Tube-load Model Research Articles

Transmission line model as a digital twin for abdominal aortic aneurysm patients

We investigated the potential of the transmission line model as a digital twin of aneurysmal aorta by comparatively analyzing how a uniform lossless tube-load model were fitted to the carotid and femoral artery tonometry waveforms pertaining to (i) 79 abdominal aortic aneurysm (AAA) patients vs their matched controls (CON) and (ii) 35 AAA patients before vs after endovascular aneurysm repair (EVAR). The uniform lossless tube-load model fitted the tonometry waveforms pertaining to AAA as well as CON and EVAR. In addition, the parameters in the tube-load model exhibited physiologically explainable changes: when normalized, both pulse transit time and reflection coefficient increased with AAA and decreased after EVAR, which can be explained by the increase in arterial compliance and the decrease in arterial inertance due to the aortic expansion associated with AAA. In sum, the tube-load model may have the potential as a digital twin to enable personalized AAA monitoring.

Hybrid modeling on reconstitution of continuous arterial blood pressure using finger photoplethysmography

The continuous estimation of arterial blood pressure (ABP) waveforms directly from single-channel photoplethysmography (PPG) signals will predictably change the way to monitor blood pressure in the future. This article proposed a new hybrid mathematical model for continuous blood pressure monitoring by investigating the relationship between the finger PPG signal and the radial ABP signal based on a public database. Considering potential damping factors and wave propagation/reflection in blood circulation, we combined the electrical network model with the tube-load model. The optimal range of model parameters was obtained through the system identification method to realize the individualized continuous blood pressure measurements. Compared with the invasive measurement, the hybrid model performed superior blood pressure estimation with high consistency. The estimated ABP waveforms correlated highly with the reference waveforms with an average correlation coefficient of 0.96. The mean absolute error/standard deviation of the estimated systolic blood pressure (SBP), mean arterial blood pressure (MAP), and diastolic blood pressure (DBP) were 3.0/4.4, 2.1/3.0 and 2.1/3.2 mmHg, respectively. The results met the requirements of the Association for the Advancement of Medical Instrumentation (AAMI). The hybrid model is expected to be embedded in small wearable devices to directly estimate the continuous blood pressure waveforms at the radial artery site through the PPG signals, pioneering the synchronous non-sensing monitoring on blood pressure and blood flow.

A Vibrating-String Model for Closed-Loop Wave Transmission and Reflection Between the Aorta and Periphery

Abstract A tube-load model is used to reconstruct aortic pressure waveform from peripheral pressure waveform. Yet, the reconstructed aortic pressure waveform is greatly affected by load impedance used. In this work, a vibrating-string model for closed-loop wave transmission and reflection between the aorta and periphery is developed to examine the roles of all the parameters involved in aortic pressure waveform. The arterial pulsatile wave theory gives rise to the standard one-dimensional wave equation for a vibrating string. A vibrating-string model based on radial displacement of the arterial wall is developed to relate aortic pressure waveform to peripheral pressure waveform, relate load impedance to input impedance, and derive theoretical expressions for associated clinical indices. The vibrating-string model is extended to incorporate blood velocity and is further connected to the left ventricle (LV) to study the role of the LV in aortic pressure waveform. The difference between the vibrating-string model and the tube-load model is also examined. Load impedance is identified as an indispensable independent parameter for reconstruction of aortic pressure waveform with accuracy, and its physiologically realistic harmonic dependence can only be obtained from the measured input impedance. The derived expressions for clinical indices interpret some clinical findings and underscore the role of harmonics in clinical indices. Some misconceptions in the tube-load model are revealed, including load impedance and characteristic impedance. This work clarifies the role of harmonics-dependence of load impedance and harmonics of aortic pressure waveform in determining clinical indices.

