A computational model of the circulating renin-angiotensin system and blood pressure regulation

Understanding the Mechanisms and Treatment of Heart Failure: Quantitative Systems Pharmacology Models with a Focus on SGLT2 Inhibitors and Sex-Specific Differences

Heart failure (HF), which is a major clinical and public health challenge, commonly develops when the myocardial muscle is unable to pump an adequate amount of blood at typical cardiac pressures to fulfill the body's metabolic needs, and compensatory mechanisms are compromised or fail to adjust. Treatments consist of targeting the maladaptive response of the neurohormonal system, thereby decreasing symptoms by relieving congestion. Sodium-glucose co-transporter 2 (SGLT2) inhibitors, which are a recent antihyperglycemic drug, have been found to significantly improve HF complications and mortality. They act through many pleiotropic effects, and show better improvements compared to others existing pharmacological therapies. Mathematical modeling is a tool used to describe the pathophysiological processes of the disease, quantify clinically relevant outcomes in response to therapies, and provide a predictive framework to improve therapeutic scheduling and strategies. In this review, we describe the pathophysiology of HF, its treatment, and how an integrated mathematical model of the cardiorenal system was built to capture body fluid and solute homeostasis. We also provide insights into sex-specific differences between males and females, thereby encouraging the development of more effective sex-based therapies in the case of heart failure.

His and her mathematical models of physiological systems

Beyond the reproductive system and reproductive behaviors, men and women exhibit major differences in many organ systems, including the anatomy of the brain, the activities of the stress and immune systems, and the metabolic and cardiovascular functions. A comprehensive understanding of the impact of these sex differences on health and disease is crucial to the development of effective sex-based therapies. Mathematical modeling has the potential of facilitating and contributing to advancing the understanding of sex differences in health and disease. Indeed, explosion of mathematical models have been developed in recent decades for different aspects of human physiology and pathophysiology. This review contains a survey of sex-specific mathematical models of physiological systems, describe insights that have been revealed in those modeling studies, and discuss future opportunities.

Cardiac and renal function interactions in heart failure with reduced ejection fraction: A mathematical modeling analysis

Congestive heart failure is characterized by suppressed cardiac output and arterial filling pressure, leading to renal retention of salt and water, contributing to further volume overload. Mathematical modeling provides a means to investigate the integrated function and dysfunction of heart and kidney in heart failure. This study updates our previously reported integrated model of cardiac and renal functions to account for the fluid exchange between the blood and interstitium across the capillary membrane, allowing the simulation of edema. A state of heart failure with reduced ejection fraction (HF-rEF) was then produced by altering cardiac parameters reflecting cardiac injury and cardiovascular disease, including heart contractility, myocyte hypertrophy, arterial stiffness, and systemic resistance. After matching baseline characteristics of the SOLVD clinical study, parameters governing rates of cardiac remodeling were calibrated to describe the progression of cardiac hemodynamic variables observed over one year in the placebo arm of the SOLVD clinical study. The model was then validated by reproducing improvements in cardiac function in the enalapril arm of SOLVD. The model was then applied to prospectively predict the response to the sodium-glucose co-transporter 2 (SGLT2) inhibitor dapagliflozin, which has been shown to reduce heart failure events in HF-rEF patients in the recent DAPAHF clinical trial by incompletely understood mechanisms. The simulations predict that dapagliflozin slows cardiac remodeling by reducing preload on the heart, and relieves congestion by clearing interstitial fluid without excessively reducing blood volume. This provides a quantitative mechanistic explanation for the observed benefits of SGLT2i in HF-rEF. The model also provides a tool for further investigation of heart failure drug therapies.

A Computer Model of Oxygen Dynamics in the Cortex of the Rat Kidney at the Cell-Tissue Level

The renal cortex drives renal function. Hypoxia/reoxygenation are primary factors in ischemia-reperfusion (IR) injuries, but renal oxygenation per se is complex and awaits full elucidation. Few mathematical models address this issue: none captures cortical tissue heterogeneity. Using agent-based modeling, we develop the first model of cortical oxygenation at the cell-tissue level (RCM), based on first principles and careful bibliographical analysis. Entirely parameterized with Rat data, RCM is a morphometrically equivalent 2D-slice of cortical tissue, featuring peritubular capillaries (PTC), tubules and interstitium. It implements hemoglobin/O2 binding-release, oxygen diffusion, and consumption, as well as capillary and tubular flows. Inputs are renal blood flow RBF and PO2 feeds; output is average tissue PO2 (tPO2). After verification and sensitivity analysis, RCM was validated at steady-state (tPO2 37.7 ± 2.2 vs. 36.9 ± 6 mmHg) and under transients (ischemic oxygen half-time: 4.5 ± 2.5 vs. 2.3 ± 0.5 s in situ). Simulations confirm that PO2 is largely independent of RBF, except at low values. They suggest that, at least in the proximal tubule, the luminal flow dominantly contributes to oxygen delivery, while the contribution of capillaries increases under partial ischemia. Before addressing IR-induced injuries, upcoming developments include ATP production, adaptation to minutes-hours scale, and segmental and regional specification.

