# Myocardial ischemic effects on cardiac electro-mechanical activity

B.V. Rathish Kumar
Department of Mathematics and Statistics, Indian Institue of Technology
Kanpur, India
Meena Pargaei
Department of Mathematics and Statistics, Indian Institue of Technology
Kanpur, India
Govt. P. G. College Champawat, Uttarakhand, India,
Luca F. Pavarino
Department of Mathematics, University of Pavia,
Simone Scacchi
Department of Mathematics, University of Milan

###### Abstract

In this work, we investigated the effect of varying strength of Hyperkalemia
and Hypoxia, in a human cardiac tissue with a local ischemic subregion, on the
electrical and mechanical activity of healthy and ischemic zones of the
cardiac muscle. The Monodomain model in a deforming domain is taken with the
addition of mechanical feedback and stretch activated channel current coupled
with the ten Tusscher human ventricular membrane model. The equations of
finite elasticity are used to describe the deformation of the cardiac tissue.
The resulting coupled electro-mechanical PDEs-ODEs non-linear system is solved
numerically using finite elements in space and finite difference method in
time. We examined the effect of local ischemia on the cardiac electrical and
mechanical activity in different cases. We concluded that the spread of
Hyperkalemic or Hypoxic region alters the electro-mechanical coupling in terms
of the action potential ($v$), intracellular calcium ion concentration
$[Ca^{+2}]_{i}$, active tension, ($T_{A}$), stretch ($\lambda$), stretch rate
($\frac{d\lambda}{dt}$). With the increase in the size of the ischemic region
by factor of five, approximately $45\%$ variation in the stretch rate
$\frac{d\lambda}{dt}$ is noticed. It is also shown that ischemia affects the
deformation (expansion and contraction) of the heart.

##  1 Introduction

The heart is considered as a muscular pump which is connected to the systemic
and pulmonary vascular systems. The main purpose of the heart and vascular
system is to sustain an appropriate supply of nutrients as oxygenated blood
and metabolic substrates to the body. The left atrium and ventricle chambers
pump the blood from pulmonary veins to the aorta, whereas the right atrium and
ventricle chambers pump the blood from vena cava to pulmonary arteries. The
cardiac cycle consists of two phases: diastole, during which the ventricles
relax and the heart fills with blood; and systole, during which the ventricles
contract and pump blood out of the heart to arteries. To this end, the heart
needs three types of cells: 1\. SA node or pacemaker cells, which produce an
electrical signal; 2. Conductors to spread the pacemaker signals; and 3.
Contractile cells, to mechanically pump blood. Pacemaker cells start the
electrical sequence of depolarization and repolarization. The electrical
signal generated from the SA node travels to the ventricles via the
atrioventricular node(AV node), the bundle of His, the right and left bundle
branches and Purkinje fibers. As the depolarization signal reaches the
contractile cells they start to contract and as the repolarization signal
reaches the myocardial cells, they start to relax. In this way the electrical
signals and the mechanical pumping action of the heart are connected via
excitation-contraction coupling. Heart’s electrical and mechanical activity
can be deduced from the ECG pattern. P wave represents the atrial
depolarization, QRS complex due to the ventricular depolarization also start
of the ventricular contraction, T wave indicate the ventricular repolarization
and the beginning of ventricular relaxation.

There exist one factor, called stretch, which is responsible for the
activation and inactivation of the mechanically controlled ion channels. In
1997, Hu & Sachs [HS97] revealed the stretch activated channels (SACs) in
cardiac cells. Positive response of the SACs contribute for heart to stretch.
Cardiac muscle stretch also leads to the increase in active tension.

For the finite elasticity material, there are various models for the strain
energy function based on the different laws. For example, in [GCM95a] Guccione
et al. used the transversely isotropic exponential law strain energy function
and the orthotropic and isotropic laws have been used in [NP04] and [CBP+14a],
respectively. Various models for the development of the active tension and the
description of the dynamics of calcium ions and the cross bridges bindings
have been proposed in [CBP+14b, DGKK13, EPPH13, KNP09, KOMM10, MW07, RLRB+14,
SMCCS06, WGRS11, PW09, LNA+12a, JGT10, GLC+11, GCRT10, DORB+13, CHLT13,
AHZ13a]. In [HMTK98], Hunter et al. presented the model based on the review of
experimental data for the mechanics of active and passive cardiac muscle. They
concluded that it could be interpreted as the four state variable model. (i)
myocardial tissue passive elasticity, (ii) tension dependence on the troponin
C and $Ca^{+2}$ bindings, (iii) tropomyosin movement kinetics and the actively
cycling cross-bridges binding sites and the length dependence of this process,
and, (iv) cross-bridge tension development kinetics under variations of the
myofilament length. In [NS07], developed a biophysically and dynamically
stable model for the cardiac myocyte of rat at the room temperature. This
model is capable of determining the changes in the concentration of
intracellular calcium ion in response to the stretch. In [LNA+12b], presented
a multiscale electro-mechanical model for the murince heart which is capable
of investigating the effects of excitation-contraction coupling. In this
paper,they show a strong relation between the tension and the velocity to
explain the differences between the single cell tension and the whole organ
pressure transient.

For the cardiac electrical activity models at the tissue level, Bidomain and
Monodomain models have been considered in various researches [CHLT13, DORB+13,
GCRT10, JGT10, WGRS11, DGKK13, MW07, RLRB+14, EPPH13, KNP09, KOMM10]. For the
ionic membrane models related to the different species has been used in
[CHLT13, DORB+13, GCRT10, JGT10, WGRS11, DGKK13, KNP09, KOMM10, MW07, RLRB+14,
EPPH13] etc.

Following the electro-mechanical coupling approach presented in [AHZ13a,
DORB+13, GCRT10, WBG07, RLRB+14, PCGW10, NQRB12, PW09, NSH05, KNP09, GK10a,
DGKK13, AANQ11] we need to impose the electrical models on the deformed domain
to propose the coupling between the electrical and mechanical model. Then by
the Lagrangian framework again this model is reformulated into the reference
domain.

There exist coupled cardiac electro-mechanical models which explain the
traveling of electric signal into the cardiac muscle and the contraction-
relaxation process. These models consists : 1) the electrical model at the
tissue, which consists, the system of non-linear parabolic reaction-diffusion
equations of the degenerate type, 2) the cell level model to describe the flow
of ionic currents via cellular membrane consist, system of ordinary
differential equations (ODEs), 3) deformation of cardiac tissue is modeled by
the quasi-static finite elasticity model, 4)intracellular calcium dynamics and
the cross bridges bindings are described by the active tension model which
consists of a system of non-linear ODEs. Existence and uniqueness for the
coupled electro-mechanical model is given in [ABQRB15, BMST18].

Cardiac arrhythmia involves the abnormality in the pattern of the heartbeat,
if the electrical impulse fails to start from the SA node or if there is some
abnormality in the impulse initiated from the SA node. Myocardial ischemia
takes place when there is abnormality in the blood flow to heart and in oxygen
supply to the heart. It is one of the main causes of sudden death. Cardiac
arrhythmia and heart attack can also occur due to the myocardial ischemia. Due
to this myocardial ischemia metabolism, electrophysiological and mechanical
changes appear which results in the reduction of action potential duration
(APD), action-potential upstroke, resting-potential shift, reduction in
conduction velocity (CV), active tension, stretch along the fiber and the
stretch rate in the ischemic cells or tissue [2, 3, 4].

Various experimental and numerical simulation work have been carried out to
analyze the electrophysiological and metabolic properties of the healthy and
ischemic cells and tissue [DMZ+16, JW89, PC87, TRET07, CFPS07, RHR+08].
Mathematical and computational studies have been focused on investigating
various physiological effects in order to understand the relationship between
the electrophysiological and metabolic parameters. It can be analyzed from the
past studies that these variations are basically due to, a) Hyperkalemia which
affects the cardiac cell resting potential due to the increase in the
concentration of extracellular potassium [SSCF90, PWM+10], b) Hypoxia defined
as the reduced oxygen supply, it reduces the cell metabolism and ADP to ATP
concentration ratio in the cell get changed. Successively, this change affects
the opening of the particular ATP-dependent potassium channels [WL94, WVL92].
This ATP-dependent potassium channels is also modulated by the mechanical
environment.