Personalized aortic pressure waveform estimation from brachial pressure waveform using an adaptive transfer function

Background and objectiveThe aortic pressure waveform (APW) provides reliable information for the diagnosis of cardiovascular disease. APW is often measured using a generalized transfer function (GTF) applied to the peripheral pressure waveform acquired noninvasively, to avoid the significant risks of invasive APW acquisition. However, the GTF ignores various physiological conditions, which affects the accuracy of the estimated APW. To solve this problem, this study utilized an adaptive transfer function (ATF) combined with a tube-load model to achieve personalized and accurate estimation of APW from the brachial pressure waveform (BPW). MethodsThe proposed method was validated using APWs and BPWs from 34 patients. The ATF was defined using a tube-load model in which pulse transit time and reflection coefficients were determined from, respectively, the diastolic-exponential-pressure-decay of the APW and a piece-wise constant approximation. The root-mean-square-error of overall morphology, mean absolute errors of common hemodynamic indices (systolic blood pressure, diastolic blood pressure and pulse pressure) were used to evaluate the ATF. ResultsThe proposed ATF performed better in estimating diastolic blood pressure and pulse pressure (1.63 versus 1.94 mmHg, and 2.37 versus 3.10 mmHg, respectively, both P < 0.10), and produced similar errors in overall morphology and systolic blood pressure (3.91 versus 4.24 mmHg, and 2.83 versus 2.91 mmHg, respectively, both P > 0.10) compared to GTF. ConclusionUnlike the GTF which uses fixed parameters trained on existing clinical datasets, the proposed method can achieve personalized estimation of APW. Hence, it provides accurate pulsatile hemodynamic measures for the evaluation of cardiovascular function.

Improving the accuracy and robustness of carotid-femoral pulse wave velocity measurement using a simplified tube-load model

Arterial stiffness, as measured by pulse wave velocity, for the early non-invasive screening of cardiovascular disease is becoming ever more widely used and is an independent prognostic indicator for a variety of pathologies including arteriosclerosis. Carotid-femoral pulse wave velocity (cfPWV) is regarded as the gold standard for aortic stiffness. Existing algorithms for cfPWV estimation have been shown to have good repeatability and accuracy, however, further assessment is needed, especially when signal quality is compromised. We propose a method for calculating cfPWV based on a simplified tube-load model, which allows for the propagation and reflection of the pulse wave. In-vivo cfPWV measurements from 57 subjects and numerical cfPWV data based on a one-dimensional model were used to assess the method and its performance was compared to three other existing approaches (waveform matching, intersecting tangent, and cross-correlation). The cfPWV calculated using the simplified tube-load model had better repeatability than the other methods (Intra-group Correlation Coefficient, ICC = 0.985). The model was also more accurate than other methods (deviation, 0.13 ms−1) and was more robust when dealing with noisy signals. We conclude that the determination of cfPWV based on the proposed model can accurately and robustly evaluate arterial stiffness.

Noninvasive estimation of aortic pressure waveform based on simplified Kalman filter and dual peripheral artery pressure waveforms

Background and ObjectiveAortic pressure (Pa) is important for the diagnosis of cardiovascular disease. However, its direct measurement is invasive, not risk-free, and relatively costly. In this paper, a new simplified Kalman filter (SKF) algorithm is employed for the reconstruction of the Pa waveform using dual peripheral artery pressure waveforms. MethodsPa waveforms obtained in a previous study were collected from 25 patients. Simultaneously, radial and femoral pressure waveforms were generated from two simulation experiments, using transfer functions. In the first, the transfer function is a known finite impulse response; and in the second, it is derived from a tube-load model. To analyze the performance of the proposed SKF algorithm, variable amounts of noise were added to the observed output signal, to give a range of signal-to-noise ratios (SNRs). Additionally, central aortic, brachial and femoral pressure waveforms were simultaneously collected from 2 Sprague-Dawley rats and the measured and reconstructed Pa waveforms were compared. ResultsThe proposed SKF algorithm outperforms canonical correlation analysis (CCA), which is the current state-of-the-art blind system identification method for the non-invasive estimation of central aortic blood pressure. It is also shown that the proposed SKF algorithm is more noise-tolerant than the CCA algorithm over a wide range of SNRs. ConclusionThe simulations and animal experiments illustrate that the proposed SKF algorithm is accurate and stable in the face of low SNRs. Improved methods for estimating central blood pressure as a measure of cardiac load adds to their value as a prognostic and diagnostic tool.

