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Isolated post-capillary pulmonary hypertension (Ipc-PH) occurs due to left heart failure, which contributes to 1 out of every 9 deaths in the United States. In some patients, through unknown mechanisms, Ipc-PH transitions to combined pre-/post-capillary PH (Cpc-PH) and is associated with a dramatic increase in mortality. Altered mechanical forces and subsequent biological signaling in the pulmonary vascular bed likely contribute to the transition from Ipc-PH to Cpc-PH. However, even in a healthy pulmonary circulation, the mechanical forces in the smallest vessels (the arterioles, capillary bed, and venules) have not been quantitatively defined. This study is the first to examine this question via a computational fluid dynamics model of the human pulmonary arteries, arterioles, venules, and veins. Using this model, we predict temporal and spatial dynamics of cyclic stretch and wall shear stress with healthy and diseased hemodynamics. In the normotensive case for large vessels, numerical simulations show that large arteries have higher pressure and flow than large veins, as well as more pronounced changes in area throughout the cardiac cycle. In the microvasculature, shear stress increases and cyclic stretch decreases as vessel radius decreases. When we impose an increase in left atrial pressure to simulate Ipc-PH, shear stress decreases and cyclic stretch increases as compared to the healthy case. Overall, this model predicts pressure, flow, shear stress, and cyclic stretch that providing a way to analyze and investigate hypotheses related to disease progression in the pulmonary circulation.

Chronic thromboembolic pulmonary hypertension (CTEPH) is caused by recurrent or unresolved pulmonary thromboemboli, leading to perfusion defects and increased arterial wave reflections. CTEPH treatment aims to reduce pulmonary arterial pressure and reestablish adequate lung perfusion, yet patients with distal lesions are inoperable by standard surgical intervention. Instead, these patients undergo balloon pulmonary angioplasty (BPA), a multi-session, minimally invasive surgery that disrupts the thromboembolic material within the vessel lumen using a catheter balloon. However, there still lacks an integrative, holistic tool for identifying optimal target lesions for treatment. To address this insufficiency, we simulate CTEPH hemodynamics and BPA therapy using a multiscale fluid dynamics model. The large pulmonary arterial geometry is derived from a computed tomography (CT) image, whereas a fractal tree represents the small vessels. We model ring- and web-like lesions, common in CTEPH, and simulate normotensive conditions and four CTEPH disease scenarios; the latter includes both large artery lesions and vascular remodeling. BPA therapy is simulated by simultaneously reducing lesion severity in three locations. Our predictions mimic severe CTEPH, manifested by an increase in mean proximal pulmonary arterial pressure above 20 mmHg and prominent wave reflections. Both flow and pressure decrease in vessels distal to the lesions and increase in unobstructed vascular regions. We use the main pulmonary artery (MPA) pressure, a wave reflection index, and a measure of flow heterogeneity to select optimal target lesions for BPA. In summary, this study provides a multiscale, image-to-hemodynamics pipeline for BPA therapy planning for inoperable CTEPH patients., Comment: 41 pages, 9 figures, 4 tables

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.

Imbalance in the autonomic nervous system can lead to orthostatic intolerance manifested by dizziness, lightheadedness, and a sudden loss of consciousness (syncope); these are common conditions, but they are challenging to diagnose correctly. Uncertainties about the triggering mechanisms and the underlying pathophysiology have led to variations in their classification. This study uses machine learning to categorize patients with orthostatic intolerance. We use random forest classification trees to identify a small number of markers in blood pressure, and heart rate time-series data measured during head-up tilt to (a) distinguish patients with a single pathology and (b) examine data from patients with a mixed pathophysiology. Next, we use Kmeans to cluster the markers representing the time-series data. We apply the proposed method analyzing clinical data from 186 subjects identified as control or suffering from one of four conditions: postural orthostatic tachycardia (POTS), cardioinhibition, vasodepression, and mixed cardioinhibition and vasodepression. Classification results confirm the use of supervised machine learning. We were able to categorize more than 95% of patients with a single condition and were able to subgroup all patients with mixed cardioinhibitory and vasodepressor syncope. Clustering results confirm the disease groups and identify two distinct subgroups within the control and mixed groups. The proposed study demonstrates how to use machine learning to discover structure in blood pressure and heart rate time-series data. The methodology is used in classification of patients with orthostatic intolerance. Diagnosing orthostatic intolerance is challenging, and full characterization of the pathophysiological mechanisms remains a topic of ongoing research. This study provides a step toward leveraging machine learning to assist clinicians and researchers in addressing these challenges. Graphical abstract Machine learning tools utilized to analyze heart rate (HR) and blood pressure (BP) time-series data from syncope and control patients. Results show that machine learning can provide accurate classification of disease groups for 98% of patients and we identified two subgroups within the control patients differentiated by their BP response.

