Sensitivity analysis and model assessment: mathematical models for arterial blood flow and blood pressure

Unraveling autonomic cardiovascular control complexity during orthostatic stress: Insights from a mathematical model

Understanding cardiovascular control mediated by the autonomic system remains challenging due to its inherent complexity. Consequently, syndromes such as orthostatic intolerance continue to evoke debates regarding the underlying pathophysiological mechanisms. This study develops a comprehensive mathematical model simulating the control of the sympathetic branch of the cardiovascular system in individuals with normal and abnormal responses to the head-up-tilt test. We recruited four young women: one control, one with vasovagal syncope, one with orthostatic hypertension, and one with orthostatic hypotension, exposing them to an orthostatic head-up tilt test (HUTT) employing non-invasive methods to measure electrocardiography and continuous blood pressure. Our work encompasses a compartmental model formulated using a system of ordinary differential equations. Using heart rate as input, we predict blood pressure, flow, and volume in compartments representing the veins, arteries, heart, and the sympathetic branch of the baroreflex control system. The latter is modulated by high- and low-pressure baroreceptor afferents activated by changes in blood pressure induced by the HUTT. Sensitivity analysis, parameter subset selection, and optimization are employed to estimate patient-specific parameters associated with autonomic performance. The model has seven sensitive and identifiable parameters with significant physiological relevance that can serve as biomarkers for patient classification. Results show that the model can reproduce a spectrum of blood pressure responses successfully, fitting the trajectory displayed by the experimental data. The controller exhibits behavior that emulates the operation of the sympathetic system. These encouraging findings underscore the potential of computational methods in evaluating pathologies associated with autonomic nervous system control, warranting further exploration and novel approaches.

The importance of incorporating ventricular-ventricular interaction (VVI) in the study of pulmonary hypertension

Ventricular ventricular interaction (VVI) affects blood volume and pressure in the right and left ventricles of the heart due to the location and balance of forces on the septal wall separating the ventricles. In healthy patients, the pressure of the left ventricle is considerably higher than the right, resulting in a septal wall that bows into the right ventricle. However, in patients with pulmonary hypertension, the pressure in the right ventricle increases significantly to a point where the pressure is similar to or surpasses that of the left ventricle during portions of the cardiac cycle. For these patients, the septal wall deviates towards the left ventricle, impacting its function. It is possible to study this effect using mathematical modeling, but existing models are nonlinear, leading to a system of algebraic differential equations that can be challenging to solve in patient-specific optimizations of clinical data. This study demonstrates that a simplified linearized model is sufficient to account for the effect of VVI and that, as expected, the impact is significantly more pronounced in patients with pulmonary hypertension.

Guidelines for mechanistic modeling and analysis in cardiovascular research

Computational, or in silico, models are an effective, noninvasive tool for investigating cardiovascular function. These models can be used in the analysis of experimental and clinical data to identify possible mechanisms of (ab)normal cardiovascular physiology. Recent advances in computing power and data management have led to innovative and complex modeling frameworks that simulate cardiovascular function across multiple scales. While commonly used in multiple disciplines, there is a lack of concise guidelines for the implementation of computer models in cardiovascular research. In line with recent calls for more reproducible research, it is imperative that scientists adhere to credible practices when developing and applying computational models to their research. The goal of this manuscript is to provide a consensus document that identifies best practices for in silico computational modeling in cardiovascular research. These guidelines provide the necessary methods for mechanistic model development, model analysis, and formal model calibration using fundamentals from statistics. We outline rigorous practices for computational, mechanistic modeling in cardiovascular research and discuss its synergistic value to experimental and clinical data.

Computational modeling for the quantitative assessment of cardiac autonomic response to orthostatic stress

Objective.The autonomic nervous system (ANS) plays a critical role in regulating not only cardiac functions but also various other physiological processes, such as respiratory rate, digestion, and metabolic activities. The ANS is divided into the sympathetic and parasympathetic nervous systems, each of which has distinct but complementary roles in maintaining homeostasis across multiple organ systems in response to internal and external stimuli. Early detection of ANS dysfunctions, such as imbalances between the sympathetic and parasympathetic branches or impairments in the autonomic regulation of bodily functions, is crucial for preventing or slowing the progression of cardiovascular diseases. These dysfunctions can manifest as irregularities in heart rate, blood pressure regulation, and other autonomic responses essential for maintaining cardiovascular health. Traditional methods for analyzing ANS activity, such as heart rate variability (HRV) analysis and muscle sympathetic nerve activity recording, have been in use for several decades. Despite their long history, these techniques face challenges such as poor temporal resolution, invasiveness, and insufficient sensitivity to individual physiological variations, which limit their effectiveness in personalized health assessments.Approach.This study aims to introduce the open-loop Mathematical Model of Autonomic Regulation of the Cardiac System under Supine-to-stand Maneuver (MMARCS) to overcome the limitations of existing ANS analysis methods. The MMARCS model is designed to offer a balance between physiological fidelity and simplicity, focusing on the ANS cardiac control subsystems' input-output curve. The MMARCS model simplifies the complex internal dynamics of ANS cardiac control by emphasizing input-output relationships and utilizing sensitivity analysis and parameter subset selection to increase model specificity and eliminate redundant parameters. This approach aims to enhance the model's capacity for personalized health assessments.Main results.The application of the MMARCS model revealed significant differences in ANS regulation between healthy (14 females and 19 males, age: 42 ± 18) and diabetic subjects (8 females and 6 males, age: 47 ± 14). Parameters indicated heightened sympathetic activity and diminished parasympathetic response in diabetic subjects compared to healthy subjects (p < 0.05). Additionally, the data suggested a more sensitive and potentially more reactive sympathetic response among diabetic subjects (p < 0.05), characterized by increased responsiveness and intensity of the sympathetic nervous system to stimuli, i.e. fluctuations in blood pressure, leading to more pronounced changes in heart rate, these phenomena can be directly reflected by gain parameters and time response parameters of the model.Significance.The MMARCS model represents an innovative computational approach for quantifying ANS functionality. This model guarantees the accuracy of physiological modeling while reducing mathematical complexity, offering an easy-to-implement and widely applicable tool for clinical measurements of cardiovascular health, disease progression monitoring, and home health monitoring through wearable technology.

