Figures
Abstract
The myocardial ischemic border zone is associated with the initiation and sustenance of arrhythmias. The profile of ionic concentrations across the border zone play a significant role in determining cellular electrophysiology and conductivity, yet their spatial-temporal evolution and regulation are not well understood. To investigate the changes in ion concentrations that regulate cellular electrophysiology, a mathematical model of ion movement in the intra and extracellular space in the presence of ionic, potential and material property heterogeneities was developed. The model simulates the spatial and temporal evolution of concentrations of potassium, sodium, chloride, calcium, hydrogen and bicarbonate ions and carbon dioxide across an ischemic border zone. Ischemia was simulated by sodium-potassium pump inhibition, potassium channel activation and respiratory and metabolic acidosis. The model predicted significant disparities in the width of the border zone for each ionic species, with intracellular sodium and extracellular potassium having discordant gradients, facilitating multiple gradients in cellular properties across the border zone. Extracellular potassium was found to have the largest border zone and this was attributed to the voltage dependence of the potassium channels. The model also predicted the efflux of from the ischemic region due to electrogenic drift and diffusion within the intra and extracellular space, respectively, which contributed to
depletion in the ischemic region.
Citation: Niederer S (2013) Regulation of Ion Gradients across Myocardial Ischemic Border Zones: A Biophysical Modelling Analysis. PLoS ONE 8(4): e60323. https://doi.org/10.1371/journal.pone.0060323
Editor: Vladimir E. Bondarenko, Georgia State University, United States of America
Received: November 5, 2012; Accepted: February 24, 2013; Published: April 5, 2013
Copyright: © 2013 Steven Niederer. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: The work was supported by UK Engineering and Physical Sciences Research Council EP/F043929/1, British Heart Foundation (PG/11/101/29212) and Boston Scientific. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The author has the following interests. This study was partly funded by Boston Scientific. There are no patents, products in development or marketed products to declare. This does not alter the author's adherence to all the PLOS ONE policies on sharing data and materials, as detailed online in the guide for authors.
Introduction
Myocardial ischemia is caused by reduced perfusion to regions of the heart leading to a localised reduction in supply of metabolites, limited waste removal and compromised ionic homeostasis. The first 10 minutes of ischemia are associated with an increased risk of arrhythmias peaking after 5–6 minutes [1]. During this period arrhythmias are commonly initiated within the border zone (BZ) separating viable, well perfused, tissue and the ischemic, underperfused, region [2]–[4]. Ischemia causes an increase in extracellular potassium (), intra and extracellular proton concentrations (
and
, respectively), intracellular sodium (
) and intracellular calcium (
) concentrations [5]. The dominant mechanisms for these changes have been attributed to a shift in the ATP/ADP ratio, which inhibits the Sodium-Potassium ATPase pump (
) and increases the conductance of ATP-inactivated
channels; respiratory acidosis causing an increase in
; and metabolic acidosis, where a shift towards anaerobic respiration increases the production of
in the cell [5]. Inherently, these changes in ionic concentrations in the ischemic region lead to gradients in properties across the BZ, creating electrophysiological heterogeneities that are thought to favour the occurrence of arrhythmias [6]–[8].
Experimentally, the development of gradients of extracellular pH () and
[9], [10] have been well characterised using ion sensitive electrodes. Intracellular metabolite gradients have been characterised by fluorescent NADH [11] and biopsy [12] measurements. However, less is known on the gradients of intracellular ions, in particular
,
and
, nor are the mechanisms that underpin the spatial and temporal evolution of these ion concentration gradients well characterised or understood. This study aims to investigate the spatial-temporal evolution of ion gradients across ischemic BZ and the primary regulators of the BZ size and rate of development.
Previous measurements of ion concentrations and metabolites across the BZ have either been performed at multiple locations but at a limited number of time points [10], [12] or have tracked the time evolution of ion concentrations but only from a limited number of locations [13], [14]. Furthermore, these measurements have only been able to characterise a subset of ions of interest across the BZ. The need to track the evolution of multiple ionic species in space and time to understand the gradients of cellular electrophysiology across the BZ motivates the use of biophysical computational modelling. Previous models of electrophysiology during acute regional ischemia have simulated the effects of these spatial gradients but have not simulated their time evolution [15]–[17]. More recent work has simulated the time evolution of gradients [18], but have not considered other ionic gradients, the effects of nonlinear interactions between
and other ions, the effect of
diffusion in the intracellular space or the effects of potential gradients on ion diffusion.
In this study a new model of cardiac tissue electrophysiology is developed to investigate the spatial-temporal evolution of ionic concentrations across the ischemic BZ, during the first 5 minutes of reduced perfusion. The proposed model extends the conventional bidomain equations to explicitly link membrane potential to ionic concentrations and enforces ionic species conservation. A model of ion regulation across the cell membrane is then developed, parameterized, validated and coupled to the tissue model. This combined model is then used to investigate the spatial and temporal dispersion of ions across the BZ.
Methods
To model the evolution of ionic concentrations in the presence of multiple ionic gradients, electric gradients and heterogeneous tissue properties requires the development of a new set of equations for modelling the myocardium. In the next section the equations to model the BZ are derived and a model of passive cell membrane ion regulation is developed and validated. The changes to the cell membrane model to simulate inhibition,
channel activation, respiratory acidosis and metabolic acidosis are then described.
Tissue Model Derivation
Consistent with previous models of cardiac tissue electrophysiology the myocardium is represented as a two phase medium, with each point in the domain containing a fraction of intra and extracellular space. This gives rise to the scaling variables.(1)where
is the volume of the
space in a unit of
myocardium volume,
is the volume fraction of space
(with
corresponding to the extra (
) or intra (
) cellular space),
is the interspace surface area per unit volume myocardium and
is area of the interspace surface. The intracellular volume fraction (
) can be separated into sub volume fractions representing the cytosol (
), mitochondria (
) or the sarcoplasmic reticulum (
) to model distinct subcellular spaces, as described below. At each point in space there exists an intracellular potential (
), an extracellular potential (
), a transmembrane potential (
) and an intra and extracellular ion concentration for each of the ion species in the model. The movement of each ion species in each domain is driven by diffusion, due to a gradient in ion concentration, and drift, due to a gradient in the electric field. This movement is described by the Nernst-Plank equations
(2)where
is the concentration of unbound ion
in space
,
is the charge of ion
,
is the effective diffusion of ion
in space
,
is Faraday's constant,
is the gas constant,
is the absolute temperature and
is the potential in the space
. The conventional Nernst-Plank equations are adapted to represent the movement of ions across the cell membrane between the intra and extracellular spaces. This gives
(3)where
is the flux of ion
across the cell membrane.
is defined in units of mM ms
mm. The surface separating the two regions is modelled as a simple capacitor and defining the transmembrane potential to scale with intracellular charge gives
(4)where
is the intracellular charge per unit cell membrane area and
is the membrane capacitance per unit cell membrane area. The charge on either side of the membrane is assumed to be equal but opposite, giving
(5)Separating the concentration of each ion species in the intra and extracellular space into ions that are membrane bound and make up the membrane charge (
) and those that are in solution gives.
