We discuss here the use of non-local models in space and fractional order operators in the characterisation of structural complexity and the modeling of propagation in heterogeneous biological tissues. In the specific, we consider the application of space-fractional operators in the context of cardiac electrophysiology, where the lack of clear separation of scales of the highly heterogeneous myocardium triggers peculiar features such as the dispersion of action potential duration, that have been observed experimentally, but cannot be described by the standard monodomain or bidomain models. We describe the methodology and compare the results of a standard monodomain model with results of a model with a non-local component in space

Cusimano, N.

Del Teso, F.

Gerardo-Giorda, L.

English

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5th International Conference on Computational and Mathematical Biomedical Engineering - CMBE201710–12 April 2017, United StatesP. Nithiarasu, A.M. Robertson (Eds.)MODELLING CARDIAC STRUCTURAL HETEROGENEITY VIASPACE-FRACTIONAL DIFFERENTIAL EQUATIONSNicole Cusimano1, Félix Del Teso2, and Luca Gerardo-Giorda11BCAM - Basque Center for Applied Mathematics, Alameda Mazarredo 14, 48009 Bilbao, Spain,{ncusimano,lgerardo}@bcamath.org2NTNU - Norwegian University for Science and Technology, Alfred Getz’ vei 1, 7034 Trondheim, Norway,felix.delteso@ntnu.noSUMMARYWe discuss here the use of non-local models in space and fractional order operators in the characteri-sation of structural complexity and the modeling of propagation in heterogeneous biological tissues.In the specific, we consider the application of space-fractional operators in the context of cardiac elec-trophysiology, where the lack of clear separation of scales of the highly heterogeneous myocardiumtriggers peculiar features such as the dispersion of action potential duration, that have been observedexperimentally, but cannot be described by the standard monodomain or bidomain models. We de-scribe the methodology and compare the results of a standard monodomain model with results of amodel with a non-local component in space.Key words: Cardiac electrophysiology, Monodomain model, fractional Laplacian1 INTRODUCTIONThe mechanical activity of cardiac chambers is governed by the excitation patterns produced bythe electrophysiology of the heart itself. Accuracy in the description of such an activity is thus ofparamount importance when attempting to capture the overall cardiac dynamics.Mathematical models of electrical propagation in excitable media are typically developed via thehomogenisation principle, namely, under the assumption that microscopic inhomogeneities in themedium have a negligible effect on the transport phenomena observed at the macroscopic scale. Inhighly heterogeneous structures, such as cardiac or neural tissue, where there is no clear separationof scales, this hypothesis is questionable. In fact, experimental data point at peculiar features ofthe electrophysiological dynamics, such as wide action potential foot [1] and a marked dispersionof action potential duration (APD) [2], that cannot be captured by standard models. Alternativemodeling strategies are thus needed to provide additional insight into the effect produced by structuralheterogeneity on electrical pulse propagation.In the last few decades, mathematical models involving differential operators of non-integer orderhave been considered in a variety of disciplines (such as physics, engineering, chemistry, rheology,economics) with the aim of reproducing transport phenomena whose characteristics significantly de-viate from the classical Markovian and Gaussian features, typical of standard diffusion models. Al-though the interest in fractional operators linked to practical applications is increasingly growing, thesuccessful implementations of fractional models to model real life phenomena are still scarce.To the best of our knowledge, the work by Bueno-Orovio et al. [3] is the first example of using aspace-fractional mathematical model in cardiac electrophysiology. The biophysical justification be-hind the use of such a fractional operator for this particular application is based on potential electricfield theory. The inhomogeneities present on a variety of length scales in biological tissue give riseto secondary sources that add up to the primary source field corresponding to the assumption of auniform and infinite volume conductor. These secondary sources can be seen as a dipole modulationof the electrical potential associated with a point source in a homogeneous tissue (monopole). By1126using Riesz potential theory, the authors in [3] showed that a fractional model can be interpreted asa smooth transition between monopole and dipole behaviour, with increasing degree of heterogene-ity as the order of the fractional operator decreases. Although capturing peculiar features of actionpotential propagation in heterogeneous media, and showing good agreement with experimental data,their numerical simulations are restricted to one dimensional intervals.The numerical methodology introduced in [3] is based on the approximation of eigenpairs of thenonlocal operator. If its extension to two and three dimensional cartesian domains is pretty straight-forward, its application to real geometries is not practical, as eigenvalues and eigenfunctions are notknown analytically in these cases.Motivated by the promising results provided by the use of space-fractional differential operators, andby the lack of treatment of non cartesian geometries, we developed in [5] a method to approximatefractional operators on general bounded domains with a variety of boundary conditions. This methodallows us to devise a nonlocal model of electrical wave propagation that is consistent with the physicalintuition and interpretation of the problem, and is flexible enough to be considered on realistic cardiacgeometries.2 METHODOLOGYMathematical models of electrical signal propagation in cardiac electrophysiology consist of suit-able spatially distributed formulations of specified cell models reproducing the response of a singleexcitable