Understanding sex differences in long-term blood pressure regulation: insights from experimental studies and computational modeling

Sex differences in blood pressure and the prevalence of hypertension are found in humans and animal models. Moreover, there has been a recent explosion of data concerning sex differences in nitric oxide, the renin-angiotensin-aldosterone system, inflammation, and kidney function. These data have the potential to reveal the mechanisms underlying male-female differences in blood pressure control. To elucidate the interactions among the multitude of physiological processes involved, one may apply computational models. In this review, we describe published computational models that represent key players in blood pressure regulation, and highlight sex-specific models and their findings.

Modeling Sex Differences in the Renin Angiotensin System and the Efficacy of Antihypertensive Therapies

The renin angiotensin system is a major regulator of blood pressure and a target for many anti-hypertensive therapies; yet the efficacy of these treatments varies between the sexes. We use published data for systemic RAS hormones to build separate models for four groups of rats: male normotensive, male hypertensive, female normotensive, and female hypertensive rats. We found that plasma renin activity, angiotensinogen production rate, angiotensin converting enzyme activity, and neutral endopeptidase activity differ significantly among the four groups of rats. Model results indicate that angiotensin converting enzyme inhibitors and angiotensin receptor blockers induce similar percentage decreases in angiotensin I and II between groups, but substantially different absolute decreases. We further propose that a major difference between the male and female RAS may be the strength of the feedback mechanism, by which receptor bound angiotensin II impacts the production of renin.

Exercise training normalizes renal blood flow responses to acute hypoxia in experimental heart failure: role of the α1-adrenergic receptor

Recent data suggest that exercise training (ExT) is beneficial in chronic heart failure (CHF) because it improves autonomic and peripheral vascular function. In this study, we hypothesized that ExT in the CHF state ameliorates the renal vasoconstrictor responses to hypoxia and that this beneficial effect is mediated by changes in α1-adrenergic receptor activation. CHF was induced in rabbits. Renal blood flow (RBF) and renal vascular conductance (RVC) responses to 6 min of 5% isocapnic hypoxia were assessed in the conscious state in sedentary (SED) and ExT rabbits with CHF with and without α1-adrenergic blockade. α1-adrenergic receptor expression in the kidney cortex was also evaluated. A significant decline in baseline RBF and RVC and an exaggerated renal vasoconstriction during acute hypoxia occurred in CHF-SED rabbits compared with the prepaced state (P < 0.05). ExT diminished the decline in baseline RBF and RVC and restored changes during hypoxia to those of the prepaced state. α1-adrenergic blockade partially prevented the decline in RBF and RVC in CHF-SED rabbits and eliminated the differences in hypoxia responses between SED and ExT animals. Unilateral renal denervation (DnX) blocked the hypoxia-induced renal vasoconstriction in CHF-SED rabbits. α1-adrenergic protein in the renal cortex of animals with CHF was increased in SED animals and normalized after ExT. These data provide evidence that the acute decline in RBF during hypoxia is caused entirely by the renal nerves but is only partially mediated by α1-adrenergic receptors. Nonetheless, α1-adrenergic receptors play an important role in the beneficial effects of ExT in the kidney.

Implementation of a model of bodily fluids regulation

The classic model of blood pressure regulation by Guyton et al. (Annu Rev Physiol 34:13–46, 1972a; Ann Biomed Eng 1:254–281, 1972b) set a new standard for quantitative exploration of physiological function and led to important new insights, some of which still remain the focus of debate, such as whether the kidney plays the primary role in the genesis of hypertension (Montani et al. in Exp Physiol 24:41–54, 2009a; Exp Physiol 94:382–388, 2009b; Osborn et al. in Exp Physiol 94:389–396, 2009a; Exp Physiol 94:388–389, 2009b). Key to the success of this model was the fact that the authors made the computer code (in FORTRAN) freely available and eventually provided a convivial user interface for exploration of model behavior on early microcomputers (Montani et al. in Int J Bio-med Comput 24:41–54, 1989). Ikeda et al. (Ann Biomed Eng 7:135–166, 1979) developed an offshoot of the Guyton model targeting especially the regulation of body fluids and acid–base balance; their model provides extended renal and respiratory functions and would be a good basis for further extensions. In the interest of providing a simple, useable version of Ikeda et al.’s model and to facilitate further such extensions, we present a practical implementation of the model of Ikeda et al. (Ann Biomed Eng 7:135–166, 1979), using the ODE solver Berkeley Madonna.