Uncertainty quantification of the impact of peripheral arterial disease on abdominal aortic aneurysms in blood flow simulations

Peripheral arterial disease (PAD) and abdominal aortic aneurysms (AAAs) often coexist and pose significant risks of mortality, yet their mutual interactions remain largely unexplored. Here, we introduce a fluid mechanics model designed to simulate the haemodynamic impact of PAD on AAA-associated risk factors. Our focus lies on quantifying the uncertainty inherent in controlling the flow rates within PAD-affected vessels and predicting AAA risk factors derived from wall shear stress. We perform a sensitivity analysis on nine critical model parameters through simulations of three-dimensional blood flow within a comprehensive arterial geometry. Our results show effective control of the flow rates using two-element Windkessel models, although specific outlets need attention. Quantities of interest like endothelial cell activation potential (ECAP) and relative residence time are instructive for identifying high-risk regions, with ECAP showing greater reliability and adaptability. Our analysis reveals that the uncertainty in the quantities of interest is 187% of that of the input parameters. Notably, parameters governing the amplitude and frequency of the inlet velocity exert the strongest influence on the risk factors' variability and warrant precise determination. This study forms the foundation for patient-specific simulations involving PAD and AAAs which should ultimately improve patient outcomes and reduce associated mortality rates.

Biventricular Interaction During Acute Left Ventricular Ischemia in Mice: A Combined In-Vivo and In-Silico Approach

Computational models provide an efficient paradigm for integrating and linking multiple spatial and temporal scales. However, these models are difficult to parameterize and match to experimental data. Recent advances in both data collection and model analyses have helped overcome this limitation. Here, we combine a multiscale, biventricular interaction model with mouse data before and after left ventricular (LV) ischemia. Sensitivity analyses are used to identify the most influential parameters on pressure and volume predictions. The subset of influential model parameters are calibrated to biventricular pressure-volume loop data (n = 3) at baseline. Each mouse underwent left anterior descending coronary artery ligation, during which changes in fractional shortening and RV pressure-volume dynamics were recorded. Using the calibrated model, we simulate acute LV ischemia and contrast outputs at baseline and in simulated ischemia. Our baseline simulations align with the LV and RV data, and our predictions during ischemia complement recorded RV data and prior studies on LV function during myocardial infarction. We show that a model with both biventricular mechanical interaction and systems-level cardiovascular dynamics can quantitatively reproduce in-vivo data and qualitatively match prior findings from animal studies on LV ischemia.

A Mechanistic Model for Long COVID Dynamics

Long COVID, a long-lasting disorder following an acute infection of COVID-19, represents a significant public health burden at present. In this paper, we propose a new mechanistic model based on differential equations to investigate the population dynamics of long COVID. By connecting long COVID with acute infection at the population level, our modeling framework emphasizes the interplay between COVID-19 transmission, vaccination, and long COVID dynamics. We conducted a detailed mathematical analysis of the model. We also validated the model using numerical simulation with real data from the US state of Tennessee and the UK.

Modeling the XBB strain of SARS-CoV-2: Competition between variants and impact of reinfection

XBB, an Omicron subvariant of SARS-CoV-2 that began to circulate in late 2022, has been dominant in the US since early 2023. To quantify the impact of XBB on the progression of COVID-19, we propose a new mathematical model which describes the interplay between XBB and other SARS-CoV-2 variants at the population level and which incorporates the effects of reinfection. We apply the model to COVID-19 data in the US that include surveillance data on the cases and variant proportions from the New York City, the State of New York, and the State of Washington. Our fitting and simulation results show that the transmission rate of XBB is significantly higher than that of other variants and the reinfection from XBB may play an important role in shaping the pandemic/epidemic pattern in the US.

Mechanistic-mathematical modeling of intracranial pressure (ICP) profiles over a single heart cycle. The fundament of the ICP curve form

The intracranial pressure (ICP) curve with its different peaks has been comprehensively studied, but the exact physiological mechanisms behind its morphology has not been revealed. If the pathophysiology behind deviations from the normal ICP curve form could be identified, it could be vital information to diagnose and treat each single patient. A mathematical model of the hydrodynamics in the intracranial cavity over single heart cycles was developed. A Windkessel model approach was generalized but the unsteady Bernoulli equation was utilized for blood flow and CSF flow. This is a modification of earlier models using the extended and simplified classical Windkessel analogies to a model that is based on mechanisms rooted in the laws of physics. The improved model was calibrated with patient data for cerebral arterial inflow, venous outflow, cerebrospinal fluid (CSF), and ICP over one heart cycle from 10 neuro-intensive care unit patients. A priori model parameter values were obtained by considering patient data and values taken from earlier studies. These values were used as an initial guess for an iterated constrained-ODE (ordinary differential equation) optimization problem with cerebral arterial inflow data as input into the system of ODEs. The optimization routine found patient-specific model parameter values that produced model ICP curves that showed excellent agreement with clinical measurements while model venous and CSF flow were within a physiologically acceptable range. The improved model and the automated optimization routine gave better model calibration results compared to previous studies. Moreover, patient-specific values of physiologically important parameters like intracranial compliance, arterial and venous elastance, and venous outflow resistance were determined. The model was used to simulate intracranial hydrodynamics and to explain the underlying mechanisms of the ICP curve morphology. Sensitivity analysis showed that the order of the three main peaks of the ICP curve was affected by a decrease in arterial elastance, a large increase in resistance to arteriovenous flow, an increase in venous elastance, or a decrease in resistance to CSF flow in the foramen magnum; and the frequency of oscillations were notably affected by intracranial elastance. In particular, certain pathological peak patterns were caused by these changes in physiological parameters. To the best of our knowledge, there are no other mechanism-based models associating the pathological peak patterns to variation of the physiological parameters.

