Addendum: Action potential propagation and block in a model of atrial tissue with myocyte-fibroblast coupling

The analytical theory of our earlier study (Mortensen et al. (2021), Mathematical Medicine and Biology, 38(1), pp. 106-131) is extended to address the outstanding cases of fibroblast barrier distribution and myocyte strait distribution. In particular, closed-form approximations to the resting membrane potential and to the critical parameter values for propagation are derived for these two non-uniform fibroblast distributions and are in good agreement with numerical estimates.


Introduction
The spatial distribution of fibroblasts within myocardial tissue, and the electrical coupling between fibroblasts and cardiomyocytes are significant but poorly-understood factors in triggering and sustaining cardiac arrhythmias. In our recent article (Mortensen et al., 2021) entitled "Action potential propagation and block in a model of atrial tissue with myocyte-fibroblast coupling", hereinafter referred to as MGSS, we proposed that electrical propagation in fibrous tissue may be understood conceptually on the basis of three elementary fibroblast distributions: (C1) uniform, (C2) fibroblast barrier, and (C3) myocyte strait, illustrated in Fig. 1. Using direct numerical simulations, we then estimated primary action potential biomarkers including conduction velocity, peak potential and triangulation index and found that propagation block occurs at certain critical values of the parameters defining each of the elementary distributions. Based on a fast-slow scale analysis (Simitev & Biktashev, 2006;Biktashev et al., 2008;Simitev & Biktashev, 2011), we demonstrated that the boundary of absolute refractoriness in myocytefibroblast tissue is determined primarily by the value of the myocyte potential ahead of a propagating pulse and used this to obtain a simple analytic expression that captures with remarkable accuracy the block of propagation in the case of a uniform fibroblast distribution (C1).
The purpose of this addendum is to extend the analytical theory of MGSS to the remaining two ele-2 of 6 P. MORTENSEN ET AL.

C1a C1b
C1a C1b C3 C2 x y C2 C3 x y x y x y FIG. 1: A schematic illustration of fibroblast distributions C1 (uniform), C2 (fibroblast barrier) and C3 (myocyte strait) as they may appear in a border zone between intact myocardium and fibrosis. Red shades indicate high fibroblast density. To the left is an image of a border zone adapted by permission from BMJ Publishing Group (Yamamura et al., 2018). Propagation fronts are considered plane waves locally so coordinate systems are attached to the idealised regions only and propagation is along the x axis.
mentary fibroblast distribution cases -(C2) fibroblast barriers and (C3) myocyte straits. In the following, we derive and report closed-form approximations to the resting membrane potentials which allows us to estimate, also in closed-form, the critical parameter values separating electrical propagation from failure in these two non-uniform distributions. Our motivation for this analysis, the background literature on the topic and the notation used here are identical to those introduced in MGSS unless explicitly stated otherwise. Hence, we refer the reader to our earlier paper MGSS for these and further details.