(6)where
and
are the ions in solution and membrane bound ions, respectively. By assuming charge neutrality for the ions within the solute gives:
(7)Multiplying Eqn 6 by the ion species charge and Faraday's constant, then summing over all ion species in space
gives.
(8)
Converting the concentration of ions per unit volume in domain to charge per unit cell membrane area and introducing a static charge term (
) that characterises all charge not attributable to
, gives
(9)
Combining with Eqn 8 gives(10)
This defines the charge on the membrane as equal to the unbalanced charge in space . Using the charge balance Eqn 5, gives
(11)
Differentiating Eqn 11 with respect to time, substituting in Eqn 3 and recognising that all transmembrane fluxes are balanced, provides(12)this ensures that there is no net charge accumulation in any unit volume of myocardium. Defining the relationship between the intra and extracellular potentials gives
(13)Using Eqn 4 and the definition of charge (Eqn 10), then gives the algebraic definition of the transmembrane potential:
(14)
Rearranging Eqn 13 and substituting in the definition of from Eqn 14 allows
to be defined in terms of
and
. Combining this definition of
with Eqns 14, 13, 12, 3 and 2 then represents a closed set of equations. These equations are equivalent to the bidomain equations in the case of a single charge carrier and no gradient in ion concentrations, as shown below.
In cardiac myocytes many important ions, including and
are heavily buffered, both within the cell and the extracellular space. To account for buffering the free and buffer bound fraction of
are calculated. In general, as
the effect of ions bound to the cell membrane will not be included in the buffering equations for simplicity. This gives
(15)where
are the unbound ions and
are the ions bound to buffers. At this time all buffers will be treated as rapid and to a single representative buffer species, giving
(16)where
,
and
are the concentration, binding affinity and Hill coefficient, respectively, for the buffer of ion
in space
. Assuming that ions bound to buffers are immobile, only
is used to calculate the diffusion and drift of ions in Eqn 3, and similarly in Eqn 12. As binding of ions to a buffer implicitly removes a charged binding site located on a static protein, Eqn 14 remains unchanged.
Due to the complex anatomy of the cardiac myocyte many sub volumes exist within the cell that affect ionic concentrations. The sum of the volume fraction () values must be less than, but do not have to be equal to, one, allowing the model to represent any volume fractions that are not directly accessible by ions. In particular the
variable can represent all space in the cell or can be substituted for
representing the volume fraction of the cytosol (a sub volume of
). This allows the effects of SR, mitochondrial or other subcellular structure volumes on intracellular ionic concentrations to be accounted for in the model.
Equation summary.
The modelled equations are given by(17)
where it is important to note that for non buffered ions
and
.
Consistency with bidomain equations.
Imposing the implicit assumptions of the bidomain equations that charge carriers are not buffered, ion concentrations are homogenous and charge is carried by a single carrier to Eqn 18, the bidomain equations can be derived. Assuming homogenous ion concentrations and considering the case of the intracellular space reduces Eqn 18 to(18)
Differentiating Eqn 14 for the intracellular space gives(19)substituting Eqn 18 into Eqn 19 gives
(20)As ion concentrations are homogenous conductivity is defined as
(21)
Then Eqn 20 reduces to(22)
Applying the single charge carrier assumption and converting from ionic flux to current gives the first bidomain equation(23)
The second bidomain equation is readily derived from applying the homogenous ion concentration assumption to Eqn 12 and multiplying by Faraday's constant (to convert from conserving ion flux to current) giving(24)
Substituting in the definition of conductivity from Eqn 21 then gives the second bidomain equation(25)
Modelling the Membrane Fluxes
Cardiac electrophysiology is predominantly determined by the movement of ,
and
. For charge neutrality
must also be included in the model. To simulate the evolution of acidosis requires the inclusion of
,
and
in the model. All of these ions (and
) were modelled in the intra and extracellular space, with
and
being buffered. The goal of the model, in this study, was not to track the propagation of the action potential but to simulate the gradients of ions that exist over the BZ. These ion gradients were modelled based on the diastolic properties of the cell. This assumption was also a requirement to enable the simulation of minutes, while remaining computationally tractable.
The membrane ion transport pathways are described first for ,
and
. Ion specific channels are then described that balance the flux of each ion species. The channel and transporter densities were determined by imposing zero net flux for each ion species, using the relative densities of
and
transporters recorded experimentally, and intra and extracellular ionic concentrations and membrane potential values derived from the literature. Where possible experimental data was taken preferentially from rabbit or guinea pig data at body temperature. This limited number of constraints then allowed the model transporters and channel densities to be uniquely determined.
Sodium regulation.
The model of regulation included representations of the
, sodium calcium exchanger (
), sodium hydrogen exchanger (
), sodium bicarbonate co-transporter (
) and a lumped sodium channel (
).
was modelled using the thermodynamically consistent equation set proposed by Smith and Crampin [19]. This model was subsequently revised by Terkildsen et al., [20] and this parameter set that was used here. The model for
was fitted to guinea pig data as limited rabbit data was available. However, the maximum flux was rescaled to match rabbit data, as described below.
The model was taken from Weber et al., [21]. The model has been fitted to rabbit experimental data at 37°C. The
model was based on the model developed by Crampin and Smith [22] and reparameterized by Niederer and Smith [23]. In this study extracellular
and
regulation of
were included. This model was fitted predominantly to sheep Purkinje data [24], although the
dependence of
remains relatively consistent between species [25]. The
model was taken from Crampin and Smith [22]. The model assumes
is electro neutral, which is true for only part of the
population [25]. There was not sufficient data to fully characterise the electrogenic and electro neutral forms of
, hence the electro neutral model was used. Background
flux across residual open fast
and persistent
channels is limited when the cell is quiescent. However, some flux is still present [26] and a simple lumped background ionic flux equation was used, given by
(26)to model the residual
flux across any open
channels. The same equation form was used for modelling all background ion channels.
Proton regulation.
In the proposed model was regulated by
, described above, chloride-hydroxide exchanger (
), hydrolysis and buffering. Background
leak or other
exchangers were not considered in the general model of
regulation, described here, but do include models of
-lactate exchange and intracellular metabolism derived
sources in the model of ischemia, described below. The
model comes from Niederer et al., [27] and was fitted to guinea pig data at 37
C. The hydrolysis of
into
and
was governed by
(27)where
and
are the forward and reverse rates of hydrolysis. Hydrolysis occurs in both the intra and extracellular space and the rate constants were assumed to be the same in both domains.
buffering in the intra and extracellular space is due to mobile and static
buffers and
[28]. The intra and extracellular buffering of
were assumed to be instantaneous and represented by a single population of buffers. To reduce the model size, partial differential equations were only solved for the total concentration of
or
. The concentration of free ions were then calculated by
(28)where
and
represent the concentration of the buffer and the binding affinity, respectively. This buffer model was used for both intra and extracellular
, with a separate set of parameters for each ion.
Calcium regulation.