There are several research papers in which they have studied about ischemia in
the animals. [FBS+85, Car99, FKF+91, Cor94, MW07, PKZ+11, WTJC+86, SSCF90].
Since it is difficult to get the human data, and hence it is challenging to
extend the properties from animals to human. But it can be achieved via
computational modeling. Even though many human models have been built and
analyzed via healthy cell data their appropriateness for the ischemia
computation is not known. So, it is essential to examine the effect of varying
ischemic parameters in the ischemic region. In [AP96, STO+00, TSO+00], studies
have been done for the global ischemia in human under different ischemic
conditions. In [PKP19], the authors has discussed the effect of local cardiac
ischemia on the electrical activity of the ischemic region and the neighboring
healthy region in a human cardiac tissue.

For the electro-mechanical coupling models, the bio-electrical activity
experiences three main feedback from the mechanical deformation:

  1. 1.

conductivity feedback: the influence of the deformation gradient on the
conductivity coefficients of the electric current flow model;

  2. 2.

convection feedback: the influence of deformation gradient and deformation
rate on the electric current flow model;

  3. 3.

ionic feedback: the influence of stretch-activated membrane channels on the
ionic current.

In [CFPS16] authors considered these three cases together and study their
effects in a strongly coupled anisotropic cardiac electro-mechanical model. X.
Jie et al. [JGT10] used a electromechanical model for the ventricle of rabbit
to analyze the arrhythmia originates from the ischemia induced
electrophysiological and the mechanical changes. As per our literature survey
no one has considered the electro-mechanical model of the human cardiac tissue
to analyze the local ischemia effects on the electro-mechanical activity.

In this study, we will discuss the influence of local ischemic region on
electrical and mechanical activity in the deforming human heart. For the
electrical activity at cell level, we will use the ten Tusscher human
ventricular membrane model [TTNNP04]. We will consider the above three cases
(i), (ii), (iii) and study the effect in deformed Monodomain model. We will
also show the effect of ischemia on this model. We will discuss the two types
of ischemia namely Hyperkalemia and Hypoxia. We will discuss the influence of
these two types of local ischemia, by changing the corresponding parameters in
the 2D tissue level model, on the electrical and mechanical activity in terms
of the action potential (AP), activation time (AT), repolarization time (RT),
action potential duration (APD) and intracellular calcium ion concentration
$[Ca^{+2}]_{i}$, active tension, ($T_{A}$), stretch ($\lambda$), stretch rate
($\frac{d\lambda}{dt}$). We will also see the effect of the spread of ischemic
zone in both the ischemic subregion and the healthy region on the electrical
and mechanical properties.

In the next section, we will describe the cardiac electro-mechanical model and
also discuss about the ischemic parameters. We will also present the modeling
of local ischemic zone. In the next section, we will present the fully
discrete system using finite elements in space and backward Euler method in
time. In section 4, we will discuss the numerical results.

##  2 Electro-mechanical models of cardiac tissue

Cardiac electro-mechanical model is a combination of electrical and mechanical
models. In this section, we will describe the mechanical and electrical models
of cardiac tissue.

###  2.1 Cardiac tissue mechanical model

Let $\hat{\Omega}$ is the undeformed cardiac domain and $\Omega(t)$ is the
deformed cardiac domain at time ${}^{\prime}t^{\prime}$ with $X=(X_{1},X_{2})$
and $x=(x_{1},x_{2})$ are the spatial coordinates respectively.

Define the deformation map, $\psi_{t}(X)=\psi(X,t)=x$ from $\hat{\Omega}$ to
$\Omega(t)$ and $u(X,t)=x-X$ as the displacement vector.

In this work, cardiac tissue is considered as a nonlinear elastic material.
The deformation gradient tensor$F(X,t)=\\{F_{i,j}\\}=\\{\frac{\partial
x_{i}}{\partial X_{j}},i,j=1,2,3\\}$,
$J(X,t)=detF(X,t)$
Cauchy-Green deformation tensor $C=F^{T}F$,
Lagrange-Green strain tensor $\epsilon=\frac{1}{2}(C-1)$.
Also the deformed body satisfy the following steady state force equilibrium
equation described as

$\displaystyle div(FS(u,X))=0,X\in\hat{\Omega},$ (1)

where $S$ is the second Piola-Kirchoff stress tensor. Prescribed displacement
is imposed on the Dirichlet boundary
$x(X,t)=\hat{x}(X),X\in\hat{\partial\Omega_{d}}$ and Neumann boundary with no
traction force $n^{t}FS(u(X,t),X)=0,X\in\hat{\partial\Omega_{n}}$.

The second Piola-Kirchoff stress tensor $S$, as taken in the studies [HMTK98,
KBK+03, VM00, CFPS16], is the combination of three components namely,

$\displaystyle S=S^{A}+S^{P}+S^{V},$

where active component $S^{A}$ generated biochemically, passive elastic
component $S^{P}$ and volumetric component $S^{V}$.

Next, $S^{P,V}$ is defined as follows

$\displaystyle S_{ij}^{P,V}=\frac{1}{2}\bigg{(}\frac{\partial
W^{P,V}}{\partial\epsilon_{ij}}+\frac{\partial
W^{P,V}}{\partial\epsilon_{ji}}\bigg{)},i,j=1,2,$ (2)

where $W^{P,V}$ are the passive and volumetric strain energy functions,
variety of these has been proposed in [CHM01, GCM95b, HO09, GK10b, SNCH00,
RNH05, SNYH06, UMM00] and $\epsilon$ is the Green-Lagrange strain.

In this work, cardiac tissue is modeled as an orthotropic hyperelastic
material. The passive strain energy function for a hyperelastic material is
given as in [EPPH13]:

$\displaystyle W^{P}=\frac{1}{2}(e^{Q}-1),$ $\displaystyle
Q=\sum_{i=l,t}b_{i}(I_{4i}-1)^{2}+b_{lt}I_{8lt}^{2},$

where $I_{4l}=\hat{b}_{l}^{T}C\hat{b}_{l}$,
$I_{4t}=\hat{b}_{t}^{T}C\hat{b}_{t}$ and
$I_{8lt}=\hat{b}_{l}^{T}C\hat{b}_{t}$,

and volumetric strain energy function for a hyperelastic material is given as

$\displaystyle W^{V}=K(J-1)^{2},$

where $K$ is a positive bulk modulus.

Now, the active component of second Piola-Kirchoff stress tensor $S^{A}$ is
defined in the form of active tension originated along the myocytes.

An isometric contraction of the cardiac tissue, generates an active tension
without changing length of the cardiac tissue. Mechanical model of active
tension is modeled by the myofilaments dynamics activated by calcium. We
consider, as suggested in [CFPS16], that the active force moves only in the
fiber direction. Therefore, active Cauchy stress is given as

$\displaystyle\sigma^{A}(x,t)=J^{-1}T_{A}a_{l}(x)\otimes a_{l}(x),$ (3)

where $a_{l}=\frac{F\hat{a}_{l}}{|F\hat{a}_{l}|}$ is a unit vector parallel to
the local fiber direction and $T_{A}$ is the active tension in the deformed
domain.

The active component of second Piola-Kirchoff stress tensor $S^{A}$ is defined
as

$\displaystyle S^{A}=JF^{-1}\sigma^{A}F^{-T},$ (4)

and the stretch rate along the fiber is defined as

$\displaystyle\lambda=\sqrt{\hat{a}_{l}^{T}C\hat{a}_{l}}.$ (5)

The active tension is dependent on calcium dynamics, stretch and stretch rate
i.e. $T_{A}=T_{A}(Ca_{i},\lambda,\frac{d\lambda}{dt})$, as discussed in
[LNA+12b] is given by the system of ODEs:

$\displaystyle\frac{d(trpn)}{dt}=k_{trpn}\bigg{(}\bigg{(}\frac{Ca_{i}}{Ca_{50}(1+\beta(\lambda-1))}\bigg{)}^{n_{trpn}}\bigg{)},$
$\displaystyle\frac{dXb}{dt}=k_{Xb}\bigg{(}(trpn_{50})(trpn)^{n_{Xb}}(1-Xb)-\frac{1}{(trpn_{50})(trpn)^{n_{Xb}}}Xb\bigg{)},$
$\displaystyle\frac{dq_{i}}{dt}=A_{i}\frac{d\lambda}{dt}-\alpha_{i}q_{i},i=1,2,$
$\displaystyle T_{A}=g(q)h(\lambda)Xb,q=q_{1}+q_{2}.$ (6)

where, function $g(q)$ is the effect on tension is defined as :

$g(q)=\left\\{\begin{array}[]{ll}\frac{aq+1}{1-q}&q\leq 0\\\
\frac{1+(a+2)Q}{1+q}&q>0\\\ \end{array}\right.$

.