A semi-physiologically based pharmacokinetic pharmacodynamic model for glycyrrhizin-induced pseudoaldosteronism and prediction of the dose limit causing hypokalemia in a virtual elderly population

Glycyrrhizin (GL) is a widely used food additive which can cause severe pseudoaldosteronism at high doses or after a long period of consumption. The aim of the present study was to develop a physiologically based pharmacokinetic (PBPK) pharmacodynamic (PD) model for GL-induced pseudoaldosteronism to improve the safe use of GL. Since the major metabolite of GL, glycyrrhetic acid (GA), is largely responsible for pseudoaldosteronism via inhibition of the kidney enzyme 11β-hydroxysteroiddehydrogenase 2 (11β-HSD 2), a semi-PBPK model was first developed in rat to predict the systemic pharmacokinetics of and the kidney exposure to GA. A human PBPK model was then developed using parameters either from the rat model or from in vitro studies in combination with essential scaling factors. Kidney exposure to GA was further linked to an Imax model in the 11β-HSD 2 module of the PD model to predict the urinary excretion of cortisol and cortisone. Subsequently, activation of the mineralocorticoid receptor in the renin-angiotensin-aldosterone-electrolyte system was associated with an increased cortisol level. Experimental data for various scenarios were used to optimize and validate the model which was finally able to predict the plasma levels of angiotensin II, aldosterone, potassium and sodium. The Monte Carlo method was applied to predict the probability distribution of the individual dose limits of GL causing pseudoaldosteronism in the elderly according to the distribution of sensitivity factors using serum potassium as an indicator. The critical value of the dose limit was found to be 101 mg with a probability of 3.07%.

A model-based approach to investigating the pathophysiological mechanisms of hypertension and response to antihypertensive therapies: extending the Guyton model

Reproducibly differential responses to different classes of antihypertensive agents are observed among hypertensive patients and may be due to interindividual differences in hypertension pathology. Computational models provide a tool for investigating the impact of underlying disease mechanisms on the response to antihypertensive therapies with different mechanisms of action. We present the development, calibration, validation, and application of an extension of the Guyton/Karaaslan model of blood pressure regulation. The model incorporates a detailed submodel of the renin-angiotensin-aldosterone system (RAAS), allowing therapies that target different parts of this pathway to be distinguished. Literature data on RAAS biomarker and blood pressure responses to different classes of therapies were used to refine the physiological actions of ANG II and aldosterone on renin secretion, renal vascular resistance, and sodium reabsorption. The calibrated model was able to accurately reproduce the RAAS biomarker and blood pressure responses to combinations of dual-RAAS agents, as well as RAAS therapies in combination with diuretics or calcium channel blockers. The final model was used to explore the impact of underlying mechanisms of hypertension on the blood pressure response to different classes of antihypertensive agents. Simulations indicate that the underlying etiology of hypertension can impact the magnitude of response to a given class of therapy, making a patient more sensitive to one class and less sensitive others. Given that hypertension is usually the result of multiple mechanisms, rather than a single factor, these findings yield insight into why combination therapy is often required to adequately control blood pressure.

Hormonal regulation of salt and water excretion: a mathematical model of whole kidney function and pressure natriuresis

We present a lumped-nephron model that explicitly represents the main features of the underlying physiology, incorporating the major hormonal regulatory effects on both tubular and vascular function, and that accurately simulates hormonal regulation of renal salt and water excretion. This is the first model to explicitly couple glomerulovascular and medullary dynamics, and it is much more detailed in structure than existing whole organ models and renal portions of multiorgan models. In contrast to previous medullary models, which have only considered the antidiuretic state, our model is able to regulate water and sodium excretion over a variety of experimental conditions in good agreement with data from experimental studies of the rat. Since the properties of the vasculature and epithelia are explicitly represented, they can be altered to simulate pathophysiological conditions and pharmacological interventions. The model serves as an appropriate starting point for simulations of physiological, pathophysiological, and pharmacological renal conditions and for exploring the relationship between the extrarenal environment and renal excretory function in physiological and pathophysiological contexts.