A simplified model of intracellular cardiac was developed assuming that
in the intracellular space reaches an approximate equilibrium over the time scales of interest. Furthermore, SERCA ATPase function was modelled with a Hill coefficient of one as opposed to two, to allow the definition of
to remain deterministic. The intracellular space was assumed to consist of a sarcoplasmic reticulum (SR) and a cytosolic space. The subsarcolemmal and dyadic space are small and are also likely to be in equilibrium with the cytosolic
, so were not included in the model. The
dynamics were described by
(29)
(30)
(31)
(32)
(33)where
is the SR
,
is the flux of calcium out of the SR,
is the uptake of
by SERCA,
is the maximum SERCA flux,
is the diffusion permeability of the SR membrane,
is the binding coefficient of
to SERCA,
and
are the volume fractions of the SR and cytosol, respectively,
is the L-type calcium channel and
is the background
channel. In this model the background and L-type
channels were modelled as a single lumped generic
channel. Introducing cytosolic buffering, ignoring the effects of SR buffering and setting
as one and assuming that the cytosol and the SR are in equilibrium then gives
(35)
(36)and collecting terms gives
As is always negative the cubic always has at least one positive real root for possible values of
. The value of
was then found using the root finding method first proposed by Francois Viete in 1600 and reused more recently by Faber and Rudy [29]:
(41)
The model of dynamics assumes that all
buffers were static and that the transport of ions via mobile buffers was accounted for in the effective diffusion parameters of free
. It is possible to extend the proposed model to include mobile buffers but they were assumed to play a secondary role in the current model.
was assumed to be buffered by a single species and was modelled using the same framework described above for
(Eqn 28).
Chloride regulation.
In this model homeostasis was maintained by the
and the
-
exchanger (
), which bring
into the cell, and a
channel that allows
to flow out of the cell. The
model is described above. The
model was taken from Crampin and Smith [22] and was developed using guinea pig data at 37
C. The
channel uses the conventional background channel formulation. A linear
dependence of the background
current was added to the model based on observations from Komukai et al., [30].
Potassium regulation.
In dynamic action potential models of cardiac electrophysiology there are a large number of channels [31] that bring
into the cell. This influx was balanced by the
pump, described above, that extrudes
. For the passive membrane model, all of the
channels were lumped into a single background current (
) formulation that was set to balance the flux of
on
. It was assumed that the membrane potential and
reversal potential are the dominant factors affecting this channel and other forms of regulation have not been considered.
Model Parameters
For each transmembrane ion pathway described above, all kinetic, binding affinity and membrane potential dependencies were taken from the original models. Here the definition of geometrical parameters, ionic concentrations, buffering parameters and the density/scaling of each transmembrane pathway are motivated from data in the literature.
Geometrical parameters.
The extracellular space is estimated to be between [32]–[35],
and
[33], [36] of the volume of the heart in rabbit, cat, and rat hearts, respectively. This gave an
value of
leading to an
value of
. The surface to volume ratio of a cell is reported as
m
[37], [38] in rat and rabbit myocytes, corresponding to a
value of
mm
. The relative SR volume was set to
of intracellular volume, giving an
value of
, based on reported values of
of cell volume [37], [39], [40] in mouse, rat and swine. The relative mitochondrial volume (
) was set to
based on an estimated mitochondrial cell volume fraction of
[37], [40]. The cytosol volume fraction was set to
of the intracellular space, resulting in an
value of
. To account for the effects of subscellular domains on intracellular ionic concentrations in the model, all references to
in Eqn 18 were replaced by
. A summary of geometrical parameters is given in Table 1.
Intracellular ionic concentrations.
has been measured using SBF1 fluorescence and
sensitive electrodes. A significant range of values have been reported from
mM [26], [41]–[44] to
mM [34], [45], [46]. Early measurements of intracellular ionic concentrations were performed using ion sensitive electrodes. These experiments measure ion activity and not ionic concentration and are often performed in multi-cellular preparations, confounding measurements. For these reasons
in quiescent myocytes was set to
mM, consistent with recent calibrated fluorescent measurements in isolated rabbit myocytes [26], [44].
No fluorescence dye is routinely used for measuring . Using ion sensitive electrodes Lee et al., [45] were able to calibrate their measurements of ion activity in rabbit myocytes using an estimated
activity coefficient of
, giving a value of
mM. This compares with a range of
mM calculated by applying the Lee et al.,
ion activity coefficient to ion activity measurements in rabbit, cat and guinea pig [34], [43], [47], [48]. Alternate measurement using flame emission spectrometry by Powell et al., [49] measured
in rat myocytes, giving a concentration of
mM. Given the lower values of the two calibrated measurements,
was set to
mM.
No dye is routinely used for measuring concentration in cardiac cells, however,
can be measured using ion sensitive electrodes.
activity has been reported as
mM [35], [48], [50], [51] in sheep, rabbit and guinea pig heart cells. Estimations of
from total tissue
concentrations have resulted in values of
mM [35] and
mM [34] in rabbit cells and
mol/g dry wt (or
mM using the
mM per
mol/g dry wt scaling factor from Bers [52]) in rat. The higher value of
mM may be attributed to the higher extracellular space used in these calculations (
as compared to
). Considering the relative convergence of values
was set to
mM.
Resting free is measured using calibrated fluorescence measurements. These measurements range from
nM in rabbit and guinea pig preparations [53]–[57]. Given this consistency the
will be set to
nM. SR
concentration is calculated from integrating the current across the cell membrane following the release of
from the SR in response to caffeine. These measurements show two populations with high values in rat (
M [58]–[61]), canine (
M [62]), rabbit (87–106
M [60], [61]) and ferret (
M [63]), compared to lower measurements in guinea pig (
M [58], [64]). Given that the majority of species have a higher reported concentration, including rabbit, simulations were run with SR
load set to
M. The buffering of
can be described by Hill equation(s), mass action equation(s) or a constant buffering power. To compromise between biophysics and complexity, buffering was modelled by a single Hill equation. Hove-Madsen and Bers [65] fitted
buffering in rabbit myocytes using two Hill curves; however, the lower affinity buffer will not play a significant role at passive diastolic
concentrations. For this reason the high affinity site, with cooperativity reduced from
to unity, was used to model
buffering, giving a buffer concentration of 208.98
M (converted using a scaling factor of
from Bers [52]) and an affinity of
M. This model of
buffering in rabbit is similar to the concentration/affinity values of
M [66],
M [67] and
M [68] values measured in other species.
has been measured using
sensitive electrodes and fluorescence dyes. Measurement of
consistently falls within the range of
[57], [69]–[71] in either HEPES or
buffered solutions. In the model
was set to 7.1.
are heavily buffered in the cytosol by intrinsic buffers and
. Here the buffering of
by
was modelled explicitly and the intrinsic buffers were assumed to be in rapid equilibrium. Leem et al., [70] measured (and modelled)
buffering by two populations of buffers with binding affinity pK values of
and
and concentrations of
mM and
mM. Zanbioni et al., [25] differentiated between mobile and fixed buffers and found that the fixed buffers had a consistent concentration of
mM and binding affinity (pK) value of
across rat, rabbit and guinea pig, while the mobile buffers had a constant
value of
. Fitting a single buffering curve to these two models over a pH range of
gives concentrations of
mM and pK values of
, which gave a value of
mM and pK value of
for this model.
Extracellular ionic concentrations.