Parameter inference in a computational model of haemodynamics in pulmonary hypertension

Pulmonary hypertension (PH), defined by a mean pulmonary arterial pressure (mPAP) greater than 20 mmHg, is characterized by increased pulmonary vascular resistance and decreased pulmonary arterial compliance. There are few measurable biomarkers of PH progression, but a conclusive diagnosis of the disease requires invasive right heart catheterization (RHC). Patient-specific cardiovascular systems-level computational models provide a potential non-invasive tool for determining additional indicators of disease severity. Using computational modelling, this study quantifies physiological parameters indicative of disease severity in nine PH patients. The model includes all four heart chambers, the pulmonary and systemic circulations. We consider two sets of calibration data: static (systolic and diastolic values) RHC data and a combination of static and continuous, time-series waveform data. We determine a subset of identifiable parameters for model calibration using sensitivity analyses and multi-start inference and perform posterior uncertainty quantification. Results show that additional waveform data enables accurate calibration of the right atrial reservoir and pump function across the PH cohort. Model outcomes, including stroke work and pulmonary resistance-compliance relations, reflect typical right heart dynamics in PH phenotypes. Lastly, we show that estimated parameters agree with previous, non-modelling studies, supporting this type of analysis in translational PH research.

A Novel Strategy to Fit and Validate Physiological Models: A Case Study of a Cardiorespiratory Model for Simulation of Incremental Aerobic Exercise

Applying complex mathematical models of physiological systems is challenging due to the large number of parameters. Identifying these parameters through experimentation is difficult, and although procedures for fitting and validating models are reported, no integrated strategy exists. Additionally, the complexity of optimization is generally neglected when the number of experimental observations is restricted, obtaining multiple solutions or results without physiological justification. This work proposes a fitting and validation strategy for physiological models with many parameters under various populations, stimuli, and experimental conditions. A cardiorespiratory system model is used as a case study, and the strategy, model, computational implementation, and data analysis are described. Using optimized parameter values, model simulations are compared to those obtained using nominal values, with experimental data as a reference. Overall, a reduction in prediction error is achieved compared to that reported for model building. Furthermore, the behavior and accuracy of all the predictions in the steady state were improved. The results validate the fitted model and provide evidence of the proposed strategy's usefulness.

Biventricular interaction during acute left ventricular ischemia in mice: a combined in-vivo and in-silico approach

Computational models provide an efficient paradigm for integrating and linking multiple spatial and temporal scales. However, these models are difficult to parameterize and match to experimental data. Recent advances in both data collection and model analyses have helped overcome this limitation. Here, we combine a multiscale, biventricular interaction model with mouse data before and after left ventricular (LV) ischemia. Sensitivity analyses are used to identify the most influential parameters on pressure and volume predictions. The subset of influential model parameters are calibrated to biventricular pressure-volume loop data (n=3) at baseline. Each mouse underwent left anterior descending coronary artery ligation, during which changes in fractional shortening and RV pressure-volume dynamics were recorded. Using the calibrated model, we simulate acute LV ischemia and contrast outputs at baseline and in simulated ischemia. Our baseline simulations align with the LV and RV data, and our predictions during ischemia complement recorded RV data and prior studies on LV function during myocardial infarction. We show that a model with both biventricular mechanical interaction and systems level cardiovascular dynamics can quantitatively reproduce in-vivo data and qualitatively match prior findings from animal studies on LV ischemia.

Sensitivity Analysis of a Mathematical Model Simulating the Post-Hepatectomy Hemodynamics Response

Recently a lumped-parameter model of the cardiovascular system was proposed to simulate the hemodynamics response to partial hepatectomy and evaluate the risk of portal hypertension (PHT) due to this surgery. Model parameters are tuned based on each patient data. This work focuses on a global sensitivity analysis (SA) study of such model to better understand the main drivers of the clinical outputs of interest. The analysis suggests which parameters should be considered patient-specific and which can be assumed constant without losing in accuracy in the predictions. While performing the SA, model outputs need to be constrained to physiological ranges. An innovative approach exploits the features of the polynomial chaos expansion method to reduce the overall computational cost. The computed results give new insights on how to improve the calibration of some model parameters. Moreover the final parameter distributions enable the creation of a virtual population available for future works. Although this work is focused on partial hepatectomy, the pipeline can be applied to other cardiovascular hemodynamics models to gain insights for patient-specific parameterization and to define a physiologically relevant virtual population.

Global Sensitivity Analysis of Four Chamber Heart Hemodynamics Using Surrogate Models

Computational Fluid Dynamics (CFD) is used to assist in designing artificial valves and planning procedures, focusing on local flow features. However, assessing the impact on overall cardiovascular function or predicting longer-term outcomes may requires more comprehensive whole heart CFD models. Fitting such models to patient data requires numerous computationally expensive simulations, and depends on specific clinical measurements to constrain model parameters, hampering clinical adoption. Surrogate models can help to accelerate the fitting process while accounting for the added uncertainty. We create a validated patient-specific four-chamber heart CFD model based on the Navier-Stokes-Brinkman (NSB) equations and test Gaussian Process Emulators (GPEs) as a surrogate model for performing a variance-based global sensitivity analysis (GSA). GSA identified preload as the dominant driver of flow in both the right and left side of the heart, respectively. Left-right differences were seen in terms of vascular outflow resistances, with pulmonary artery resistance having a much larger impact on flow than aortic resistance. Our results suggest that GPEs can be used to identify parameters in personalized whole heart CFD models, and highlight the importance of accurate preload measurements.

Postural orthostatic tachycardia syndrome explained using a baroreflex response model

Patients with postural orthostatic tachycardia syndrome (POTS) experience an excessive increase in heart rate (HR) and low-frequency (∼0.1 Hz) blood pressure (BP) and HR oscillations upon head-up tilt (HUT). These responses are attributed to increased baroreflex (BR) responses modulating sympathetic and parasympathetic signalling. This study uses a closed-loop cardiovascular compartment model controlled by the BR to predict BP and HR dynamics in response to HUT. The cardiovascular model predicts these quantities in the left ventricle, upper and lower body arteries and veins. HUT is simulated by letting gravity shift blood volume (BV) from the upper to the lower body compartments, and the BR control is modelled using set-point functions modulating peripheral vascular resistance, compliance, and cardiac contractility in response to changes in mean carotid BP. We demonstrate that modulation of parameters characterizing BR sensitivity allows us to predict the persistent increase in HR and the low-frequency BP and HR oscillations observed in POTS patients. Moreover, by increasing BR sensitivity, inhibiting BR control of the lower body vasculature, and decreasing central BV, we demonstrate that it is possible to simulate patients with neuropathic and hyperadrenergic POTS.

