Examining temporal changes in model-optimized parameters using longitudinal hemodynamic measurements.

BACKGROUND: We previously applied hemodynamic data to personalize a mathematical model of the circulation expressed as physically interpretable parameters. The aim of this study was to identify patterns in the data that could potentially explain the estimated parameter changes. This included investigating whether the parameters could be used to track the effect of physical activity on high blood pressure. Clinical trials have repeatedly detected beneficial changes in blood pressure after physical activity and uncovered changes in lower level phenotypes (such as stiffened or high-resistance blood vessels). These phenotypes can be characterized by parameters describing the mechanical properties of the circulatory system. These parameters can be incorporated in and contextualized by physics-based cardiovascular models of the circulation, which in combination can become tools for monitoring cardiovascular disease progression and management in the future. METHODS: Closed-loop and open-loop models of the left ventricle and systemic circulation were previously optimized to data from a pilot study with a 12-week exercise intervention period. Basal characteristics and hemodynamic data such as blood pressure in the carotid, brachial and finger arteries, as well as left-ventricular outflow tract flow traces were collected in the trial. Model parameters estimated for measurements made on separate days during the trial were used to compute parameter changes for total peripheral resistance, systemic arterial compliance, and maximal left-ventricular elastance. We compared the changes in these cardiovascular model-based estimates to changes from more conventional estimates made without the use of physics-based models by correlation analysis. Additionally, ordinary linear regression and linear mixed-effects models were applied to determine the most informative measurements for the selected parameters. We applied maximal aerobic capacity (measured as VO2max ) data to examine if exercise had any impact on parameters through regression analysis and case studies. RESULTS AND CONCLUSIONS: Parameter changes in arterial parameters estimated using the cardiovascular models correlated moderately well with conventional estimates. Estimates based on carotid pressure waveforms gave higher correlations (0.59 and above when p < 0.05 ) than those for finger arterial pressure. Parameter changes over the 12-week study duration were of similar magnitude when compared to short-term changes after a bout of intensive exercise in the same parameters. The short-term changes were computed from measurements made immediately before and 24 h after a cardiopulmonary exercise test used to measure VO2max . Regression analysis indicated that changes in VO2max did not account for any substantial amount of variability in total peripheral resistance, systemic arterial compliance, or maximal left-ventricular elastance. On the contrary, changes in stroke volume contributed to far more explained variability. The results suggest that more research is required to be able to accurately track exercise-induced changes in the vasculature for people with pre-hypertension and hypertension using lumped-parameter models.

Global sensitivity analysis with multifidelity Monte Carlo and polynomial chaos expansion for vascular haemodynamics.

Computational models of the cardiovascular system are increasingly used for the diagnosis, treatment, and prevention of cardiovascular disease. Before being used for translational applications, the predictive abilities of these models need to be thoroughly demonstrated through verification, validation, and uncertainty quantification. When results depend on multiple uncertain inputs, sensitivity analysis is typically the first step required to separate relevant from unimportant inputs, and is key to determine an initial reduction on the problem dimensionality that will significantly affect the cost of all downstream analysis tasks. For computationally expensive models with numerous uncertain inputs, sample-based sensitivity analysis may become impractical due to the substantial number of model evaluations it typically necessitates. To overcome this limitation, we consider recently proposed Multifidelity Monte Carlo estimators for Sobol' sensitivity indices, and demonstrate their applicability to an idealized model of the common carotid artery. Variance reduction is achieved combining a small number of three-dimensional fluid-structure interaction simulations with affordable one- and zero-dimensional reduced-order models. These multifidelity Monte Carlo estimators are compared with traditional Monte Carlo and polynomial chaos expansion estimates. Specifically, we show consistent sensitivity ranks for both bi- (1D/0D) and tri-fidelity (3D/1D/0D) estimators, and superior variance reduction compared to traditional single-fidelity Monte Carlo estimators for the same computational budget. As the computational burden of Monte Carlo estimators for Sobol' indices is significantly affected by the problem dimensionality, polynomial chaos expansion is found to have lower computational cost for idealized models with smooth stochastic response.