Analytic approximations of propagation through fibroblast barriers and myocyte straits
Fibrous atrial tissue is modelled in MGSS as a 2D continuum of atrial myocytes where a fixed number of identical fibroblasts, n(x, y), are connected in parallel at every Cartesian point (x, y) via an inter-cell conductance G gap . In particular, the "fibroblast barrier" distribution (C2) is defined by and the "myocyte strait" distribution (C3) is defined by and these are illustrated in Fig. 1. Here ∆ x and ∆ y are the widths of barriers and straits, respectively, N is a positive integer, and H(·) is the Heaviside step function. The propagation of electrical excitation in atrial tissues with such distributions is then described by a boundary value problem for the monodomain equations (2.1) of MGSS. Further, in MGSS we demonstrated that the boundary of absolute refractoriness is determined to a good approximation by the value of the myocyte potential ahead of a propagating pulse (prefront voltage), V α m . Note that, superscripts in V α m , and further below in V 0 m , do not denote exponentiation.
We now consider action potentials travelling along the x axis in fully rested tissue with fibroblast distributions (2.1). Then, the prefront voltage V α m is identical to the steady state myocyte potential which we proceed to determine. Distributions (2.1) are functions of a single variable denoted by s for brevity, with s being x in the case of a fibroblast barrier and y in the case of a myocyte strait, respectively. Since these functions are also even, the steady state version of equations (2.1) of MGSS can be reduced in both fibroblast distribution cases to a one-dimensional, time-independent system posed on the real half-line Here, χ is the cell surface-to-volume ratio, σ ss with s = x, y are the relevant diagonal component of the conductivity tensor and their values are listed in Table 1 of MGSS. To make the problem analytically tractable, we have also linearized the equations near the uncoupled resting potentials V 0 m = −81 mV and V 0 f = −46 mV of the original myocyte and fibroblast models of Courtemanche et al. (1998) and Morgan et al. (2016) as detailed in section 4.5 of MGSS, with G m and G f being the coefficients to the leadingorder Taylor series terms. Equation (2.2b) is easily solved for the fibroblast resting potential V α f , and we are left with a boundary-value problem for a single second-order linear inhomogeneous ordinary differential equation for the myocyte resting potential V α m . Since the distributions n(s) are piecewise constant, this resulting equation has constant coefficients in each of the intervals s ∈ [0, ∆ s/2] and s ∈ [∆ s/2, ∞). Imposing continuity and smoothness matching conditions at s = ∆ s/2 and boundary conditions (2.2c), we find the solution Here the constants K 1 , K 2 , K 3 and λ take different values in the intervals s ∈ [0, ∆ s/2] and s ∈ [∆ s/2, ∞) and for the different choices of fibroblast distributions (2.1) as listed in Table 1 with further notation defined as The critical curves ∆ s(N) partitioning the generic parameter planes π(N, ∆ s) of the problem into regions of propagation and no-propagation are determined from the conditions for fibroblast barrier and myocyte strait cases, respectively, as detailed in the formulation of equation (4.12) of MGSS. By the first of boundary conditions (2.2c), the profile V α m (s) has an absolute extremum at s = 0 equal to V α m (s = 0) = 2K 1 + K 3 . Hence, in the case of the fibroblast barrier distribution (2.1a), condition (2.5) becomes √ κ 1 κ 6 (κ 3 κ 7 + κ 8 This equation can be solved exactly for ∆ s. Indeed, the only terms that depend on ∆ s are κ 7 and κ 8 , and noting from (2.4) that κ 7 = (κ 2 8 − 1) 1/2 , we solve the equation for κ 8 . Using the definition of κ 8 , we find a closed-form approximation for the critical curve in the case of the fibroblast barrier distribution (2.1a) Since K 1 does not vanish, we note that equation (2.6) cannot have any solutions precisely when E h = K 3 . The latter defines a vertical asymptote for the critical curve and can be solved explicitly to find (2.8) This is in agreement with equation (4.12) of MGSS for the uniform distribution (C1) as the latter can be thought of as an infinitely wide fibroblast barrier. For the myocyte strait distribution (2.1b) the approximation for the critical curve takes the form This is obtained from the condition min s∈[0,∞) V α m (s) = 2K 1 + K 3 = E h in a similar way to the fibroblast barrier case but rewriting first in terms of κ 11 rather than κ 8 . This curve does not have a vertical asymptote as V 0 m = E h . Figures 2 and 3 show a comparison of the derived closed-form approximations with values from the direct numerical simulations reported in MGSS for both fibroblast distributions. The discrepancy between the expression for the resting myocyte potential (2.3) and the numerical values is due to retaining only the linear terms in the Taylor expansions leading to the right-hand sides of the first two of equations (2.2). The expressions for the critical curves (2.7) and (2.9) are compared to the corresponding numerical curves shown in Figures 6(a) and 8(a) of MGSS, respectively, and their accuracy is additionally affected by the asymptotic reduction procedure used in MGSS to separate the description of fronts from the description of steady state equilibrium. However, the linearisation errors and the asymptotic

Conclusion
Two archetypal non-uniform spatial distributions of myocyte-fibroblast coupling were considered here. Approximations to the resting potentials of the coupled cells and to the distribution parameters at which action potential propagation is blocked were derived in explicit analytic form and are in good correspondence with values from direct numerical simulations. The results of the addendum are significant as they provide theoretical underpinning of realistic 2D and 3D computational studies where high fibroblast density as opposed to collagen accumulation leads to resting depolarization and spatial distribution of refractoriness (McDowell et al., 2011) and to the generation of complex fractionated atrial electrograms (Ashihara et al., 2012). Further, closed-form approximations of propagation in inhomogeneous medium such as the ones derived here can be used to estimate poorly constrained values of histological and electrophysiological parameters of myocardial tissue.