A modified approach to determine the six cardiac bidomain conductivities

Abstract

Accurate values for the six cardiac bidomain conductivities are crucial for meaningful computational studies of conduction in cardiac tissue, and are yet to be determined by experimental means. Although previous studies have proposed an approach using a multi-electrode array to measure potentials, from which the conductivities can be determined, it has been found that the conductivities cannot be retrieved consistently when the noise in the potentials varies. This paper presents a protocol, which not only has been shown to retrieve the conductivities to a reasonable accuracy, but does so under the presence of a more appropriate additive Gaussian noise model, while using fewer computational resources. Through repetitions of the protocol, a comparison of two pre-fabricated 128 electrode arrays, one array with a square arrangement of electrodes and the other with a rectangular arrangement, was made against a 75-electrode array proposed in previous studies. Results indicated that the two pre-fabricated arrays were generally more capable of obtaining the cardiac conductivities to a higher degree of accuracy than the 75-electrode array. The 128-electrode rectangular array was orientated such that the length of the array first ran along the direction of the fibres, then was reorientated such that the length of the array ran perpendicular to the direction of the fibres. The 128-electrode rectangular array, when orientated in this manner, was more capable of retrieving the conductivities than the remainder of the arrays tested, and thus we suggest this arrangement be used during experimental trials.

1. Introduction

A lack of understanding of cardiac tissue’s electrical properties is one of the major challenges in being able to realistically simulate cardiac electrophysiological behaviour [1–3]. Such simulations can facilitate an understanding that often cannot be achieved through experimental means alone. Insights gained through simulations of myocardial ischaemia, defibrillation, ST-segment deviation or ventricular fibrillation are essential for expanding the current knowledge and can perhaps even inform or guide clinical decision making in the future. However, for these simulations to be useful, a close match must exist between the system’s true state and the output of the simulation [4].

Such electrophysiological simulations are often performed through the use of the bidomain model [5], which takes into account differing electrical properties of the two interpenetrating domains of cardiac tissue [6,7], extracellular ($e$) space and intracellular ($i$) space, through the averaging of the electrical properties of each domain. It is known that [8] cardiac tissue is arranged as fibres in a laminar structure, which in turn are stacked with a slight offset relative to one another along a line between the outer wall of the heart and the inner wall of the heart. Along with accounting for the rotation of the fibres due to this phenomenon, the bidomain model assumes that electrical current propagates in three directions: along the direction of the fibres (longitudinal $\left(l\right)$), across the direction of the fibres within the sheet (transverse $\left(t\right)$), and normal to sheets of the fibres (normal $\left(n\right)$). Consequently, there are six conductivity values that are required to characterise the electrical behaviour of the cardiac tissue in each of the orthogonal directions within each space: $g_{il}{g}_{it},g_{in},g_{el},g_{et}$ and $g_{en}$.

However, despite efforts over the last fifty years, difficulties associated with making and interpreting experimental measurements have prevented researchers from determining conductivity parameters that are consistent and account for the anisotropic nature of cardiac tissue. Under the assumption that the conductivity values are identical in the transverse and normal direction, experimentalists were able to determine four of the six conductivity values of the left ventricle [9–11]. However, a lack of consensus exists about the accepted set of four experimental conductivity values due to the significant variance between the different studies. Other studies have also shown that all six conductivity values are necessary to accurately model the electric field in the left ventricular tissue [12].

Studies in the past decade have proposed various techniques for retrieving all six bidomain conductivities, through either experimental and theoretical means [13,14], while several other studies were able to experimentally retrieve a subset of the conductivities [15–17]. However, no study has fully retrieved these values through purely experimental means [3].

Caldwell, Hooks, Trew and colleagues have presented pioneering work where an 11 needle array of plunge needles with $1-4\text{mm}$ electrode spacings and 137 recording sites was designed, fabricated, and used to demonstrate the anisotropic nature of the tissue [13,18]. These studies were able to determine the ratios between the conductivities under the assumption of a monodomain medium. However, this method could not obtain the six bidomain conductivities from the monodomain conductivities.

Trew and colleagues [14] also utilised a $4\text{mm}\times 4\text{mm}\times 4\text{mm}$ array with 325 electrodes to obtain experimental measurements. A comparison between the potentials obtained through virtually constructed cardiac tissue and the experimental results allowed for the retrieval of the intracellular conductivities. The extracellular conductivities, however, were provided by previous experimental results for the sums of the monodomain conductivities [13]. Costa, Frank and colleagues [19] have proposed a technique to obtain the six bidomain conductivities from the monodomain conductivities, which are themselves obtained through 1D cable simulations. More recent publications from Barr, Pollard and Waits [20–22] provide methods to separate the extracellular and intracellular contributions; however, they are limited by practical issues. Various other advances in this area and techniques to retrieve four of the six conductivities are detailed in Ref. [3].

In the past decade, Johnston and Johnston [23] have designed a 75-electrode array that can be utilised to obtain potential cardiac measurements. By employing a mathematical model, a solution method and a two-pass inversion technique to determine cardiac conductivities from potential measurements, it was shown in the simulation study that all six of the bidomain conductivities could be determined to a reasonable degree of accuracy. However, in addition to the inversion process being computationally demanding, our recent analysis of this two-pass inversion approach has shown that the conductivities yielded are inconsistent between repetitions.

This paper presents a novel protocol that can be utilised to theoretically validate the use of a particular electrode array to retrieve the bidomain conductivities, while accounting for more accurate sensor noise. The protocol has been developed to enable a more thorough understanding of the retrieved conductivities, while ensuring that the results are consistent between repetitions. The novel protocol also has the added benefit of not being as computationally demanding as previous techniques.

To obviate the necessity of building a bespoke electrode array capable of obtaining cardiac potential measurements, we propose utilising a pre-fabricated array. However, it is uncertain if these pre-fabricated arrays can retrieve the cardiac conductivities to a similar degree of accuracy to those proposed previously [23]. For this reason, we plan to employ the protocol to theoretically determine if the pre-fabricated arrays can retrieve accurate cardiac conductivities.

This paper is structured as follows: Section 2 introduces the model as well as the solution technique, Section 3 justifies the use of the modified protocol for inversions, and in Section 4, repetitions of the modified protocol are used to compare various pre-fabricated arrays with the 75-electrode array. Section 5 presents a discussion of the results from Section 4, as well as makes recommendations for the choice and deployment of the most suitable of the pre-fabricated electrode arrays to be used during experimental trials. The paper concludes with a summary of the work presented, as well as possible future avenues of research in this area.