Key points Inflammation in response to bacterial endotoxin challenge impacts physiological functions, including cardiovascular, thermal and pain dynamics, although the mechanisms are poorly understood. We develop an innovative mathematical model incorporating interaction pathways between inflammation and physiological processes observed in response to an endotoxin challenge. We calibrate the model to individual data from 20 subjects in an experimental study of the human inflammatory and physiological responses to endotoxin, and we validate the model against human data from an independent study. Using the model to simulate patient responses to different treatment modalities reveals that a multimodal treatment combining several therapeutic strategies gives the best recovery outcome. Abstract Uncontrolled, excessive production of pro-inflammatory mediators from immune cells and traumatized tissues can cause systemic inflammatory conditions such as sepsis, one of the ten leading causes of death in the USA, and one of the three leading causes of death in the intensive care unit. Understanding how inflammation affects physiological processes, including cardiovascular, thermal and pain dynamics, can improve a patient's chance of recovery after an inflammatory event caused by surgery or a severe infection. Although the effects of the autonomic response on the inflammatory system are well-known, knowledge about the reverse interaction is lacking. The present study develops a mathematical model analyzing the inflammatory system's interactions with thermal, pain and cardiovascular dynamics in response to a bacterial endotoxin challenge. We calibrate the model with individual data from an experimental study of the inflammatory and physiological responses to a one-time administration of endotoxin in 20 healthy young men and validate it against data from an independent endotoxin study. We use simulation to explore how various treatments help patients exposed to a sustained pathological input. The treatments explored include bacterial endotoxin adsorption, antipyretics and vasopressors, as well as combinations of these. Our findings suggest that the most favourable recovery outcome is achieved by a multimodal strategy, combining all three interventions to simultaneously remove endotoxin from the body and alleviate symptoms caused by the immune system as it fights the infection.

Anovulation refers to a menstrual cycle characterized by the absence of ovulation. Exogenous hormones such as synthetic progesterone and estrogen have been used to attain this state to achieve contraception. However, large doses are associated with adverse effects such as increased risk for thrombosis and myocardial infarction. This study utilizes optimal control theory on a modified menstrual cycle model to determine the minimum total exogenous estrogen/progesterone dose, and timing of administration to induce anovulation. The mathematical model correctly predicts the mean daily levels of pituitary hormones LH and FSH, and ovarian hormones E2, P4, and Inh throughout a normal menstrual cycle and reflects the reduction in these hormone levels caused by exogenous estrogen and/or progesterone. Results show that it is possible to reduce the total dose by 92% in estrogen monotherapy, 43% in progesterone monotherapy, and that it is most effective to deliver the estrogen contraceptive in the mid follicular phase. Finally, we show that by combining estrogen and progesterone the dose can be lowered even more. These results may give clinicians insights into optimal formulations and schedule of therapy that can suppress ovulation.AUTHOR SUMMARYHormonal contraceptives composed of exogenous estrogen and/or progesterone are commonly administered artificial means of birth control. Despite many benefits, adverse side effects associated with high doses such as thrombosis and myocardial infarction, cause hesitation to usage. Our study presents an improved mathematical model for hormonal control of the menstrual cycle and applies optimal control theory to minimize total exogenous estrogen and/or progesterone dose, and determine timing of administration that lead to contraception. We observe a reduction in dosage of about 92% in estrogen monotherapy and 43% in progesterone monotherapy. Our simulations show that it is most effective to deliver the estrogen contraceptive in the mid follicular phase. In addition, we illustrate that combination therapy significantly lower doses further. Our findings may give clinicians insights into optimal dosing scheme for contraception.

Detection and monitoring of patients with pulmonary hypertension, defined as a mean blood pressure in the main pulmonary artery above 25 mmHg, requires a combination of imaging and hemodynamic measurements. This study demonstrates how to combine imaging data from microcomputed tomography images with hemodynamic pressure and flow waveforms from control and hypertensive mice. Specific attention is devoted to developing a tool that processes computed tomography images, generating subject-specific arterial networks in which one-dimensional fluid dynamics modeling is used to predict blood pressure and flow. Each arterial network is modeled as a directed graph representing vessels along the principal pathway to ensure perfusion of all lobes. The one-dimensional model couples these networks with structured tree boundary conditions representing the small arteries and arterioles. Fluid dynamics equations are solved in this network and compared to measurements of pressure in the main pulmonary artery. Analysis of microcomputed tomography images reveals that the branching ratio is the same in the control and hypertensive animals, but that the vessel length-to-radius ratio is significantly lower in the hypertensive animals. Fluid dynamics predictions show that in addition to changed network geometry, vessel stiffness is higher in the hypertensive animal models than in the control models.

Pulmonary hypertension (PH), i.e., high blood pressure in the lungs, is a serious medical condition that can damage the right ventricle of the heart and ultimately lead to heart failure. Standard diagnostic procedures are based on right-heart catheterization, which is an invasive technique that can potentially have serious side effects. Recent methodological advancements in fluid dynamics modelling of the pulmonary blood circulation system promise to mathematically predict the blood pressure based on non-invasive measurements of the blood flow. Thus, subsequent to PH diagnostication, further investigations would no longer require catheterization. However, in order for these alternative techniques to be applicable in the clinic, accurate model calibration and parameter estimation are paramount. Medical interventions taken to combat high blood pressure (as predicted from the mathematical model) alter the underlying cardiovascular physiology, thus interfering with the parameter estimation procedure. In the present study, we have carried out a series of cardiovascular simulations to assess the reliability of cardiovascular physiological parameter estimation in the presence of medical interventions. Our principal result is that if the closed-loop effect of medical interventions is accounted for, the model calibration provides accurate parameter estimates. This finding has important implications for the applicability of cardio-physiological modelling in the clinical practice.