The concentration of the majority of ions in the extracellular space have been measured in canine [46], [72], rat [36], [73] and cat [74] hearts. These measurements provide a consistent range of ion concentrations for ,
and
, giving
as
mM,
as
mM and
as
mM. In the model
was set to
mM,
was set to
mM, consistent with measurements in guinea pig hearts [75] and rabbit atrium [76] and
was set to
mM. The
values reported range from
mM, however, these values do not differentiate between buffered and ionized
. The properties of
buffering in the extracellular space are not well characterised and are generally ignored in previous cardiac cell models. To approximate the buffering properties of extracellular
using a simple single species steady state mass action model, with no cooperative binding, requires two parameters, the concentration of the buffer and the binding affinity. Assuming the ratio of free
in the extracellular space to bound ions is similar to serum [77] and assuming that the primary buffer of
in the extracellular space are phospholipids, then extracellular
buffering will have a binding affinity of
mM [78], within the range observed across multiple species [79], and a buffer concentration of
mM based on a free
concentration of
mM and assuming
of
are bound to buffers [77].
was set to
, to be consistent with the majority of experimental studies [57], [69]–[71] and measurements across a range of species [80]. Limited measurements were available to model
buffering, however, Yan and Kleber [81] reported 39 mM of buffered
in the extracellular space. By assuming similar binding affinities for the intra and extracellular buffers and that
is
, then gave a concentration of extracellular proton buffers of
mM. A summary of ion concentrations and buffering parameters are given in Table 2 and Table 3, respectively.
The concentration of was calculated using the parameters proposed and measured for guinea pig ventricular myocytes at
C by Leem and Vaughan-Jones [82]. The concentration of
in the extracellular solution was calculated using
(42)where
is the solubility of
, set to
mM mmHg
from human measurements at pH
and
C [83],
is the fraction of air that is
, set to
at baseline and
is atmospheric pressure, set to
mmHg
. This gives a partial pressure of
(
) of
mmHg, consistent with although slightly higher than the
3 mmHg measured in rabbit hearts [84]. The hydration of
was modelled by a mass action reaction (Eqn 27). The
and
values were measured and modelled by Leem and Vaughan-Jones in guinea pig myocytes at
C [82], giving values of
s
and
s
, respectively, resulting in an equilibrium constant of
. The rate of hydration of
was assumed to be similar in both the intra and extracellular space. The movement of
across the cell membrane was modelled by Fick's law. Previous models have used permittivity values of
mms
[82], based on measurements in red blood cells. This value was reused despite its lack of species and cell type consistency, as there were no recent studies characterising the permeability in cardiac myocytes and the high permeability leads
to be close to equilibrium between the intra and extracellular spaces.
Current, exchanger and pump densities.
The model of each transmembrane ion pathway () was separated into a kinetic regulatory component (
) dependent on transmembrane potential and ionic concentrations, and a scalar (
) representing either the maximum flux or channel conduction of the pathway. The flux across a pathway (
) is then given by
(43)
The values were determind by calculating all of the
values (excluding
,
and
as
and
were unknown) using the ionic concentrations in Table 2 and assuming a membrane potential of −80 mV. The remaining unknown
and ionic concentrations were then determined from a limited number of measurements and by enforcing a zero net flux condition described by
(44)
As the parameters for hydrolysis and permeability of the cell membrane to (
,
and
) were derived from the literature, the constraints on
and
were used to determine the concentrations of
and
, respectively. Measurements of transmembrane
influx gave
as
mM ms
,
as
mM ms
,
as
mM ms
and
as
mM ms
[85]. The
flux was assumed to be equal to the sum of all
influx. As the model does not include
-
-2-
co-transporter (NaK2Cl) or
-
exchanger (NaMg), primarily due to the limited data to constrain the kinetics,
was reduced from the flux measured by Despa et al., [85] to 1.13
mM ms
. Setting these
values defines
,
,
,
,
and
. In the experimental and modelling work by the Vaughan-Jones group [70], [86] the relative size of
to
is
at
, this gave an estimated ratio of
. Scaling
determined from [85] by 0.16 provided an estimate of
of
mM ms
. This allows
and
to be defined.
must be balanced by
and as
and
were known, this flux was used to set
. Combining the
and
concentrations with the defined hydrolysis parameters and the known value of
then gave
. Knowing
and
gave
and hence
. Similarly,
was calculated from
and
,which gave
. Finally,
was calculated using
and
, calculated using the derived
value. By automating this parameter derivation process the model parameters could be updated to ensure static ion concentrations, and hence membrane potential, for any perturbation in model parameters or ionic concentrations. Parameters for all simulations in this study were derived following this process.
Intracellular calcium dynamics.
The intracellular regulation of was treated as an equilibrium system. This resulted in the SR effectively acting as an additional buffer on
. The parameters
,
,
,
and
were defined from enforcing a zero net flux constraint and experimental measurements. Measurements of
are similar in rat and rabbit [65] and have been reported as 0.260–0.350
M in rabbit [65], [87] and 250–280
M in rat [65], [87], [88]. A value of 0.3
M was used in the model.
is reported as 0.001–0.012
Mms
in rabbit, mouse and rat [54], [89]–[91]. A value of 0.01
Mms
was used in the model. Enforcing a zero net flux constraint on the SR gave
as 0.038
Mms
, comparable with values of 0.03–0.08
Mms
measured in rabbit [65], [87], [92] with units converted using scale factors from Bers [52].
Membrane Model Validation
To validate the passive membrane model, a series of tests based on the response of the cell models intracellular ionic concentrations and transmembrane potential to changes in extracellular ionic concentrations or inhibition of major ion transporters was performed. In order to maximise the number of tests the model was compared against data from cardiac cells, regardless of temperature, species or preparation type. This maximised the number of tests but meant that a quantitative comparison was not valid and so only a qualitative comparison was performed. A summary of experimental data used in the validation is provided in Table 4. Fig. 1 shows the comparison between the model and data. From the 72 simulations performed 43 matched the experimental data, data was not available or was inconsistent for 19, changes were too small to be measured in 5 and the model did not match experiments in 5 cases.
Arrow direction shows model prediction for the change in intracellular concentrations and membrane potential for a change in extracellular ion concentrations or inhibition of membrane transporter as indicated by the panel label. The color of the arrow indicates how the model compares with experimental data, summarised in Table 4. Black arrows indicate where the model matches experimental data, white arrows indicate where there is no or inconsistent data, gray arrows indicate where no change was observed experimentally and striped arrows indicate where the model does not match experimental data. Data was considered inconsistent if different studies reported opposite changes in intracellular ion concentration or membrane potential in response to a change in extracellular ionic concentrations (for example the change in membrane potential following a decrease in ). In cases where one study reported no change in intracellular ion concentration or membrane potential and another found a change, it was assumed that the change was correct (for example the change in
in response to an increase or a decrease in
). For example in panel A) corresponding to a decrease in
(as indicated by the panel label), the experimental data comes from row 1 of Table? 4. The model predicts that
decreases and that
and
increase, consistent with experimental measurements and the arrows are shaded black. There is no consistent observed change in the transmembrane potential so the
arrow remains white. No data is available for the change in
in response to a decrease in
so the
arrow remains white. The model predicts a decrease in
yet experimental measurements found an increase, hence the
arrow is striped.