For more details refer to [LNA+12b].

###  2.2 Cardiac tissue electrical model on a deforming domain: Monodomain
model

We assume that the cardiac tissue is insulated. The influence of the cardiac
tissue deformation on the Monodomain model in the strong coupling framework is
due to three different mechano-electrical feedback: a) presence of deformation
gradient $F$ in the conductivity coefficients structure, b) presence of
deformation gradient $F$ and the deformation rate in the conductive term, and
c) presence of stretch $\lambda$ in the $I_{ion}$.

Monodomain model on the deformed domain $\Omega(t)$ is defined as follows:

$\displaystyle\frac{\partial v}{\partial t}-div(D(x)\nabla
v)+I_{ion}(v,w,c,\lambda)=I_{app},x\in\Omega(t),0\leq t\leq T$ (7)
$\displaystyle\frac{\partial w}{\partial t}-g(v,w)=0,x\in\Omega(t),0\leq t\leq
T$ (8) $\displaystyle\frac{\partial c}{\partial
t}-f(v,w,c)=0,x\in\Omega(t),0\leq t\leq T$ (9) $\displaystyle
v(x,0)=v_{0}(x,0),\hskip 14.22636ptw(x,0)=w_{0}(x,0),\hskip
14.22636ptc(x,0)=c_{0}(x,0),x\in\Omega$ (10) $\displaystyle n^{T}D(x)\nabla
v=0,x\in\partial\Omega,0\leq t\leq T$ (11)

where,

$\displaystyle
D(x)={\sigma_{l}}a_{l}(x){a_{l}}^{T}(x)+{\sigma_{t}}a_{t}(x){a_{t}}^{T}(x),$

where, $a_{l}(x)$ is a unit vector along the local fiber direction and
$a_{t}(x)$ is a unit vector transverse to the fiber direction and they both
are orthogonal to each other. $\sigma_{l}$ and $\sigma_{t}$ are the
conductivity coefficients in the corresponding directions.

$\displaystyle
D(x)={\sigma_{t}}I+({\sigma_{l}}-{\sigma_{t}})a_{l}(x){a_{l}}^{T}(x).$

Let $\hat{v}(X,t),\hat{w}(X,t),\hat{c}(X,t)$ and $\hat{D}(X,t)$ are the action
potential, gating, concentration variables on the undeformed domain
$\hat{\Omega}$. Then

$\displaystyle\frac{\partial\hat{v}}{\partial t}=\frac{\partial v}{\partial
t}+\nabla v.\frac{\partial\psi_{t}}{\partial t},(\text{convective term})$
$\displaystyle\nabla v(x,t)=F^{-T}Grad\hat{v}(X,t),$ $\displaystyle dx=JdX,$

Thus the monodomain system on the reference domain $\hat{\Omega}$ is given as
follows:

$\displaystyle J\bigg{(}\frac{\partial\hat{v}}{\partial
t}-F^{-T}\nabla\hat{v}.\frac{d\psi_{t}}{dt}\bigg{)}-div(JF^{-1}DF^{-T}\nabla\hat{v})+JI_{ion}(\hat{v},\hat{w},\hat{c},\lambda)=JI_{app},\hskip
20.00003ptX\in\hat{\Omega},$

coupled with the system of ODEs for the gating and concentration variables
$w(x,t)$ and $c(x,t)$ respectively, given by

$\displaystyle\frac{\partial w}{\partial t}-g(v,w)=0,$ $\displaystyle
X\in\hat{\Omega},T$ (12) $\displaystyle\frac{\partial c}{\partial
t}-f(v,w,c)=0,$ $\displaystyle X\in\hat{\Omega}.$ (13)

Initial conditions:

$\displaystyle\hat{v}(X,0)=\hat{v}_{0}(X),\hskip 5.69054ptX\in\hat{\Omega},$
$\displaystyle w(X,0)=w_{0}(X),\hskip 14.22636ptc(X,0)=c_{0}(X),\hskip
2.84526ptX\in\hat{\Omega}.$ (14)

Boundary condition:

$\displaystyle n^{T}F^{-1}D(x)F^{-T}\nabla\hat{v}=0,\hskip
14.22636pt\text{on}\hskip 14.22636pt\partial\hat{\Omega}\times(0,T).$ (15)

Now, we compute the conductivity tensor $\hat{D}(X,t)$ on the reference domain
is given as:

$\displaystyle\hat{D}(X,t)=F^{-1}D(x(X,t))F^{-T}(X,t),\hskip
22.76219ptX\in\hat{\Omega}$

We know the conductivity tensor in the deformed domain $\Omega(t)$ is defined
as

$\displaystyle
D(x)={\sigma_{t}}I+({\sigma_{l}}-{\sigma_{t}})a_{l}(x){a_{l}}^{T}(x).$

Now, let $\hat{a}_{l}(X)$ is a unit vector parallel to the local fiber
direction in the reference domain, then in the deformed domain the unit vector
$a_{l}(x)$ along the local fiber direction is obtained by

$\displaystyle a_{l}$ $\displaystyle=\frac{F\hat{a}_{l}}{\|F\hat{a}_{l}\|},$
$\displaystyle=\frac{F\hat{a}_{l}}{\sqrt{\hat{a}_{l}^{T}C\hat{a}_{l}}}$

Thus,

$\displaystyle
a_{l}a_{l}^{T}=\frac{F\hat{a}_{l}\hat{a}_{l}^{T}F^{T}}{{\hat{a}_{l}^{T}C\hat{a}_{l}}}$

###  2.3 Ionic model

$I_{ion}(v,w,c,\lambda)$, $g(v,w)$ and $f(v,w,c)$ are defined by the cell
level model. In this work we are using the human ventricular cell level model,
ten Tusscher model [TTNNP04] (TT06). Dynamics of the ionic currents is modeled
by the 17 ODEs. The ionic current flow $I_{sac}$ due to the stretch activated
channels and $K_{ATP}$ current is also taken into account. Thus the total
ionic current is given by

$\displaystyle
I_{ion}=I_{Na}+I_{K1}+I_{to}+I_{Kr}+I_{Ks}+I_{CaL}+I_{NaCa}+I_{NaK}+$
$\displaystyle\hskip
17.07182ptI_{pCa}+I_{pK}+I_{bCa}+I_{bNa}+I_{KATP}+I_{sac},$ (16)

where, the fast sodium $(I_{Na})$, L-type calcium$(I_{CaL})$, transient
outward$(I_{to})$ , rapid delayed rectifier$(I_{Kr})$ and slow delayed
rectifier $(I_{Ks})$, and inward rectifier currents $(I_{K1})$, $I_{NaCa}$ is
$Na/Ca^{2+}$ exchange current, $I_{NaK}$ is $Na/K^{+}$ pump current, $I_{pCa}$
and $I_{pK}$ are plateau $Ca^{2+}$ and $K^{+}$ currents, $I_{bCa}$ and
$I_{bK}$ are background $Ca^{2+}$ and $K^{+}$ currents and $I_{KATP}$ is
the$K_{ATP}$ current, all are modeled as in [TTNNP04], $I_{KATP}$ which
accounts for Hypoxia is defined as in [AP96]:

$\displaystyle
I_{KATP}=G_{KATP}f_{ATP}\bigg{(}\frac{[K^{+}]_{o}}{5.4}\bigg{)}^{0.3}\frac{1}{40+3.5e^{0.025v}}(v-E_{k}),$
(17)

where, $G_{KATP}$ is the conductance and $E_{k}$ is the resting potential of
potassium.

$I_{sac}$, the stretch activated current [AHZ13b] is given as follows :

$\displaystyle I_{sac}=I_{sac,n}+I_{Ko},$ $\displaystyle
I_{sac,n}=I_{sac,Na}+I_{sac,K},$

where, $I_{sac,n}$ is the non-specific current defined as follows:

$\displaystyle
I_{sac,Na}=G_{sac}\gamma_{slsac}(v-v_{Na})\bigg{(}-\frac{v_{R}+85}{v_{R}-65}\bigg{)},$
$\displaystyle I_{sac,K}=G_{sac}\gamma_{slsac}(v-v_{K}).$

The specific $K^{+}$ stretch-dependent current is given by

$\displaystyle
I_{Ko}=G_{Ko}\frac{\gamma_{sl,Ko}}{1+exp(-(10+v)/45)}(v-v_{K}),$

where, $\gamma_{sl,Ko}=3(\lambda-1)+0.7$.