From September-December 2017, the Mathematical Biosciences Institute at Ohio State University hosted a series of workshops on control theory in biology and medicine, including workshops on control and modulation of neuronal and motor systems, control of cellular and molecular systems, control of disease / personalized medicine across heterogeneous populations, and sensorimotor control of animals and robots. This special issue presents tutorials and research articles by several of the participants in the MBI workshops.

This study uses Bayesian inference to quantify the uncertainty of model parameters and haemodynamic predictions in a one-dimensional pulmonary circulation model based on an integration of mouse haemodynamic and micro-computed tomography imaging data. We emphasize an often neglected, though important source of uncertainty: in the mathematical model form due to the discrepancy between the model and the reality, and in the measurements due to the wrong noise model (jointly called ‘model mismatch’). We demonstrate that minimizing the mean squared error between the measured and the predicted data (the conventional method) in the presence of model mismatch leads to biased and overly confident parameter estimates and haemodynamic predictions. We show that our proposed method allowing for model mismatch, which we represent with Gaussian processes, corrects the bias. Additionally, we compare a linear and a nonlinear wall model, as well as models with different vessel stiffness relations. We use formal model selection analysis based on the Watanabe Akaike information criterion to select the model that best predicts the pulmonary haemodynamics. Results show that the nonlinear pressure–area relationship with stiffness dependent on the unstressed radius predicts best the data measured in a control mouse.

Objective: Postural Orthostatic Tachycardia Syndrome (POTS) is associated with the onset of tachycardia upon postural change. The current diagnosis involves the measurement of heart rate (HR) and blood pressure (BP) during head-up tilt (HUT) or active standing test. A positive diagnosis is made if HR changes with more than 30 bpm (40 bpm in patients aged 12–19 years), ignoring all of the BP and most of the HR signals. This study examines 0.1 Hz oscillations in systolic arterial blood pressure (SBP) and HR signals providing additional metrics characterizing the dynamics of the baroreflex. Methods: We analyze data from 28 control subjects and 28 POTS patients who underwent HUT. We extract beat-to-beat HR and SBP during a 10 min interval including 5 minutes of baseline and 5 minutes of HUT. We employ Uniform Phase Empirical Mode Decomposition (UPEMD) to extract 0.1 Hz stationary modes from both signals and use random forest machine learning and $\boldsymbol{k}$ -means clustering to analyze the outcomes. Results show that the amplitude of the 0.1 Hz oscillations is higher in POTS patients and that the phase response between the two signals is shorter (p Conclusion: POTS is associated with an increase in the amplitude of SBP and HR 0.1 Hz oscillation and shortening of the phase between the two signals. Significance: The 0.1 Hz phase response and oscillation amplitude metrics provide new markers that can improve POTS diagnostic augmenting the existing diagnosis protocol only analyzing the change in HR.

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 three 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.

This paper presents an optimal control approach to modeling effects of cardiovascular regulation during head-up tilt (HUT). Many patients who suffer from dizziness or light-headedness are administered a head-up tilt test to explore potential deficits within the autonomic control system, which maintains the cardiovascular system at homeostasis. This system is complex and difficult to study in vivo, and thus we propose to use mathematical modeling to achieve a better understanding of cardiovascular regulation during HUT. In particular, we show the feasibility of using optimal control theory to compute physiological control variables, vascular resistance and cardiac contractility, quantities that cannot be measured directly, but which are useful to assess the state of the cardiovascular system. A non-pulsatile lumped parameter model together with pseudo- and clinical data are utilized in the optimal control problem formulation. Results show that the optimal control approach can predict time-varying quantities regulated by the cardiovascular control system. Our results compare favorable to our previous study using a piecewise linear spline approach, less a priori knowledge is needed, and results were obtained at a significantly lower computational cost.

This study uses a one-dimensional fluid dynamics arterial network model to infer changes in hemodynamic quantities associated with pulmonary hypertension in mice. Data for this study include blood flow and pressure measurements from the main pulmonary artery for 7 control mice with normal pulmonary function and 5 mice with hypoxia-induced pulmonary hypertension. Arterial dimensions for a 21-vessel network are extracted from micro-CT images of lungs from a representative control and hypertensive mouse. Each vessel is represented by its length and radius. Fluid dynamic computations are done assuming that the flow is Newtonian, viscous, laminar, and has no swirl. The system of equations is closed by a constitutive equation relating pressure and area, using a linear model derived from stress-strain deformation in the circumferential direction assuming that the arterial walls are thin, and also an empirical nonlinear model. For each dataset, an inflow waveform is extracted from the data, and nominal parameters specifying the outflow boundary conditions are computed from mean values and characteristic timescales extracted from the data. The model is calibrated for each mouse by estimating parameters that minimize the least squares error between measured and computed waveforms. Optimized parameters are compared across the control and the hypertensive groups to characterize vascular remodeling with disease. Results show that pulmonary hypertension is associated with stiffer and less compliant proximal and distal vasculature with augmented wave reflections, and that elastic nonlinearities are insignificant in the hypertensive animal.