Diffusion Parameters
The tissue model required the definition of diffusion parameters for each of the ionic species in the intra and extracellular space. The diffusion parameters used in the model were all for the apparent diffusivity of free ions, lumping the effects of any mobile buffers, tortuosity and gap junctions into a single diffusion parameter. Using Eqn 21 the conductivity parameters were shown to be directly related to the conductivity parameters in the bidomain equations. From the review of bidomain conductivities and relative values in the intra and extracellular space by Roth [93] a conductivity value of 0.25 Sm was taken for both the intra and extracellular space. This conductivity, along with the ionic concentrations from Table 2, were used to derive the diffusion parameters. The apparent diffusion of
and
are
mm
ms
and
mm
ms
in the >intracellular space [86] and were assumed to be similar in the extracellular space. The apparent diffusion constant of
(including the effects of mobile buffers) has been estimated from experimental measurements in previous modelling studies as
mm
ms
[94]–[96]. Here a value of
mm
ms
[94] was used and the effect of diffusion in the SR was not included, as it was assumed to be non contiguous between cells. It was assumed that
diffusion is limited by buffering, resulting in the same value in the intra and extracellular space.
,
and
were all assumed to have a common diffusion coefficient in either the intra or extracellular space as these ions are only nominally buffered and are only affected by gap junctions and tortuosity. Diffusion of
was estimated from previous modelling/experimental studies of
in tissue as
mm
ms
[97], and was assumed to be the same in the intra and extracellular space. Solving Eqn 21 for the diffusion of intra and extracellular
,
and
gives the diffusion parameters summarised in Table 5.
Finally, the effects of on gap junction conductivity were included. Measurements in cell pairs by Swietach et al., [98] have demonstrated that the permeability of gap junctions has a biphasic dependence on the
concentration. To introduce these effects into the model the intracellular diffusion constants were scaled by
(45)where
/
and
/
are the cooperativity and binding affinity for the activation and deactivation of gap junctions by protons, respectively and
scales all intracellular diffusion constants. Using the measured parameters from end-to-end cell pairs in Swietach et al., [98]
,
,
mM and
mM. In this model of proton effects on gap junction permeability there is an implicit assumption that the gap junction permeability plays a dominant role in defining diffusion. At this stage no other regulators of permeability were included in the model, notably
regulation is absent but this can readily be included in the modelling framework as required.
Simulating Ischemia
In this study the effect of inhibition,
activation and respiratory and metabolic acidosis during ischemia on cell ionic homeostasis were considered. This list is not exhaustive and absent factors are discussed below. Ischemia was modelled by respiratory acidosis, metabolic acidosis, increased
channel conductance and decreased
function. The specific time course of each of these changes is poorly characterised. Previous modelling studies have assumed that changes in cell function with ischemia have evolved linearly [99] or as a nonlinear function of prescribed metabolite concentrations [20]. To avoid any undue bias from the arbitrary selection of a time course, all changes are initially considered instantaneous.
NaK inhibition and channel activation were modelled by scaling the respective fluxes. Respiratory acidosis was assumed to result from an imbalance of production and washout of
. To simulate this, an (implicitly electro neutral) intracellular source of
was introduced into the ischemic region. There was no flux of
out of the extracellular space and this resulted in a build up of
in both the intra and extracellular space in the ischemic region.
To model metabolic acidosis required the introduction of an additional intracellular source. However, introducing a source of cations into the cell compromised the conservation of charge constraint. To provide an electro neutral source of
required the concurrent introduction of a source of anions that match the production of
inside the cell. It was implicitly assumed that the anions entered the cell in an electro neutral form with an
bound, and the anions and
separate within the cell due to metabolic processes. The source of
in ischemia is likely to be due to increased ATP production through glycolysis [100]; in the absence of oxygen this also results in increased lactate production. As lactate readily dissociates from
it was assumed that the anion source that matches
flux has similar characteristics to lactate including transmembrane regulation via the lactate-
membrane exchanger MCT1 [101]. Given the ambiguity in structure, MCT1 was modelled as an ordered exchanger, using the kinetic parameters from Vinnakota and Beard [102] and the maximum influx value of
mMms
recorded in guinea pig myocytes [103]. In the absence of metabolic acidosis, lactate concentration was assumed to be nominal, consistent with the small flux of lactate observed in rabbit hearts under normal conditions [104]. Model simulations were performed on a 1D strand with length 32 mm, oriented in the preferred conduction or fibre direction, with a transition between ischemic and viable tissue at 16 mm to ensure the simulation captured the BZ width with nominal boundary condition artefacts. The transition between viable tissue and ischemic tissue was approximated by a Hill equation with a Hill coefficient set to ensure a steep transition between the ischemic region and viable tissue over
m, consistent with the rapid drop in oxygen pressure across the BZ in swine [105].
Numerical Methods for Tissue Model
The nonlinear equations were solved using a fully implicit finite difference scheme with a line search Newton-Raphson method. The transmembrane flux component of the Jacobian was calculated using finite differencing with the remainder calculated analytically. The Jacobian was inverted directly using MatLab and was only recalculated if the residual fails to decrease or convergence was not reached within Newton-Raphson iterations.
The dependence of the BZ width on the spatial and temporal discretizations was determined to test for numerical convergence. The width of the transition for each ion concentration between the viable and ischemic region, referred to as the BZ width for each ion, was calculated by fitting a Hill curve to each ionic profile. The width of the BZ for each ion was then calculated as the distance between and
of the change in concentration.
A convergence analysis was performed and a mesh discretization of m and a time step of
ms was used. Increasing the spatial discretization by a factor of
results in a maximum change in BZ width of any ion of
mm. Decreasing the time step to
ms increases the maximum BZ width of any ion by
mm.
Results
The individual effects of each of the four components of ischemia were first demonstrated. A combined model of ischemia was then developed and the width and magnitude of the changes in ionic concentrations across the ischemic BZ were predicted. The effect of movement of ,
and
within and between intra and extracellular spaces across the BZ were calculated to show the net movement of ions across the BZ.
Simulating Individual Components of Ischemia
The level of inhibition of , the activation of
and the level of
and
production in the cell during ischemia is not known. To investigate the effects of each of these aspects of ischemia, they were each individually introduced into the model at five levels of severity, for x
mm along a
mm strand. Fig. 2 shows the effect of each component on the change in
,
,
and
across the ischemic BZ. The range of alterations in ion concentrations is shown by the shaded regions. The minimum and maximum changes in each component are shown by dashed lines and the mid change is shown by the solid line.
(yellow) was inhibited by up to 100% (no flux) in
increments,
(red) was scaled by up to
in
increments, metabolic acidosis was simulated by introducing an
flux in five increments up to a maximum value of
Mms
and respiratory acidosis (blue) was simulated by a intracellular
flux increased in five increments up to a maximum value of
Mms
.
Effect of 50–250% increase in (red shaded region enclosed by dashed line) and a 150% increase in
(red line) on A)
, B)
, C)
and D)
. Effect of 20–100% maximum
flux (purple shaded region enclosed by dashed line) and 60% maximum
flux (purple line) on E)
, F)
, G)
and H)
. Effect of 20–100% NaK inhibition (yellow shaded region enclosed by dashed line) and a 60% NaK inhibition (yellow line) on I)
, J)
, K)
and L)
. Effect of 20–100% maximum
flux (blue shaded region enclosed by dashed line) and a 60% maximum
flux (blue line) on M)
, N)
, O)
and P)
.