The two types of ischemia, Hyperkalemia and Hypoxia will be incorporated into
the model by varying the parameters corresponding to each of the two types of
ischemia.

Human Cardiac tissue is considered as a square $ABCD$, and the ischemic region
is taken as a small square $A_{1}B_{1}C_{1}D_{1}$. In this study, local
ischemia is modeled by varying the ischemic parameter value in the sub-domain
$A_{1}B_{1}C_{1}D_{1}$ only and using the normal value of the parameter in the
remaining part of the cardiac tissue $ABCD$. Also considering the ischemia as
mild, moderate and severe by taking the $[K^{+}]_{o}$ parameter value in the
corresponding range in the domain $A_{1}B_{1}C_{1}D_{1}$ only. Then, we
compare the electrophysiological properties of the ischemic and non-ischemic
region points say $(x,y)$ and $(x^{\prime},y^{\prime})$ respectively (see
Fig.(1)).

Figure 1: : Cardiac tissue $ABCD$ with single ischemic subregion $A1B1C1D1$,
$(x,y)$ and $(x^{\prime},y^{\prime})$ are the points inside and outside the
ischemic subregion.

##  3 Weak Formulation

First of all we define the strong form the model. Monodomain model on the
deformed domain $\Omega(t)$ is defined as follows:

(elasticity equation) $\displaystyle-\nabla.(\nabla u\sigma(x,\gamma))=f,$
$\displaystyle x\in\Omega(t),a.e.t\in(0,T)$ (18) (Tissue level model)
$\displaystyle\frac{\partial v}{\partial t}-div(D(x)\nabla
v)+I_{ion}(v,w,c,\lambda)=I_{app},$ $\displaystyle x\in\Omega(t),0\leq t\leq
T$ (19) (cell level model) $\displaystyle\frac{\partial w}{\partial
t}=G(v,w),$ $\displaystyle x\in\Omega(t),0\leq t\leq T$ (20)
$\displaystyle\frac{\partial c}{\partial t}=h(v,w,c),$ $\displaystyle
x\in\Omega(t),0\leq t\leq T$ (21) $\displaystyle\frac{\partial\gamma}{\partial
t}=S(\gamma,w),$ $\displaystyle x\in\Omega(t),0\leq t\leq T$ (22)
$\displaystyle v(x,0)=v_{0}(x,0),\hskip 5.69054ptw(x,0)=w_{0}(x,0),\hskip
5.69054ptc(x,0)=c_{0}(x,0),$ $\displaystyle x\in\Omega(t)$ (23) $\displaystyle
n^{T}D(x)\nabla v=0,$ $\displaystyle x\in\partial\Omega,0\leq t\leq T.$ (24)

Now, we will define the general form of the ionic model so that the weak
formulation of the above model equations can be written.

The general form of ionic current
$I_{ion}:\mathbb{R}\times\mathbb{R}^{d}\times(0,\infty)^{n}\to\mathbb{R}$ is
defined as follows:

$\displaystyle I_{ion}(v,w,c):=\sum_{i=1}^{n}Q_{i}(v,w,logc_{i})+R(v,w,c),$
(25)

where,

$\displaystyle Q_{i}\in
C^{1}(\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}),$ $\displaystyle
0<\underline{\rm
H}(w)\leq\frac{\partial}{\partial\zeta}Q_{i}(v,w,\zeta)\leq\overline{\rm
H}(w),$ $\displaystyle|\frac{\partial}{\partial v}Q_{i}(v,w,0)|\leq L(w),$
(26)

$\underline{\rm H},\overline{\rm H},L\in C^{0}(\mathbb{R}^{d},\mathbb{R}_{+})$
and

$\displaystyle R\in
C^{0}(\mathbb{R}\times\mathbb{R}^{d}\times(0,\infty)^{n})\cap
Lip(\mathbb{R}\times[0,1]^{d}\times(0,+\infty)^{m}).$ (27)

The following system of ODEs describing the dynamics of gating variables,

$\displaystyle\frac{\partial w_{i}}{\partial t}=G_{i}(v,w_{i}),\hskip
14.22636pti=1,2,...,d.$ (28)

where, $G_{i}$ has the particular form as

$\displaystyle G_{i}(v,w):=\alpha_{i}(v)(1-w_{i})-\beta_{i}(v)w_{i},\hskip
14.22636pti=1,2,...,d,$

and $\alpha_{i}$ and $\beta_{i}$ are positive rational exponential functions
in $v$.

Assumption, $\forall\hskip 5.69054pti=1,2,...,d$,

$\displaystyle G_{i}:\mathbb{R}^{2}\to\mathbb{R}\hskip 14.22636pt\textbf{is
locally Lipschitz continuous},$ $\displaystyle G_{i}(v,0)\geq 0\hskip
14.22636pt\forall v\in\mathbb{R},$ $\displaystyle G_{i}(v,1)\leq 0\hskip
14.22636pt\forall v\in\mathbb{R}.$ (29)

ODEs describing the dynamics of concentration variables is given by

$\displaystyle\frac{\partial c_{j}}{\partial
t}=h_{j}(v,w,c_{j}):=Q_{j}(v,w,c_{j})+R_{j}(v,w,c_{j}),\hskip
14.22636ptj=1,2,...,n,$ (30)

where ,

$\displaystyle R_{j}\in
C^{0}(\mathbb{R}\times\mathbb{R}^{d}\times(0,\infty)^{n})\cap
Lip(\mathbb{R}\times[0,1]^{k}\times(0,+\infty)^{m}),\hskip
14.22636ptj=1,2,...,n.$ (31)

The function $S$ is given by
$S(\gamma,w)=\beta(\sum_{j=1}^{k}\eta_{j}w_{j}-\eta_{0}\gamma)$, where $\beta$
and $\eta_{j}$ are the positive pathological parameters. Now, we will define
the weak formulation of the above defined model.

###  3.1 Weak formulation

Definition: A weak solution of the problem (18)-(24) is a set
$(u,v,w,c,\gamma)$ such that $u\in L^{2}(0,T;H^{1}(\Omega)^{3})$, $v\in
L^{2}(0,T;H^{1}(\Omega))$, $v_{t}\in L^{2}(0,T;(H^{1}(\Omega))^{\prime})$,
$w\in C(0,T;L^{2}(\Omega)^{m})$, $c\in C(0,T;L^{2}(\Omega)^{n})$, $z\in
C(0,T;L^{2}(\Omega))$, and it satisfy the following:

$\forall\phi_{1}\in H^{1}(\Omega)^{3},\phi_{2}\in H^{1}(\Omega),\phi_{3}\in
L^{2}(\Omega)^{m},\phi_{4}\in L^{2}(\Omega)^{n},\phi_{5}\in L^{2}(\Omega)$
a.e. $t\in(0,T)$ such that:

$\displaystyle\int_{\Omega}\nabla
u\sigma(x,\gamma):\nabla\phi_{1}=\int_{\Omega}f\phi_{1}dx+\int_{\partial\Omega}u\phi_{1}dx,$
(32) $\displaystyle\int_{\Omega}\frac{\partial v}{\partial
t}\phi_{2}+\int_{\Omega}D(x,\nabla u)\nabla
v\nabla\phi_{2}+\int_{\Omega}I_{ion}(v,w,c)\phi_{2}=\int_{\Omega}I_{app}\phi_{2},$
(33) $\displaystyle\int_{\Omega}\frac{\partial w}{\partial
t}\phi_{4}=\int_{\Omega}G(v,w)\phi_{3},$ (34)
$\displaystyle\int_{\Omega}\frac{\partial c}{\partial
t}\phi_{4}=\int_{\Omega}h(v,w,c)\phi_{4},$ (35)
$\displaystyle\int_{\Omega}\frac{\partial\gamma}{\partial
t}\phi_{5}=\int_{\Omega}S(\gamma,w)\phi_{5}.$ (36)

Note: $H^{1}(\varOmega)=\\{u:u\in L^{2}(\varOmega),\frac{\partial u}{\partial
x_{i}}\in L^{2}(\varOmega),i=1,2\\}$,
Note: $L^{2}(\Omega)=\\{u:\int_{\varOmega}u^{2}<\infty\\}$.

##  4 Numerical Scheme

The above electro-mechanical model is solved via finite element method for
space discretization and implicit-explicit (IE) Euler method for the time
discretization.