Tube-load model: A clinically applicable pulse contour analysis method for estimation of cardiac stroke volume

Background and Objectives: Accurate, reproducible, and reliable real-time clinical measurement of stroke volume (SV) is challenging. To accurately estimate arterial mechanics and SV by pulse contour analysis, accounting for wave reflection, such as by a tube-load model, is potentially important. This study tests for the first time whether a dynamically identified tube-load model, given a single peripheral arterial input signal and pulse transit time (PTT), provides accurate SV estimates during hemodynamic instability.Methods: The model is tested for 5 pigs during hemodynamic interventions, using either an aortic flow probe or admittance catheter for a validation SV measure. Performance is assessed using Bland-Altman and polar plot analysis for a series of long-term state-change and short-term dynamic events.Results:The overall median bias and limits of agreement (2.5th, 97.5th percentile) from Bland-Altman analysis were -10% [-49, 36], and -1% [-28,20] for state-change and dynamic events, respectively. The angular limit of agreement (maximum of 2.5th, 97.5th percentile) from polar-plot analysis for state-change and dynamic interventions was 35.6∘, and 35.2∘, respectively.Conclusion: SV estimation agreement and trending performance was reasonable given the severity of the interventions. This simple yet robust method has potential to track SV within acceptable limits during hemodynamic instability in critically ill patients, provided a sufficiently accurate PTT measure.

Estimating Central Blood Pressure From a Single Peripheral Pressure Measurement

Abstract This paper presents a set of criteria used to identify the parameters describing a tube-load model of the arterial system to estimate central blood pressure (BP). The criteria are generalizable to accommodate for inter- and intra-subject variability encountered in the ICU. The proposed single measurement transfer function (SMTF) requires only a single peripheral pressure measurement, commonly available in the ICU, for central pressure estimation, removing the need for an additional measurement of pulse transit time, common to other central BP estimation models. The method was tested using data from six (6) porcine experiments where septic shock was induced and subsequent treatment was performed. Systolic pressure (SP), pulse pressure (PP) and root-mean-squared (RMSE) errors relative to invasive measurements of aortic pressure were used to assess accuracy. The SMTF method produced mean errors ≤5mmHg across all metrics with 84.4, 88.7 and 63.2% of SP, PP, and RMSE, respectively, within this bound of measured central pressure. Peripheral BP accuracy was also assessed as it is commonly used as a surrogate for central BP in clinical settings. Finally, two alternate methods utilizing the same model equations with additional inputs, one using the measured pulse transit time and the other minimizing the RMSE with measured aortic pressure, were implemented to compare the SMTF accuracy to best case outputs given the model equations.

Reconstruction of Continuous Brachial Arterial Pressure From Continuous Finger Arterial Pressure Using a Two-Level Optimization Strategy.

We attempt to reconstruct brachial arterial pressure (BAP) waves from finger arterial pressure waves measured using the vascular unloading technique without arm-cuff calibration. A novel method called two-level optimization (TOP) strategy is proposed as follows. We first derive a simplified transfer function (TF) based on a tube-load model with only two parameters to be estimated, a coefficient B and a time delay ∆t. Then, at level one, two minimization problems are formulated to estimate the optimal coefficient Bopt and time delay ∆topt. Then, we can derive an optimal TF hopt(t). However, this derivation requires true (or reference) BAP waves. Therefore, at level two, we apply multiple linear regression (MLR) to further model the relationship between the derived optimal parameters and subjects' physiologic parameters. Hence, eventually, one can estimate coefficient BMLR and time delay ∆tMLR from subject's physiologic parameters to derive the MLR-based TF hMLR(t) for the BAP reconstruction. Twenty-one volunteers were recruited for the data collection. The mean ± standard deviation of the root mean square errors between the reference BAP waves and the BAP waves reconstructed by hopt(t), hMLR(t), and a generalized transfer function (GTF) were 3.46 ± 1.42mmHg, 3.61 ± 2.28mmHg, and 6.80 ± 3.73mmHg (significantly larger with p < 0.01), respectively. The proposed method can be considered as a semi-individualized TF which reconstructs significantly better BAP waves than a GTF. The proposed TOP strategy can potentially be useful in more general reconstruction of proximal BP waves.