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.

Heart transplant patients are followed with periodic right heart catheterizations (RHCs) to identify post-transplant complications and guide treatment. Post-transplant positive outcomes are associated with a steady reduction of right ventricular and pulmonary arterial pressures, toward normal levels of right-side pressure (about 20mmHg) measured by RHC. This study shows more information about patient progression is obtained by combining standard RHC measures with mechanistic computational cardiovascular systems models. This study shows: to understand how cardiovascular system models can be used to represent a patient's cardiovascular state, and to use these models to track post-transplant recovery and outcome. To obtain reliable parameter estimates comparable within and across datasets, we use sensitivity analysis, parameter subset selection, and optimization to determine patient specific mechanistic parameter that can be reliably extracted from the RHC data. Patient-specific models are identified for ten patients from their first post-transplant RHC and longitudinal analysis is done for five patients. Results of sensitivity analysis and subset selection show we can reliably estimate seven non-measurable quantities including ventricular diastolic relaxation, systemic resistance, pulmonary venous elastance, pulmonary resistance, pulmonary arterial elastance, pulmonary valve resistance and systemic arterial elastance. Changes in parameters and predicted cardiovascular function post-transplant are used to evaluate cardiovascular state during recovery in five patients. Of these five patients, only one patient showed inconsistent trends during recovery in ventricular pressure-volume relationships and power output. At a four-year recovery time point this patient exhibited biventricular failure along with graft dysfunction while the remaining four exhibited no cardiovascular complications., Comment: 53 Pages (including supplement), 9 figures in manuscript, 9 figures in supplement

This study performs parameter inference in a partial differential equations system of pulmonary circulation. We use a fluid dynamics network model that takes selected parameter values and mimics the behaviour of the pulmonary haemodynamics under normal physiological and pathological conditions. This is of medical interest as it enables tracking the progression of pulmonary hypertension. We show how we make the fluids model tractable by reducing the parameter dimension from a 55D to a 5D problem. The Delayed Rejection Adaptive Metropolis algorithm, coupled with constraint non‐linear optimization, is successfully used to learn the parameter values and quantify the uncertainty in the parameter estimates. To accommodate for different magnitudes of the parameter values, we introduce an improved parameter scaling technique in the Delayed Rejection Adaptive Metropolis algorithm. Formal convergence diagnostics are employed to check for convergence of the Markov chains. Additionally, we perform model selection using different information criteria, including Watanabe Akaike Information Criteria.

During the last decade, there has been an increasing interest in the coupling between the acute inflammatory response and the Hypothalamic–Pituitary–Adrenal (HPA) axis. The inflammatory response is activated acutely by pathogen- or damage-related molecular patterns, whereas the HPA axis maintains a long-term level of the stress hormone cortisol which is also anti-inflammatory. A new integrated model of the interaction between these two subsystems of the inflammatory system is proposed and coined the integrated inflammatory stress (ITIS) model. The coupling mechanisms describing the interactions between the subsystems in the ITIS model are formulated based on biological reasoning and its ability to describe clinical data. The ITIS model is calibrated and validated by simulating various scenarios related to endotoxin (LPS) exposure. The model is capable of reproducing human data of tumor necrosis factor alpha, adrenocorticotropic hormone (ACTH) and cortisol and suggests that repeated LPS injections lead to a deficient response. The ITIS model predicts that the most extensive response to an LPS injection in ACTH and cortisol concentrations is observed in the early hours of the day. A constant activation results in elevated levels of the variables in the model while a prolonged change of the oscillations in ACTH and cortisol concentrations is the most pronounced result of different LPS doses predicted by the model.

Delay differential equations (DDEs) are widely used in mathematical modeling to describe physical and biological systems. Delays can impact model dynamics, resulting in oscillatory behavior. In physiological systems, this instability may signify (i) an attempt to return to homeostasis or (ii) system dysfunction. In this study, we analyze a nonlinear, nonautonomous, nonhomogeneous open-loop neurological control model describing the autonomic nervous system response to the Valsalva maneuver. Unstable modes have been identified as a result of parameter interactions between the sympathetic delay and time-scale. In a two-parameter bifurcation analysis, we examine both the homogeneous and nonhomogeneous systems. Discrepancies between solutions result from the presence of the forcing functions which stabilize the system. We use analytical methods to determine stability regions for the homogeneous system, identifying transcendental relationships between the parameters. We also use computational methods to determine stability regions for the nonhomogeneous system. The presence of a Hopf bifurcation within the system is discussed and solution types from the sink and stable focus regions are compared to two control patients and a patient with postural orthostatic tachycardia syndrome (POTS). The model and its analysis support the current clinical hypotheses that patients suffering from POTS experience altered nervous system activity., Mathematical Biosciences, 2019

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.