Cardiac tissue is discretized with rectangular grid $T_{h_{e}}$ (grid size
$h_{e}$) for the electrical monodomain model (2.2) and $T_{h_{m}}$ (grid size
$h_{m}$) for the mechanical model (1), where $T_{h_{e}}$ is a refinement of
$T_{h_{m}}$. We will then use quadrilateral finite elements in space for both
the electrical and mechanical models. For time discretization of both
electrical and mechanical models we will use equal time steps $dt$.

We will solve this coupled electro-mechanical model in two steps:

1) Time discretization of the ODEs to find $w$, $c$ then solve the mechanical
model to find the deformed system.

2) Solve the electrical monodomain model to compute $v$.

1) a) Compute $w^{n+1},c^{n+1}$, for a given $v^{n},w^{n},c^{n}$, by solving
(8) and (9) via Implicit- explicit Euler method:

$\displaystyle w^{n+1}=w^{n}+dtG(v^{n},w^{n+1}),$ (37) $\displaystyle
c^{n+1}=c^{n}+dtf(v^{n},w^{n+1},c^{n});$ (38)

b) Now, use the obtained $w^{n+1},c^{n+1}$ to solve the mechanical model (1)-
(2.1) and compute the new deformed coordinates $x^{n+1}$ in terms of the
deformation gradient tensor $F^{n+1}$.

2)Solve the electrical monodomain model using the linear finite elements in
space and the IE Euler method in time, for a given $w^{n+1},c^{n+1}$ and
$F^{n+1}$ and compute the action potential $v^{n+1}$.

Define

$\displaystyle V_{h}=\\{v\in H^{1}(\Omega):v\hskip 0.56905pt\text{is
continuous in}\hskip 0.56905pt\Omega:{v}_{\mid E}\in Q_{1}(E),\forall E\in
T_{h_{e}}\\},$ $\displaystyle W_{h}=\\{u\in H^{1}(\Omega)^{3}:u\hskip
0.56905pt\text{is continuous in}\hskip 0.56905pt\Omega:{u}_{\mid E}\in
Q_{1}(E),\forall E\in T_{h_{m}}\\}.$

After solving the ODEs, solve the mechanical equation using the FEM
(mechanical discretization, $T_{h_{m}}$) and calculate the deformation
gradient at the next level.

Choose $\\{\xi_{i}\\}_{i=1}^{N}$ and $\\{\psi_{i}\\}_{i=1}^{N}$ as the basis
functions in the space $V_{h}$ and $W_{h}$ respectively. Then we can write the
following:

$\displaystyle u_{n}=\sum_{i=0}^{n}a_{u,i}\boldsymbol{\psi}_{n},$
$\displaystyle v_{n}=\sum_{i=0}^{n}a_{v,i}\xi_{n},$

The mechanical equation using the finite element method (mechanical
discretization, $T_{h_{m}}$) becomes as follows:

$\displaystyle\int_{T_{h_{m}}}S\nabla\Big{(}\sum_{i=1}^{N}u_{i,t}\psi_{i}(x)\Big{)}:\nabla\psi_{j}=0.$

and then solve it using the Newton’s method.

Also, solve the monodomain equation using the finite element method
(electrical discretization, $T_{h_{e}}$) in space, the weak form of the
monodomain equation becomes as follows:

$\displaystyle\int_{T_{h_{e}}}\Big{(}\sum_{i=1}^{N}v_{i,t}\xi_{i}(x)\Big{)}\xi_{j}+\int_{T_{h_{e}}}D(x,\nabla
u)\nabla\Big{(}\sum_{i=1}^{N}v_{i}\xi_{i}(x)\Big{)}\nabla\xi_{j}=\int_{T_{h_{e}}}I_{ion}^{h}\xi_{j},$

$\displaystyle\sum_{i=0}^{n}a^{\prime}_{v,i}\int_{\Omega}\xi_{i}\xi_{n}+\sum_{i=0}^{n}a_{v,i}\int_{\Omega}D(x,\nabla
u_{i,n})\nabla\xi_{i}\nabla\xi_{n}+\int_{\Omega}I_{ion}(a_{v,i},a_{w,i},a_{c,i})\xi_{n}=\int_{\Omega}I_{app}\xi_{n}.$

This is an ODE in time which is now solved using the IE Euler method.

##  5 Results and Discussion

We will present the results computed from the numerical simulation performed
via Matlab Ra2015 library. Cardiac tissue is taken as a two dimensional slab
of size $[0,1]^{2}$. The conductivity tensor in two-dimension we are using is
as follows:

$\displaystyle D=\begin{bmatrix}\sigma_{l}&0&\\\ 0&\sigma_{t}\end{bmatrix}$

We need to choose an appropriate grid system so that the numerical solution is
computed up-to acceptable accuracy. Such a goal is achieved through grid
validation test. Prior to the grid validation we will briefly discuss the
discontinuity treatment of $I_{ion}$ expression due to ischemic subregions
inside a healthy cardiac tissue. While Hyperkalemic ischemic zones results in
the discontinuity of $[K^{+}]_{o}$ parameter at the interface of healthy and
ischemic parts of a cardiac tissue the Hypoxic ischemic zones results in the
discontinuity of $f_{ATP}$ parameter at the interface. Here these
discontinuities are treated to a desired order of accuracy by a simple
regularization effect through appropriate interpolating polynomials. These
local interpolating polynomials of high order accuracy are constructed
following the procedure discussed by Alsoudani and Bogle in [AB14] using
refined localized data around the discontinuity. Different grid systems
consisting of (a) 441 (b) 625 (c) 841 (d) 1089 and (e) 2601 degrees of freedom
(dofs) have been considered. We compare the action potential obtained using
these grid systems (as in [PKP19]). In all these cases only a marginal
variation (less than 2–3% for 441 and 625 grids and 0.5% for others) is found.

The parameters value in the model is taken from the reference [CFPS16].
Applied stimulus $I_{app}$ is taken as 200 $mA/cm^{3}$ to start the
depolarization process [CFPS16]. The initial condition for the action
potential and all the gating variables of the TT06 ionic model are taken as
the corresponding resting values while the boundary of the domain is assumed
to be insulated.
Activation time (AT) is defined as the unique time $\tau_{a}(x)$ during the
upstroke phase of the action potential when $v(x,\tau_{a}(x))=-50$.
Repolarization time (RT) is defined as the unique time $\tau_{r}(x)$ during
the repolarization phase of the action potential when
$v(x,\tau_{r}(x))=0.9v_{R}$. Action potential duration (APD) is defined as
$\tau_{r}(x)-\tau_{a}(x)$.

First of all, we will present the results for the simulation of the four
situations S1, S2, S3 and S4 in the cardiac tissue which are, (S1) without the
mechanical feedback, (S2) consider the mechanical feedback into the diffusion
term of the Monodomain model but without $I_{sac}$ current and the convective
term, (S3) Take the mechanical feedback in the diffusion term and the
$I_{sac}$ current, and, (S4) Take the mechanical feedback in the diffusion
term with $I_{sac}$ current and the convective term. We will also examine the
impact on the temporal variation of the action potential $(v)$, intracellular
calcium concentration ($[Ca^{+2}]_{i}$), active tension $(T_{A})$, stretch
along the fiber $(\lambda)$ and stretch rate $(\frac{d\lambda}{dt})$ at the
points (describe later) in the cardiac tissue with the change of Hyperkalemia
and Hypoxia both. We will also examine the increased ischemic region size
effect onto the non-ischemic region in terms of its electro-mechanical
properties.

###  5.1 (Case 1) Cardiac tissue without ischemic subregion

####  5.1.1 Effect of S2, S3 and S4 situations on electrical and mechanical
waveforms

We presented the action potential $(v)$, intracellular calcium concentration
($[Ca^{+2}]_{i}$), active tension $(T)$, stretch along the fiber $(\lambda)$
and stretch rate $(\frac{d\lambda}{dt})$ at the points (0.1875, 0.1875) (M1)
and (0.5, 0.5) (M2) of the human cardiac tissue (see Fig. (2), in the
subsequent simulations M1 will be in the ischemic zone). The simulations are
plotted in Fig (2). From Fig. (2(a)) and (2(b)), it can be examined that
action potential$(v)$ for the situations S1 and S2 i.e. without mechanical
feedback and the addition of mechanical feedback without $I_{sac}$
respectively, had a little difference. This implies that the addition of
mechanical feedback had a little effect on the cardiac electrical activity in
terms of ventricular action potential profile and hence on QRS complex of ECG.