2. Bidomain model and solution method

2.1. Model geometry and governing equations

In the current study, the ventricular muscle is represented by a slab geometry to reduce the complexity of the solution techniques [23–25]. The model is constructed using a block of cardiac tissue, 2 cm in the $x$ and $y$ directions $(-1\le x\le 1,-1\le y\le 1$), with a thickness of 1 cm extending from a plane at $z=0$, representing the epicardium, to $z=1$, representing the endocardium. The endocardium is also assumed to be in contact with a pool of blood whose thickness extends to infinity. The model is constructed under the assumption that the epicardium’s cardiac fibres are aligned with the $x$-axis [26]. This defines the longitudinal direction of the cardiac tissue. We also define the transverse direction of cardiac tissue as the $y$-axis and the normal direction (orthogonal to the transverse and longitudinal directions) as the $z$-axis.

The following bidomain governing equations describe the distribution of the intracellular and extracellular cardiac potentials ${\phi }_i$ and ${\phi }_e$, respectively [5],

$$\nabla \cdot M_i\nabla {\phi }_i=\frac{\beta }{R}\left({\phi }_i-{\phi }_e\right),$$ (1)

$$\nabla \cdot M_e\nabla {\phi }_e=-\frac{\beta }{R}\left({\phi }_i-{\phi }_e\right)-I_c,$$ (2)

where $\beta$ is the ratio of surface cell area to cell volume, $R$ is the specific resistance of the membrane separating the intracellular and extracellular domains and $I_c$ is the current per unit volume applied in the extracellular space. The conductivity tensors, ${\mathbf{M}}_q(q=i,e)$, account for the anisotropy that arises partly due to the differing conductivities in the three orthogonal directions, as well as from the rotation of the cardiac fibres from the epicardium to the endocardium. Assuming that the rotation of the cardiac fibres between the epicardium and the endocardium varies linearly with depth through an angle $\alpha$ [27,28], the conductivity tensors, ${\mathbf{M}}_q$, for a rectangular block of cardiac tissue are of the form [5],

$$M_q=\left[\begin{array}{ccc}\left(g_{ql}-g_{qt}\right)c^2+g_{qt}& \left(g_{ql}-g_{qt}\right)cs& 0\\ \left(g_{ql}-g_{qt}\right)cs& \left(g_{ql}-g_{qt}\right)s^2+g_{ql}& 0\\ 0& 0& g_{qn}\end{array}\right],$$ (3)

where $q=i$ or $e,c=\text{cos} \left(\alpha z\right)$ and $s=\text{sin} \left(\alpha z\right)$.

Due to the blood being a source-free region, its electrical potential, ${\phi }_b$, is governed by Laplace’s equation

$${\nabla }^2{\phi }_b(x,y,z)=0.$$ (4)

2.2. Boundary conditions

Working under the assumption that the heart is resting in a medium that is non-conductive gives

$$\frac{\partial {\phi }_e}{\partial z}=\frac{\partial {\phi }_i}{\partial z}=0\hspace{1em}\text{at}\hspace{1em}z=0.$$ (5)

Assuming that there is the continuity of the extracellular potential and current at the interface between the myocardial tissue and the blood at the endocardium gives

$${\phi }_b={\phi }_e\hspace{1em}\text{and}\hspace{1em}{g}_b\frac{\partial {\phi }_b}{\partial z}=g_{en}\frac{\partial {\phi }_e}{\partial z}\hspace{1em}\text{at}\hspace{1em}z=1,$$ (6)

where $g_b$ is the conductivity of the blood.

Furthermore, it is assumed that the intracellular space is insulated by the extracellular space at the epicardium; thus,

$$\frac{\partial {\phi }_i}{\partial z}=0\hspace{1em}\text{at}\hspace{1em}z=1.$$ (7)

The $x$ and $y$ boundaries are insulated at the boundaries of the domain, which gives the final set of boundary conditions,

$$M_q\nabla {\phi }_q\cdot n=0,\text{and}\nabla {\phi }_b\cdot n=0,\text{at}x,y=\pm 1,$$ (8)

where $\mathbf{n}$ is the outward pointing normal from the boundary.

Lastly, it is worth noting that,

$${\phi }_b\to 0\hspace{1em}\text{as}\hspace{1em}z\to \infty .$$ (9)

2.3. Solution technique

The bidomain equations, (1) and (2), can be partly solved analytically through the expansion of each of ${\phi }_i$ and ${\phi }_e$ as a Fourier series,

$${\phi }_q(x,y,z)=\sum _{k=0}^{\infty }\sum _{n=0}^{\infty }{C}_{kn}^{q}cosk\pi \mathit{\text{ycos}}n\pi x+D_{kn}^{q}cosk\pi \mathit{\text{ysin}}n\pi x+E_{kn}^q\mathit{\text{sink}}\pi \mathit{\text{ycos}}n\pi x+F_{kn}^q\mathit{\text{sink}}\pi \mathit{\text{ysin}}n\pi x,$$ (10)

where $q=i$ or $e$ and $C_{kn}^q,D_{kn}^q,E_{kn}^q$ and $F_{kn}^q$ are unknown coefficients.

The expansion is substituted into Equations (1) and (2), resulting in two sets of four ordinary differential equations in terms of the unknown coefficients $C_{kn}^q\left(z\right),D_{kn}^q\left(z\right),E_{kn}^q\left(z\right)$ and $F_{kn}^q\left(z\right)$. A one-dimensional finite difference scheme is used to determine these coefficients, yielding a banded system of linear algebraic equations, which is solved numerically through the use of standard techniques [29]. A detailed explanation of the full solution technique is provided in previous papers [25,30].

2.4. Model parameters

The value of the ratio of surface cell area to cell volume $\left(\beta \right)$ used is that of Weidmann [31] with $\beta =2000{\text{cm}}^{-1}$, while the specific resistance of the cell membrane $\left(R\right)$ is taken to be $9100\Omega {\text{cm}}^2$, consistent with values found in the literature [9,25,32]. The direct current being injected into the tissue $\left(I_0\right)$ has a magnitude of $1\mu {\text{Amm}}^{-3}$, similar in magnitude to values proposed and used in previous simulation studies of the myocardium [25,32]. Finally, $\alpha$ and $g_b$ are fixed at angle 120 and 6.7 mS/cm, respectively, as used in previous studies [24,25].