A computational analysis of the long-term regulation of arterial pressure

The asserted dominant role of the kidneys in the chronic regulation of blood pressure and in the etiology of hypertension has been debated since the 1970s. At the center of the theory is the observation that the acute relationships between arterial pressure and urine production-the acute pressure-diuresis and pressure-natriuresis curves-physiologically adapt to perturbations in pressure and/or changes in the rate of salt and volume intake. These adaptations, modulated by various interacting neurohumoral mechanisms, result in chronic relationships between water and salt excretion and pressure that are much steeper than the acute relationships. While the view that renal function is the dominant controller of arterial pressure has been supported by computer models of the cardiovascular system known as the "Guyton-Coleman model", no unambiguous description of a computer model capturing chronic adaptation of acute renal function in blood pressure control has been presented. Here, such a model is developed with the goals of: 1. representing the relevant mechanisms in an identifiable mathematical model; 2. identifying model parameters using appropriate data; 3. validating model predictions in comparison to data; and 4. probing hypotheses regarding the long-term control of arterial pressure and the etiology of primary hypertension. The developed model reveals: long-term control of arterial blood pressure is primarily through the baroreflex arc and the renin-angiotensin system; and arterial stiffening provides a sufficient explanation for the etiology of primary hypertension associated with ageing. Furthermore, the model provides the first consistent explanation of the physiological response to chronic stimulation of the baroreflex.

A computational analysis of the long-term regulation of arterial pressure

The asserted dominant role of the kidneys in the chronic regulation of blood pressure and in the etiology of hypertension has been debated since the 1970s. At the center of the theory is the observation that the acute relationships between arterial pressure and urine production—the acute pressure-diuresis and pressure-natriuresis curves—physiologically adapt to perturbations in pressure and/or changes in the rate of salt and volume intake. These adaptations, modulated by various interacting neurohumoral mechanisms, result in chronic relationships between water and salt excretion and pressure that are much steeper than the acute relationships. While the view that renal function is the dominant controller of arterial pressure has been supported by computer models of the cardiovascular system known as the “Guyton-Coleman model”, no unambiguous description of a computer model capturing chronic adaptation of acute renal function in blood pressure control has been presented. Here, such a model is developed with the goals of: 1. representing the relevant mechanisms in an identifiable mathematical model; 2. identifying model parameters using appropriate data; 3. validating model predictions in comparison to data; and 4. probing hypotheses regarding the long-term control of arterial pressure and the etiology of primary hypertension. The developed model reveals: long-term control of arterial blood pressure is primarily through the baroreflex arc and the renin-angiotensin system; and arterial stiffening provides a sufficient explanation for the etiology of primary hypertension associated with ageing. Furthermore, the model provides the first consistent explanation of the physiological response to chronic stimulation of the baroreflex.

An integrated physiology-based model for the interaction of RAA system biomarkers with drugs

The renin angiotensin aldosterone system (RAAS) is a paramount target for the pharmacological treatment of cardiovascular diseases. As modeling and simulation techniques are becoming increasingly utilized in cardiovascular research, our aim was to develop a physiology-based model that describes the effect of different drugs at different doses on the RAAS and integrates physiology-based description drug pharmacokinetics (PK). First, a basic RAAS model was developed in which equations for drug effects were included and missing parameters estimated. Next, a physiology-based PK model for enalapril and enalaprilat was developed and coupled to the RAAS model. Simulation of the effects of angiotensin-converting enzyme inhibitors, angiotensin receptor blockers, and aliskiren administration on angiotensins I and II did not reveal significant overestimation or underestimation. For all drugs, the error numerics were acceptable. The model also encompassed the PK of intravenous and oral enalapril and its conversion to enalaprilat. In summary, we report a physiology-based model for the interaction of the RAAS biomarkers angiotensin I and II with enalapril, benazepril, aliskiren, and losartan that allows for an adequate description of the RAAS response after single administration of the drugs. Such a comprehensive description may lead to a better understanding of the effects of pharmacological interventions in the RAAS.