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.

Contraceptive drugs intended for family planning are used by the majority of married or in-union women in almost all regions of the world. The two most prevalent types of hormones associated with contraception are synthetic estrogens and progestins. Hormonal based contraceptives contain a dose of a synthetic progesterone (progestin) or a combination of a progestin and a synthetic estrogen. In this study we use mathematical modeling to understand better how these contraceptive paradigms prevent ovulation, special focus is on understanding how changes in dose impact hormonal cycling. To explain this phenomenon, we added two autocrine mechanisms essential to achieve contraception within our previous menstrual cycle models. This new model predicts mean daily blood concentrations of key hormones during a contraceptive state achieved by administering progestins, synthetic estrogens, or a combined treatment. Model outputs are compared with data from two clinical trials: one for a progestin only treatment and one for a combined hormonal treatment. Results show that contraception can be achieved with synthetic estrogen, with progestin, and by combining the two hormones. An advantage of the combined treatment is that a contraceptive state can be obtained at a lower dose of each hormone. The model studied here is qualitative in nature, but can be coupled with a pharmacokinetic/pharamacodynamic (PKPD) model providing the ability to fit exogenous inputs to specific bioavailability and affinity. A model of this type may allow insight into a specific drug’s effects, which has potential to be useful in the pre-clinical trial stage identifying the lowest dose required to achieve contraception., Author summary This study presents a mathematical model for hormonal control of the menstrual cycle of adult women with special emphasis on the effects of oral contraceptive drugs. Our model predicts daily blood levels of ovarian and pituitary hormones in close agreement with data found in the biological literature for normally cycling women. In particular, we study reproductive hormones which are produced by the pituitary gland in the brain and which promote the development of ovarian follicles and the production of ovarian hormones. In turn, the ovarian hormones affect the synthesis and release of the pituitary hormones. We use this model to test the effects of different oral contraceptive treatments. We show that the administration of synthetic progesterone or of synthetic estrogen have a contraceptive effect by preventing ovulation. We illustrate that low doses of each drug given together are most effective at achieving contraception. In addition, model simulations indicate how quickly a combined contraceptive treatment produces a non-ovulatory menstrual cycle and how fast the cycle returns to normal after the treatment ends. If we couple our model with a model for absorption and metabolism of oral contraceptive drugs, the resulting model may help discover minimal effective doses of these drugs and may lead to patient-specific dosing strategies.

Patient-specific models for diagnostics and treatment planning require reliable parameter estimation and model predictions. Mathematical models of physiological systems are often formulated as systems of nonlinear ordinary differential equations (ODEs) with many parameters and few options for measuring all state variables. Consequently, it can be difficult to determine which parameters can reliably be estimated from available data. This investigation highlights pitfalls associated with practical parameter identifiability and subset selection. The latter refer to the process associated with selecting a subset of parameters that can be identified uniquely by parameter estimation protocols. The methods will be demonstrated using five examples of increasing complexity, as well as with patient specific model predicting arterial blood pressure. This study demonstrates that methods based on local sensitivities are preferable in terms of computational cost and model fit when good initial parameter values are available, but that global methods should be considered when initial parameter value are not known or poorly understood. For global sensitivity analysis, Morris screening provides results in terms of parameter sensitivity ranking at a much lower computational cost.

This study develops non-pulsatile and pulsatile models for the prediction of blood flow and pressure during head-up tilt. This test is used to diagnose potential pathologies within the autonomic control system, which acts to keep the cardiovascular system at homeostasis. We show that mathematical modeling can be used to predict changes in cardiac contractility, vascular resistance, and arterial compliance, quantities that cannot be measured but are useful to assess the system’s state. These quantities are predicted as time-varying parameters modeled using piecewise linear splines. Having models with various levels of complexity formulated with a common set of parameters, allows us to combine long-term non-pulsatile simulations with pulsatile simulations on a shorter time-scale. We illustrate results for a representative subject tilted head-up from a supine position to a $$60^{\circ }$$ angle. The tilt is maintained for 5 min before the subject is tilted back down. Results show that if volume data is available for all vascular compartments three parameters can be identified, cardiovascular resistance, vascular compliance, and ventricular contractility, whereas if model predictions are made against arterial pressure and cardiac output data alone, only two parameters can be estimated either resistance and contractility or resistance and compliance.

Objective: Characteristic impedance (Zc) is an important component in the theory of hemodynamics. It is a commonly used metric of proximal arterial stiffness and pulse wave velocity. Calculated using simultaneously measured dynamic pressure and flow data, estimates of characteristic impedance can be obtained using methods based on frequency or time domain analysis. Applications of these methods under different physiological and pathological conditions in species with different body sizes and heart rates show that the two approaches do not always agree. In this study, we have investigated the discrepancies between frequency and time domain estimates accounting for uncertainties associated with experimental processes and physiological conditions. Approach: We have used published data measured in different species including humans, dogs, and mice to investigate: (a) the effects of time delay and signal noise in the pressure-flow data, (b) uncertainties about the blood flow conditions, (c) periodicity of the cardiac cycle versus the breathing cycle, on the frequency and time domain estimates of Zc, and (d) if discrepancies observed under different hemodynamic conditions can be eliminated. Main results and Significance: We have shown that the frequency and time domain estimates are not equally sensitive to certain characteristics of hemodynamic signals including phase lag between pressure and flow, signal to noise ratio and the end of systole retrograde flow. The discrepancies between two types of estimates are inherent due to their intrinsically different mathematical expressions and therefore it is impossible to define a criterion to resolve such discrepancies. Considering the interpretation and role of Zc as an important hemodynamic parameter, we suggest that the frequency and time domain estimates should be further assessed as two different hemodynamic parameters in a future study.