Age and sex-dependent sensitivity analysis of a common carotid artery model.

The common carotid artery (CCA) is an accessible and informative site for assessing cardiovascular function which makes it a prime candidate for clinically relevant computational modelling. The interpretation of supplemental information possible through modelling is encumbered by measurement uncertainty and population variability in model parameters. The distribution of model parameters likely depends on the specific sub-population of interest and delineation based on sex, age or health status may correspond to distinct ranges of typical parameter values. To assess this impact in a 1D-CCA-model, we delineated specific sub-populations based on age, sex and health status and carried out uncertainty quantification and sensitivity analysis for each sub-population. We performed a structured literature review to characterize sub-population-specific variabilities for eight model parameters without consideration of health status; variations for a healthy sub-populations were based on previously established references values. The variabilities of diameter and distensibility found in the literature review differed from those previously established in a healthy population. Model diameter change and pulse pressure were most sensitive to variations in distensibility, while pressure was most sensitive to resistance in the Windkessel model for all groups. Uncertainties were lower when variabilities were based on a healthy sub-population; however, the qualitative distribution of sensitivity indices was largely similar between the healthy and general population. Average sensitivity of the pressure waveform showed a moderate dependence on age with decreasing sensitivity to distal resistance and increasing sensitivity to distensibility and diameter. The female population was less sensitive to variations in diameter but more sensitive to distensibility coefficient than the male population. Overall, as hypothesized input variabilities differed between sub-populations and resulted in distinct uncertainties and sensitivities of the 1D-CCA-model outputs, particularly over age for the pressure waveform and between males and females for pulse pressure.

Monitoring variability in parameter estimates for lumped parameter models of the systemic circulation using longitudinal hemodynamic measurements.

BACKGROUND: Physics-based cardiovascular models are only recently being considered for disease diagnosis or prognosis in clinical settings. These models depend on parameters representing the physical and physiological properties of the modeled system. Personalizing these parameters may give insight into the specific state of the individual and etiology of disease. We applied a relatively fast model optimization scheme based on common local optimization methods to two model formulations of the left ventricle and systemic circulation. One closed-loop model and one open-loop model were applied. Intermittently collected hemodynamic data from an exercise motivation study were used to personalize these models for data from 25 participants. The hemodynamic data were collected for each participant at the start, middle and end of the trial. We constructed two data sets for the participants, both consisting of systolic and diastolic brachial pressure, stroke volume, and left-ventricular outflow tract velocity traces paired with either the finger arterial pressure waveform or the carotid pressure waveform. RESULTS: We examined the feasibility of separating parameter estimates for the individual from population estimates by assessing the variability of estimates using the interquartile range. We found that the estimated parameter values were similar for the two model formulations, but that the systemic arterial compliance was significantly different ([Formula: see text]) depending on choice of pressure waveform. The estimates of systemic arterial compliance were on average higher when using the finger artery pressure waveform as compared to the carotid waveform. CONCLUSIONS: We found that for the majority of participants, the variability of parameter estimates for a given participant on any measurement day was lower than the variability both across all measurement days combined for one participant, and for the population. This indicates that it is possible to identify individuals from the population, and that we can distinguish different measurement days for the individual participant by parameter values using the presented optimization method.

Parameter estimation for closed-loop lumped parameter models of the systemic circulation using synthetic data.