2.5. Bidomain conductivity values

However, to date, no study has been able to measure all six of the bidomain conductivities experimentally, datasets have been published in the literature [33–35], based on partly experimental and theoretical work that provide estimated values for the six conductivities. However, there is a notable variation between these datasets. Other published datasets present only a subset of the six bidomain conductivities. A comprehensive review of conductivity datasets can be found in Ref. [3]. The conductivity set that is used in the upcoming studies is given by Johnston et al. [36], and was chosen as it was developed from a survey of the literature. The nominal values for the conductivities within this dataset are as follows, where units are in $\text{mS}/\text{cm}:g_{il}=2.40,g_{it}=0.24$, $g_{\text{in}}=0.10,g_{el}=2.40,g_{et}=1.60$ and $g_{en}=1.00$. Along with the conductivity values, a space constant, defined in terms of the various parameters of the bidomain model, is often used to suggest current electrode placements [32,37]:

$${\lambda }_w=\sqrt{\frac{R}{\beta }\frac{g_{iw}{g}_{ew}}{g_{iw}+g_{ew}}},$$ (11)

where $w=l$, $t$, or $n$ and ${\lambda }_w$ is the space constant in a given direction.

3. Retrieving cardiac conductivities

3.1. Electrode array

Several multi-electrode arrays, proposed in previous studies, have been shown to retrieve the bidomain conductivities through simulations of experimental measurements of cardiac potentials [6,25,38,39]. Although, to date, no study has been able to measure all six of the bidomain conductivities experimentally [3]. Johnston and Johnston [30,39] utilised a $2\text{mm}\times 2\text{mm}\times 2\text{mm}$ multi-electrode array consisting of 25 micro-needles, each containing three electrodes arranged in three planes (75 electrodes in total) (Fig. 1). This array was theoretically shown to retrieve all six of the conductivities; however, the array was unable to retrieve the intracellular conductivities as accurately as the extracellular conductivities. Given the time and expense associated with manufacturing such an electrode array [3], in this work, we have considered using a pre-existing array. NeuroNexus Matrix Arrays (NeuroNexus, Ann Arbor, MI, USA), though initially designed for use in brain tissue, are suitable for use across various biological tissues. Two electrode arrays are considered in this study; both consist of four forks, with each fork consisting of four shanks and each shank consisting of eight electrodes, thus resulting in 128 electrodes (Fig. 2). Two electrodes are used for the current application, while the remainder are used as measuring electrodes. The spacing between the shanks is $400\mu \text{m}$, with a $200\mu \text{m}$ spacing between electrodes on a shank, and either a $400\mu \text{m}$ or $600\mu \text{m}$ spacing between forks depending on the array chosen. For ease of reference, these arrays are labelled as either the 400-400-200 array or the 400-600-200 array, respectively. Due to the extra degree of freedom given by the asymmetry of the 400-600-200 array, we want to study the effects of orientating this array such that the $600-\mu \text{m}$ spacings are along the longitudinal direction ($x$-direction); this configuration will be labelled the ‘600-400-200 array’. We stress that the 400-600-200 array and the 600-400-200 array are identical but are orientated differently. For all the arrays, a ‘closely-spaced’ pair of electrodes is initially used to apply current. These are shown as red Sr and Sn electrodes in Figs. 1 and 2. This configuration is chosen as per previous studies [32] that suggest the distance between the two current injection electrodes should be less than the space constant when determining the extracellular conductivities. Using this specific electrode configuration, the first set of measurements is made (first pass). Then the current injection electrodes are placed diagonally opposite to each other such that the distance between these electrodes is greater than the space constant - the ‘widely-spaced’ configuration. This configuration is shown as the green Sr and Sn electrodes in Figs. 1 and 2. Using the wide-spaced configuration, a second set of measurements are made (the second pass). During the first pass, a high proportion of current flows through the extracellular space [32], thus resulting in the accurate retrieval of the extracellular conductivities. Whereas during the second pass, a proportion of the current is re-directed into the intracellular space, thus aiding with the retrieval of the intracellular conductivities [32].

Fig. 1.

The $2\text{mm}\times 2\text{mm}\times 2\text{mm}$ 75-electrode multi-electrode array, comprised of three electrodes per shank in three ‘layers’, that is used for making the potential measurements [30,39]. Sr and Sn, respectively, mark current source and sink electrodes. Current electrodes used during the first and second passes are marked red and green, respectively. The spacing between the electrodes and also between the shanks is 0.5 mm. The ‘closely–spaced’ electrodes on the shanks in the inner square are used in the first pass. The second pass uses all the electrodes.

Fig. 2.

Schematics of two NeuroNexus128-electrode array configurations, compromised of four forks, with each shank containing eight measuring electrodes. Distance between the forks can either be $400\mu \text{m}$ or $600\mu \text{m}$ depending on the array chosen. All electrodes, apart from those applying current, are used to measure cardiac potentials marked in blue. Current source and sink electrodes, labelled $\text{Sr}$ and $\text{Sn}$, used during the first and second passed are marked red and green, respectively.

3.2. Inversion algorithm

Theoretically testing to see whether the NeuroNexus electrode arrays can retrieve accurate values for the bidomain conductivities first involves simulating a set of measured potentials through a forward solution of the bidomain model (Section 2). Noise is added to the measured potentials to simulate experimental measurements, from which the two-pass inversion process is then applied to retrieve the original conductivity parameters from the ‘noisy’ potentials.

3.2.1. Introducing noise

In previous work, multiplicative noise was introduced to the synthetic potential measurements at each electrode on the array [24,30,39] in order to mimic noise caused by uncorrelated sources. For example, geometric noise caused by the placement of the electrode array relative to the tissue, circuitry noise and inherent noise caused by equipment could all affect the potential measurements. It is typical for the noise in these scenarios to be independent of the system’s state. However, for multiplicative noise, if the magnitude of the potential were small, the noise being introduced would also consequently be small, thus allowing for the possibility to not incorporate the inherent levels of noise found in the system as a whole. Here, additive noise will be used to mimic experimental noise, as per convention in systems of a similar nature [40–42]. A Gaussian distribution with a mean of zero and a defined standard deviation is used to obtain noise samples, which are added to the exact synthetic potentials found on each of the measuring electrodes via the bidomain model. The resulting set of potentials is labelled a ‘measurement set’. Refer to Section 3.8 for more detail. Here, 30–200 measurement sets are synthetically produced, using the same noise distribution but with different noise samples. The synthetic potential measurements from all the measurement sets are averaged for each electrode, resulting in a set of potentials (‘averaged measurement set’) used to obtain the cardiac conductivities. The influence of the number of measurement sets chosen on the accuracy of the retrieved conductivities, and the standard deviation of the distribution will be discussed in detail in Sections 4.1.2 and 3.8, respectively.

