Mechano-calcium and mechano-electric feedbacks in the human cardiomyocyte analyzed in a mathematical model

Abstract

Experiments on animal hearts (rat, rabbit, guinea pig, etc.) have demonstrated that mechano-calcium feedback (MCF) and mechano-electric feedback (MEF) are very important for myocardial self-regulation because they adjust the cardiomyocyte contractile function to various mechanical loads and to mechanical interactions between heterogeneous myocardial segments in the ventricle walls. In in vitro experiments on these animals, MCF and MEF manifested themselves in several basic classical phenomena (e.g., load dependence, length dependence of isometric twitches, etc.), and in the respective responses of calcium transients and action potentials. However, it is extremely difficult to study simultaneously the electrical, calcium, and mechanical activities of the human heart muscle in vitro. Mathematical modeling is a useful tool for exploring these phenomena. We have developed a novel model to describe electromechanical coupling and mechano-electric feedbacks in the human cardiomyocyte. It combines the 'ten Tusscher-Panfilov' electrophysiological model of the human cardiomyocyte with our module of myocardium mechanical activity taken from the 'Ekaterinburg-Oxford' model and adjusted to human data. Using it, we simulated isometric and afterloaded twitches and effects of MCF and MEF on excitation-contraction coupling. MCF and MEF were found to affect significantly the duration of the calcium transient and action potential in the human cardiomyocyte model in response to both smaller afterloads as compared to bigger ones and various mechanical interventions applied during isometric and afterloaded twitches.

Keywords: Electromechanical coupling; Human myocardium; Mechano-electric feedback.

Fig. 1

Ionic membrane currents and intracellular calcium homeostasis in the TP + M model. Calcium currents: i_CaL—L-type Ca²⁺ current; i_bCa—background Ca²⁺ current. Potassium currents: i_K1—inward rectifier K⁺ current; i_to—transient outward current; i_Kr, i_Ks—rapid and slow delayed rectifier current; i_pK—plateau K⁺ current. Sodium currents: i_Na—fast Na⁺ current; i_bNa—background Na⁺ current. Pumps and exchangers: i_pCa—sarcolemmal Ca²⁺ pump current; i_NaK—Na⁺–K⁺ pump current; i_NaCa—Na⁺–Ca²⁺ exchanger (NCX) current. Calcium translocations: I_rel—Ca²⁺ release from the sarcoplasmic reticulum (SR) via ryanodine receptors to the subspace (SS); I_xfer—Ca²⁺ diffusion from SS to the cytoplasm; I_leak—a small Ca²⁺ leakage from the SR to the cytoplasm; I_up—Ca²⁺ pumping from the cytoplasm to the SR, where Ca²⁺ is partially buffered (Buffer in SR). Cytoplasmic buffering is divided into two compartments: Ca²⁺–troponin C complex formation (Ca–TnC) inherited from [28] and buffering by other intracellular ligands (Buffer). The figure is modified from the diagram in the Physiome Model Repository (https://models.physiomeproject.org/exposure/a7179d94365ff0c9c0e6eb7c6a787d3d/ten_tusscher_model_2006_IK1Ko_M_units.cellml/view) based on [20]

Fig. 2

Rheological scheme of virtual cardiac sample in the TP + M model. It consists of a contractile element, which is a generator of active force, three elastic and two viscous elements. Contractile element in the model reproduces sarcomeres in the cardiomyocyte

Fig. 3

Simulation of isometric contractions at different sample lengths in the TP + M model. The initial length of the sample was decreased from 95% to 80%L_max (see inset from dark to light grey lines). a The steady-state signals for isometric force (as compared to the force F₀ generated at 95%L_max initial length), sarcomere length, and membrane potential are shown for all lengths considered. b The steady-state signals for the concentration of calcium–troponin C complexes ([Ca–TnC]), intracellular Ca²⁺ concentration ([Ca²⁺]_i) and Na⁺–Ca²⁺ exchange current (i_NaCa) for 85% and 95%L_max initial lengths. Dashed line in the intracellular Ca²⁺ concentration panel is for the numerical experiment where mechano-electric feedbacks were eliminated from the model by imposing isometric conditions on the sarcomere (see text for details)

Fig. 4

Main characteristics of the isometric cycles in the TP + M model. a “Length–Force” diagrams depicting the relationship between length and generated force obtained in a series of isometric contractions with length decreased from reference length L₀ = L_init (at which the virtual sample generates maximum isometric force F₀) to 84%L₀. b Length dependence of isometric twitch temporal characteristics: TTP—time to peak twitch; t₅₀, t₇₀—time to 50% and 70% force decay from peak force in isometric cycles. c Length dependence of action potential duration at 90% of repolarization (APD₉₀) in isometric cycles