A detailed physiologically based model to simulate the pharmacokinetics and hormonal pharmacodynamics of enalapril on the circulating endocrine Renin-Angiotensin-aldosterone system

The renin-angiotensin-aldosterone system (RAAS) plays a key role in the pathogenesis of cardiovascular disorders including hypertension and is one of the most important targets for drugs. A whole body physiologically based pharmacokinetic (wb PBPK) model integrating this hormone circulation system and its inhibition can be used to explore the influence of drugs that interfere with this system, and thus to improve the understanding of interactions between drugs and the target system. In this study, we describe the development of a mechanistic RAAS model and exemplify drug action by a simulation of enalapril administration. Enalapril and its metabolite enalaprilat are potent inhibitors of the angiotensin-converting-enzyme (ACE). To this end, a coupled dynamic parent-metabolite PBPK model was developed and linked with the RAAS model that consists of seven coupled PBPK models for aldosterone, ACE, angiotensin 1, angiotensin 2, angiotensin 2 receptor type 1, renin, and prorenin. The results indicate that the model represents the interactions in the RAAS in response to the pharmacokinetics (PK) and pharmacodynamics (PD) of enalapril and enalaprilat in an accurate manner. The full set of RAAS-hormone profiles and interactions are consistently described at pre- and post-administration steady state as well as during their dynamic transition and show a good agreement with literature data. The model allows a simultaneous representation of the parent-metabolite conversion to the active form as well as the effect of the drug on the hormone levels, offering a detailed mechanistic insight into the hormone cascade and its inhibition. This model constitutes a first major step to establish a PBPK-PD-model including the PK and the mode of action (MoA) of a drug acting on a dynamic RAAS that can be further used to link to clinical endpoints such as blood pressure.

Renal renin secretion as regulator of body fluid homeostasis

The renin-angiotensin system is essential for body fluid homeostasis and blood pressure regulation. This review focuses on the homeostatic regulation of the secretion of active renin in the kidney, primarily in humans. Under physiological conditions, renin secretion is determined mainly by sodium intake, but the specific pathways involved and the relations between them are not well defined. In animals, renin secretion is a log-linear function of sodium intake. Close associations exist between sodium intake, total body sodium, extracellular fluid volume, and blood volume. Plasma volume increases by about 1.5 mL/mmol increase in daily sodium intake. Several lines of evidence indicate that central blood volume may vary substantially without measurable changes in arterial blood pressure. At least five intertwining feedback loops of renin regulation are identifiable based on controlled variables (blood volume, arterial blood pressure), efferent pathways to the kidney (nervous, humoral), and pathways operating via the macula densa. Taken together, the available evidence favors the notion that under physiological conditions (1) volume-mediated regulation of renin secretion is the primary regulator, (2) macula densa mediated mechanisms play a substantial role as co-mediator although the controlled variables are not well defined so far, and (3) regulation via arterial blood pressure is the exception rather than the rule. Improved quantitative analyses based on in vivo and in silico models are warranted.

Virtual patients and sensitivity analysis of the Guyton model of blood pressure regulation: towards individualized models of whole-body physiology

Mathematical models that integrate multi-scale physiological data can offer insight into physiological and pathophysiological function, and may eventually assist in individualized predictive medicine. We present a methodology for performing systematic analyses of multi-parameter interactions in such complex, multi-scale models. Human physiology models are often based on or inspired by Arthur Guyton's whole-body circulatory regulation model. Despite the significance of this model, it has not been the subject of a systematic and comprehensive sensitivity study. Therefore, we use this model as a case study for our methodology. Our analysis of the Guyton model reveals how the multitude of model parameters combine to affect the model dynamics, and how interesting combinations of parameters may be identified. It also includes a "virtual population" from which "virtual individuals" can be chosen, on the basis of exhibiting conditions similar to those of a real-world patient. This lays the groundwork for using the Guyton model for in silico exploration of pathophysiological states and treatment strategies. The results presented here illustrate several potential uses for the entire dataset of sensitivity results and the "virtual individuals" that we have generated, which are included in the supplementary material. More generally, the presented methodology is applicable to modern, more complex multi-scale physiological models.

Embedding a cardiac pulsatile model into an integrated model of the cardiovascular regulation for heart failure followup

The analysis of followup data from patients suffering from heart failure is a difficult task, due to the complex and multifactorial nature of this pathology. In this paper, we present a coupled model, integrating a pulsatile heart into a model of the short to long-term regulations of the cardiovascular system. An interface method is proposed to couple these models, which present significantly different time scales. Results from a sensitivity analysis of the original and integrated models are proposed with simulations reproducing the main effects of the short- and long-term responses of an acute decompensated heart failure episode on a patient undergoing cardiac resynchronization therapy.