3.3. Tikhonov regularisation

Difficulty arises in retrieving the conductivities due to non-linear dependence between the bidomain conductivities and the cardiac potentials in the forward model, which is represented in the following form,

$$G(m)=\phi ,$$ (12)

where $\mathbf{G}$ is the forward model, $\mathbf{m}={\left[g_{il},g_{it},g_{in},g_{el},g_{et},g_{en}\right]}^T$ and $\mathit{\phi }$ is the vector of measured potentials. Due to the presence of noise in experimental measurements of cardiac potentials, it is necessary to minimise the following Tikhonov functional [43] to obtain an approximation for $\mathbf{m}$

$$\parallel G(m)-\phi {\parallel }_2^2+{\gamma }^2\parallel m{\parallel }_2^2,$$ (13)

where $\gamma$ is the regularisation parameter. The functional is minimised using a SolvOpt solver [44], which employs the modified Shor’s r-algorithm to minimise non-linear multivariate functions using the method of exact penalisation [45]. The SolvOpt solver is modified to include an efficient stopping criterion as well as a technique for choosing the initial step size [44]. Additionally, a constraint that the conductivities must be positive is also applied. The inversion process is terminated once the relative error in the functional ((13)) is less than ${10}^{-6}$ for two successive iterations.

3.4. First pass of the inversion algorithm

The measuring electrodes (as shown in Figs. 1 and 2) are used to generate noisy potentials via the procedure outlined above or are used to measure experimental potentials. The potentials are then used to simultaneously fit $g_{il,}{g}_{it},g_{in},g_{el}{g}_{et}$ and $g_{en}$ using the solver. As in Refs. [23,24], initial values for the conductivities are taken to be ${10}^{-3}\text{S}/\text{cm}$ except for $g_{in}$ which is taken to be ${10}^{-4}\text{S}/\text{cm}$. During the first pass, the Tikhonov functional (13), minimised by the solver is given by,

$$f_1=\sum _{i=1}^E{\left[{\phi }_M(i)-{\phi }_C(i)\right]}^2+{\gamma }^2\left[g_{il}^2+g_{it}^2+g_{in}^2+g_{el}^2+g_{et}^2+g_{en}^2\right],$$ (14)

where ${\phi }_M$ and ${\phi }_C$ are measured and calculated potentials at electrode $i$ and $E$ is the number of measuring electrodes. The regularisation parameter $\gamma$ taken to be ${10}^{-2}$ when the conductivity units are S/cm, as in previous studies [28,30]. This process yields a set of six bidomain conductivities.

3.5. Second pass of the inversion algorithm

Like the first pass, cardiac potentials are either simulated using additive noise or measured using the second pass current injection electrodes (Figs. 1 and 2). The second pass aims to refine the intracellular conductivities measured in the first pass; thus only $g_{il},g_{it}$ and $g_{in}$ are retrieved, with the remainder of the conductivities held constant at their values found in the first pass. The Tikhonov functional to be minimised for the second pass is given by,

$$f_2=\sum _{i=1}^E{\left[{\phi }_M(i)-{\phi }_C(i)\right]}^2+{\gamma }^2\left[g_{il}^2+g_{it}^2+g_{in}^2\right].$$ (15)

The second pass is used to refine the intracellular conductivities.

3.6. The original protocol for retrieving conductivities

The original protocol, used by Johnston and Johnston [23] with the 75-electrode array (Fig. 1), involved making cardiac potential measurements on the array, equivalent to a measurement set, from which inversion was performed immediately. Thus any variability in the potential measurements due to noise was carried into the inversion algorithm and thus into the inversion results. This process was repeated numerous times, and the inversion results were averaged to yield a single set of six bidomain conductivities. Percentage relative error was used to determine the accuracy of the retrieved conductivity parameter,

$$\text{Relative}\text{Error}\%=\frac{|\text{Retrieved}\text{Value}-\text{Exact}\text{Value}|}{\text{Exact}\text{Value}}\times 100\text{\%}.$$ (16)

we have recently found that there is a lack of consistency between sets of inversion. As an example, repetitions of the protocol to retrieve five sets of the extracellular cardiac conductivities yielded relative errors in $g_{el}$, $g_{et}$ and $g_{en}$ that ranged from 0.1% to 0.75%, 0.09%-0.6% and 0.09%–0.28%, respectively. Variations in the retrieved intracellular conductivities were even more pronounced, with relative errors typically having a range of 3%–7%. These results were obtained for 5% noise using 100 sets of potential measurements on the 75-electrode array. We believe this variation is due to the introduction of multiplicative noise, where particular noise sets might tend to favour the retrieval of a particular conductivity value while other noise sets favour a different conductivity value. A more thorough understanding of the spread of conductivities could be achieved through repetitions of the original protocol; however, this is infeasible due to the computational costs involved, as will be discussed in Section 4.2.

3.7. Introducing a modified protocol for retrieving conductivities

1. Using the first pass current electrodes, potentials are either measured experimentally or simulated on an electrode array using the bidomain model and a nominal set of conductivity values, thus giving a single measurement set.

2. If potentials are simulated, noise samples are added to these synthetic potentials to mimic an experimental measurement set. In the case experimental measurements are available, disregard this step.

3. Steps 1 and 2 are repeated a number of times, thus yielding a number of measurement sets.

4. The potentials from all the measurement sets are averaged at each electrode, giving an averaged measurement set.

5. The first pass inversion is performed on the average measurement set of potentials to retrieve the six cardiac conductivities.

6. Steps 1–4 are repeated using the second pass current electrodes. The second pass inversion is performed, with the extracellular conductivities fixed at their values obtained from step 5. This refines the values of the intracellular conductivities.

