Simulation of the Effects of Extracellular Calcium Changes Leads to a Novel Computational Model of Human Ventricular Action Potential With a Revised Calcium Handling.

The importance of electrolyte concentrations for cardiac function is well established. Electrolyte variations can lead to arrhythmias onset, due to their important role in the action potential (AP) genesis and in maintaining cell homeostasis. However, most of the human AP computer models available in literature were developed with constant electrolyte concentrations, and fail to simulate physiological changes induced by electrolyte variations. This is especially true for Ca2+, even in the O'Hara-Rudy model (ORd), one of the most widely used models in cardiac electrophysiology. Therefore, the present work develops a new human ventricular model (BPS2020), based on ORd, able to simulate the inverse dependence of AP duration (APD) on extracellular Ca2+ concentration ([Ca2+]o), and APD rate dependence at 4 mM extracellular K+. The main changes needed with respect to ORd are: (i) an increased sensitivity of L-type Ca2+ current inactivation to [Ca2+]o; (ii) a single compartment description of the sarcoplasmic reticulum; iii) the replacement of Ca2+ release. BPS2020 is able to simulate the physiological APD-[Ca2+]o relationship, while also retaining the well-reproduced properties of ORd (APD rate dependence, restitution, accommodation and current block effects). We also used BPS2020 to generate an experimentally-calibrated population of models to investigate: (i) the occurrence of repolarization abnormalities in response to hERG current block; (ii) the rate adaptation variability; (iii) the occurrence of alternans and delayed after-depolarizations at fast pacing. Our results indicate that we successfully developed an improved version of ORd, which can be used to investigate electrophysiological changes and pro-arrhythmic abnormalities induced by electrolyte variations and current block at multiple rates and at the population level.


Supplementary
. Left: APD rate dependence comparison with other human ventricular AP models. Right: APD vs [Ca 2+ ] o relationship in ORd (light blue, (5)), BPS2020 (dark blue), ORd CiPA (red, (10)) and Grandi model (green, (11)).  Table S2. Root mean square error to quantify the distance between experimental data and simulations with the ORd and the BPS2020 models.

Automatic parameter optimization
After the introduction of the new I CaL formulation and the achieved physiological inverse dependence between the extracellular Ca 2+ concentration ([Ca 2+ ] o ) and action potential duration (APD), an automatic optimization (similar to (12)) was performed to tune the model in order to fit the experimental data for APD rate dependence and restitution from (5), without losing the correct APD- The cost function of the optimization procedure was based on quantitative APD data (

Root Mean Square Error
ORd model BPS2020 model Figure 3A APD rate dependence (ms) 13.5 6.3 Figure 3A APD restitution (ms) 15.5 14.6 Figure 3B I Kr Block rate dependence (ms) 69.4 52.7 Figure 3B I Ks Block rate dependence (ms) 36.6 33.7  During the optimization the model was paced for 100 beats in each iteration. The final values of the state variables in each iteration were taken as initial values for the next one. Simulations with the optimized parameters, to quantify the goodness of fit, were run for 1000s to completely ensure steadystate achievement.

Cost function
In our optimization procedure, we minimized the cost function where each term is formulated as: The parameters chosen for automatic optimization are multiplicative factors for the following current: I Kr (bGKr for the conductance), I K1 (bGK1 for the conductance and kslope_IK1 for the steady state rectification slope), I NaCa (bGncx for the conductance), J up (cJup for the conductance), I NaL (bINaL for conductance), J diff (τJdiff for the diffusion time constant), I Cab (gICab for the conductance), I CaL (cPCa for conductance) and kCDI.

Search method and stop criterion
The initial values of the selected parameters were obtained from a previous modified version of the O'Hara-Rudy model (13) and used to start the automatic optimization.
The minimization of the cost function was based on the trust region reflective algorithm, using the Matlab built-in lsqnonlin function (14). The automatic optimization stopped when the number of iteration reached 150 or the minimum change in variables for finite-difference gradients was smaller than 0.1.

Model equations
We present the equations we changed in the BPS2020 with respect to the ORd model. For I Kr the equation was taken from (10) with the adjustment of the conductance (as reported in section 2.1.6) and for I NaF from (15) with the modification presented in section 2.1.10. The equations follow the naming convention of the ORd model. Membrane potential is reported as .

BPS2020 Human Model Concentrations and Buffers
In all the equations of the concentration balances have been substituted with .