Modelling Ischemia
Inherently, there is no single mode of ischemia and the relative contribution of acidosis, inhibition or
activation will depend on the residual flow, age, gender, disease state, location of ischemic region and species under study. To provide a representative case to study, 5 minutes of ischemia were simulated in the presence of all four ischemic mechanisms that match representative results from the literature.
Partial pressure measurements of in canine ischemic models show a
increase in
partial pressure following occlusion, depending on the level of flow inhibition, after
minutes. The elevation in
was approximately linear over time and hence in the model it was assumed
concentration increases by
increase in the first
minutes of ischemia [106], [107]. This corresponded to an increase in the concentration of
in the model from
mM to
mM. In the model respiratory acidosis was caused by an increase in
flux that cannot be vented from the extracellular space. Although it is recognised that the decrease in
due to metabolic acidosis will also contribute to elevated
, initially the
flux was set at a
level to achieve an increase in
concentration to
mM, which resulted in a decrease of
to
. During ischemia
decreases rapidly before plateauing after approximately
minutes. The decrease in the initial
minutes has been reported to fall between
to
pH units in rat [108], ferret, [109] and guinea pig [110] preparations. Respiratory and metabolic acidosis will both contribute to this drop in
. In the model setting the level of intracellular
flux to
caused a
pH unit drop and an increase in
to
mM. Combined metabolic and respiratory acidosis caused a
unit drop in pH and an increase in
to
mM.
In studies of ischemia tends to increase linearly with time. In the rat heart ischemia caused an increase in
by
over
minutes [111], by
after
minutes [112], from
mM to
mM during
minutes ischemia [113], by
,
,
and
after 9,
,
and
minutes, respectively [114], by two fold over
minutes [115], from
mM to
mM over
minutes [116], by
over
minutes [117], by
over
minutes [118], no change over
minutes [108] and by
over
minutes [119]. Changes in
in guinea pig hearts during ischemia is controversial with reports of a decrease from
mM to
mM over
minutes [120] and of an increase of
over
minutes [121]. Only considering the cases where
increases, as these represent a repeatable consensus result, the range of expected increases in
over a
minute period is
, assuming a linear increase with time. These cluster into two groups with ranges
and
. In the model reducing the maximum flux of
by
caused an increase in
to
mM (
) in the absence of acidosis or
mM (
) in the presence of the acidotic components of ischemia, described above.
Given the inhibition of and levels of respiratory and metabolic acidosis a sweep of
activation values was performed in the presence of these changes to achieve the desired level of extracellular
accumulation. During ischemia,
increases over three characteristic phases, with the first phase occurring during the initial
minutes of ischemia prior to reaching a plateau from minutes
to
, before increasing again. In guinea pigs, ischemia caused
to increase from
to
mM over
minutes [122] or
to
mM over
minutes [123]. In swine, ischemia caused
to increase from
mM to
mM after
minutes [124], [125]. In rat, ischemia caused an increase from
mM to 8mM over
minutes [110], although other groups have seen a biphasic change in
in rat with an increase from
mM to
mM before falling back to
mM then continuing to rise again, observed over the first
minutes [126]. In rabbit, ischemia caused
to increase from
mM to
mM over
minutes [10], [110], [125]. In canine,
increases from
mM to
mM after
minutes [127], [128]. These results indicate an increase in
from
mM to
mM after
minutes and assuming
of this rise occurs in the first
minutes [127], and combined with the 5 minute data gives an estimate of
after 5 minutes of ischemia as
mM. In the model an increase in
of
was used to achieve an elevation of
to
mM, resulting in an increase in the membrane potential of
mV to
mV. This is consistent with measurements in cat (
mV over
minutes) [129], [130], sheep (
mV over
minutes) [131], guinea pigs (
mV over
minutes) [132] and mice (
mV over
minutes) [133], but less than measurements in rabbit (
mV over
minutes [10]) and guinea pig (
mV over
minutes [120] or
mV over
minutes [134]–[136]). The broad variation in membrane potential changes was not unexpected given the range of changes in
and
reported and in simulations an intermediate value has been achieved.
The individual and combined effects of the four components of ischemia on the temporal evolution of the maximum change in ionic concentrations and the spatial concentration and potential distribution profile after minutes of simulated ischemia are shown in Fig. 3. As ischemia progressed,
continued to increase. The early rise was attributed to
inhibition; the later increases were due to metabolic and respiratory acidosis. The early rise in
was significantly affected by
inhibition with
activation playing a greater role as ischemia progresses and the membrane potential diverges from the
reversal potential. The decrease in
was solely due to acidosis with no impact from
activation or
inhibition.
elevation was contributed to principally by
inhibition, while increased
decreased
. Ischemia caused an initial drop followed by a sustained rise in
.
activation caused an early hyperpolarisation of the membrane potential, which was subsequently countered by the depolarising effects of
inhibition, with acidosis having a limited effect.
Complete ischemia, inhibition,
inhibition, respiratory acidosis and metabolic acidosis are represented by black, yellow, red, blue and purple lines, respectively. Column 1 shows the profile of ionic concentrations and membrane potential after 5 minutes and column 2 shows the evolution of the change in magnitude in ionic concentrations and potential across the border zone with time.
Figure 4 plots the width of the BZ for each ion with striped bars corresponding to extracellular space and the darker the bar the more significant the concentration gradient relative to the initial concentration. This plot shows that had a significantly wider BZ with greater magnitude than
. For pH regulation,
had a narrower BZ compared to the significantly wider
BZ, which may indicate the facilitation of proton transport via
diffusion.
Solid bars indicate intracellular gradient and striped bars indicate extracellular gradient.
Extracellular Potassium Gradients
The BZ width was significantly wider than other BZ ion widths and notably larger than the
BZ width. To determine the cause of this extended
BZ the source of the cumulative changes in
due to transmembrane flux, drift or diffusion over the 5 minutes of ischemia were calculated and plotted in Fig. 5. This showed that only the transmembrane flux had a significant gradient across the ischemic region. Separating the transmembrane flux into the
and
components then identified the
channel, which includes the ATP-inactivated
current, as the cause of this gradient. The gradient of IKb was due to the extensive membrane potential gradient into the ischemic region (see Fig. 3). To confirm that this was the cause of the
BZ width, the membrane potential was calculated as normal, but an additional clamped membrane potential was calculated at each time step. The clamped membrane potential had the same maximum and minimum values as the correct membrane potential but instead of a smooth gradient across the BZ it had a sharp transition over the BZ. A comparison of this clamped and the control membrane potentials is shown in the Fig. 5C. The effects of using a clamped membrane potential to calculate the
flux or the
current on the cumulative changes in
after 5 minutes of simulated ischemia are plotted in Fig. 5B, demonstrating that by removing the gradient in the membrane potential experienced by
there is a significant narrowing in the
BZ.