Global sensitivity analysis in physiologically-based pharmacokinetic/pharmacodynamic models of inhaled and opioids anesthetics and its application to generate virtual populations

The integration between physiologically-based pharmacokinetics (PBPK) models and pharmacodynamics (PD) models makes it possible to describe the absorption, distribution, metabolism and excretion processes of drugs, together with the concentration-response relationship, being a fundamental framework with wide applications in pharmacology. Nevertheless, the enormous complexity of PBPK models and the large number of parameters that define them leads to the need to study and understand how the uncertainty of the parameters affects the variability of the models output. To study this issue, this paper proposes a global sensitivity analysis (GSA) to identify the parameters that have the greatest influence on the response of the model. It has been selected as study cases the PBPK models of an inhaled anesthetic and an analgesic, along with two PD interaction models that describe two relevant clinical effects, hypnosis and analgesia during general anesthesia. The subset of the most relevant parameters found adequately with the GSA method has been optimized for the generation of a virtual population that represents the theoretical output variability of various model responses. The generated virtual population has the potential to be used for the design, development and evaluation of physiological closed-loop control systems.

Systems analysis of the mechanisms governing the cardiovascular response to changes in posture and in peripheral demand during exercise

Blood flows and pressures throughout the human cardiovascular system are regulated in response to various dynamic perturbations, such as changes to peripheral demands in exercise, rapid changes in posture, or loss of blood from hemorrhage, via the coordinated action of the heart, the vasculature, and autonomic reflexes. To assess how the systemic and pulmonary arterial and venous circulation, the heart, and the baroreflex work together to effect the whole-body responses to these perturbations, we integrated an anatomically-based large-vessel arterial tree model with the TriSeg heart model, models capturing nonlinear characteristics of the large and small veins, and baroreflex-mediated regulation of vascular tone and cardiac chronotropy and inotropy. The model was identified by matching data from the Valsalva maneuver (VM), exercise, and head-up tilt (HUT). Thirty-one parameters were optimized using a custom parameter-fitting tool chain, resulting in an unique, high-fidelity whole-body human cardiovascular systems model. Because the model captures the effects of exercise and posture changes, it can be used to simulate numerous clinical assessments, such as HUT, the VM, and cardiopulmonary exercise stress testing. The model can also be applied as a framework for representing and simulating individual patients and pathologies. Moreover, it can serve as a framework for integrating multi-scale organ-level models, such as for the heart or the kidneys, into a whole-body model. Here, the model is used to analyze the relative importance of chronotropic, inotropic, and peripheral vascular contributions to the whole-body cardiovascular response to exercise. It is predicted that in normal physiological conditions chronotropy and inotropy make roughly equal contributions to increasing cardiac output and cardiac power output during exercise. Under upright exercise conditions, the nonlinear pressure-volume relationship of the large veins and sympathetic-mediated venous vasoconstriction are both required to maintain preload to achieve physiological exercise levels. The developed modeling framework is built using the open Modelica modeling language and is freely distributed.

Transmission rates and environmental reservoirs for COVID-19 - a modeling study

The coronavirus disease 2019 (COVID-19) remains a global pandemic at present. Although the human-to-human transmission route for this disease has been well established, its transmission mechanism is not fully understood. In this paper, we propose a mathematical model for COVID-19 which incorporates multiple transmission pathways and which employs time-dependent transmission rates reflecting the impact of disease prevalence and outbreak control. Applying this model to a retrospective study based on publicly reported data in China, we argue that the environmental reservoirs play an important role in the transmission and spread of the coronavirus. This argument is supported by our data fitting and numerical simulation results for the city of Wuhan, for the provinces of Hubei and Guangdong, and for the entire country of China.

Sensitivity analysis and uncertainty quantification of 1-D models of pulmonary hemodynamics in mice under control and hypertensive conditions

Pulmonary hypertension (PH), defined as an elevated mean blood pressure in the main pulmonary artery (MPA) at rest, is associated with vascular remodeling of both large and small arteries. PH has several sub-types that are all linked to high mortality rates. In this study, we use a one-dimensional (1-D) fluid dynamics model driven by in vivo measurements of MPA flow to understand how model parameters and network size influence MPA pressure predictions in the presence of PH. We compare model predictions with in vivo MPA pressure measurements from a control and a hypertensive mouse and analyze results in three networks of increasing complexity, extracted from micro-computed tomography (micro-CT) images. We introduce global scaling factors for boundary condition parameters and perform local and global sensitivity analysis to calculate parameter influence on model predictions of MPA pressure and correlation analysis to determine a subset of identifiable parameters. These are inferred using frequentist optimization and Bayesian inference via the Delayed Rejection Adaptive Metropolis (DRAM) algorithm. Frequentist and Bayesian uncertainty is computed for model parameters and MPA pressure predictions. Results show that MPA pressure predictions are most sensitive to distal vascular resistance and that parameter influence changes with increasing network complexity. Our outcomes suggest that PH leads to increased vascular stiffness and decreased peripheral compliance, congruent with clinical observations.

Global sensitivity analysis informed model reduction and selection applied to a Valsalva maneuver model

In this study, we develop a methodology for model reduction and selection informed by global sensitivity analysis (GSA) methods. We apply these techniques to a control model that takes systolic blood pressure and thoracic tissue pressure data as inputs and predicts heart rate in response to the Valsalva maneuver (VM). The study compares four GSA methods based on Sobol' indices (SIs) quantifying the parameter influence on the difference between the model output and the heart rate data. The GSA methods include standard scalar SIs determining the average parameter influence over the time interval studied and three time-varying methods analyzing how parameter influence changes over time. The time-varying methods include a new technique, termed limited-memory SIs, predicting parameter influence using a moving window approach. Using the limited-memory SIs, we perform model reduction and selection to analyze the necessity of modeling both the aortic and carotid baroreceptor regions in response to the VM. We compare the original model to systematically reduced models including (i) the aortic and carotid regions, (ii) the aortic region only, and (iii) the carotid region only. Model selection is done quantitatively using the Akaike and Bayesian Information Criteria and qualitatively by comparing the neurological predictions. Results show that it is necessary to incorporate both the aortic and carotid regions to model the VM.

