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.

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.

Impact of baseline coronary flow and its distribution on fractional flow reserve prediction.

Model-based prediction of fractional flow reserve (FFR) in the context of stable coronary artery disease (CAD) diagnosis requires a number of modelling assumptions. One of these assumptions is the definition of a baseline coronary flow, ie, total coronary flow at rest prior to the administration of drugs needed to perform invasive measurements. Here we explore the impact of several methods available in the literature to estimate and distribute baseline coronary flow on FFR predictions obtained with a reduced-order model. We consider 63 patients with suspected stable CAD, for a total of 105 invasive FFR measurements. First, we improve a reduced-order model with respect to previous results and validate its performance versus results obtained with a 3D model. Next, we assess the impact of a wide range of methods to impose and distribute baseline coronary flow on FFR prediction, which proved to have a significant impact on diagnostic performance. However, none of the proposed methods resulted in a significant improvement of prediction error standard deviation. Finally, we show that intrinsic uncertainties related to stenosis geometry and the effect of hyperemic inducing drugs have to be addressed in order to improve FFR prediction accuracy.

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.

Bond Graph Model of Cerebral Circulation: Toward Clinically Feasible Systemic Blood Flow Simulations.

We propose a detailed CellML model of the human cerebral circulation that runs faster than real time on a desktop computer and is designed for use in clinical settings when the speed of response is important. A lumped parameter mathematical model, which is based on a one-dimensional formulation of the flow of an incompressible fluid in distensible vessels, is constructed using a bond graph formulation to ensure mass conservation and energy conservation. The model includes arterial vessels with geometric and anatomical data based on the ADAN circulation model. The peripheral beds are represented by lumped parameter compartments. We compare the hemodynamics predicted by the bond graph formulation of the cerebral circulation with that given by a classical one-dimensional Navier-Stokes model working on top of the whole-body ADAN model. Outputs from the bond graph model, including the pressure and flow signatures and blood volumes, are compared with physiological data.

Optimization of topological complexity for one-dimensional arterial blood flow models.

As computational models of the cardiovascular system are applied in modern personalized medicine, maximizing certainty of model input becomes crucial. A model with a high number of arterial segments results in a more realistic description of the system, but also requires a high number of parameters with associated uncertainties. In this paper, we present a method to optimize/reduce the number of arterial segments included in one-dimensional blood flow models, while preserving key features of flow and pressure waveforms. We quantify the preservation of key flow features for the optimal network with respect to the baseline networks (a 96-artery and a patient-specific coronary network) by various metrics and quantities like average relative error, pulse pressure and augmentation pressure. Furthermore, various physiological and pathological states are considered. For the aortic root and larger systemic artery pressure waveforms a network with minimal description of lower and upper limb arteries and no cerebral arteries, sufficiently captures important features such as pressure augmentation and pulse pressure. Discrepancies in carotid and middle cerebral artery flow waveforms that are introduced by describing the arterial system in a minimalistic manner are small compared with errors related to uncertainties in blood flow measurements obtained by ultrasound.

Roadmap for cardiovascular circulation model.

Computational models of many aspects of the mammalian cardiovascular circulation have been developed. Indeed, along with orthopaedics, this area of physiology is one that has attracted much interest from engineers, presumably because the equations governing blood flow in the vascular system are well understood and can be solved with well-established numerical techniques. Unfortunately, there have been only a few attempts to create a comprehensive public domain resource for cardiovascular researchers. In this paper we propose a roadmap for developing an open source cardiovascular circulation model. The model should be registered to the musculo-skeletal system. The computational infrastructure for the cardiovascular model should provide for near real-time computation of blood flow and pressure in all parts of the body. The model should deal with vascular beds in all tissues, and the computational infrastructure for the model should provide links into CellML models of cell function and tissue function. In this work we review the literature associated with 1D blood flow modelling in the cardiovascular system, discuss model encoding standards, software and a model repository. We then describe the coordinate systems used to define the vascular geometry, derive the equations and discuss the implementation of these coupled equations in the open source computational software OpenCMISS. Finally, some preliminary results are presented and plans outlined for the next steps in the development of the model, the computational software and the graphical user interface for accessing the model.

Velocity profiles in the human ductus venosus: a numerical fluid structure interaction study.

The veins distributing oxygenated blood from the placenta to the fetal body have been given much attention in clinical Doppler velocimetry studies, in particular the ductus venosus. The ductus venosus is embedded in the left liver lobe and connects the intra-abdominal portion of the umbilical vein (IUV) directly to the inferior vena cava, such that oxygenated blood can bypass the liver and flow directly to the fetal heart. In the current work, we have developed a mathematical model to assist the clinical assessment of volumetric flow rate at the inlet of the ductus venosus. With a robust estimate of the velocity profile shape coefficient (VC), the volumetric flow rate may be estimated as the product of the time-averaged cross-sectional area, the time-averaged cross-sectional maximum velocity and the VC. The time average quantities may be obtained from Doppler ultrasound measurements, whereas the VC may be estimated from numerical simulations. The mathematical model employs a 3D fluid structure interaction model of the bifurcation formed by the IUV, the ductus venosus and the left portal vein. Furthermore, the amniotic portion of the umbilical vein, the right liver lobe and the inferior vena cava were incorporated as lumped model boundary conditions for the fluid structure interaction model. A hyperelastic material is used to model the structural response of the vessel walls, based on recently available experimental data for the human IUV and ductus venous. A parametric study was constructed to investigate the VC at the ductus venosus inlet, based on a reference case for a human fetus at 36 weeks of gestation. The VC was found to be [Formula: see text] (Mean [Formula: see text] SD of parametric case study), which confirms previous studies in the literature on the VC at the ductus venosus inlet. Additionally, CFD simulations with rigid walls were performed on a subsection of the parametric case study, and only minor changes in the predicted VCs were observed compared to the FSI cases. In conclusion, the presented mathematical model is a promising tool for the assessment of ductus venosus Doppler velocimetry.

Human Ductus Venosus Velocity Profiles in the First Trimester.

The fluid dynamics in the human fetal ductus venosus in the early stage of pregnancy is not well explored. Consequently, there is an uncertainty in the interpretation of the temporal and spatial velocity variation in the ductus venosus. A robust estimation procedure for non-invasive measurement of the blood flow, based on conventional Doppler ultrasound measurements, is therefore missing. The aim of the present study was to describe the spatial and temporal velocity distribution at the ductus venosus bifurcation for boundary condition typical for fetuses at 11-13 weeks of gestation by means of a mathematical model. In particular we wanted to investigate velocity profiles at the ductus venosus inlet region in early pregnancy under normal conditions, to assess whether robust estimates of velocity profile shape coefficients may be given in order to provide noninvasive volumetric flow rate assessment in the ductus venosus. Such information will be useful in a clinical assessment of the fetus. Our model predicted a close to parabolic velocity profile in the inlet section of the ductus venosus during the cardiac cycle, with a shape factor of 0.53. Our simulations also showed that during atrial contraction (the A-wave), transient simultaneous positive and negative velocities may be observed in the same cross-section, in Womersley-like velocity profiles. Thus, as previous clinical investigators have reported these velocities as either positive or negative, our findings challenge clinical interpretation.