Further, as we add the current generated due to the stretch activated channels
i.e. $I_{sac}$, the plateau phase, repolarization phase and hence APD of
action potential gets affected. Fast upstroke, early repolarization and hence
$10\%$ decrease in APD is noticed. Next, with the addition of convective term
i.e. S4, negligible difference in action potential profile in comparison to S2
is noticed.

Since, S1 is without mechanical case, therefore, we will consider the
mechanical parameters for the situations S2, S3 and S4. Now, from Fig. (2(c))-
(2(j)), an alteration in the waveforms of the mechanical parameters is noticed
with the addition of $I_{sac}$ at the points M1 and M2 of the human cardiac
tissue. With the addition of the $I_{sac}$, waveform of $[Ca^{+2}]_{i}$ at the
points M1 and M2 get affected during the resting phase and hence the
contractile force and the contractility. While addition of convective term
does not affect the contractility.

Stretch along the fiber $(\lambda)$ and hence the stretch rate
$\frac{d\lambda}{dt}$ changes with the addition of $I_{sac}$. From Fig. (2(g))
and (2(h)), it is to be analyzed that during the systolic contraction there is
shortening of fiber i.e. $\lambda<1$ and after this, myocyte stretch
$\lambda>1$. There is almost $1\%$ change in the length of myocytes at point
M1 and $3\%$ change in the length of myocytes at point M2 relative to the
resting length.

Thus, it is concluded that ventricular contraction without $I_{sac}$ had
little effect on the action potential and hence on ECG. With the addition of
$I_{sac}$ i.e. stretch regulated electrical activities alter the AP phase as
repolarization phase and APD and hence alter the QRS complex and QT interval
of ECG. The mechanical stretch induced current $I_{sac}$ also changes the
myocardial resting $[Ca^{+2}]_{i}$ which is related to the changes in
contractile force and hence contractility. This is consistent with the
experimental studies (). Fiber length also changes due to the stretch
activated channels, which is responsible for the contraction and expansion of
the heart. In short, it is to be verified that the stretch activated channels
plays an important role in the electro-mechanical activity of HCT or in the
functioning of the heart.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 2: : $v$ (a), $[Ca^{+2}]_{i}$ (b), $T_{A}$ (c), $\lambda$ (d),
$\frac{d\lambda}{dt}(e)$ for the cases, without MEF, without $I_{sac}$, with
$I_{sac}$ and with $I_{sac}+$ CONV at M1(column1) and M2(column2).

###  5.2 (Case 2) Cardiac tissue with ischemic subregion

We consider the cardiac tissue of size $[0,1]^{2}$ with ischemic subregion as
$[0.1563,0.25]^{2}$. The ischemic parameters $[K^{+}]_{o}$ and $f_{ATP}$ are
varied in the subregion only. The impact of the change for these parameter
values are analyzed on the ischemic region and on the healthy neighborhood of
the ischemic region. In this case, we will examine the effect at points
$M1(0.1875,0.1875)$ and $M2(0.5,0.5)$. $M1$ lies inside the ischemic zone of
the cardiac tissue while $M2$ lies in the healthy neighborhood of this
ischemic region (see Fig. (3)). Now, we will examine the electrophysiological
and the mechanical properties of the cardiac tissue via action potential
$(v)$, intracellular calcium concentration $[Ca^{+2}]_{i}$, active tension
$T_{A}$, stretch along fiber $(\lambda)$ and stretch rate
$(\frac{d\lambda}{dt})$ at these two points.

Figure 3: : Cardiac tissue with one ischemic subregion, $M1$ is the point
inside the ischemic subregion and $M2$ is outside the ischemic subregion.

####  5.2.1 (a) Hyperkalemia (Effect of ischemia when only $[K^{+}]_{o}$
varies)

$[K^{+}]_{o}$ is taken in the range $5.4-20mM$ [SR97]. $[K^{+}]_{o}=5.4mM$ is
considered as the normal value. In Fig. (4) and (5), action potential $(v)$,
intracellular calcium concentration ($[Ca^{+2}]_{i}$), active tension $(T)$,
stretch along the fiber $(\lambda)$ and stretch rate $(\frac{d\lambda}{dt})$,
at the points $M1$ and $M2$ corresponding to S2, S3 and S4 for different
values of $[K^{+}]_{o}$ is drawn. In Fig. (4), effect of Hyperkalemia at the
point $M1$ in the ischemic region corresponding to S2, S3 and S4 can be easily
visualized. With the increase in the strength of Hyperkalemia, resting
potential (RP) increasingly reaches to the higher level of potential values
than that of the normal RP value. RP corresponding to S2, S3 and S4 for
$[K^{+}]_{o}=20mM$ turns out to be -50mV while the RP for the normal value of
$[K^{+}]_{o}$ is -85mV. Thus, there is approximately $40\%$ variation in the
RP when Hyperkalemia strength increases up to 20mM. For S2, the ischemic cell
repolarize faster than that of the normal cell and hence the APD also reduces
with the increase in the Hyperkalemia strength also the cell gets activated
earlier as $[K^{+}]_{o}$ reaches to 20mM. This faster repolarization and
reduction in APD at M1 is more evident with the addition of the ionic current
generated by the stretch activated channels i.e. $(I_{sac})$. While the
addition of convective term (i.e. S4) shows negligible change in AP. Changes
in the values of APD at $M1$ is given as follows:

| S2 | S3 | S4
---|---|---|---
APD (at M1 when $[K^{+}]_{o}=20$mM) | 272 | 265 | 264

From Fig. (5), we can see that at M2 there are small changes in all the phases
of the action potential for all the cases S2, S3, and, S4, when $[K^{+}]_{o}$
reaches to 20mM. Early depolarization, faster repolarization is visible. With
the addition of $(I_{sac})$ nearby healthy cell M2 goes faster to the resting
state while the addition of convective term shows negligible effect.

From Fig. (4(d))- (4(f))and (5(d))-(5(f)), at M1 and M2, changes in the
waveform of the concentration of the intracellular calcium ions
$[Ca^{+2}]_{i}$ for the cases S2, S3, and S4 are negligible when
$[K^{+}]_{o}\leq 12mM$. Changes in the mechanical waveform of $[Ca^{+2}]_{i}$
are visible when $[K^{+}]_{o}$ reaches 20mM and this change in $[Ca^{+2}]_{i}$
waveform leads to change in contraction force and hence contractility of the
cardiac tissue. It is to be noticed that at $M1$ and $M2$ the effect in the
waveform of $T_{A}$ for the cases S2, S3, and S4 is negligible when
$[K^{+}]_{o}\leq 12mM$ and the effect in the change in mechanical waveform of
$T_{A}$ is visible when $[K^{+}]_{o}$ reaches to 20mM. It is also clearly
visible that the influence of Hyperkalemia on the stretch along fiber
$\lambda$ and stretch rate, $\frac{d\lambda}{dt}$ i.e. the change in the
waveform of $\lambda$ and hence $\frac{d\lambda}{dt}$ is negligible at M1
corresponding to S2, S3 and S4. The change in the waveform of $\lambda$ and
hence $\frac{d\lambda}{dt}$ is more at the point M2 i.e. stretch along the
fiber and stretch rate is more with the increase in Hyperkalemia strength.
There is more than 3% change in the length of the myocytes relative to the
resting length at point M2.

AT, RT and APD contours for S2, S3 and S4 with $[K^{+}]_{o}=20mM$ is presented
in Fig. (6)- (8). From Fig. (6), it is visualized form the isochrone lines
that the addition of $I_{sac}$ does not affect the AT pattern. Also,
increasing the level of Hyperkalemia leads to a visible hastening in the
activation of human cardiac cells. An elliptic area is clearly visible which
activated earlier than the surrounding one. This indicates that ischemic
region generate a new wave.

Fig. (7) and (8) indicates that due to the addition of $I_{sac}$ cardiac cells
slightly repolarize faster and hence reduces APD. Also the cardiac cells near
to the ischemic region repolarize faster or RT of the ischemic cells or
neighboring to the ischemic cells reduces and hence these cells go to the
resting state earlier and this tendency increases as the Hyperkalemia gets
severe.

In Fig. (9) and (10), we presented the action potential contours in
deformation of the cardiac tissue with the increase in the strength of
Hyperkalemia, for S2 and S3, respectively. The change in the deformation of
cardiac domain with increasing the Hyperkalemic strength is clearly visible.