Physics-based models can be applied to describe mechanisms in both health and disease, which has the potential to accelerate the development of personalized medicine. The aim of this study was to investigate the feasibility of personalizing a model of systemic hemodynamics by estimating model parameters. We investigated the feasibility of estimating model parameters for a closed-loop lumped parameter model of the left heart and systemic circulation using the step-wise subset reduction method. This proceeded by first investigating the structural identifiability of the model parameters. Secondly, we performed sensitivity analysis to determine which parameters were most influential on the most relevant model outputs. Finally, we constructed a sequence of progressively smaller subsets including parameters based on their ranking by model output influence. The model was then optimized to data for each set of parameters to evaluate how well the parameters could be estimated for each subset. The subsequent results allowed assessment of how different data sets, and noise affected the parameter estimates. In the noiseless case, all parameters could be calibrated to less than 10-3% error using time series data, while errors using clinical index data could reach over 100%. With 5% normally distributed noise the accuracy was limited to be within 10% error for the five most sensitive parameters, while the four least sensitive parameters were unreliably estimated for waveform data. The three least sensitive parameters were particularly challenging to estimate so these should be prioritized for measurement. Cost functions based on time series such as pressure waveforms, were found to give better parameter estimates than cost functions based on standard indices used in clinical assessment of the cardiovascular system, for example stroke volume (SV) and pulse pressure (PP). Averaged parameter estimate errors were reduced by several orders of magnitude by choosing waveforms for noiseless synthetic data. Also when measurement data were noisy, the parameter estimation procedure based on continuous waveforms was more accurate than that based on clinical indices. By application of the stepwise subset reduction method we demonstrated that by the addition of venous pressure to the cost function, or conversely fixing the systemic venous compliance parameter at an accurate value improved all parameter estimates, especially the diastolic filling parameters which have least influence on the aortic pressure.

Effect of side branch flow upon physiological indices in coronary artery disease.

Recent efforts have demonstrated the ability of computational models to predict fractional flow reserve from coronary artery imaging without the need for invasive instrumentation. However, these models include only larger coronary arteries as smaller side branches cannot be resolved and are therefore neglected. The goal of this study was to evaluate the impact of neglecting the flow to these side branches when computing angiography-derived fractional flow reserve (vFFR) and indices of volumetric coronary artery blood flow. To compensate for the flow to side branches, a leakage function based upon vessel taper (Murray's Law) was added to a previously developed computational model of coronary blood flow. The augmented model with a leakage function (1Dleaky) and the original model (1D) were then applied to predict FFR as well as inlet and outlet flow in 146 arteries from 80 patients who underwent invasive coronary angiography and FFR measurement. The results show that the leakage function did not significantly change the vFFR but did significantly impact the estimated volumetric flow rate and predicted coronary flow reserve. As both procedures achieved similar predictive accuracy of vFFR despite large differences in coronary blood flow, these results suggest careful consideration of the application of this index for quantitatively assessing flow.

Uncertainty Quantification and Sensitivity Analysis for Computational FFR Estimation in Stable Coronary Artery Disease.

PURPOSE: The main objectives of this study are to validate a reduced-order model for the estimation of the fractional flow reserve (FFR) index based on blood flow simulations that incorporate clinical imaging and patient-specific characteristics, and to assess the uncertainty of FFR predictions with respect to input data on a per patient basis. METHODS: We consider 13 patients with symptoms of stable coronary artery disease for which 24 invasive FFR measurements are available. We perform an extensive sensitivity analysis on the parameters related to the construction of a reduced-order (hybrid 1D-0D) model for FFR predictions. Next we define an optimal setting by comparing reduced-order model predictions with solutions based on the 3D incompressible Navier-Stokes equations. Finally, we characterize prediction uncertainty with respect to input data and identify the most influential inputs by means of sensitivity analysis. RESULTS: Agreement between FFR computed by the reduced-order model and by the full 3D model was satisfactory, with a bias ([Formula: see text]) of [Formula: see text] at the 24 measured locations. Moreover, the uncertainty related to the factor by which peripheral resistance is reduced from baseline to hyperemic conditions proved to be the most influential parameter for FFR predictions, whereas uncertainty in stenosis geometry had greater effect in cases with low FFR. CONCLUSION: Model errors related to solving a simplified reduced-order model rather than a full 3D problem were small compared with uncertainty related to input data. Improved measurement of coronary blood flow has the potential to reduce uncertainty in computational FFR predictions significantly.

Modeling the differentiation of A- and C-type baroreceptor firing patterns.

