL model both in terms of RMSE and correlation coefficient (

On the one hand, the results all in all suggest that exponential aortic tapering may not be physiologically relevant for at least two reasons. First, the TL model with exponential tapering tends to fall back to the uniform TL model as it is fitted to the experimental data. Second, the TL model with exponential aortic tapering exhibited poor predictive accuracy than the uniform TL model if anatomically plausible aortic tapering was enforced. In fact, this finding may be corroborated by a prior study, which showed that the tapered TL model did not exhibit superiority to its uniform counterpart in representing the aortic impedance (Fogliardi et al.,

Typical anatomical aortic diameter data with respect to the distance from aortic inlet (black circles) and its exponential (blue solid line) and linear (red dashed line) fits. The anatomical data are from a prior work (Reymond et al.,

The results of this study suggest that the uniform TL model may be more robust and thus preferred as representation for BP wave propagation in human aorta relative to the exponentially tapered TL model. In comparison with the uniform TL model, the exponentially tapered TL model may not provide valid physiological insight on the aortic tapering, and the improvement in the goodness of fit offered by the exponential aortic tapering may only be marginal. Considering that exponential aortic tapering is relevant from physiological standpoint, future work on more rigorous investigation and refinement of exponentially tapered TL model will be rewarding.

The datasets for this manuscript are not publicly available because no IRB approval and participant consent were obtained for making the data publicly available. Requests to access the datasets should be directed to J-OH,

The study was carried out in accordance with the recommendations of the University of Alberta Health Research Ethics Board with written informed consent from all subjects. All subjects gave written informed consent in accordance with the Declaration of Helsinki. The protocol was approved by the University of Alberta Health Research Ethics Board (ID Pro00021889).

J-OH and RM designed the study. BF and MM conducted human subject study. AM, AT, RM, and J-OH analyzed the data. BF, MM, RM, and J-OH reviewed the data analysis results. AM and J-OH drafted and revised the manuscript. AT, BF, MM, and RM reviewed the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

In this study, a variant of the exponentially tapered tube-load model of the aorta developed by Fogliardi et al. (

where _{0} is the aortic inlet radius, then the aortic inertance, compliance, and characteristic impedance can be expressed as follows, according to the quadratic dependence of the aortic inertance and compliance on the radius based on the assumptions that (i) the aortic incremental Young modulus is constant and (ii) the aortic wall thickness is proportional to the aortic radius:

where _{c}(_{0}, _{0}, and _{c0} are their respective values at the aortic inlet.

The BP wave at a distance

where _{f}(_{b}(^{2} = −1), and _{f}(_{b}(

where

The relationship (A.5) can be parameterized with θ_{1} =

where