COVID-19 and underlying health conditions: A modeling investigation

We propose a mathematical model based on a system of differential equations, which incorporates the impact of the chronic health conditions of the host population, to investigate the transmission dynamics of COVID-19. The model divides the total population into two groups, depending on whether they have underlying conditions, and describes the disease transmission both within and between the groups. As an application of this model, we perform a case study for Hamilton County, the fourth-most populous county in the US state of Tennessee and a region with high prevalence of chronic conditions. Our data fitting and simulation results quantify the high risk of COVID-19 for the population group with underlying health conditions. The findings suggest that weakening the disease transmission route between the exposed and susceptible individuals, including the reduction of the between-group contact, would be an effective approach to protect the most vulnerable people in this population group.

Modeling individual differences in cardiovascular response to gravitational stress using a sensitivity analysis

The human cardiovascular (CV) system elicits a physiological response to gravitational environments, with significant variation between different individuals. Computational modeling can predict CV response, however model complexity and variation of physiological parameters in a normal population makes it challenging to capture individual responses. We conducted a sensitivity analysis on an existing 21-compartment lumped-parameter hemodynamic model in a range of gravitational conditions to 1) investigate the influence of model parameters on a tilt test CV response and 2) to determine the subset of those parameters with the most influence on systemic physiological outcomes. A supine virtual subject was tilted to upright under the influence of a constant gravitational field ranging from 0 g to 1 g. The sensitivity analysis was conducted using a Latin hypercube sampling/partial rank correlation coefficient methodology with subsets of model parameters varied across a normal physiological range. Sensitivity was determined by variation in outcome measures including heart rate, stroke volume, central venous pressure, systemic blood pressures, and cardiac output. Results showed that model parameters related to the length, resistance, and compliance of the large veins and parameters related to right ventricular function have the most influence on model outcomes. For most outcome measures considered, parameters related to the heart are dominant. Results highlight which model parameters to accurately value in simulations of individual subjects' CV response to gravitational stress, improving the accuracy of predictions. Influential parameters remain largely similar across gravity levels, highlighting that accurate model fitting in 1 g can increase the accuracy of predictive responses in reduced gravity.NEW & NOTEWORTHY Computational modeling is used to predict cardiovascular responses to altered gravitational environments. However, considerable variation between subjects and model complexity makes accurate parameter assignment for individuals challenging. This computational effort studies sensitivity in cardiovascular model outcomes due to varying parameters across a normal physiological range. This allows determination of which parameters have the largest influence on outcomes, i.e., which parameters must be most carefully selected to give accurate predictions of individual responses.

On-line parameter identification of the lumped arterial system model: A simulation study

A lumped model of the arterial system has been used in constructing a hybrid mock loop due to its real-time response. However, the parameters of the model are always from a general case and not adapted to a specific patient. In this study, we focused on on-line parameter identification of the lumped model of the arterial system that could be used for a specific patient. A five-element lumped arterial model is adopted in this study, in which the five parameters are to be determined. The aortic flow rate and the venous pressure are chosen as the inputs of the model, and aortic pressure as the output. An iterative optimization based on the established state space equations of the five-element model is used to seek the best parameter values by minimizing the difference between the model prediction and the previously obtained aortic pressure. The method is validated using simulated data from a complete numerical cardiovascular model. Results show that the method can track the dynamic variation of the parameters very well. Finally, a sensitivity analysis of the model parameters is conducted to interpret the effect of parameter changes. The good performance of the identification demonstrates the potential application of this method to customize a hybrid mock loop for a specific patient or clinically monitor the arterial vessel characteristics in real time.

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.

Identifying important parameters in the inflammatory process with a mathematical model of immune cell influx and macrophage polarization

In an inflammatory setting, macrophages can be polarized to an inflammatory M1 phenotype or to an anti-inflammatory M2 phenotype, as well as existing on a spectrum between these two extremes. Dysfunction of this phenotypic switch can result in a population imbalance that leads to chronic wounds or disease due to unresolved inflammation. Therapeutic interventions that target macrophages have therefore been proposed and implemented in diseases that feature chronic inflammation such as diabetes mellitus and atherosclerosis. We have developed a model for the sequential influx of immune cells in the peritoneal cavity in response to a bacterial stimulus that includes macrophage polarization, with the simplifying assumption that macrophages can be classified as M1 or M2. With this model, we were able to reproduce the expected timing of sequential influx of immune cells and mediators in a general inflammatory setting. We then fit this model to in vivo experimental data obtained from a mouse peritonitis model of inflammation, which is widely used to evaluate endogenous processes in response to an inflammatory stimulus. Model robustness is explored with local structural and practical identifiability of the proposed model a posteriori. Additionally, we perform sensitivity analysis that identifies the population of apoptotic neutrophils as a key driver of the inflammatory process. Finally, we simulate a selection of proposed therapies including points of intervention in the case of delayed neutrophil apoptosis, which our model predicts will result in a sustained inflammatory response. Our model can therefore provide hypothesis testing for therapeutic interventions that target macrophage phenotype and predict outcomes to be validated by subsequent experimentation.

Using experimental data and mathematical analysis, we develop a model for the inflammatory response that includes macrophage polarization between M1 and M2 phenotypes. Dysfunction of this phenotypic switch can disrupt the timely influx and egress of immune cells during the healing process and lead to chronic wounds or disease. The modulation of macrophage population has been suggested as a strategy to dampen inflammation in diseases that feature chronic inflammation, such as diabetes and atherosclerosis. It is therefore important that we learn more about which components of the system drive the population level switch in phenotype. Our model is able to reproduce the expected timing of sequential influx of neutrophils and macrophages in response to an inflammatory stimulus. Model parameters were estimated with weighted least squares fitting to in vivo experimental data from a mouse model of peritonitis while considering identifiability of parameter sets. We perform sensitivity analysis that identifies primary drivers of the system, and predict the effects of variations in these key parameters on immune cell populations.

