# Limit cycle initialization

See code in GitLab.
Author: Fernando Campos fernando.campos@kcl.ac.uk

This tutorial demonstrates how to initialize a cardiac tissue with state variables obtained from a single-cell stimulation. To run the experiments of this tutorial change directories as follows:

cd \${TUTORIALS}/02_EP_tissue/03B_study_prep_init

## Conceptual Overview

Similar to in-vitro experiments, cardiac tissue is usually stabilized by pacing it at basic cycle length (BCL) in order to reduce beat-to-beat changes. However, stabilization might require a few hundred beats which can take days or even weeks to complete in organ-scale models. Unlike tissue simulations, pacing an isolated cell until it achieves its stable cycle limit is computationally feasible. In order to save computational efforts, the single-cell model state variables ($$V_{\mathrm m}$$, channel gating variables, etc.) at the end of the pacing protocol can be stored and used to initialize a tissue model. This initialization procedure is equivalent to pacing the entire tissue model in a space-clamped mode. Although this procedure is computationally cheap, it is biologically unrealistic since it ignores any differences due to the activation sequence of a previous beat. If information on the activation sequence in the cardiac tissue is available beforehand, it can be used in combination with the single-cell initialization method to overcome this limitation.

## Problem Setup

This example will run a simulation using a 2D sheet model (1 cm x 1 cm) in which all cells are initialized with single-cell model states stored in file INIT.sv. The file contains initial conditions of all ordinary differential equations (ODEs) describing channel gating and ionic concentrations in the chosen cellular model. The initial conditions are obtained after pacing the cell model at a prescribed BCL for a number N of beats.

## Usage

Run

./run.py --help 

to see the optional arguments available (default values are indicated), such as:

--model             Model of the cellular action potential
Options: {DrouhardRoberge, tenTusscherPanfilov}
Default: DrouhardRoberge
--init-method       Method to initialize the tissue model
Options: {None, sv_init or prepace}
Default: None
--bcl               Basic cycle length for repetitive single-cell stimulation
Default: 500 ms (2 Hz)
--npls              Number of pulses applied to the single-cell with the given bcl
Default: 10

If the program is run with the --visualize option, meshalyzer will automatically load transmembrane potentials $$V_{\mathrm m}$$:

./run.py --visualize

## Key Parameters

The key parameter is --init-method. This will be set to one of three possible values in order to initialize the 2D sheet model:

• none: all cells are initialized with the cellular model default states that are hard coded in openCARP
• sv_init: all cells are initialized with the cellular model states stored in file INIT.sv. Model states are computed here by running a single-cell simulation, where the cell is paced npls (default: 10 pulses) times with the prescribed bcl (default: 500ms). Model states are taken at the end of the simulation protocol.
• prepace: unlike in the sv_init initialization method, where state variables are saved into a file at a prescribed time instant (usually at the end of a single-cell pacing protocol), in the prepace method cells are initialized with model states taken at different time instants depending on their a priori known activation time. The difference between the two methods is illustrated below:

## Interpreting Results

The difference between the results produced with each initialization option (i.e.: none, sv_init and prepace) can be appreciated by looking at the spatial distribution of $$V_{\mathrm m}$$ at time t = 0 with meshalyzer (--visualize):

• A) By default (--init-method=none), $$V_{\mathrm m}$$ at t = 0 ms will be -86.9269 mV for all cells in the tissue. This is because -86.9269 is the default initial condition of $$V_{\mathrm m}$$ in the DrouhardRoberge model (the initial condition of $$V_{\mathrm m}$$ in the tenTusscherPanfilov model is -86.2).
• B) By running with the --init-method=sv_init option, $$V_{\mathrm m}$$ at t = 0 ms will be -85.7103 mV for all cells in the tissue. This is because -85.7103 is the initial condition of $$V_{\mathrm m}$$ in file INIT.sv, all model states were obtained after pacing the DrouhardRoberge model for 10 times with a bcl of 500 ms.
• C) By running with the --init-method=prepace option, $$V_{\mathrm m}$$ at t = 0 ms will vary between -85.7108 mV (left lower corner) and -85.6336 mV (right upper corner). This is because, unlike the previous methods where all cells are initialized with the same cellular model states, cells are initialized according to model states obtained at different time instants (see above) according to the activation sequence in file LATs.dat:

Although the differences in $$V_{\mathrm m}$$ between the three initialization methods seem negligible, they can be substantial in other state variables of the ionic model with a slower time decay such as the intracellular calcium $$Ca_{\mathrm i}$$. $$Ca_{\mathrm i} = 0.30$$ if --init-method=none and $$Ca_{\mathrm i} = 0.172033$$ (see file INIT.sv) if --init-method=sv_init with the bcl = 500 ms and npls = 10. These differences increase for all state variables as the bcl is reduced.

## openCARP Parameters

The relevant parameters used in the tissue-scale simulation are shown below:

# Model of the cellular action potential
-imp_region[0].im = model

# Used by the sv_init method
-imp_region[0].im_sv_init = 'INIT.sv'

# Used by the prepace method
-prepacing_beats = 10
-prepacing_lats  = 'LATs.dat'
-prepacing_bcl   = 500

imp_region[] is used to assign a model of cellular action potential to all cells in the tissue. If the sv_init initialization method is choosen, then the tissue will be initialized with cellular model states in file INIT.sv. However, if the prepace method is choosen, then the cell model in imp_region[] will be paced npls times with the given bcl. Tissue activation times in file LATs.dat are used to guide state set-up for prepacing.

Once the initial conditions are prescribed, the tissue simulation is started. In this example, a transmembrane current of 100 uA/cm^2 is applied for a period of 1 ms to the cells located at the middle of the left side of the tissue.

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