The cardiovascular control system is continuously engaged to maintain homeostasis, but it is known to fail in a large cohort of patients suffering from orthostatic intolerance. Numerous clinical studies have been put forward to understand how the system fails, yet noninvasive clinical data are sparse, typical studies only include measurements of heart rate and blood pressure, as a result it is difficult to determine what mechanisms that are impaired. It is known, that blood pressure regulation is mediated by changes in heart rate, vascular resistance, cardiac contractility, and a number of other factors. Given that numerous factors contribute to changing these quantities, it is difficult to devise a physiological model describing how they change in time. One way is to build a model that allows these controlled quantities to change and to compare dynamics between subject groups. To do so, it requires more knowledge of how these quantities change for healthy subjects. This study compares two methods predicting time-varying changes in cardiac contractility and vascular resistance during head-up tilt. Similar to the study by Williams et al. [51] , the first method uses piecewise linear splines, while the second uses the ensemble transform Kalman filter (ETKF) [1] , [11] , [12] , [33] . In addition, we show that the delayed rejection adaptive Metropolis (DRAM) algorithm can be used for predicting parameter uncertainties within the spline methodology, which is compared with the variability obtained with the ETKF. While the spline method is easier to set up, this study shows that the ETKF has a significantly shorter computational time. Moreover, while uncertainty of predictions can be augmented to spline predictions using DRAM, these are readily available with the ETKF.

Successful clinical use of patient-specific models for cardiovascular dynamics depends on the reliability of the model output in the presence of input uncertainties. For 1D fluid dynamics models of arterial networks, input uncertainties associated with the model output are related to the specification of vessel and network geometry, parameters within the fluid and wall equations, and parameters used to specify inlet and outlet boundary conditions. This study investigates how uncertainty in the flow profile applied at the inlet boundary of a 1D model affects area and pressure predictions at the center of a single vessel. More specifically, this study develops an iterative scheme based on the ensemble Kalman filter (EnKF) to estimate the temporal inflow profile from a prior distribution of curves. The EnKF-based inflow estimator provides a measure of uncertainty in the size and shape of the estimated inflow, which is propagated through the model to determine the corresponding uncertainty in model predictions of area and pressure. Model predictions are compared to ex vivo area and blood pressure measurements in the ascending aorta, the carotid artery, and the femoral artery of a healthy male Merino sheep. Results discuss dynamics obtained using a linear and a nonlinear viscoelastic wall model.

A novel multiscale mathematical and computational model of the pulmonary circulation is presented and used to analyse both arterial and venous pressure and flow. This work is a major advance over previous studies by Olufsen and coworkers (Ottesen et al., 2003; Olufsen et al., 2012) which only considered the arterial circulation. For the first three generations of vessels within the pulmonary circulation, geometry is specified from patient-specific measurements obtained using magnetic resonance imaging (MRI). Blood flow and pressure in the larger arteries and veins are predicted using a nonlinear, cross-sectional-area-averaged system of equations for a Newtonian fluid in an elastic tube. Inflow into the main pulmonary artery is obtained from MRI measurements, while pressure entering the left atrium from the main pulmonary vein is kept constant at the normal mean value of 2 mmHg. Each terminal vessel in the network of ‘large’ arteries is connected to its corresponding terminal vein via a network of vessels representing the vascular bed of smaller arteries and veins. We develop and implement an algorithm to calculate the admittance of each vascular bed, using bifurcating structured trees and recursion. The structured-tree models take into account the geometry and material properties of the ‘smaller’ arteries and veins of radii ≥ 50µm. We study the effects on flow and pressure associated with three classes of pulmonary hypertension expressed via stiffening of larger and smaller vessels, and vascular rarefaction. The results of simulating these pathological conditions are in agreement with clinical observations, showing that the model has potential for assisting with diagnosis and treatment of circulatory diseases within the lung.

Computational fluid dynamics (CFD) models are emerging tools for assisting in diagnostic assessment of cardiovascular disease. Recent advances in image segmentation have made subject-specific modelling of the cardiovascular system a feasible task, which is particularly important in the case of pulmonary hypertension, requiring a combination of invasive and non-invasive procedures for diagnosis. Uncertainty in image segmentation propagates to CFD model predictions, making the quantification of segmentation-induced uncertainty crucial for subject-specific models. This study quantifies the variability of one-dimensional CFD predictions by propagating the uncertainty of network geometry and connectivity to blood pressure and flow predictions. We analyse multiple segmentations of a single, excised mouse lung using different pre-segmentation parameters. A custom algorithm extracts vessel length, vessel radii and network connectivity for each segmented pulmonary network. Probability density functions are computed for vessel radius and length and then sampled to propagate uncertainties to haemodynamic predictions in a fixed network. In addition, we compute the uncertainty of model predictions to changes in network size and connectivity. Results show that variation in network connectivity is a larger contributor to haemodynamic uncertainty than vessel radius and length.