BZ width. A) Reference change in
after 5 minutes of simulated ischemia caused by transmembrane flux (blue dashed line), diffusion (purple line), drift (red line) and the total change due to all causes (yellow line). The BZ is indicated by the yellow shaded region. B) Change in
in the reference model (yellow line) compared with change in
when
(purple dashed line) or
(blue dashed line) are exposed to a clamped membrane potential. The reference BZ and BZ when
is exposed to a clamped membrane potential are indicated by the yellow and purple shaded regions, respectively.
Drift and Diffusion
To investigate the relative contribution of drift and diffusion to intra region fluxes (inter or extracellular) the drift and diffusion fluxes in each region were plotted, alongside the transmembrane flux, over the length of the strand after minutes of ischemia. Fig. 6 shows the differences in drift and diffusion between
,
and
. As expected from the intra and extracellular gradients, intracellular
and
diffused into and out of the ischemic region in the intra and extracellular space, respectively. The converse was the case for
. Due to the decrease in transmembrane potential, characteristic of ischemic regions, there was a convergence of intra and extracellular potentials with the extracellular potential decreasing in the ischemic region and the intracellular potential increasing. This gradient caused positive ions to drift into the ischemic region in the extracellular space and drift out of the ischemic region in the intracellular space. The converse was true for negatively charged ions. As drift is proportional to the ionic concentration,
drift was significant in the extracellular space and
drift was significant in the intracellular space. For
, drift and diffusion operated in the same direction in the intra and extracellular space. The result was a cyclical movement of
moving into the ischemic region in the extracellular space, while moving out of the ischemic region in the intracellular space. The
movement was also circular but in the opposite direction (Fig. 6P). However, for
drift and diffusion were in opposite directions. In the extracellular space where the
concentration was low, diffusion dominated and
moved out of the ischemic region. In the intracellular space, where there was a higher concentration of
, drift dominated, also causing
ions to move out of the ischemic region. Thus ischemia caused a depletion of
in the ischemic region through both the intra and extracellular space and, contrary to previous hypothesis [13], the model suggests that intracellular
movement is the dominant path for
to leave the ischemic region.
Intra and extracellular A) , B)
and C)
ionic concentrations. D) Intra and extracellular potential. Intracellular drift and diffusion flux of E)
, F)
and G)
. Extracellular drift and diffusion of H)
, I)
and J)
. Transmembrane flux of K)
, L)
and M)
. Schematics showing the general direction of ion movement within and between the intra and extracellular space for N)
, O)
and P)
.
Discussion
In this study a new model of cardiac tissue electrophysiology was developed. The model predicted that the width of the gradient across an ischemic BZ would be significantly wider the
BZ. The cause of this difference was attributed to the voltage dependence of the
channel. The model also demonstrated that, due to electrogenic drift,
moved out of the ischemic region in both the intra and extracellular space which will lead to
depletion.
The model of ionic movement and tissue electrophysiology was developed by combining the Nernst-Plank equations with the bidomain framework. No attempt was made to explicitly validate the proposed tissue model equations due to the paucity of experimental data. However, applying simplifying assumptions with regards to ionic or voltage gradients reduces the proposed equations to the well validated bidomain equations [137], [138] or coupled reaction-diffusion equations [28], [86], respectively, providing support for the validity of the proposed modelling framework. The limited attempts at simulating the spatial temporal evolution of ionic gradients across the ischemic BZ have largely uncoupled the movement of ions and the electric field. Potse et al., [18] demonstrated that measured gradients across an ischemic BZ could be simulated using a model of
diffusion coupled to a source term. The spatially varying
gradient could then be included as a boundary condition to models of transmembrane current in the bidomain equations. This model did not include any effect of electric gradients on
movement,
movement, inter ionic species interactions or the effect of the ischemic region on any other ion gradient. Similar sets of equations to those proposed here have been used for simulating the potential gradient surrounding cells, including the Debye layer [139], electrical propagation along strands of cardiac cells [140], [141] and for modelling ion diffusion in the cable equation [142]. These previous models have either explicitly represented the intra and extracellular domains or only considered the intracellular domain but have not modelled the tissue within the bidomain framework, as derived and implemented here.
The proposed equations can be applied generally in three dimensions as opposed to the one dimensional simulations presented here. The current study does not consider the effects of anisotropy on ion or membrane potential gradients across the BZ, however, if implemented in two or three dimensions the model is capable of representing tissue anisotropy and any effects this may have on BZ gradients. By conserving ionic species the proposed equations provide a more biophysical representation of cardiac electrophysiology than the bidomain equations and can appropriately be applied to simulate a broader range of conditions. However, these benefits come at a cost. Unlike the bidomain equations, with two partial differential equations that can readily be uncoupled and solved as two sets of linear equations [143], the proposed model is nonlinear and contains two parabolic partial differential equations for each ionic species and one elliptic partial differential equation to model the electric potential. This results in a significant increase both in the complexity and number of equations that must be solved and hence comes at a significant increase in computational cost. The proposed framework can be used to simulate electrically active tissue by introducing a full action potential cell model [31]. This would require identifying and separating out the transmembrane pathways for each ionic species present in the cell model to calculate the net transmebrane flux for each ion or in Eqn 18. The internal cell model state variables, including gating variables, Markov states and intracellular
dynamics would also need to be solved, introducing a system of nonlinear ordinary differential equations at each grid point. The small time steps and increased number of degrees of freedom required to simulate electrically active tissue would further increase the computational cost of the proposed model.
A new model of ion movement across the un-stimulated cell membrane was developed. This model has been published online and is available at cellml.org. The model creation approach demonstrated two novel methods. Firstly, the ion transporter densities were uniquely constrained by a small number of experimental data sets and a zero net flux constraint. This provided a repeatable and unique method for determining model parameters under quiescent conditions and could readily be applied to any cardiac electrophysiology model to constrain model parameters. Secondly, in the development of this model a grid of experimental observations were created to provide a comprehensive validation of the model response to changes in ion concentrations and in the presence of blockers of major transporters. Although these observations only provide a qualitative comparison, they do provide a benchmark for quantifying the generality of models. Furthermore, Table 4 identified
experiments that do not appear to have been performed or remain controversial, highlighting the potential for additional experiments.
In this study the process of validating the model against experimental observations demonstrated the general capacity of the model to replicate the majority of experimental results. However, the model was unable to match ten of the
observations. Five of the experimental observations found no change in a measurement. Due to the numerical nature of the model, even very small changes in a concentration can be observed and without adding in a semi arbitrary threshold that should reflect the variability and confidence of each experimental measurement it was not possible for the model to return no change in a value. In general the five remaining failed observations can be attributed to absent mechanisms in the model or inconsistencies with the model and specific experimental setups. The model did not predict the
response to depressed
and the
response to depressed
. This is potentially due to the absence of NaK2Cl from the model, largely due to the lack of data characterising the transporter. The model was unable to replicate the decrease in
due to elevated
. This may be due to the specifics of the experimental protocol. Decreasing
, which should have the opposite effect on
, caused both increases and decreases in
in different studies indicating that the
dependence on
is sensitive to the specifics of the experimental setup. The model did not predict the change in
or
with a decrease in
. Experimental observations report an increase in
with both a decrease or an increase in
. This may indicate that the
is at some minima with respect to
, although this seems unlikely. It more likely reflects inconsistent data due to differences in experimental setups that were impossible for the model to replicate. In cardiac myocytes it is postulated that
and
compete for common buffering sites [144], [145]. The decrease in
may drain the cell of
, reducing the
bound to buffers that could then be occupied by
, resulting in a decrease in
. A common pool of
and
buffering was not present in the model and this may explain the disparity between model predictions and experimental results. Despite the inability of the model to replicate
of the
observations, the majority of the absent mechanisms are expected to play a secondary role during ischemia. NaK2Cl is a potential contributor to the elevation in
[146], although this has been questioned [131], and its absence is unlikely to affect the general model conclusions.