Tapered vs. Uniform Tube-Load Modeling of Blood Pressure Wave Propagation in Human Aorta.

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 r = 0.98+/−0.02 in the tapered tube-load model and r = 0.98+/−0.01 mmHg in the uniform tube-load model. They also exhibited frequency responses comparable to the non-parametric frequency response derived from the aortic and femoral BP waveforms in most patients. Hence, the uniform tube-load model was superior to its tapered counterpart in terms of the Akaike Information Criterion (AIC). In general, the tapered tube-load model yielded the degree of tapering smaller than what is physiologically relevant: the aortic inlet-outlet radius ratio was estimated as 1.5 on the average, which was smaller than the anatomically plausible typical radius ratio of 3.5 between the ascending aorta and femoral artery. When the tapering ratio was restricted to the vicinity of the anatomically plausible typical value, the exponentially tapered tube-load model tended to underperform the uniform tube-load model (RMSE: 3.9+/−1.1 mmHg; r = 0.97+/−0.02). It was concluded that the uniform tube-load model may be more robust and thus preferred as the representation for BP wave propagation in human aorta; compared to the uniform tube-load model, the exponentially tapered tube-load model may not provide valid physiological insight on the aortic tapering, and its efficacy on the goodness of fit may be only marginal.

Investigation of Viscoelasticity in the Relationship Between Carotid Artery Blood Pressure and Distal Pulse Volume Waveforms.

We investigated the relationship between carotid artery blood pressure (BP) and distal pulse volume waveforms (PVRs) via subject-specific mathematical modeling. We conceived three physical models to define the relationship: a tube-load model augmented with a gain (TLG), Voigt (TLV), and standard linear solid (TLS) models. We compared these models using PVRs measured via BP cuffs at an upper arm and an ankle as well as carotid artery tonometry waveform collected from 133 subjects. At both upper arm and ankle, PVR was related to carotid artery tonometry by TLV and TLS models better than by TLG model; when root-mean-squared over all the subjects, the systolic and diastolic BP errors between measured carotid artery tonometry waveform and the one estimated from distal PVR reduced from 4.3mmHg and 4.6mmHg (TLG) to 1.1mmHg and 1.0mmHg (TLS) for the upper arm (p < 0.0167), and from 2.1mmHg and 1.7mmHg (TLG) to 2.1mmHg and 1.5mmHg (TLV) for the ankle. Further, TLV and TLS models exhibited superior Akaike's Information Criterion (AIC) in both locations than TLG model. However, the difference between TLG versus TLV and TLS models associated with the ankle was not large. Therefore, the relationship of central arterial BP to arm PVR arises from both wave reflection and viscoelasticity while the relationship to ankle PVR mainly arises from wave reflection. These findings may imply that an effective subject-specific transfer function for estimating accurate central arterial BP from an arm PVR should account for the impact of viscoelasticity.

Estimation of Pulse Transit Time as a Function of Blood Pressure Using a Nonlinear Arterial Tube-Load Model.

pulse transit time (PTT) varies with blood pressure (BP) throughout the cardiac cycle, yet, because of wave reflection, only one PTT value at the diastolic BP level is conventionally estimated from proximal and distal BP waveforms. The objective was to establish a technique to estimate multiple PTT values at different BP levels in the cardiac cycle. a technique was developed for estimating PTT as a function of BP (to indicate the PTT value for every BP level) from proximal and distal BP waveforms. First, a mathematical transformation from one waveform to the other is defined in terms of the parameters of a nonlinear arterial tube-load model accounting for BP-dependent arterial compliance and wave reflection. Then, the parameters are estimated by optimally fitting the waveforms to each other via the model-based transformation. Finally, PTT as a function of BP is specified by the parameters. The technique was assessed in animals and patients in several ways including the ability of its estimated PTT-BP function to serve as a subject-specific curve for calibrating PTT to BP. the calibration curve derived by the technique during a baseline period yielded bias and precision errors in mean BP of 5.1 ± 0.9 and 6.6 ± 1.0 mmHg, respectively, during hemodynamic interventions that varied mean BP widely. the new technique may permit, for the first time, estimation of PTT values throughout the cardiac cycle from proximal and distal waveforms. the technique could potentially be applied to improve arterial stiffness monitoring and help realize cuff-less BP monitoring.