Computational model of cardiovascular response to centrifugation and lower body cycling exercise

Short-radius centrifugation combined with exercise has been suggested as a potential countermeasure against spaceflight deconditioning. Both the long-term and acute physiological responses to such a combination are incompletely understood. We developed and validated a computational model to study the acute cardiovascular response to centrifugation combined with lower body ergometer exercise. The model consisted of 21 compartments, including the upper body, renal, splanchnic, and leg circulation, as well as a four-chamber heart and pulmonary circulation. It also included the effects of gravity gradient and ergometer exercise. Centrifugation and exercise profiles were simulated and compared with experimental data gathered on 12 subjects exposed to a range of gravitational levels (1 and 1.4G measured at the feet) and workload intensities (25-100 W). The model was capable of reproducing cardiovascular changes (within ± 1 SD from the group-averaged behavior) due to both centrifugation and exercise, including dynamic responses during transitions between the different phases of the protocol. The model was then used to simulate the hemodynamic response of hypovolemic subjects (blood volume reduced by 5-15%) subjected to similar gravitational stress and exercise profiles, providing insights into the physiological responses of experimental conditions not tested before. Hypovolemic results are in agreement with the limited available data and the expected responses based on physiological principles, although additional experimental data are warranted to further validate our predictions, especially during the exercise phases. The model captures the cardiovascular response for a range of centrifugation and exercise profiles, and it shows promise in simulating additional conditions where data collection is difficult, expensive, or infeasible.NEW & NOTEWORTHY Artificial gravity combined with exercise is a potential countermeasure for spaceflight deconditioning, but the long-term and acute cardiovascular response to such gravitational stress is still largely unknown. We provide a novel mathematical model of the cardiovascular system that incorporates gravitational stress generated by centrifugation and lower body cycling exercise, and we validate it with experimental measurements from human subjects. Simulations of experimental conditions not used for model development corroborate the model's predictive capabilities.

Parameter estimation in a minimal model of cardio-pulmonary interactions

Mechanical ventilation is a widely used breathing support for patients in intensive care. Its effects on the respiratory and cardiovascular systems are complex and difficult to predict. This work first presents a minimal mathematical model representing the mechanics of both systems and their interaction, in terms of flows, pressures and volumes. The aim of this model is to get insight on the two systems' status when mechanical ventilation settings, such as positive end-expiratory pressure, are changing. The parameters of the model represent cardiac elastances and vessel compliances and resistances. As a second step, these parameters are estimated from 16 experimental datasets. The data come from three pig experiments reproducing intensive care conditions, where a large range of positive end-expiratory pressures was imposed by the mechanical ventilator. The data used for parameter estimation is limited to information available in the intensive care unit, such as stroke volume, central venous pressure and systemic arterial pressure. The model is able to satisfactorily reproduce this experimental data, with mean relative errors ranging from 1 to 26%. The model also reproduces the dynamics of the cardio-vascular and respiratory systems, and their interaction. By looking at the estimated parameter values, one can quantitatively track how the two coupled systems mechanically react to changes in external conditions imposed by the ventilator. This work thus allows real-time, model-based management of ventilator settings.

Practical identifiability analysis of a minimal cardiovascular system model

BACKGROUND AND OBJECTIVE

Parameters of mathematical models of the cardiovascular system can be used to monitor cardiovascular state, such as total stressed blood volume status, vessel elastance and resistance. To do so, the model parameters have to be estimated from data collected at the patient's bedside. This work considers a seven-parameter model of the cardiovascular system and investigates whether these parameters can be uniquely determined using indices derived from measurements of arterial and venous pressures, and stroke volume.

METHODS

An error vector defined the residuals between the simulated and reference values of the seven clinically available haemodynamic indices. The sensitivity of this error vector to each model parameter was analysed, as well as the collinearity between parameters. To assess practical identifiability of the model parameters, profile-likelihood curves were constructed for each parameter.

RESULTS

Four of the seven model parameters were found to be practically identifiable from the selected data. The remaining three parameters were practically non-identifiable. Among these non-identifiable parameters, one could be decreased as much as possible. The other two non-identifiable parameters were inversely correlated, which prevented their precise estimation.

CONCLUSIONS

This work presented the practical identifiability analysis of a seven-parameter cardiovascular system model, from limited clinical data. The analysis showed that three of the seven parameters were practically non-identifiable, thus limiting the use of the model as a monitoring tool. Slight changes in the time-varying function modeling cardiac contraction and use of larger values for the reference range of venous pressure made the model fully practically identifiable.

Non-invasive Assessment of Systolic and Diastolic Cardiac Function During Rest and Stress Conditions Using an Integrated Image-Modeling Approach

Background: The possibility of non-invasively assessing load-independent parameters characterizing cardiac function is of high clinical value. Typically, these parameters are assessed during resting conditions. However, for diagnostic purposes, the parameter behavior across a physiologically relevant range of heart rate and loads is more relevant than the isolated measurements performed at rest. This study sought to evaluate changes in non-invasive estimations of load-independent parameters of left-ventricular contraction and relaxation patterns at rest and during dobutamine stress. Methods: We applied a previously developed approach that combines non-invasive measurements with a physiologically-based, reduced-order model of the cardiovascular system to provide subject-specific estimates of parameters characterizing left ventricular function. In this model, the contractile state of the heart at each time point along the cardiac cycle is modeled using a time-varying elastance curve. Non-invasive data, including four-dimensional magnetic resonance imaging (4D Flow MRI) measurements, were acquired in nine subjects without a known heart disease at rest and during dobutamine stress. For each of the study subjects, we constructed two personalized models corresponding to the resting and the stress state. Results: Applying the modeling framework, we identified significant increases in the left ventricular contraction rate constant [from 1.5 ± 0.3 to 2 ± 0.5 (p = 0.038)] and relaxation constant [from 37.2 ± 6.9 to 46.1 ± 12 (p = 0.028)]. In addition, we found a significant decrease in the elastance diastolic time constant from 0.4 ± 0.04 s to 0.3 ± 0.03 s (p = 0.008). Conclusions: The integrated image-modeling approach allows the assessment of cardiovascular function given as model-based parameters. The agreement between the estimated parameter values and previously reported effects of dobutamine demonstrates the potential of the approach to assess advanced metrics of pathophysiology that are otherwise difficult to obtain non-invasively in clinical practice.

