Structural correlation method for model reduction and practical estimation of patient specific parameters illustrated on heart rate regulation

Sensitivity analysis of closed-loop one-chamber and four-chamber models with baroreflex

The baroreflex is one of the most important control mechanisms in the human cardiovascular system. This work utilises a closed-loop in silico model of baroreflex regulation, coupled to pulsatile mechanical models with (i) one heart chamber and 36-parameters and (ii) four chambers and 51 parameters. We perform the first global sensitivity analysis of these closed-loop systems which considers both cardiovascular and baroreflex parameters, and compare the models with their respective unregulated equivalents. Results show the reduced influence of regulated parameters compared to unregulated equivalents and that, in the physiological resting state, model outputs (pressures, heart rate, cardiac output etc.) are most sensitive to parasympathetic arc parameters. This work provides insight into the effects of regulation and model input parameter influence on clinical metrics, and constitutes a first step to understanding the role of regulation in models for personalised healthcare.

Convergence, sampling and total order estimator effects on parameter orthogonality in global sensitivity analysis

Dynamical system models typically involve numerous input parameters whose "effects" and orthogonality need to be quantified through sensitivity analysis, to identify inputs contributing the greatest uncertainty. Whilst prior art has compared total-order estimators' role in recovering "true" effects, assessing their ability to recover robust parameter orthogonality for use in identifiability metrics has not been investigated. In this paper, we perform: (i) an assessment using a different class of numerical models representing the cardiovascular system, (ii) a wider evaluation of sampling methodologies and their interactions with estimators, (iii) an investigation of the consequences of permuting estimators and sampling methodologies on input parameter orthogonality, (iv) a study of sample convergence through resampling, and (v) an assessment of whether positive outcomes are sustained when model input dimensionality increases. Our results indicate that Jansen or Janon estimators display efficient convergence with minimum uncertainty when coupled with Sobol and the lattice rule sampling methods, making them prime choices for calculating parameter orthogonality and influence. This study reveals that global sensitivity analysis is convergence driven. Unconverged indices are subject to error and therefore the true influence or orthogonality of the input parameters are not recovered. This investigation importantly clarifies the interactions of the estimator and the sampling methodology by reducing the associated ambiguities, defining novel practices for modelling in the life sciences.

Identifiability of enzyme kinetic parameters in substrate competition: a case study of CD39/NTPDase1

CD39 (NTPDase1-nucleoside triphosphate diphosphohydrolase 1) is a membrane-tethered ectonucleotidase that hydrolyzes extracellular ATP to ADP and ADP to AMP. This enzyme is expressed in a variety of cell types and tissues and has broadly been recognized within vascular tissue to have a protective role in converting "danger" ligands (ATP) into neutral ligands (AMP). In this study, we investigate the enzyme kinetics of CD39 using a Michaelis-Menten modeling framework. We show how the unique situation of having a reaction product also serving as a substrate (ADP) complicates the determination of the governing kinetic parameters. Model simulations using values for the kinetic parameters reported in the literature do not align with corresponding time-series data. This dissonance is explained by CD39 kinetic parameters previously being determined by graphical/linearization methods, which have been shown to distort the underlying error structure and lead to inaccurate parameter estimates. Modern methods of estimating these kinetic parameters using nonlinear least squares are still challenging due to unidentifiable parameter interactions. We propose a workflow to accurately determine these parameters by isolating the ADPase and ATPase reactions and estimating the respective ADPase parameters and ATPase parameters with independent data sets. Theoretically, this ensures all kinetic parameters are identifiable and reliable for future prospective model simulations involving CD39. These kinds of mathematical models can be used to understand how circulating purinergic nucleotides affect disease etiology and potentially inform the development of corresponding therapies.

Model-based assessment of cardiopulmonary autonomic regulation in paced deep breathing

Autonomic dysfunction can lead to many physical and psychological diseases. The assessment of autonomic regulation plays an important role in the prevention, diagnosis, and treatment of these diseases. A physiopathological mathematical model for cardiopulmonary autonomic regulation, namely Respiratory-Autonomic-Sinus (RSA) regulation Model, is proposed in this study. A series of differential equations are used to simulate the whole process of RSA phenomenon. Based on this model, with respiration signal and ECG signal simultaneously acquired in paced deep breathing scenario, we manage to obtain the cardiopulmonary autonomic regulation parameters (CARP), including the sensitivity of respiratory-sympathetic nerves and respiratory-parasympathetic nerves, the time delay of sympathetic, the sensitivity of norepinephrine and acetylcholine receptor, as well as cardiac remodeling factor by optimization algorithm. An experimental study has been conducted in healthy subjects, along with subjects with hypertension and coronary heart disease. CARP obtained in the experiment have shown their clinical significance.