Once again, it is crucial to note that noise is only added in the absence of experimental trials. If experimental data are available, the measurement sets will be averaged, and the appropriate inversion algorithm (first pass or second pass) will be performed to retrieve the bidomain conductivities. However, due to the lack of experimental results available, the results presented in this paper are obtained synthetically. In Section 4.1, we will justify that step 3 should be repeated 100 times.

3.7.1. Modified protocol with repeated inversions

Analysis of the modified protocol indicates that retrieved conductivities are inconsistent between repetitions, similar to results obtained through the original protocol. We believe this is due to the dependence of the retrieved conductivities on the noisy potentials. Consequently, one noise set might favour the retrieval of a particular conductivity value over others. Thus, we cannot arrive at concrete conclusions due to the variability of these synthetic results. In order to obtain a general understanding of the spread of the retrieved conductivity values, and thus provide more definitive conclusions, the inversions (Steps 1–5 and Step 6) were repeated using different instances of averaged measurement sets. Repeating the inversion allows us to understand i) the width of the spread of the retrieved conductivities, ii) possible outliers, iii) and provide an indication of the expected retrieved result. Due to the decreased computational cost of the modified protocol (discussed in Section 4.2), the first pass inversion (Steps 1–5) is able to be first repeated a number of times. The medians of the retrieved conductivities from the first pass are then used as initial values in the inversion algorithm’s second pass. The second pass (Step 6), similar to the first pass, is also repeated a number of times. Accordingly, sets of values of size $n$ for each cardiac conductivity are obtained at the end of this procedure, where $n$ is the number of inversion repetitions. For ease of reference, and to avoid confusion with the standard modified protocol, the modified protocol with repeated inversions will now be referred to as the MPRI. It is crucial to note that these repetitions of inversion in MPRI are performed with the sole purpose of acquiring a more thorough understanding of the behaviour of the retrieved conductivities, and thus allow for more definitive conclusions to be made, and would not be done in an actual experiment. Note that one of the purposes of this paper suggest a suitable electrode array to be used during experimental trials. To ensure that our conclusions are consistent and valid, we propose using MPRI in the upcoming studies. Recall that MPRI yields sets of values for each cardiac conductivity. The relative error of each value within these sets is calculated, through the use of Equation (16), and the medians of these relative errors are used for comparisons. For ease of reference, we will label this value the ‘median relative error’. Note that this is different from the ‘relative error of the median retrieved value’. It is crucial to note that though the latter can provide more desirable relative errors, we believe the former provides a better indication of the relative errors we might typically observe from one repetition of the modified protocol. Along with the median relative error, the measure of the dispersion of the retrieved conductivities from MPRI is visualised through the use of notched box plots.

3.8. Standard deviation of noise distributions

As mentioned in Section 3.2.1, studies using the original protocol [23,24,30,39] have used multiplicative noise. Typically, the noise levels introduced with multiplicative noise were 2%, 5%, 10% and 25%. A comparison platform must be established to understand the influence of utilising MPRI with additive rather than multiplicative noise. For comparisons of the protocols, a normalised root mean square error (NRMSE) is used,

$$\text{NRMSE}=\frac{\sqrt{\frac{1}{E}{\sum }_{i=1}^E{\left({\phi }_{ni}-{\phi }_{exi}\right)}^2}}{\overline{\phi }},$$ (17)

where ${\phi }_{ni}$ and ${\phi }_{exi}$ are the noisy potential and exact potential measured at the $i$th electrode respectively, $E$ is the number of measuring electrodes and $\bar{\phi }$ is the root mean square potential across all the measuring electrodes. Here,

$${\phi }_{ni}={\phi }_{exi}+N_i,$$ (18)

where $N_i$ is the noise added to the $i$ th electrode. Thus is follows that

$$\text{NRMSE}=\frac{\sqrt{\frac{1}{E}{\sum }_{i=1}^E{N}_i^2}}{\overline{\phi }}\approx \frac{\sigma }{\overline{\phi }},$$ (19)

where $\sigma$ is the standard deviation of the Gaussian noise distribution. Hence, comparisons between MPRI and the original protocol can be made, provided that the standard deviations are chosen such that the NRMSE is approximately equal to the multiplicative noise levels introduced. This is, of course, dependent on the value of $\bar{\phi }$, which in itself is dependent on the placement of the current electrodes on the electrode array during the first and second passes.

4. Results

4.1. Justification of protocol parameters

With MPRI, several aspects need to be studied to develop a standard protocol for retrieving the bidomain conductivities. Namely, we require an understanding of how the accuracy of the retrieved values of the conductivities depends on (i) the number of measurement sets (Section 3.2.1), (ii) the number of repetitions of the inversion algorithm, and (iii) the standard deviation of the noise distribution. These studies are performed on the 75-electrode array (Fig. 1) as this array has been shown in previous studies to theoretically be able to retrieve the cardiac conductivities to a reasonable accuracy using multiplicative noise [39], and thus allows us to eliminate one source of variability.

4.1.1. Determining NRMSE values

Following the procedure stated Section 3.8, the root mean square of the cardiac potentials $\left(\bar{\phi }\right)$ were obtained from first and second pass current electrode configurations across the arrays. It was found that the 1st decile and the 9th decile of $\bar{\phi }$ were 2.895 mV and 4.506 mV, respectively. For this given range of potentials, the associated values for $\sigma$ for the different NRMSE values ((19)) are provided in Table 1.

Table 1.

Standard deviation $\left(\sigma \right)$ values for associated NRMSE values in Volts.

NRMSE Values (%)	Standard Deviation $\left(\mathit{\sigma }\right)$ of Noise Distribution (Volts)
2	7.5 × 10−5
5	1.5 × 10−4
10	3.5 × 10−4
25	1.0 × 10−3

4.1.2. Number of measurement sets

This study aims to determine the number of measurement sets required such that the accuracy of the retrieved conductivities is maximised. Recall that the noise samples obtained from a Gaussian distribution are added to the synthetic potential of each measuring electrode - simulating a single experimental measurement at that given electrode - to yield a measurement set (Section 3.2.1). This process is repeated, as it would be during an experiment, and the resulting potentials at each measuring electrode are averaged. The averaged potentials across all the measuring electrodes make up the averaged measurement set. Note that we are justified in taking the mean of the potentials at each measuring electrode as they were shown to be normally distributed through the use of the Shapiro Wilk test [46]. Although using a high number of measurement sets helps eliminate noise and results in conductivities being retrieved to higher accuracy, continued experimental measurements might include drifting of potentials, and this could lead to inaccurate measurements. As a compromise, we investigate retrieving conductivities using 30, 50, 100, 150 and 200 measurement sets with MPRI. These results are presented in the top half of Table 2, where for ease of comparison, the median relative errors for each of the intracellular and extracellular conductivities are summed. The results presented are obtained with 30 repetitions of inversion for reasons discussed in Section 4.1.3 and using 5% NRMSE equivalent value. As expected, using 200 measurement sets yields the best results. We conclude that using approximately 100 measurement sets, though not as ideal as 200 measurement sets, provides an acceptable compromise in being able to reduce potential drift, while still being able to retrieve accurate cardiac conductivity values.