Modeling of cardiovascular circulation for the early detection of coronary arterial blockage

Coronary arteries are responsible for maintaining blood supply to the heart. When these arteries get blocked due to plaque deposition, the corresponding pathological condition is referred to as coronary artery disease. This disease develops gradually over the years and consequently, the function of the heart deteriorates, leading to a heart attack in many cases. As the symptoms manifest themselves only when it has become severe, detection of the disease often gets delayed. In order to detect it early and take preventive action, this work is aimed at detecting the arterial blockage in its early stage via cardiovascular modeling. To achieve this, the cardiovascular circulation has been modeled as a sixth order nonlinear system. Blood circulation in a body is viewed as an electrical system using the pressure-voltage analogy. In this case, the heart is considered as a self-excited generator. The rest of the body tissues form a systemic load. In the models reported in the literature, coronary circulation has been assumed to be a part of the systemic load. However, this circulation path has its own importance as it is responsible for the blood supply to the heart. Therefore, in our work, the coronary path is separated out from the rest of the body tissues. This enables us to explicitly model the coronary arterial resistance and thereby helps us to detect coronary arterial blockage condition by estimating this parameter from blood pressure measurements. Increase in the coronary resistance is found to reduce the left ventricular ejection fraction; this information can therefore be used as an index for coronary arterial blockage. It has been shown that the systolic function of the heart deteriorates when the resistance of the coronary path increases beyond a critical value; the situation can be related to a severe blockage condition. The model has been tested on a chosen sample of 20 subjects suffering from coronary artery disease and the results are found to be quite promising.

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.

A mathematical model of human heart including the effects of heart contractility varying with heart rate changes

Incorporating the intrinsic variability of heart contractility varying with heart rate into the mathematical model of human heart would be useful for addressing the dynamical behaviors of human cardiovascular system, but models with such features were rarely reported. This study focused on the development and evaluation of a mathematical model of the whole heart, including the effects of heart contractility varying with heart rate changes. This model was developed based on a paradigm and model presented by Ottesen and Densielsen, which was used to model ventricular contraction. A piece-wise function together with expressions for time-related parameters were constructed for modeling atrial contraction. Atrial and ventricular parts of the whole heart model were evaluated by comparing with models from literature, and then the whole heart model were assessed through coupling with a simple model of the systemic circulation system and the pulmonary circulation system. The results indicated that both atrial and ventricular parts of the whole heart model could reasonably reflect their contractility varying with heart rate changes, and the whole heart model could exhibit major features of human heart. Results of the parameters variation studies revealed the correlations between the parameters in the whole heart model and performances (including the maximum pressure and the stroke volume) of every chamber. These results would be useful for helping users to adjust parameters in special applications.

Bayesian sensitivity analysis of a 1D vascular model with Gaussian process emulators

One-dimensional models of the cardiovascular system can capture the physics of pulse waves but involve many parameters. Since these may vary among individuals, patient-specific models are difficult to construct. Sensitivity analysis can be used to rank model parameters by their effect on outputs and to quantify how uncertainty in parameters influences output uncertainty. This type of analysis is often conducted with a Monte Carlo method, where large numbers of model runs are used to assess input-output relations. The aim of this study was to demonstrate the computational efficiency of variance-based sensitivity analysis of 1D vascular models using Gaussian process emulators, compared to a standard Monte Carlo approach. The methodology was tested on four vascular networks of increasing complexity to analyse its scalability. The computational time needed to perform the sensitivity analysis with an emulator was reduced by the 99.96% compared to a Monte Carlo approach. Despite the reduced computational time, sensitivity indices obtained using the two approaches were comparable. The scalability study showed that the number of mechanistic simulations needed to train a Gaussian process for sensitivity analysis was of the order O(d), rather than O(d×103) needed for Monte Carlo analysis (where d is the number of parameters in the model). The efficiency of this approach, combined with capacity to estimate the impact of uncertain parameters on model outputs, will enable development of patient-specific models of the vascular system, and has the potential to produce results with clinical relevance.

Arterial pulse attenuation prediction using the decaying rate of a pressure wave in a viscoelastic material model

The present study examines the possibility of attenuating blood pulses by means of introducing prosthetic viscoelastic materials able to absorb energy and damp such pulses. Vascular prostheses made of polymeric materials modify the mechanical properties of blood vessels. The effect of these materials on the blood pulse propagation remains to be fully understood. Several materials for medical applications, such as medical polydimethylsiloxane or polytetrafluoroethylene, show viscoelastic behavior, modifying the original vessel stiffness and affecting the propagation of blood pulses. This study focuses on the propagation of pressure waves along a pipe with viscoelastic materials using the Maxwell and the Zener models. An expression of exponential decay has been obtained for the Maxwell material model and also for low viscous coefficient values in the Zener model. For relatively high values of the viscous term in the Zener model, the steepest part of the pulse can be damped quickly, leaving a smooth, slowly decaying wave. These mathematical models are critical to tailor those materials used in cardiovascular implants to the mechanical environment they are confronted with to repair or improve blood vessel function.

Impact of Awareness Programs on Cholera Dynamics: Two Modeling Approaches

We propose two differential equation-based models to investigate the impact of awareness programs on cholera dynamics. The first model represents the disease transmission rates as decreasing functions of the number of awareness programs, whereas the second model divides the susceptible individuals into two distinct classes depending on their awareness/unawareness of the risk of infection. We study the essential dynamical properties of each model, using both analytical and numerical approaches. We find that the two models, though closely related, exhibit significantly different dynamical behaviors. Namely, the first model follows regular threshold dynamics while rich dynamical behaviors such as backward bifurcation may arise from the second one. Our results highlight the importance of validating key modeling assumptions in the development and selection of mathematical models toward practical application.