The baroreceptor neurons serve as the primary transducers of blood pressure for the autonomic nervous system and are thus critical in enabling the body to respond effectively to changes in blood pressure. These neurons can be separated into two types (A and C) based on the myelination of their axons and their distinct firing patterns elicited in response to specific pressure stimuli. This study has developed a comprehensive model of the afferent baroreceptor discharge built on physiological knowledge of arterial wall mechanics, firing rate responses to controlled pressure stimuli, and ion channel dynamics within the baroreceptor neurons. With this model, we were able to predict firing rates observed in previously published experiments in both A- and C-type neurons. These results were obtained by adjusting model parameters determining the maximal ion-channel conductances. The observed variation in the model parameters are hypothesized to correspond to physiological differences between A- and C-type neurons. In agreement with published experimental observations, our simulations suggest that a twofold lower potassium conductance in C-type neurons is responsible for the observed sustained basal firing, where as a tenfold higher mechanosensitive conductance is responsible for the greater firing rate observed in A-type neurons. A better understanding of the difference between the two neuron types can potentially be used to gain more insight about pathophysiology and treatment of diseases related to baroreflex function, e.g. in patients with autonomic failure, a syndrome that is difficult to diagnose in terms of its pathophysiology.

A guide to uncertainty quantification and sensitivity analysis for cardiovascular applications.

As we shift from population-based medicine towards a more precise patient-specific regime guided by predictions of verified and well-established cardiovascular models, an urgent question arises: how sensitive are the model predictions to errors and uncertainties in the model inputs? To make our models suitable for clinical decision-making, precise knowledge of prediction reliability is of paramount importance. Efficient and practical methods for uncertainty quantification (UQ) and sensitivity analysis (SA) are therefore essential. In this work, we explain the concepts of global UQ and global, variance-based SA along with two often-used methods that are applicable to any model without requiring model implementation changes: Monte Carlo (MC) and polynomial chaos (PC). Furthermore, we propose a guide for UQ and SA according to a six-step procedure and demonstrate it for two clinically relevant cardiovascular models: model-based estimation of the fractional flow reserve (FFR) and model-based estimation of the total arterial compliance (CT ). Both MC and PC produce identical results and may be used interchangeably to identify most significant model inputs with respect to uncertainty in model predictions of FFR and CT . However, PC is more cost-efficient as it requires an order of magnitude fewer model evaluations than MC. Additionally, we demonstrate that targeted reduction of uncertainty in the most significant model inputs reduces the uncertainty in the model predictions efficiently. In conclusion, this article offers a practical guide to UQ and SA to help move the clinical application of mathematical models forward. Copyright © 2015 John Wiley & Sons, Ltd.

Modeling the afferent dynamics of the baroreflex control system.

In this study we develop a modeling framework for predicting baroreceptor firing rate as a function of blood pressure. We test models within this framework both quantitatively and qualitatively using data from rats. The models describe three components: arterial wall deformation, stimulation of mechanoreceptors located in the BR nerve-endings, and modulation of the action potential frequency. The three sub-systems are modeled individually following well-established biological principles. The first submodel, predicting arterial wall deformation, uses blood pressure as an input and outputs circumferential strain. The mechanoreceptor stimulation model, uses circumferential strain as an input, predicting receptor deformation as an output. Finally, the neural model takes receptor deformation as an input predicting the BR firing rate as an output. Our results show that nonlinear dependence of firing rate on pressure can be accounted for by taking into account the nonlinear elastic properties of the artery wall. This was observed when testing the models using multiple experiments with a single set of parameters. We find that to model the response to a square pressure stimulus, giving rise to post-excitatory depression, it is necessary to include an integrate-and-fire model, which allows the firing rate to cease when the stimulus falls below a given threshold. We show that our modeling framework in combination with sensitivity analysis and parameter estimation can be used to test and compare models. Finally, we demonstrate that our preferred model can exhibit all known dynamics and that it is advantageous to combine qualitative and quantitative analysis methods.