Table 2.

Sum median relative errors (RE) of intracellular and extracellular cardiac conductivities for a variety of the Number of Measurement Sets (NOMS) and Repetitions of Inversion (ROI). Results presented were obtained for a noise distribution with a standard deviation of $1.5\times {10}^{-4}\text{V}$ (5% error). Results were obtained on the 75-electrode array (Fig. 1).

		Sum RE of $g_e$ (%)	Sum RE of $g_i$ (%)
NOMS	30	0.99	9.32
	50	0.78	6.65
	100	0.46	4.36
	150	0.48	4.27
	200	0.31	3.03
ROI	10	0.40	5.38
	30	0.48	4.64
	50	0.48	4.51
	70	0.55	4.52
	100	0.53	4.54

4.1.3. Number of inversion repetitions in MPRI

The median relative errors of repetitions of inversion (ROIs), are presented in Table 2 and were obtained using 100 measurement sets and using 5% NRMSE equivalent value. We found that the measure of the dispersion of the retrieved conductivities is similar for 30 ROI and higher and that 10 ROI can provide results with similar errors to using 30 ROI. However, the dispersion of the conductivities can sometimes be misleading for higher noise levels when 10 ROI are used. Based on this, between 10 and 30 repetitions are used in what follows, depending on the specific scenario. For example, for lower noise levels, we typically use a smaller number of inversion repetitions, and for higher noise levels, we use a higher number of inversion repetitions. Again, these conclusions hold for noise distributions of various standard deviations and for both passes.

4.1.4. Consistency of results - MPRI

With MPRI established, we undertook a study to determine whether two individual iterations of MPRI produce a similar set of results. It was found that the medians and the spread of the retrieved conductivities were consistent for any two individual iterations of MPRI. Though outliers were occasionally found, these values do not often contribute significantly to the overall distribution of results.

4.2. Computational costs: the original protocol vs. the modified protocol vs. MPRI

As highlighted in the Introduction, the inversion process is computationally intensive. The original protocol involved invoking the inversion algorithm for every set of cardiac potential measurements. In contrast, the modified protocol only employs the inversion algorithm once all the cardiac potential measurements have been made, and an average measurement set has been obtained. In the modified protocol, due to the reduced variability of the averaged measurement set, the inversion algorithm also retrieves the conductivities more quickly than the original protocol. These factors allowed for many repetitions of the inversion to be performed with the modified protocol (MPRI), thus providing us with a more thorough understanding of the spread of the retrieved conductivities. Two Intel Xeon(R) E5-2630 v3 containing eight cores, each with a clock speed of 2.40 GHz, were used to run these inversions. The inversion algorithm was written in C++ and parallelised to run on multiple CPU threads using a message passing interface (MPICH2) and techniques given by Ref. [47]. Under this scenario, a first pass modified protocol inversion with 100 measurement sets typically had a run-time of approximately 3–5 min, while the second pass modified protocol inversion with 100 measurement sets typically had a run-time of 2–3 min. Thus, in total, we were able to retrieve a set of six bidomain conductivities through the modified protocol in approximately 5–8 min. For comparison, utilising the same hardware, the original protocol would yield a set of six bidomain conductivities in approximately 9–13 h. As we have justified in the previous sections, that the number of repetitions of inversion required in MPRI is approximately between 10 and 30, and the number of measurement sets is approximately 100. We found that 30 repetition of inversions in MPRI yields 30 sets of each cardiac conductivity in approximately 2–4 h. Note the variability in computational times is caused by the noise, where certain noise sets tend to faster computational times while others might not. Furthermore, lower noise levels typically led to lower computational times and vice versa. Thus, to generalise these findings, the modified protocol generally yields the six bidomain cardiac conductivities in $1/q$th of the time required to complete the original protocol, where $q$ is the number of measurement sets. Similarly, MPRI would typically yield a set of conductivities of size $n$ in $n/q$ of the time required to complete the original protocol.

4.3. Validating MPRI

One of this paper’s aims is to synthetically validate the use of the MPRI as a means to compare multi-electrode arrays. We compared the original protocol and MPRI for noise levels and NRMSE equivalent values of 2%, 5%, 10% and 25%, where the MPRI used thirty repetitions of the inversion algorithm, each using 100 measurement sets, while the original protocol used 100 measurement sets. The median relative error for each retrieved bidomain conductivity from MPRI (Table 3) is compared with the relative errors of the six conductivities obtained from the original protocol. Note that the use of the median relative error means that we are not using the best set of results retrieved from MPRI (most accurate) for comparison, nor are we using the worst set of results retrieved from MPRI (least accurate), but rather we are using the set of results that is most likely to be retrieved from one repetition of the modified protocol. Thus, this provides a consistent, verifiable method to understand the accuracy of the conductivity values that would typically be retrieved from the modified protocol, and facilities a fair comparison between the two protocols.In Table 3, it can be seen that the median relative error values from MPRI have a similar degree of accuracy as those of the original protocol. Though there are instances where the MPRI provides better results and vice versa, we believe these to be insignificant due to the inconsistency and uncertainty associated with the results of the original protocol as detailed in Section 3.6. Consequently, it would not be reasonable to definitively conclude that one protocol is superior. However, it is possible to suggest that the MPRI, and thus the modified protocol, is as adequate at retrieving cardiac conductivities as the original protocol, due to the relative errors of both protocols being of a similar magnitude. Note that, though not presented, the same set of conclusions also holds for 10% and 25% noise, and their corresponding NRMSE values, as well.

Table 3.