Modelling the cardiovascular system for assessing the blood pressure curve

A four compartment model of the cardiovascular system is developed. To allow for easy interpretation and to minimise the number of parameters, an effort was made to keep the model as simple as possible. Using a standard method (Matlab function fminsearch) to calculate the parameter values led to unacceptable run times or non-convergence. Consequently we developed an algorithm which first finds the most important model parameters and uses these as a basis for a four stage process which accurately determines all parameter values. This process is then applied to data from three ICU patients. Good agreement between the model and measured arterial pressure is demonstrated in all cases.

An integrated mathematical model of the human cardiopulmonary system: model development

Several cardiovascular and pulmonary models have been proposed in the last few decades. However, very few have addressed the interactions between these two systems. Our group has developed an integrated cardiopulmonary model (CP Model) that mathematically describes the interactions between the cardiovascular and respiratory systems, along with their main short-term control mechanisms. The model has been compared with human and animal data taken from published literature. Due to the volume of the work, the paper is divided in two parts. The present paper is on model development and normophysiology, whereas the second is on the model's validation on hypoxic and hypercapnic conditions. The CP Model incorporates cardiovascular circulation, respiratory mechanics, tissue and alveolar gas exchange, as well as short-term neural control mechanisms acting on both the cardiovascular and the respiratory functions. The model is able to simulate physiological variables typically observed in adult humans under normal and pathological conditions and to explain the underlying mechanisms and dynamics.

Mathematical Model of Cardiovascular and Metabolic Responses to Umbilical Cord Occlusions in Fetal Sheep

Fetal acidemia during labor is associated with an increased risk of brain injury and lasting neurological deficits. This is in part due to the repetitive occlusions of the umbilical cord (UCO) induced by uterine contractions. Whereas fetal heart rate (FHR) monitoring is widely used clinically, it fails to detect fetal acidemia. Hence, new approaches are needed for early detection of fetal acidemia during labor. We built a mathematical model of the UCO effects on FHR, mean arterial blood pressure (MABP), oxygenation and metabolism. Mimicking fetal experiments, our in silico model reproduces salient features of experimentally observed fetal cardiovascular and metabolic behavior including FHR overshoot, gradual MABP decrease and mixed metabolic and respiratory acidemia during UCO. Combined with statistical analysis, our model provides valuable insight into the labor-like fetal distress and guidance for refining FHR monitoring algorithms to improve detection of fetal acidemia and cardiovascular decompensation.

Contribution of collagen fiber undulation to regional biomechanical properties along porcine thoracic aorta

As major extracellular matrix components, elastin, and collagen play crucial roles in regulating the mechanical properties of the aortic wall and, thus, the normal cardiovascular function. The mechanical properties of aorta, known to vary with age and multitude of diseases as well as the proximity to the heart, have been attributed to the variations in the content and architecture of wall constituents. This study is focused on the role of layer-specific collagen undulation in the variation of mechanical properties along the porcine descending thoracic aorta. Planar biaxial tensile tests are performed to characterize the hyperelastic anisotropic mechanical behavior of tissues dissected from four locations along the thoracic aorta. Multiphoton microscopy is used to image the associated regional microstructure. Exponential-based and recruitment-based constitutive models are used to account for the observed mechanical behavior while considering the aortic wall as a composite of two layers with independent properties. An elevated stiffness is observed in distal regions compared to proximal regions of thoracic aorta, consistent with sharper and earlier collagen recruitment estimated for medial and adventitial layers in the models. Multiphoton images further support our prediction that higher stiffness in distal regions is associated with less undulation in collagen fibers. Recruitment-based models further reveal that regardless of the location, collagen in the media is recruited from the onset of stretching, whereas adventitial collagen starts to engage with a delay. A parameter sensitivity analysis is performed to discriminate between the models in terms of the confidence in the estimated model parameters.

Patient-specific assessment of cardiovascular function by combination of clinical data and computational model with applications to patients undergoing Fontan operation

The assessment of cardiovascular function is becoming increasingly important for the care of patients with single-ventricle defects. However, most measurement methods available in the clinical setting cannot provide a separate measure of cardiac function and loading conditions. In the present study, a numerical method has been proposed to compensate for the limitations of clinical measurements. The main idea was to estimate the parameters of a cardiovascular model by fitting model simulations to patient-specific clinical data via parameter optimization. Several strategies have been taken to establish a well-posed parameter optimization problem, including clinical data-matched model development, parameter selection based on an extensive sensitivity analysis, and proper choice of parameter optimization algorithm. The numerical experiments confirmed the ability of the proposed parameter optimization method to uniquely determine the model parameters given an arbitrary set of clinical data. The method was further tested in four patients undergoing the Fontan operation. Obtained results revealed a prevalence of ventricular abnormalities in the patient cohort and at the same time demonstrated the presence of marked inter-patient differences and preoperative to postoperative changes in cardiovascular function. Because the method allows a quick assessment and makes use of clinical data available in clinical practice, its clinical application is promising.

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.

A cardiovascular mathematical model of graded head-up tilt

A lumped parameter model of the cardiovascular system has been developed and optimized using experimental data obtained from 13 healthy subjects during graded head-up tilt (HUT) from the supine position to . The model includes descriptions of the left and right heart, direct ventricular interaction through the septum and pericardium, the systemic and pulmonary circulations, nonlinear pressure volume relationship of the lower body compartment, arterial and cardiopulmonary baroreceptors, as well as autoregulatory mechanisms. A number of important features, including the separate effects of arterial and cardiopulmonary baroreflexes, and autoregulation in the lower body, as well as diastolic ventricular interaction through the pericardium have been included and tested for their significance. Furthermore, the individual effect of parameter associated with heart failure, including LV and RV contractility, baseline systemic vascular resistance, pulmonary vascular resistance, total blood volume, LV diastolic stiffness and reflex gain on HUT response have also been investigated. Our fitted model compares favorably with our experimental measurements and published literature at a range of tilt angles, in terms of both global and regional hemodynamic variables. Compared to the normal condition, a simulated congestive heart failure condition produced a blunted response to HUT with regards to the percentage changes in cardiac output, stroke volume, end diastolic volume and effector response (i.e., heart contractility, venous unstressed volume, systemic vascular resistance and heart rate) with progressive tilting.