Data-driven lossy tube-load modeling of arterial tree: in-human study.

In this paper, we present and validate a data-driven method to lossy tube-load modeling of arterial tree in humans. In the proposed method, the lossy tube-load model is fitted to central aortic and peripheral blood pressure (BP) waves in the time domain. For this purpose, we employ a time-domain lossy tube-load model in which the wave propagation constant is formulated to two terms: one responsible for the alteration of wave amplitude and the other for the transport delay. Using the experimental BP data collected from 17 cardiac surgery patients, we showed that the time-domain lossy tube-load model is able to accurately represent the relation between central aortic versus upper-limb and lower-limb BP waves. In addition, the comparison of lossy versus lossless tube-load models revealed that (1) the former outperformed the latter in general with the root-mean-squared errors (RMSE) of 3.1 mm Hg versus 3.5 mm Hg, respectively (p-value &lt; 0.05), and (2) the efficacy of the former over the latter was more clearly observed in case the normalized difference in the mean central aortic versus peripheral BP was large; when the difference was &gt;5% of the underlying mean BP, lossy and lossless models showed the RMSE of 2.7 mm Hg and 3.7 mm Hg, respectively (p-value &lt; 0.05).

Improved pulse wave velocity estimation using an arterial tube-load model.

Pulse wave velocity (PWV) is the most important index of arterial stiffness. It is conventionally estimated by noninvasively measuring central and peripheral blood pressure (BP) and/or velocity (BV) waveforms and then detecting the foot-to-foot time delay between the waveforms wherein wave reflection is presumed absent. We developed techniques for improved estimation of PWV from the same waveforms. The techniques effectively estimate PWV from the entire waveforms, rather than just their feet, by mathematically eliminating the reflected wave via an arterial tube-load model. In this way, the techniques may be more robust to artifact while revealing the true PWV in absence of wave reflection. We applied the techniques to estimate aortic PWV from simultaneously and sequentially measured central and peripheral BP waveforms and simultaneously measured central BV and peripheral BP waveforms from 17 anesthetized animals during diverse interventions that perturbed BP widely. Since BP is the major acute determinant of aortic PWV, especially under anesthesia wherein vasomotor tone changes are minimal, we evaluated the techniques in terms of the ability of their PWV estimates to track the acute BP changes in each subject. Overall, the PWV estimates of the techniques tracked the BP changes better than those of the conventional technique (e.g., diastolic BP root-mean-squared errors of 3.4 versus 5.2 mmHg for the simultaneous BP waveforms and 7.0 versus 12.2 mmHg for the BV and BP waveforms (p <; 0.02)). With further testing, the arterial tube-load model-based PWV estimation techniques may afford more accurate arterial stiffness monitoring in hypertensive and other patients.

Editorial: engineering approaches to study cardiovascular physiology: modeling, estimation, and signal processing

EDITORIAL article Front. Physiol., 05 November 2012Sec. Computational Physiology and Medicine Volume 3 - 2012 | https://doi.org/10.3389/fphys.2012.00425

Tube-Load Model Parameter Estimation for Monitoring Arterial Hemodynamics

A useful model of the arterial system is the uniform, lossless tube with parametric load. This tube-load model is able to account for wave propagation and reflection (unlike lumped-parameter models such as the Windkessel) while being defined by only a few parameters (unlike comprehensive distributed-parameter models). As a result, the parameters may be readily estimated by accurate fitting of the model to available arterial pressure and flow waveforms so as to permit improved monitoring of arterial hemodynamics. In this paper, we review tube-load model parameter estimation techniques that have appeared in the literature for monitoring wave reflection, large artery compliance, pulse transit time, and central aortic pressure. We begin by motivating the use of the tube-load model for parameter estimation. We then describe the tube-load model, its assumptions and validity, and approaches for estimating its parameters. We next summarize the various techniques and their experimental results while highlighting their advantages over conventional techniques. We conclude the review by suggesting future research directions and describing potential applications.