Percentage relative errors (%) of the set of conductivities retrieved by original protocol compared with the median relative errors of cardiac conductivities retrieved via MPRI. Conductivities were retrieved using the 75-electrode array for 2% and 5% noise and their corresponding NRMSE levels.

Noise/NRMSE	Protocol Used	$g_{el}$	$g_{et}$	$g_{en}$	$g_{il}$	$g_{it}$	$g_{in}$
2%	Original	0.07	0.06	0.06	0.06	1.06	1.17
	Modified	0.07	0.09	0.09	0.39	1.00	0.93
5%	Original	0.18	0.13	0.21	1.14	2.44	4.80
	Modified	0.16	0.23	0.17	0.77	1.98	2.05

4.4. Comparison of the arrays

We performed simulations to compare the original 75-electrode array (Fig. 1) with the NeuroNexus arrays (Fig. 2) using MPRI. Thirty inversion repetitions, each with 100 measurement sets, were performed for noise samples obtained from distributions with the 2%, 5%, 10% and 25% NRMSE values. Figs. 3 and 4 show the relative errors for the retrieved conductivity values in the form of a notched box plot. Here the narrowing of the box indicates a 95% confidence interval of the median calculated using a Gaussian-based asymptotic approximation [48]. The lower and upper ‘whiskers’ represent the first and fourth quartile, respectively, with the box representing the interquartile range (second and third quartile). Possible outliers are denoted as points, and the line between the boxes represents the median. A ‘flipped’ appearance of the notch indicates that the 95% confidence interval lies outside the first or third quartile. Thus the size of the notch is a measure of the uncertainty in the median.Figs. 3 and 4 show that the extracellular conductivities are all retrieved to a comparable accuracy to each other and to a higher accuracy than the intracellular conductivities, with relative errors for each extracellular conductivity of approximately 0.07% and 0.15% for 2% NRMSE and 5% NRMSE, respectively, depending on the array used to retrieve the conductivities. With regards to retrieval of the intracellular conductivities, we found that $g_{il}$ is obtained more accurately than the remainder of the intracellular conductivities, with relative errors of approximately 0.5% and 0.75% for 2% NRMSE and 5% NRMSE, respectively. The accuracy of $g_{it}$ and $g_{in}$ was comparable, with relative errors of approximately 0.75–1.25% and 1.5–2.5% for 2% NRMSE and 5% NRMSE, respectively. The results for an NRMSE equivalent of 2% noise (Fig. 3(a)) indicate that the NeuroNexus arrays are more likely to retrieve $g_{el}$ and $g_{et}$ to a higher accuracy than the 75-electrode array. For $g_{\text{en}}$ although the medians of the 400-600-200 array and the 75-electrode array are similar, analysis of the certainty of the median of the 400-600-200 array suggests it is better than the 75-electrode array at retrieving this particular conductivity value. Note that a change of orientation of the 400-600-200 array (the 600-400-200 array) has increased the accuracy of the retrieval of $g_{el}$ and $g_{en}$ compared to the 400-600-200 array. Analysis of the retrieved intracellular conductivities for 2% NRMSE (Fig. 3(b)) indicates that the NeuroNexus arrays are as capable, if not better, at retrieving the three intracellular conductivities than the 75-electrode array except in the retrieval of $g_{it}$ through the use of the 400-400-200 and 600-400-200 array whose fourth quartile ranges are notably larger than the remainder of the arrays. A similar set of conclusions to those above was found for an NRMSE equivalent of 5%. Under this noise level, we do note that the range of the 4th quartile of the 400-600-200 array suggests that it is not as capable of retrieving $g_{en}$ as the 75-electrode array (Fig. 4(b)). However, upon analysis of the certainty of the medians through examination of the notches, we believe that the 400-400-200 array is generally as capable of retrieving this particular conductivity compared to the 75-electrode array. Note that only 2% and 5% NRMSE box plots are presented, as conclusions for 10% and 25% NRMSE are similar. The relative errors for 10% NRMSE were approximately 1.8–3.1 times greater than those observed during the first pass for a 5% NRMSE, while being between 2 and 4.5 times greater for the second pass. Similarly, the relative errors for a 25% NRMSE were generally 5–7 times greater for the extracellular conductivities, and 6–9 times greater for the intracellular conductivities than those observed for a 5% NRMSE.

Fig. 3.

Notched box plots of the relative errors of retrieved cardiac (a) extracellular and (b) intracellular conductivity values for an NRMSE equivalent of 2% on the NeuroNexus Arrays and the 75-electrode array.

Fig. 4.

Notched box plots of the relative errors of retrieved cardiac (a) extracellular and (b) intracellular conductivity values for an NRMSE equivalent of 5% on the NeuroNexus Arrays and the 75-electrode array.