Models simulations found a BZ width for the different ions of between mm (
) and
mm (
). The variation in the width of each ion across the BZ means that there was no single BZ width predicted by the model. The majority of ions transitioned from viable to ischemic concentrations over
mm, with
and
notable outliers. The model predictions are consistent with previous measurements of the BZ between
and
mm [10], [11], [147]–[149]. However, the model did not include the sharp gradients in metabolites over
mm [11] or the effects of regions of reduced perfusion affecting mechanics over 20 mm distance from the ischemic region [150].
The model predicted that and
have significantly different BZ widths (Fig. 4). It was expected that the coupling of
and
via
would result in concordant gradients in these two ions across the BZ. The extended
gradient was attributed to the voltage dependence of the
(Fig 5). Unlike
,
is regulated by two electrogenic transmembrane pathways
and
, whereas
is regulated by multiple exchangers that are electro neutral, or have attenuated voltage dependence, compared to ion channels. This discordance in
and
gradients will have a significant effect on the gradient of action potential morphology across the BZ. Elevation of
depolarises the cell and shortens the action potential duration, whereas elevation of
causes a reduction in action potential duration [151], [152]. The combined effect of the two gradients will be a slower change in resting membrane potential on the length scale of the
gradient and a much more rapid transition in action potential duration due to the faster change in
in conjunction with the change in
over the BZ. The temporal evolution of the two gradients are also distinct, with a sustained constant increase in
over the first
minutes, while
increases rapidly for
minutes before plateauing (Fig. 3). These simulation results are consistent with experimental measurements of ionic concentrations [111], [152] and changes in ECG morphology, which report an early elevation of resting membrane potential, followed by a decrease in action potential duration [153] during early ischemia. These spatial-temporal increases in ion concentrations and secondary effects on electrophysiology will increase tissue heterogeneity and have the capacity to play an important role in the BZ arrhythmogenic substrate.
Experimental measurements have reported a decrease in during ischemia [152]. Loss of
has been attributed to a combination of increased
conduction and
inhibition [152], [154]. However, depletion of
due to transmembrane
movement, in the absence of a potential gradient, would cause an intracellular flux of
into the ischemic zone, due to diffusion, to replenish
, mitigating the effects of changes of
transmembrane flux on
. The model proposed here demonstrated that this is not necessarily the case. The model predicted a significant electrogenic
flux in the intracellular space out of the ischemic zone (Fig. 6), resulting in a net efflux of
out of the ischemic zone both in the intra and extracellular space. The movement of
in the intracellular space out of the ischemic zone would further deplete
in the ischemic region and exacerbate
loss, but may limit
accumulation in the extracellular space.
Limitations
The model is inherently an approximation and hence represents a finite set of known cellular properties and changes that occur during ischemia. In particular, the model treated all buffers as static and rapid, the effects of protons on channel, exchanger and co-transports were not considered, ischemic changes were instantaneous and the model did not include all possible changes or pathways that may affect ionic homeostasis during ischemia.
The tissue model assumed that and
buffers are static, rapid and made up of a single population of binding sites. It is known that some of the buffers for both
and
are mobile, these could be introduced into the model framework as an additional concentration but these effects were approximated, without the additional computational cost of adding a additional ionic concentration, by using effective diffusion constants. The equilibrium assumption was likely to be valid in the current model due to the long time scales of interest; however, simulation of cardiac action potentials would require the re-evaluation of this assumption. It is also known that
[28] and
[155] are buffered by multiple proteins with distinct binding kinetics but over the range of concentrations simulated these multiple buffer species were unlikely to have significant effects.
The model of intracellular dynamics assumes that the SR and cytosolic
concentrations remain in equilibrium, which is clearly not the case during an action potential. The model of intracellular
dynamics provided a numerically efficient representation of intracellular
buffering and SR
uptake. The use of a Hill coefficient of one for SERCA as opposed to the more common and biophysical value of two removed the need for an additional differential equation to model
or the solution of a set of nonlinear equations to model
regulation. Furthermore, over the range of
values studied it was possible to adjust the maximum SERCA flux to minimize discrepancies between a model with a Hill coefficient of one or two.
It is well recognised that play an important role in regulating cellular electrophysiology [156]. Experimental and modelling studies have demonstrated the effects of
on ryanodine receptor opening probabilities,
,
channels,
buffering and SERCA [22], [157]. The majority of the effects of
on
regulation are unlikely to play a significant role in determining the spatial and temporal
and
. However, inhibition of
potentially contributes to
gradients but this will be secondary to the effects of
inhibition.
The model treats all changes for ischemia as instantaneous. The time dependence of inhibition of , activation of
, increased
flux or increased
flux are not known and can only be approximated. A linear ramp in ischemic changes has been used previously, but this fails to consider the possibility that changes occur over different time scales. In order to minimise ambiguity in model simulations an instantaneous change in
,
,
flux and
flux was chosen.
To limit the scope of this study, the effects of cell swelling and the effects of this on changes in ion concentrations [122] were not included in the model. However, previous, modelling studies have found these effects to not significantly alter accumulation and may not fundamentally alter the study conclusions [20]. Mitani and Shattock [146] identified the Na dependent potassium channel as a potential contributor to the elevation of
. However, the effects of an increase in any
current are captured by the elevation in
that does not necessarily need to be the sole result of an increase in conduction in the ATP inactivated
channel. Furthermore, the model does not include an increased
channel conductance during ischemia. This has been reported during ischemia in the form of increased permeability of
through the persistent
current [116] and the ATP activated
channel [158]. However, other groups have found a limited impact of the persistent
channel in
accumulation during ischemia [159] and previous modelling studies have shown that its inclusion is not required to capture the salient features of ischemia in the single cell [20]. For these reasons a potential increase in the
channel conductance was not included in this study.
Summary
A new mathematical framework was derived for simulating cardiac tissue electrophysiology with ion species conservation. The model was used to simulate the movement of ions due to transmembrane channels, pumps and transporters, diffusion and drift in the intra and extracellular space. The model predicted that 1) the sodium BZ is approximately a quarter of the length of the potassium BZ and this is due to the effects of the membrane potential gradient on and 2) that during ischemia there is a gross movement of potassium ions out of the ischemic region in both the intra and extracellular space due to the effects of drift, which will lead to a depletion of
from the ischemic region.
Author Contributions
Conceived and designed the experiments: SAN. Performed the experiments: SAN. Analyzed the data: SAN. Contributed reagents/materials/analysis tools: SAN. Wrote the paper: SAN.
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