This study develops a lumped cardiovascular-respiratory system-level model that incorporates patient-specific data to predict cardiorespiratory response to hypercapnia (increased CO(2) partial pressure) for a patient with congestive heart failure (CHF). In particular, the study focuses on predicting cerebral CO(2) reactivity, which can be defined as the ability of vessels in the cerebral vasculature to expand or contract in response CO(2) induced challenges. It is difficult to characterize cerebral CO(2) reactivity directly from measurements, since no methods exist to dynamically measure vasomotion of vessels in the cerebral vasculature. In this study we show how mathematical modeling can be combined with available data to predict cerebral CO(2) reactivity via dynamic predictions of cerebral vascular resistance, which can be directly related to vasomotion of vessels in the cerebral vasculature. To this end we have developed a coupled cardiovascular and respiratory model that predicts blood pressure, flow, and concentration of gasses (CO(2) and O(2)) in the systemic, cerebral, and pulmonary arteries and veins. Cerebral vascular resistance is incorporated via a model parameter separating cerebral arteries and veins. The model was adapted to a specific patient using parameter estimation combined with sensitivity analysis and subset selection. These techniques allowed estimation of cerebral vascular resistance along with other cardiovascular and respiratory parameters. Parameter estimation was carried out during eucapnia (breathing room air), first for the cardiovascular model and then for the respiratory model. Then, hypercapnia was introduced by increasing inspired CO(2) partial pressure. During eucapnia, seven cardiovascular parameters and four respiratory parameters was be identified and estimated, including cerebral and systemic resistance. During the transition from eucapnia to hypercapnia, the model predicted a drop in cerebral vascular resistance consistent with cerebral vasodilation.

Dynamic cerebral autoregulation, that is the transient response of cerebral blood flow to changes in arterial blood pressure, is currently assessed using a variety of different time series methods and data collection protocols. In the continuing absence of a gold standard for the study of cerebral autoregulation it is unclear to what extent does the assessment depend on the choice of a computational method and protocol. We use continuous measurements of blood pressure and cerebral blood flow velocity in the middle cerebral artery from the cohorts of 18 normotensive subjects performing sit-to-stand manoeuvre. We estimate cerebral autoregulation using a wide variety of black-box approaches (ARI, Mx, Sx, Dx, FIR and ARX) and compare them in the context of reproducibility and variability. For all autoregulation indices, considered here, the ICC was greater during the standing protocol, however, it was significantly greater (Fisher's Z-test) for Mx (p < 0.03), Sx (p, 4 figures

Mathematical models have long been used for prediction of dynamics in biological systems. Recently, several efforts have been made to render these models patient specific. One way to do so is to employ techniques to estimate parameters that enable model based prediction of observed quantities. Knowledge of variation in parameters within and between groups of subjects have potential to provide insight into biological function. Often it is not possible to estimate all parameters in a given model, in particular if the model is complex and the data is sparse. However, it may be possible to estimate a subset of model parameters reducing the complexity of the problem. In this study, we compare three methods that allow identification of parameter subsets that can be estimated given a model and a set of data. These methods will be used to estimate patient specific parameters in a model predicting baroreceptor feedback regulation of heart rate during head-up tilt. The three methods include: structured analysis of the correlation matrix, analysis via singular value decomposition followed by QR factorization, and identification of the subspace closest to the one spanned by eigenvectors of the model Hessian. Results showed that all three methods facilitate identification of a parameter subset. The ”best” subset was obtained using the structured correlation method, though this method was also the most computationally intensive. Subsets obtained using the other two methods were easier to compute, but analysis revealed that the final subsets contained correlated parameters. In conclusion, to avoid lengthy computations, these three methods may be combined for efficient identification of parameter subsets.

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, whereas 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 into the underlying pathophysiology facilitating development of targeted interventions improving baroreflex function in diseased individuals, e.g. in patients with autonomic failure, a syndrome that is difficult to diagnose in terms of its pathophysiology., Comment: Keywords: Baroreflex model, mechanosensitivity, A- and C-type afferent baroreceptors, biophysical model, computational model