Next, the effect of increase in the size of the ischemic region from
$[0.1563,0.25]^{2}$ to $[0.0938,0.3125]^{2}$ on action potential,
intracellular calcium ions $[Ca^{+2}]_{i}$, active tension $T_{A}$, stretch
along the fiber $\lambda$ and stretch rate, $\frac{d\lambda}{dt}$ are
presented in Fig. (11) for the case $[K^{+}]_{o}=20$mM. Clearly the spread of
severe Hyperkalemic ischemic zone further raises already increased resting
potential levels, at $M1$, from -50mV to -45mV respectively. Also the growth
in ischemic size leads to faster repolarization with a reduced APD. There is
almost $9\%$ further drop in APD is noticed with a factor of five times
increase in the ischemic subregion size. From the action potential plot
corresponding to point $M2$, it is visible that there is almost $5\%$ drop in
APD for this neighboring healthy cell. Thus, it is amply clear that in the
proximity of ischemic subregion the CEA in terms of action potential and hence
the ECG of healthy cardiac tissue is affected with the severity of
Hyperkalemia. This effect in the ischemic and healthy cells will increase with
the spread of ischemic region.

It is also noticed that there is variation in the waveform of $[Ca^{+2}]_{i}$
at M1 while it is going to rest, but at M2 the waveform changes during all the
phases. The variation in the stretch along the fiber and the stretch rate is
more in the neighboring region point M2 with the expansion of the ischemic
region while the change at the point M1 inside the ischemic region is
negligible. In short, expanding the ischemic region size affect the mechanical
properties of the neighboring healthy cells more intensely.

Hence, it could be concluded that addition of $I_{sac}$ leads to ECG change
and mechanical contraction as it reduces the APD and affects the contractility
and the length of length of myocytes of healthy and ischemic parts of the
human cardiac tissue. It is also observed that the severity of Hyperkalemia
leads to reduction in APD, elevation in resting potential, affects the
contractile force and contractility and the stretch activated channels, hence
affects the QRS complex and QT interval of ECG and the mechanical contraction
or can say the electro-mechanical activity of healthy and ischemic regions of
human cardiac tissue. All these effects on the electro-mechanical activity of
a human heart gets more intense as the ischemic regions expands or more
cardiac cells becomes ischemic.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

Figure 4: : $v$ (first row), $[Ca^{+2}]_{i}$ (second row), $T_{A}$ (third
row), $\lambda$ (fourth row), $\frac{d\lambda}{dt}$ (fifth row) for the cases
without $I_{sac}$ (first column), with $I_{sac}$ (second column) and with
$I_{sac}+$ CONV (fourth column) at M1, for different values of $[K^{+}]_{o}$,
data1 ($[K^{+}]_{o}=5.4$), data2 ($[K^{+}]_{o}=12$), data3
($[K^{+}]_{o}=20$)).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

Figure 5: : $v$ (first row), $[Ca^{+2}]_{i}$ (second row), $T_{A}$ (third
row), $\lambda$ (fourth row), $\frac{d\lambda}{dt}$ (fifth row) for the cases
without $I_{sac}$ (first column), with $I_{sac}$ (second column) and with
$I_{sac}+$ CONV (fourth column) at M2, for different values of $[K^{+}]_{o}$,
data1 ($[K^{+}]_{o}=5.4$), data2 ($[K^{+}]_{o}=12$, data3 ($[K^{+}]_{o}=20$).)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6: : AT for the cases without $I_{sac}$ (first column), with $I_{sac}$
(second column) and with $I_{sac}+$ CONV (third column) for
$[K^{+}]_{o}=5.4$mM (first row) and $[K^{+}]_{o}=20$mM (second row).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: : RT for the cases without $I_{sac}$ (first column), with $I_{sac}$
(second column) and with $I_{sac}+$ CONV (third column) for
$[K^{+}]_{o}=5.4$mM (first row) and $[K^{+}]_{o}=20$mM (second row).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8: : APD for the cases without $I_{sac}$ (first column), with $I_{sac}$
(second column) and with $I_{sac}+$ CONV (third column) for
$[K^{+}]_{o}=5.4$mM (first row) and $[K^{+}]_{o}=20$mM (second row).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 9: : Action potential contours in a deforming domain for the cases
without $I_{sac}$ for different values of $[K^{+}]_{o}$ = 5.4 mM (first
column), $[K^{+}]_{o}$=12 mM (second column), $[K^{+}]_{o}$ = 20 mM (third
column), at time t=90 (first row), t=150 (second row), t=240 (third row).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 10: : Action potential contours in a deforming domain for the cases
with $I_{sac}$ for different values of $[K^{+}]_{o}$ =5.4 (first column),
$[K^{+}]_{o}$=12 (second column), $[K^{+}]_{o}$ = 20 (third column), at time
t=90 (first row), t=150 (second row), t=240 (third row).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 11: : $v$(first row), $[Ca^{+2}]_{i}$(second row), $T_{A}$(third row),
$\lambda$ (fourth row) and $\frac{d\lambda}{dt}$ (fifth row), at M1(first
column), M2 (second column), for the case with $I_{sac}$ (second column) for
$[K^{+}]_{o}=20$, with two different sizes of ischemic regions, data1
($[0.1563,0.25]^{2}$), data2 ($[0.0938,0.3125]^{2}$).

Note: As it is concluded that the addition of $I_{sac}$ plays an important
role in the electro-mechanical activity of a human cardiac tissue. So, next
results will be with $I_{sac}$ case only.

####  5.2.2 (a) Hypoxia (Effect of ischemia when only $f_{ATP}$ varies)

$f_{ATP}$ is taken in the range $0.1-0.5\%$. Influence of the $f_{ATP}$ values
(oxygen level) on the action potential and the mechanical parameters
$[Ca^{+2}]_{i}$, active tension $(T_{A})$, stretch along the fiber
$(\lambda_{l})$ and stretch rate $(\frac{d\lambda_{l}}{dt})$ at the two points
$M1$ and $M2$ (described above) of the cardiac tissue are estimated and
presented in Fig. (12). From Fig. (12(a)), it can be seen that as the oxygen
level in the ischemic region of the cardiac tissue decreases (or $f_{ATP}$
value increased), delay in the closing of L-type $Ca^{+2}$ channels is
noticed, and further rapid close of the $K^{+}$ channels takes place,
therefore the plateau phase and then the repolarization phase of the cardiac
action potential get affected. Therefore, APD also reduces with the increase
in the strength of Hypoxia. While from the Fig. (12(b)), it can be seen that
influence of the increase in $f_{ATP}$ values is negligible on the
neighborhood point M2 on the non-ischemic region of cardiac tissue.

Further from figures (12(c)) - (12(j)), it is visulaized that there is
negligible change in the waveform of the mechanical parameters, intracellular
calcium concentration ($[Ca^{+2}]_{i}$), active tension $(T_{A})$, stretch
along the fiber $(\lambda_{l})$ and stretch rate $(\frac{d\lambda_{l}}{dt})$
at the points $M1$ and $M2$.

From Fig. (13(a))-(13(g)), it is clear that the activation time remain
unaffected by $f_{ATP}$ and hence the isochrones of AT corresponding to these
cases turn out to be same in the entire domain including those in ischemic
region. The front near the ischemic region is quasi elliptic and becomes flat
after that.

RT isochrones show that as we increase the $f_{ATP}$ values in the ischemic
region i.e. as the oxygen level in the ischemic region of the cardiac tissue
decreases, the cardiac cells in the ischemic region repolarize earlier i.e.
the repolarization time of ischemic cell decreases with the increase in
$f_{ATP}$ values while the repolarization time for the non-ischemic cardiac
cells remains same. The loss of quasi-ellipticity and the folding of
repolarization time isochrones especially in the ischemic subregion depict an
accelerated repolarization wave propagation in the human cardiac tissue. The
isochrone lines for APD shows that APD in the ischemic region reduces as we
are increasing $f_{ATP}$ values.