5. Discussion

5.1. Array orientation and number of electrodes

Previous studies [23,24,30,39] indicate that the retrieved intracellular values are typically not as accurate as the extracellular values - a conclusion consistent with results presented in Figs. 3 and 4. This is partially due to the lack of sensitivity of the cardiac potentials to the intracellular conductivities, compared to the extracellular conductivities [23,36]. Thus, due to the greater magnitude of the relative errors of the intracellular conductivities, we suggest that it may be prudent to place a greater emphasis, in the following discussion, on the ability of an array to retrieve the intracellular conductivities. Let us now consider the retrieval of each of the intracellular conductivities in turn. First, observe that the 600-400-200 NeuroNexus array is best at retrieving $g_{il}$, (Figs. 3 (b) and 4(b)). Based on this result, we could speculate that the greater the length of the array along the longitudinal direction, the greater the opportunity for the current to travel through the intracellular space in that direction, consequently allowing for the accurate retrieval of the intracellular conductivity of that direction. On the other hand, note that the 600-400-200 array spans 1.8 mm in the longitudinal direction, while the 75-electrode spans 2 mm in this direction. Although these arrays have similar coverage, we suspect that the increased number of measuring electrodes allows the NeuroNexus array to sample the potentials more densely. The other two NeuroNexus arrays span 1.2 mm in the longitudinal direction and are as capable of retrieving $g_{il}$ as the 75-electrode array. Though the coverage of these arrays in the longitudinal direction is less than the 75-electrode array, we suspect that the increased number of measuring electrodes of the NeuroNexus arrays counteract their lack of coverage. Based on these results, we further hypothesise that a relationship might exist between the number of measuring electrodes plus the distance an array spans in a particular direction and the accuracy of the retrieved intracellular conductivity in that direction. We speculate that this notion may explain why the 400-400-200 and 400-600-200 arrays are as capable of retrieving $g_{il}$ as the 75-electrode array. Shifting our attention to the retrieval of $g_{it}$, we note that the 400-600-200 array is best at retrieving $g_{it}$ (Figs. 3(b) and 4(b)). In this case, the array is oriented such that there is a 1.8 mm span in the transverse direction. Note also that the arrays that span 1.2 mm in the transverse direction (the 400-400-200 and the 600-400-200 arrays) are unable to obtain conductivities to a similar accuracy as the 75-electrode array (2 mm span), even though these arrays are populated with more measuring electrodes. These observations reinforce the interpretations from the previous paragraph. This also leads us to believe that perhaps the retrieval of this particular conductivity is more sensitive to the coverage of an array along the transverse direction. This idea, however, is yet to be investigated further. Finally, investigating the retrieval of $g_{in}$, we observe that all the NeuroNexus arrays (the 400-400-200, 400-600-200 and 600-400-200 arrays) are equally capable of retrieving this particular conductivity and perform better than the 75-electrode array. As all NeuroNexus arrays tested in this study span 1.4 mm in the normal direction, it could be that the increased capability of the NeuroNexus arrays to retrieve $g_{in}$ is due to greater coverage in the normal direction (compared with 1 mm for the 75-electrode array), or the increased number of measuring electrodes present in the normal direction within these arrays, or both. In summary, we have found that the NeuroNexus arrays can retrieve the bidomain conductivities to a similar degree, if not a higher degree, of accuracy than the 75-electrode array. It would appear that a relationship exists between the greater dimension the array spans in a particular direction and the accuracy of the retrieved intracellular conductivity in that direction. Furthermore, there may also be a connection between a greater number of measuring electrodes within an array and increased capability of the array to retrieve intracellular cardiac conductivities.

5.2. Experimental recommendations

As noted in Section 5.1, the 400-600-200/600-400-200 array is superior at retrieving $g_{il}$ and $g_{it}$, while being as capable of retrieving $g_{in}$ as the other arrays tested. Thus, we suggest making two sets of measurements during the second pass using this array: the first set of measurements should be made with the $600-\mu \text{m}$ spacings placed along the longitudinal direction, to obtain $g_{ib}$ and the second set of measurements, where the $600-\mu \text{m}$ spacings are along the transverse direction, to obtain $g_{it}$. This approach would yield two different $g_{in}$ values, which we suggest be averaged. Due to the similarity of the extracellular results of the 400-600-200 and the 600-400-200 arrays, especially for higher noise levels (as indicated by Fig. 4(a)), and due to the lower magnitude of relative errors, we suggest that either configuration (400-600-200 or 600-400-200 array) could be used to make the first pass measurements.

5.3. Limitations

It has to be noted that these conclusions are theoretically based and not yet validated by experimental means. Furthermore, fibre rotation, the surface to volume ratio of cells and membrane resistance were chosen based on previous studies and are not known for humans. For example, values for the surface to volume ratio of cells are in the range of 1000–5000 ${\text{cm}}^{-1}$ [3,49] Although investigations suggest that changes in these parameters would not drastically alter this paper’s conclusions, this is yet to be thoroughly investigated. This work has also been carried out under various other assumptions. It has been assumed, as it has been in previous studies [12,27,30,39], that angle of inclination of the fibres relative to the epicardial surface is zero, and the rotation of fibres varies linearly with depth. These assumptions are standard in studies involving the bidomain model and are approximately representative of ventricular cardiac tissue, as shown in Ref. [27]. It has also been assumed that no significant injury currents are caused when the array is inserted into the cardiac tissue [50]. Furthermore, the model geometry considered during this study, the slab model, lacks the curvature typically found in cardiac tissue, although given that the arrays span at most 1.8 mm of the tissue, this is likely to be a reasonable assumption. Another drawback pertaining to the model geometry considered during this study is the assumption of an infinite blood medium. However, studies performed using more realistic geometries under ischaemic conditions [51,52] have suggested that potential distributions of these models are similar to those given by the slab model. Thus, it would be reasonable to conclude the conductivities retrieved through the slab model would be similar to those obtained through the use of more realistic models of the left ventricle. Although once again, this notion could be investigated further.

6. Conclusion

This paper has presented a modified protocol that utilised a more appropriate variant of sensor noise (additive noise) as a means to retrieve cardiac conductivities. Using a variant of the modified protocol (MPRI), where inversion repetitions were performed, we were able to theoretically validate the use of an electrode array to retrieve cardiac conductivities in experimental trials. It was found that the expected results of the modified protocol, obtained through taking the values from MPRI that give the median relative error, are of a similar accuracy to those retrieved through the use of the original protocol. It has also been shown that due to the modified protocol being computationally more efficient, repetitions of the modified protocol can be performed, and thus, we were able to obtain the spread of retrieved conductivities obtained from the repetitions, from which more robust conclusions could be made. Due to the expense and time involved in acquiring bespoke electrode arrays, we proposed utilising two-prefabricated NeuroNexus arrays; these were the 400-400-200 array and the 600-400-200/400-600-200 array, where the $600-\mu \text{m}$ spacings were oriented along the longitudinal and transverse directions, respectively. Conductivities obtained through these arrays were compared against a control - the 75-electrode array - through the use of MPRI. We were able to retrieve extracellular conductivities to a similar or higher accuracy with the NeuroNexus arrays than with the control, and these conclusions were more pronounced for higher noise levels. Investigations of the retrieved intracellular conductivities suggest that the 600-400-200/400-600-200 array is more suitable for experimental trials than the remainder of the arrays tested, perhaps due to its larger number of electrodes and the fact that it can be oriented to place the longer side to match the direction of the desired conductivity. Future studies could investigate the effects of considering a more realistic representation of ventricular tissue - perhaps an ellipsoidal representation or a realistic representation obtained through scans. Having developed a thoroughly tested protocol for theoretically validating an array’s potential for retrieving cardiac conductivities, future work could look into employing the protocol to identify alternate source-sink electrode placements that could perhaps yield conductivities of even higher accuracy. Experimental trials are also planned. These will utilise the wider of the two NeuroNexus arrays to obtain cardiac potentials, from which cardiac conductivities will be retrieved.