A better understanding of the biomechanical properties of the arterial wall provides important insight into arterial vascular biology under normal (healthy) and pathological conditions. This insight has potential to improve tracking of disease progression and to aid in vascular graft design and implementation. In this study, we use linear and nonlinear viscoelastic models to predict biomechanical properties of the thoracic descending aorta and the carotid artery under ex vivo and in vivo conditions in ovine and human arteries. Models analyzed include a four-parameter (linear) Kelvin viscoelastic model and two five-parameter nonlinear viscoelastic models (an arctangent and a sigmoid model) that relate changes in arterial blood pressure to the vessel cross-sectional area (via estimation of vessel strain). These models were developed using the framework of Quasilinear Viscoelasticity (QLV) theory and were validated using measurements from the thoracic descending aorta and the carotid artery obtained from human and ovine arteries. In vivo measurements were obtained from ten ovine aortas and ten human carotid arteries. Ex vivo measurements (from both locations) were made in eleven male Merino sheep. Biomechanical properties were obtained through constrained estimation of model parameters. To further investigate the parameter estimates we computed standard errors and confidence intervals and we used analysis of variance to compare results within and between groups. Overall, our results indicate that optimal model selection depends on the arterial type. Results showed that for the thoracic descending aorta (under both experimental conditions) the best predictions were obtained with the nonlinear sigmoid model, while under healthy physiological pressure loading the carotid arteries nonlinear stiffening with increasing pressure is negligible, and consequently, the linear (Kelvin) viscoelastic model better describes the pressure-area dynamics in this vessel. Results comparing biomechanical properties show that the Kelvin and sigmoid models were able to predict the zero-pressure vessel radius; that under ex vivo conditions vessels are more rigid, and comparatively, that the carotid artery is stiffer than the thoracic descending aorta; and that the viscoelastic gain and relaxation parameters do not differ significantly between vessels or experimental conditions. In conclusion, our study demonstrates that the proposed models can predict pressure-area dynamics and that model parameters can be extracted for further interpretation of biomechanical properties.

This paper combines a generalized viscoelastic model with a one-dimensional (1D) fluid dynamics model for the prediction of blood flow, pressure, and vessel area in systemic arteries. The 1D fluid dynamics model is derived from the Navier–Stokes equations for an incompressible Newtonian flow through a network of cylindrical vessels. This model predicts pressure and flow and is combined with a viscoelastic constitutive equation derived using the quasilinear viscoelasticity theory that relates pressure and vessel area. This formulation allows for inclusion of an elastic response as well as an appropriate creep function allowing for the description of the viscoelastic deformation of the arterial wall. Three constitutive models were investigated: a linear elastic model and two viscoelastic models. The Kelvin and sigmoidal viscoelastic models provide linear and nonlinear elastic responses, respectively. For the fluid domain, the model assumes that a given flow profile is prescribed at the inlet, that flow is c...

The objective of this study was to develop a personalized inflammatory model and estimate subject-specific parameters that could be related to changes in heart rate variability (HRV), a measure that can be obtained non-invasively in real time. An inflammatory model was developed and calibrated to measurements of interleukin-6 (IL-6), tumor necrosis factor (TNF-alpha), interleukin-8 (IL-8) and interleukin-10 (IL-10) over 8 hours in 20 subjects administered a low dose of lipopolysaccharide. For this model, we estimated 11 subject-specific parameters for all 20 subjects. Estimated parameters were correlated with changes in HRV, computed from ECG measurements using a built-in HRV module available in Labchart. Results revealed that patients could be separated into two groups expressing normal and abnormal responses to endotoxin. Abnormal responders exhibited increased HRV, most likely as a result of increased vagal firing. The observed correlation between the inflammatory response and HRV brings us a step further towards understanding if HRV predictions can be used as a marker for inflammation. Analyzing HRV parameters provides an easy, non-invasively obtained measure that can be used to assess the state of the subject, potentially translating to identifying a non-invasive marker that can be used to detect the onset of sepsis.

This study presents an analysis of a cerebral autoregulation (CA) model developed by Ursino and Lodi Ursino and Lodi (1997). We have used this model to analyze non-invasive measurements of cerebral blood flow velocity (CBFV) and arterial blood pressure obtained during postural change from sitting to standing for a healthy young subject. This paper includes a sensitivity analysis, ranking model parameters from the most to the least sensitive, and an analysis (using a methodology called subset selection) that allows identification of correlations among model parameters. Finally, we estimated patient specific parameters using the Levenberg-Marquardt optimization method minimizing the least square errors between computed and measured values of CBFV.

This study shows how sensitivity analysis and subset selection can be employed in a cardiovascular model to estimate total systemic resistance, cerebrovascular resistance, arterial compliance, and time for peak systolic ventricular pressure for healthy young and elderly subjects. These quantities are parameters in a simple lumped parameter model that predicts pressure and flow in the systemic circulation. The model is combined with experimental measurements of blood flow velocity from the middle cerebral artery and arterial finger blood pressure. To estimate the model parameters we use nonlinear optimization combined with sensitivity analysis and subset selection. Sensitivity analysis allows us to rank model parameters from the most to the least sensitive with respect to the output states (cerebral blood flow velocity and arterial blood pressure). Subset selection allows us to identify a set of independent candidate parameters that can be estimated given limited data. Analyses of output from both methods allow us to identify five independent sensitive parameters that can be estimated given the data. Results show that with the advance of age total systemic and cerebral resistances increase, that time for peak systolic ventricular pressure is increases, and that arterial compliance is reduced. Thus, the method discussed in this study provides a new methodology to extract clinical markers that cannot easily be assessed noninvasively.