cell to an applied electrical stimulus. Cell models describe the temporal evolution of thetransmembrane potential u and the changes in u caused by the opening and closing of the various ionchannels present in the cell membrane, driving the movement of ions into and out of the cell. A clas-sical approach adopted in order to account for pulse propagation is to introduce spatial dependencein the model via the monodomain formulation. The latter consists of a coupled ODE-PDE systemwhose PDE component is a nonlinear parabolic equation (see, e.g. [4]).The space-fractional formulation of the monodomain model can obtained by replacing the diffusionterm in the parabolic part with a nonlocal operator. Here we consider the fractional Laplacian of orders ∈ (0, 1), (−∆)s, and the resulting system reads:χ!Cm∂u∂t+ Iion(u, z)"= −D(−∆)su+ Istimdzdt= f(u, z),(1)where χ is the cell surface-to-volume ratio, Cm is the membrane capacitance per unit area, z is asuitable vector of secondary variables used in the description of the dynamics of the ion channels inthe cell membrane, Iion is the sum of all transmembrane ionic currents, D is a constant conductivity,Istim is the electrical stimulus, and f is a vector-valued function describing the temporal evolutionof z. System (1) is completed by suitable initial conditions and homogeneous Neumann boundaryconditions to model an insulated domain.The crucial issue with fractional order differential operators is that they are naturally defined on theentire space Rn, n ≥ 1. However, in the majority of practical cases one needs to mathematicallymodel quantities that are defined on a bounded domain Ω ⊂ Rn only. The main challenge is howto suitably restrict, adapt, or interpret the definition of a fractional operator so that it preserves itsnonlocal character, all while providing a well-posed problem on Ω.Based on the spectral definition of the fractional Laplacian and via the heat semi-group formalism,we propose in [5] a novel numerical approach for the discretisation of fractional powers of the Lapla-cian on bounded and possibly irregular domains, coupled with homogeneous Dirichlet, Neumann, orRobin boundary conditions. Combining finite elements and a suitable quadrature rule, the method iswell suited for treating complex domains like the cardiac chambers. The main focus in [5] is the dis-cretisation of (−∆)su for a given function u : Ω → R, but our approach provides a natural frameworkfor the numerical treatment of fractional parabolic problems on bounded domains such as (1).11273 PRELIMINARY RESULTSPromising results have been obtained in one spatial dimension in [6], where the authors investigatethe effect of changing the order of the fractional Laplacian on action potential shape and spatialpropagation. We present here some preliminary results on a two dimensional slice of cardiac tissuereconstructed from CT scan. The ionic model we implemented is the Rogers-McCulloch variant ofthe FitzHugh-Nagumo one [7], featuring a resting potential of 0 mV, a peak potential of 100 mV, andan APD of around 100 msec. In this first set of tests we consider a constant conductivity in the wholedomain. In Figure 1, we plot the spatial domain Ω (left), and the time course of the action potentialcomputed with the fractional monodomain (with s = 0.75) at five different points labeled from A toE (right): APD dispersal and wide foot of the depolarization phase are visible.0 20 40 60 80 100 120Time (ms)020406080100Potential (mV)ABCDEFigure 1: Left: computational domain. Right: action potential course in points A, B, C, D, and EIn Figure 2, we compare, in terms of APD, the results of a standard monodomain and a spatialfractional monodomain. In particular, we compute APD90, namely the APD computed at 90% re-polarization. The use of fractional monodomain allows us to capture a pronounced dispersion ofAPD90, a feature expected in highly heterogeneous structures [1] that the standard formulation of themonodomain fails to exhibit.Figure 2: APD: comparison between standard (left) and fractional (right) monodomain.4 CONCLUSIONSThe fractional monodomain, being nonlocal, shows great potential in modeling the effect of cardiacmicrostructure on the electrical propagation in the myocardium. The theoretical approach followed in[5] and the numerical method developed therein, allow us to set up the problem on general bounded1128domains and to treat numerically system (1) in this general setting. We show how this strategy canbe implemented on bounded geometries of practical interest, coupled with physically meaningfulboundary conditions.REFERENCES[1] M.S. Spach, R.C. Barr Effects of cardiac microstructure on propagating electrical waveforms.Circ. Res. 2000:86:e23-e28.[2] B. Hanson, P. Sutton, N. Elameri, M. Gray, H. Critchley, J.S. Gill, P. Taggart. Interaction ofactivation-repolarization coupling and restitution properties in humans. Circulation: Arrhyth-mia and Electrophysiology. 2009 Apr 1;2(2):162-70.[3] A. Bueno-Orovio, D. Kay, V. Grau, B. Rodriguez, K. Burrage. Fractional diffusion models ofcardiac electrical propagation: Role of structural heterogeneity in dispersion of repolarization.J. R. Soc. Interface. 2014; 11(97):20140352.[4] F. Sachse. Computational Cardiology. (2004) Springer, Berlin.[5] N. Cusimano, F. del Teso, L. Gerardo-Giorda, G. Pagnini. Discretizations of the fractionalLaplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions. Inpreparation. 2017.[6] N. Cusimano, A. Bueno-Orovio, I. Turner, K. Burrage. On the order of the fractional Laplacianin determining the spatio-temporal evolution of a space-fractional model of cardiac electrophys-iology. PLoS ONE. 2015; 10(12):e0143938.[7] J.M. Rogers, A.D. McCulloch. A collocation - Galerkin finite element model of car-diac action potential propagation, IEEE Trans Biomed Eng. 1994; 41(8): 743–757, doi:10.1109/10.310090.1129

Modeling cardiac structural heterogeneity via space-fractional differential equations

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Modeling cardiac structural heterogeneity via space-fractional differential equations

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