In Fig. (14(a))-(14(i)), we presented the action potential contours in
deforming cardiac tissue with the increase in the strength of Hypoxia. We can
visualize the change in the deformation of cardiac domain with the increasing
Hypoxic strength. Thus, change in oxygen level affects the contraction and
expansion of ventricle of human heart.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 12: :$v$(first row), $[Ca^{+2}]_{i}$(second row), $T_{A}$(third row),
$\lambda$ (fourth row) and $\frac{d\lambda}{dt}$ (fifth row), at M1(first
column), M2 (second column), for different values of $f_{ATP}$. data1
($f_{ATP}=0$), data2 ($f_{ATP}=0.1\%$), data3 ($f_{ATP}=0.3\%$), data4
($f_{ATP}=0.5\%$).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 13: :AT (first column), RT (second column), APD (third column) for
different values of $f_{ATP}=0$(first row), $f_{ATP}=0.1\%$(second row),
$f_{ATP}=0.3\%$(third row), $f_{ATP}=0.5\%$(fourth row).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 14: :AP contours in a deforming domain for different values of
$f_{APT}$ =0 (first row), $f_{APT}=0.3\%$ (second row), $f_{APT}=0.5\%$ (third
row), at time t=90 (first column), t=150 (seconf column), t=240 (third
column).

Next, as we increase the size of the ischemic region from $[0.1563,0.25]^{2}$
to $[0.0938,0.3125]^{2}$, action potential at the points M1 and M2 get
influenced as shown in Fig (15). From the Fig (15(a)), we can conclude that,
at $M_{1}$ (ischemic region point), with the increase in the size of the
ischemic region there is further delay in the closing of L-type calcium
channels. Thus, a delay in plateau phase and then early repolarization of the
action potential takes place. APD also reduces with increase in the size of
the ischemic region. From the action potential plot corresponding to point
$M_{2}$ in Fig. (15(b)), a delay in plateau phase and early repolarization of
the action potential and reduction in the APD of these neighboring cells is
also clearly visible. At $M1$ there is $11\%$ drop in APD with a factor of
five times increase in the ischemic region size. Depending on the distance of
the neighboring points from the ischemic region, more than $5\%$ drop in APD
with a factor of five times increase in the ischemic region size is noticed.

Now, from Fig. (15(c)) - (15(j)), it is visible that increasing the size of
the ischemic region also considerably affects the waveforms of the mechanical
parameters $[Ca^{+2}]_{i}$, $T_{A}$, $\lambda$ and $\frac{d\lambda}{dt}$
corresponding to the point M1 than those of the point M2. From the figures
(15(g)) and (15(h)), it can be noticed that an increase in the size of the
ischemic region by a factor of five leads to additional shortening of fiber
length during systolic contraction and an enhanced stretch in the fibers post
systolic contraction. At $M1$ $3-4\%$ change in the length of the myocytes
relative to the resting length is noticed and at $M_{2}$ only a $1-2\%$ change
in the length of the myocytes relative to the resting length is noticed.
Therefore, the stretch rate $\frac{d\lambda}{dt}$ also changes at the points
$M_{1}$ and $M_{2}$. There is approximately $45\%$ variation in the stretch
rate $\frac{d\lambda}{dt}$ at $M1$ and approximately $5-25\%$ variation in the
neighboring points is detected.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 15: :$v$(first row), $[Ca^{+2}]_{i}$(second row), $T_{A}$(third row),
$\lambda$ (fourth row) and $\frac{d\lambda}{dt}$ (fifth row), at M1(first
column), M2 (second column), for the case with $I_{sac}$ (second column) for
$f_{ATP}=0.3\%$, with two different sizes of ischemic regions, data1
($[0.1563,0.25]^{2}$), data2 ($[0.0938,0.3125]^{2}$).

Thus, the cardiac electro-mechanical activity mathematically generated by PDE-
ODE system leads to tracing of ionic channel dynamics, intracellular calcium
ion abnormalities including the tissue structural influences. It may be noted
that the ionic channels corresponding to each cardiac cell adds to the
generation of cardiac action potential and also forms the basis for calcium
induction, calcium release and cardiac mechanical contraction. The change in
the action potential phase and hence in ECG pattern affects both the
electrical and mechanical function of heart. These numerical results help in
tracing the affected action potential phase, leading to a disturbance in ECG
pattern. It also helps in identification of corresponding affected ionic
channel dynamics. Thus, the current analysis can provide a useful guideline
for the treatment of Hyperkalemia and Hypoxia.

Thus, the heart failure due to the cardiac disease from the abnormal
functioning of the coupled electrical and mechanical activity can be traced
through the virtual reconstruction of ionic channels and intracellular calcium
ion abnormalities with cardiac structural reconstruction through the proposed
numerical modeling. The ionic channels corresponding to each cardiac cell adds
to the generation of cardiac action potential and also forms the basis for
calcium induction, calcium release and cardiac mechanical contraction. The
change in the action potential phase and hence in ECG pattern affects both the
electrical and mechanical function of heart.

##  6 Conclusion

Local ischemia in the deformed human cardiac tissue is modeled by varying the
ischemic parameters value in the ischemic subregion of the cardiac tissue
domain. Two types of ischemic effects, namely, Hyperkalemia and Hypoxia, are
considered in this work. Monodomain model in a deforming domain is taken with
the TT06 human cell level model. The coupled electro-mechanical PDEs-ODEs non-
linear system of equations are solved numerically using linear finite elements
in space and backward Euler finite difference scheme in time. We examine the
cardiac electrical and mechanical activity in terms of the action potential
($v$) and intracellular calcium ion concentration $[Ca^{+2}]_{i}$, active
tension, ($T_{A}$), stretch ($\lambda$), stretch rate ($\frac{d\lambda}{dt}$),
in several cases for local ischemia. We investigated the effect of varying
strength of Hyperkalemia and Hypoxia in the ischemic subregion of cardiac
tissue, on the electrical and mechanical activity of healthy and ischemic
zones in the cardiac muscle. We also investigated the impact of increasing the
size of the ischemic region on the electrical and mechanical parameters of
neighboring cells.

(a) With the severity of Hyperkalemia the concentration of extracellular
potassium ions increase leading to, (i) Fall in availability of sodium
channels and a decrease in inward sodium current, (ii) AP comes to a resting
state significantly prior to reaching the normal resting potential level, and,
(iii) Increased potassium conductance leading to shortening of RT in turn
shortens QT interval is noticed.

These changes in cellular ionic dynamics not only further alters the AT, RP,
RT and APD of affected cells with the spread of Hyperkalemic region but also
increasingly alters the AP of healthy cells in its vicinity. Also, this spread
of Hyperkalemic region alters the waveform of the mechanical parameters
$[Ca^{+2}]_{i},T_{A},\lambda$ ,$\frac{d\lambda}{dt}$ of the ischemic and the
neighboring healthy cells.

Thus, the severity of Hyperkalemia leads to reduction in APD, elevation in
resting potential, affects the contractile force and contractility and the
stretch activated channels, hence affects the QRS complex and QT interval of
ECG and the mechanical contraction or can say the electro-mechanical activity
of healthy and ischemic regions of human cardiac tissue. All these effects on
the electro-mechanical activity of a human heart gets more intense as the
ischemic regions expands or more cardiac cells becomes ischemic.

(b) Severity of Hypoxic ischemia, with a reduction in intracellular ATP
concentration, affect both influx of calcium ions and efflux of potassium ions
and alters the normal functioning of calcium channels and balanced potassium
channels. The increase in intracellular $Ca^{+2}$ levels completely disturbs
the Phase1, Phase2 and Phase 3 of AP. As the Hypoxically affected region
increases, the Plateau phase, repolarization phase and the APD corresponding
to the AP of healthy cells in its vicinity are affected and the ionic dynamics
of Hypoxically already degenerated cells get further disturbed. It is also
visible that increasing the size of the ischemic region by a factor of five
considerably affects the waveforms of the mechanical parameters
$[Ca^{+2}]_{i}$, $T_{A}$, $\lambda$ and $\frac{d\lambda}{dt}$. There is
approximately $45\%$ variation in the stretch rate $\frac{d\lambda}{dt}$ at
$M1$ and approximately $5-25\%$ variation in the neighboring points is
detected.

Thus, the change in the action potential phase and hence in ECG pattern
affects both the electrical and mechanical function of heart. These numerical
results help in tracing the affected action potential phase, leading to a
disturbance in ECG pattern. It also helps in identification of corresponding
affected ionic channel dynamics. Thus, the current analysis can provide a
useful guideline for the treatment of Hyperkalemia and Hypoxia.

### Financial disclosure

We would like to thank the DST for support through Inspire Fellowship, ID no.
is IF130906. We would also like to thanks the ICTP-INdAM.

### CONFLICTS OF INTEREST

The authors declare no potential conflict of interests.

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