A model-based analysis of autonomic nervous function in response to the Valsalva maneuver

The Valsalva maneuver (VM) is a diagnostic protocol examining sympathetic and parasympathetic activity in patients with autonomic dysfunction (AD) impacting cardiovascular control. Because direct measurement of these signals is costly and invasive, AD is typically assessed indirectly by analyzing heart rate and blood pressure response patterns. This study introduces a mathematical model that can predict sympathetic and parasympathetic dynamics. Our model-based analysis includes two control mechanisms: respiratory sinus arrhythmia (RSA) and the baroreceptor reflex (baroreflex). The RSA submodel integrates an electrocardiogram-derived respiratory signal with intrathoracic pressure, and the baroreflex submodel differentiates aortic and carotid baroreceptor regions. Patient-specific afferent and efferent signals are determined for 34 control subjects and 5 AD patients, estimating parameters fitting the model output to heart rate data. Results show that inclusion of RSA and distinguishing aortic/carotid regions are necessary to model the heart rate response to the VM. Comparing control subjects to patients shows that RSA and baroreflex responses are significantly diminished. This study compares estimated parameter values from the model-based predictions to indices used in clinical practice. Three indices are computed to determine adrenergic function from the slope of the systolic blood pressure in phase II [α (a new index)], the baroreceptor sensitivity (β), and the Valsalva ratio (γ). Results show that these indices can distinguish between normal and abnormal states, but model-based analysis is needed to differentiate pathological signals. In summary, the model simulates various VM responses and, by combining indices and model predictions, we study the pathologies for 5 AD patients.NEW & NOTEWORTHY We introduce a patient-specific model analyzing heart rate and blood pressure during a Valsalva maneuver (VM). The model predicts autonomic function incorporating the baroreflex and respiratory sinus arrhythmia (RSA) control mechanisms. We introduce a novel index (α) characterizing sympathetic activity, which can distinguish control and abnormal patients. However, we assert that modeling and parameter estimation are necessary to explain pathologies. Finally, we show that aortic baroreceptors contribute significantly to the VM and RSA affects early VM.

Cardiovascular regulation in response to multiple hemorrhages: analysis and parameter estimation

Mathematical models can provide useful insights explaining behavior observed in experimental data; however, rigorous analysis is needed to select a subset of model parameters that can be informed by available data. Here we present a method to estimate an identifiable set of parameters based on baseline left ventricular pressure and volume time series data. From this identifiable subset, we then select, based on current understanding of cardiovascular control, parameters that vary in time in response to blood withdrawal, and estimate these parameters over a series of blood withdrawals. These time-varying parameters are first estimated using piecewise linear splines minimizing the mean squared error between measured and computed left ventricular pressure and volume data over four consecutive blood withdrawals. As a final step, the trends in these splines are fit with empirical functional expressions selected to describe cardiovascular regulation during blood withdrawal. Our analysis at baseline found parameters representing timing of cardiac contraction, systemic vascular resistance, and cardiac contractility to be identifiable. Of these parameters, vascular resistance and cardiac contractility were varied in time. Data used for this study were measured in a control Sprague-Dawley rat. To our knowledge, this is the first study to analyze the response to multiple blood withdrawals both experimentally and theoretically, as most previous studies focus on analyzing the response to one large blood withdrawal. Results show that during each blood withdrawal both systemic vascular resistance and contractility decrease acutely and partially recover, and they decrease chronically across the series of blood withdrawals.

Practical identifiability and uncertainty quantification of a pulsatile cardiovascular model

Mathematical models are essential tools to study how the cardiovascular system maintains homeostasis. The utility of such models is limited by the accuracy of their predictions, which can be determined by uncertainty quantification (UQ). A challenge associated with the use of UQ is that many published methods assume that the underlying model is identifiable (e.g. that a one-to-one mapping exists from the parameter space to the model output). In this study we present a novel workflow to calibrate a lumped-parameter model to left ventricular pressure and volume time series data. Key steps include using (1) literature and available data to determine nominal parameter values; (2) sensitivity analysis and subset selection to determine a set of identifiable parameters; (3) optimization to find a point estimate for identifiable parameters; and (4) frequentist and Bayesian UQ calculations to assess the predictive capability of the model. Our results show that it is possible to determine 5 identifiable model parameters that can be estimated to our experimental data from three rats, and that computed UQ intervals capture the measurement and model error.

How to choose biomarkers in view of parameter estimation

In numerous applications in biophysics, physiology and medicine, the system of interest is studied by monitoring quantities, called biomarkers, extracted from measurements. These biomarkers convey some information about relevant hidden quantities, which can be seen as parameters of an underlying model. In this paper we propose a strategy to automatically design biomarkers to estimate a given parameter. Such biomarkers are chosen as the solution of a sparse optimization problem given a user-supplied dictionary of candidate features. The method is in particular illustrated with two realistic applications, one in electrophysiology and the other in hemodynamics. In both cases, our algorithm provides composite biomarkers which improve the parameter estimation problem.

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.