Mathematical modeling of antihypertensive therapy

Abstract

Hypertension is a multifactorial disease arising from complex pathophysiological pathways. Individual characteristics of patients result in different responses to various classes of antihypertensive medications. Therefore, evaluating the efficacy of therapy based on in silico predictions is an important task. This study is a continuation of research on the modular agent-based model of the cardiovascular and renal systems (presented in the previously published article). In the current work, we included in the model equations simulating the response to antihypertensive therapies with different mechanisms of action. For this, we used the pharmacodynamic effects of the angiotensin II receptor blocker losartan, the calcium channel blocker amlodipine, the angiotensin-converting enzyme inhibitor enalapril, the direct renin inhibitor aliskiren, the thiazide diuretic hydrochlorothiazide, and the β-blocker bisoprolol. We fitted therapy parameters based on known clinical trials for all considered medications, and then tested the model's ability to show reasonable dynamics (expected by clinical observations) after treatment with individual drugs and their dual combinations in a group of virtual patients with hypertension. The extended model paves the way for the next step in personalized medicine that is adapting the model parameters to a real patient and predicting his response to antihypertensive therapy. The model is implemented in the BioUML software and is available at https://gitlab.sirius-web.org/virtual-patient/antihypertensive-treatment-modeling.

Keywords: agent-based modular model; antihypertensive therapy; blood pressure regulation; cardiovascular system; mathematical modeling; renal system.

FIGURE 1

Modular agent-based model of the cardiovascular (purple) and renal (green) systems, including the pharmacodynamic module (yellow). Blue arrows indicate directed connections between the renal and cardiovascular sub-diagrams. For the visual simplicity, we added transition nodes (buses) for connections between modules. Target points of antihypertensive agents are marked in red (details are given in the section “Modeling antihypertensive effects” below). Abbreviations in the labels of target points: CCB, calcium channel blocker; ACE, angiotensin-converting enzyme.

FIGURE 2

Illustration of the Frank-Starling law. (A) Relationship curves between ventricular performance (i.e., stroke volume, SV) and ventricular end-diastolic volume (EDV) depending on the myocardial contractility. An increase in contractility results in a shift of the curve upward and to the left (greater SV for a given level of EDV and lower EDV at any level of SV, curves 1 to 2, points A to B), while depression of contractility leads to a shift downward and to the right (curves 1 to 3, points A to D). The scheme shows the relative levels of EDV that cause dyspnea and pulmonary edema, as well as the levels of ventricular performance required during rest and walking. For more details, see the study by Braunwald et al., 1967. [Reproduced from (Braunwald et al., 1967). Copyright © 2022 Massachusetts Medical Society. Reprinted with permission from Massachusetts Medical Society]. (B) Simulation of the left ventricular (LV) Frank-Starling curves in the model. The curves were obtained using the normal equilibrium state by changing the oxygen demand $R{O}_2$ from 4.2 ml/s (≈250 ml/min) to 16.7 ml/s (≈1,000 ml/min) and varying the LV inotropic state $K_{L0}$ above and below the normal value of 0.55. (C) At a critically low value of $K_{L0}=0.2$ and a normal value of $R{O}_2=4.2$ ml/sec, the model demonstrates a reduced systolic blood pressure (SBP) and a persistent increase in pulmonary venous pressure (PVP), which after 30 h of the model experiment results in a state interpreted as cardiogenic shock. After this state, the mathematical system becomes unstable and further simulation results are not subject to analysis, since it is assumed that the virtual patient is dead.

FIGURE 3

Algorithm for generating a virtual patient. The mean arterial pressure (MAP), cardiac output (CO) and hematocrit (Hct) values are taken from the real patient’s history or randomly selected from physiological ranges in accordance with the patient’s diseases. When the estimation of the renal submodel parameters is completed, the equilibrium values of the total blood volume (V) and the concentration of angiotensin II bound to the AT1 receptors (AT1_ANGII) are passed to the cardiovascular submodel. The latter is then calibrated so that the simulated MAP and CO equilibrium values, and the constant Hct value, coincide with the values set in the renal submodel. When the calibration of both submodels is finished, the equilibrium values are combined into a virtual patient.

FIGURE 4

Distribution of physiological parameters in the virtual population (n = 186). Designations in the figure: AAR, afferent arteriolar resistance; ANGII, angiotensin II concentration; BMI, body mass index; DBP, diastolic blood pressure; EAR, efferent arteriolar resistance; GFR, glomerular filtration rate; Hct, hematocrit; He, hemoglobin; HR, heart rate; LV, left ventricular; LV aPFR, LV active peak filling rate; LV EF, LV ejection fraction; LV ePFR, LV early peak filling rate; LV EDP, LV end-diastolic pressure; LV EDV, LV end-diastolic volume; LV ESV, LV end-systolic volume; LV PSP, LV peak systolic pressure; LV SV, LV stroke volume; PAC, plasma aldosterone concentration; PRA, plasma renin activity; PRC, plasma renin concentration; RVR, renal vascular resistance; SBP, systolic blood pressure; SVR, systemic vascular resistance; TBW, total body water.

FIGURE 5

Comparison of simulated reduction in systolic blood pressure (SBP) and diastolic blood pressure (DBP) with clinical measurements obtained for aliskiren 150 or 300 mg, amlodipine 5 mg, bisoprolol 5 mg, enalapril 20 mg, HCTZ 12.5 mg, and losartan 50 or 100 mg. The data are presented as mean ± SD. The red bars denote the baseline data. The blue bars correspond to the values after therapy. To simplify the figure, we left in the references only the first author and the year of the relevant research.

FIGURE 6

Simulation of plasma RAAS parameters (PRA, plasma renin activity; PRC, plasma renin concentration; Ang I, angiotensin I; Ang II, angiotensin II; PAC, plasma aldosterone concentration) and renal hemodynamic parameters (GFR, glomerular filtration rate; RBF, renal blood flow; RVR, renal vascular resistance) in the virtual hypertensive population at baseline (red) and after 4 weeks of treatment (blue) with aliskiren (150 or 300 mg), amlodipine (5 mg), bisoprolol (5 mg), enalapril (20 mg), HCTZ (12.5 mg), and losartan (50 or 100 mg). The data are given as mean ± SD.

FIGURE 7

Simulation of the left ventricular (LV) parameters (LV SV, LV stroke volume; LV EF, LV ejection fraction; LV EDP, LV end-diastolic pressure; LV PSP, LV peak systolic pressure; LV EDV, LV end-diastolic volume; LV ESV, LV end-systolic volume; LV ePFR, LV early peak filling rate; LV aPFR, LV active peak filling rate) in the virtual hypertensive population at baseline (red) and after 4 weeks of treatment (blue) with aliskiren (150 or 300 mg), amlodipine (5 mg), bisoprolol (5 mg), enalapril (20 mg), HCTZ (12.5 mg), and losartan (50 or 100 mg). The data are given as mean ± SD.

FIGURE 8

Simulated changes in systolic and diastolic blood pressure and heart rate obtained for aliskiren 150 mg (Al150), aliskiren 300 mg (Al300), enalapril 20 mg (E20), losartan 100 mg (L100), losartan 50 mg (L50), amlodipine 5 mg (Aml5), bisoprolol 5 mg (B5), hydrochlorothiazide 12.5 mg (H12.5), and combinations of one RAAS inhibitor with one drug from other antihypertensive classes. All drugs give a statistically significant (p < 0.0001, Kolmogorov-Smirnov test) decrease in blood pressure, while statistically significant changes in heart rate are marked with “*”. The data are presented as mean ± SD.