Fig. 5

Modified afterloaded contractions recorded in experiments with muscle samples from the left ventricular myocardium of a patient with dilated cardiomyopathy. From top to bottom: intracellular calcium transient, muscle length (ML), and force. The muscle is allowed to contract against predefined loads and as the end-systolic shortening is reached it is restretched at constant velocity to its initial length. “0” indicates isometric contractions; “3”, an afterloaded contraction against an afterload of 40% of maximal isometric force; and “5”, an isotonic contraction against passive resting force (With permission from [55])

Fig. 6

Simulation of modified afterloaded contractions with quick muscle restretching (Fig. 5) in the TP + M model. The mode of cardiac muscle contraction is simulated in the following way. The muscle is allowed to contract against different loads (decreased in a from dark to light grey lines) in the same manner as in the isotonic afterloaded mode. Then, at the moments of maximum (end-systolic) shortening (shown by triangles) the muscle is forced to stretch at a velocity much higher than that of its lengthening in the full afterloaded cycle (as shown further for the virtual sample in Fig. 7). Thus, the sample quickly returns to its initial length (L_init = 90%L_max in this simulation) and then relaxes isometrically. Dotted lines are for isometric contraction, dashed lines are for preloaded twitch. a, b Active force normalized to peak isometric force at L_init; length of the virtual sample (in % of L_init). c, d Intracellular Ca²⁺ concentration ([Ca²⁺]_i) and membrane potential

Fig. 7

Simulation of a series of isotonic afterloaded contractions in the TP + M model. The initial virtual sample length L_init is equal to 90%L_max. F_isom is a peak active isometric force at L_init. Each afterload was applied following a steady-state isometric contraction (dotted lines) varying from a high afterload of 0.9F_isom (black lines) to a low afterload of 0.1F_isom (light grey lines). a Time-dependent signals of the generated force (F/F_isom), sample length, sarcomere length and membrane potential for various loads applied. b Time-dependent signals of the concentration of calcium–troponin C complexes ([Ca–TnC]), intracellular Ca²⁺ transient ([Ca²⁺]_i), Na⁺–Ca²⁺ exchange current (i_NaCa) and inward rectifier K⁺ current (i_K1) for isometric and afterloaded contractions at low afterload

Fig. 8

The load dependence index (LDI) is set as the ratio t_isot/t_isom

Fig. 9

Load-dependent characteristics in the TP + M model for two initial virtual sample lengths L_init: 90% and 95%L_max. The abscissa plots the force values normalized to the maximum isometric value F_isom corresponding to the initial length L_init. a The load dependence index (LDI) calculated as shown in Fig. 8. b Action potential duration calculated at 90% repolarization (APD₉₀)

Fig. 10

“Length–Force” diagram depicting relationship between end-systolic shortening and end-systolic force obtained in a series of isotonic afterloaded contractions at initial length L_init = 90%L_max (solid line) compared with relationship between length and generated force obtained in a series of isometric contractions (dashed line), where length decreases from reference length L₀ = L_init (at which the virtual sample generates maximum isometric force F₀) to 88%L₀

Fig. 11

Simulation of a quick change in the load during an isotonic afterloaded cycle in the TP + M model. The initial virtual sample length L_init is equal to 90%L_max. Dotted lines are for isometric contraction. Dashed lines show afterloaded contractions under different loads (20, 40 and 60% of peak isometric force F_isom at L_init). Solid lines show the traces for quick increase (a) and decrease (b) in the load approximately in the middle of the isotonic plateau. From top to bottom in both a and b: force, sample length, intracellular Ca²⁺ transient ([Ca²⁺]_i) and membrane potential changes. c, d The value tCa₇₀ is time from peak Ca²⁺ transient to 70% decay. APD₉₀ is action potential duration at 90% repolarization

Fig. 12

Simulation of quick releases of the virtual sample during the isometric cycle in the TP + M model. The initial length L_init is equal to 90%L_max. The sample is quickly released to 95%L_init at 90 and 240 ms after the stimulus and is held at a new length until the end of the cycle. a–d Virtual sample length, force development, intracellular Ca²⁺ transient ([Ca²⁺]_i) and membrane potential in the simulation

Fig. 13

Simulation of different modes of contractions in the TP + M model with the same protocol as applied to rat single cardiac myocytes [66]. According to this protocol, one low-loaded shortening follows one isometric force generation for 90%L_max initial length with the pacing rate of 1 Hz. Stimulation moments are labeled on the time axes. a The steady-state signals for developed force (as compared to the peak isometric force F_isom). b The steady-state signals for sample length. c Ca²⁺ transients for the last isometric twitch (dashed line) and low-loaded shortening (dotted line) are superimposed