Detailed ionic models of cardiac cells are difficult for numerical simulations because they consist of a large number of equations and contain small parameters. The presence of small parameters, however, may be used for asymptotic reduction of the models. Earlier results have shown that the asymptotics of cardiac equations are non-standard. Here we apply such a novel asymptotic method to an ionic model of human atrial tissue in order to obtain a reduced but accurate model for the description of excitation fronts. Numerical simulations of spiral waves in atrial tissue show that wave fronts of propagating action potentials break-up and self-terminate. Our model, in particular, yields a simple analytical criterion of propagation block, which is similar in purpose but completely different in nature to the `Maxwell rule' in the FitzHugh-Nagumo type models. Our new criterion agrees with direct numerical simulations of break-up of re-entrant waves.
Background: The arrhythmogenic role of atrial complex morphology has not yet been clearly elucidated. We hypothesized that bumpy tissue geometry can induce action potential duration (APD) dispersion and wavebreak in atrial fibrillation (AF). Methods and Results: We simulated 2D-bumpy atrial model by varying the degree of bumpiness, and 3D-left atrial (LA) models integrated by LA computed tomographic (CT) images taken from 14 patients with persistent AF. We also analyzed wave-dynamic parameters with bipolar electrograms during AF and compared them with LA-CT geometry in 30 patients with persist…
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.
We study the interplay between traveling action potentials and spatial inhomogeneities in the FitzHugh-Nagumo model to investigate possible mechanisms for the occurrence of fibrillatory states in the atria of the heart. Different dynamical patterns such as ectopic foci, localized and meandering spiral waves are found depending on the characteristics of the inhomogeneities. Their appearance in dependence of the size and strength of the inhomogeneities is quantified by phase diagrams. Furthermore it is shown that regularly paced waves in a region R, that is connected by a small bridge connection…
The electrical coupling between myocytes and fibroblasts and the spacial distribution of fibroblasts within myocardial tissues are significant factors in triggering and sustaining cardiac arrhythmias but their roles are poorly understood. This article describes both direct numerical simulations and an asymptotic theory of propagation and block of electrical excitation in a model of atrial tissue with myocyte-fibroblast coupling. In particular, three idealised fibroblast distributions are introduced: uniform distribution, fibroblast barrier and myocyte strait, all believed to be constituent blo…
Interruptions in nonlinear wave propagation, commonly referred to as wave breaks, are typical of many complex excitable systems. In the heart they lead to fatal rhythm disorders, the so-called arrhythmias, which are one of the main causes of sudden death in the industrialized world. Progress in the treatment and therapy of cardiac arrhythmias requires a detailed understanding of the triggers and dynamics of these wave breaks. In particular, two very important questions are: 1) What determines the potential of a wave break to initiate re-entry? and 2) How do these breaks evolve such that the sy…
Tissues are characterized by layers of functional units such as cells and extracellular matrix (ECM). Nevertheless, how dynamics at interlayer interfaces help transmit cellular forces in tissues remains overlooked. Here, we investigate a multi-layer system where a layer of epithelial cells is seeded upon an elastic substrate in contact with a hard surface. Our experiments show that, upon a cell extrusion event in the cellular layer, long-range wave propagation emerges in the substrate only when the two substrate layers were weakly attached to each other. We then derive a theoretical model whic…
The response of the living tissue to the effects of strong heating or cooling can cause the blood flow rate to vary by an order of magnitude. A mathematical model for the freezing of living tissue is formulated which takes into account the nonlocal temperature dependence of the blood flow rate when the temperature distribution in the tissue is substantially nonuniform, as in cryosurgery.
Standard rheometers assess mechanical properties of viscoelastic samples up to 100~Hz, which often hinders the assessment of the local-scale dynamics. We demonstrate that high-frequency analysis can be achieved by inducing broadband Rayleigh waves and monitoring their media-dependent propagation using Optical Coherence Tomography. Here, we present a new broadband wave analysis based on two-dimensional Fourier Transformation. We validated this method by comparing the mechanical parameters to monochromatic excitation data and a standard oscillatory test. Our method allows for high-frequency mech…
The heartbeat is mediated between cardiac cells by waves of electrical depolarisation. During cardiac arrhythmias, electrical activity was found to be organised in scroll waves which rotate around a dynamical filament curve. In this thesis, a curved-space approach is used to mathematically capture anisotropy of wave propagation. We derive for the first time the covariant laws of motion for traveling wave fronts and scroll wave filaments in anisotropic excitable media such as cardiac tissue. We show that locally varying anisotropy yields non-zero Riemann tensor components, which may alter the s…
Making use of the poroelastic theory for hydrated polymeric matrices, the ultrasound (US) propagation in a gel medium filled by spherical cells is studied . The model describes the connection between the poroelastic structure of living means and the propagation behavior of the acoustic waves. The equation of fast compressional wave, its phase velocity and its attenuation as a function of the elasticity, porosity and concentration of the cells into the gel external matrix are investigated. The outcomes of the theory agree with the measurements done on PVA gel scaffolds inseminated by porcine li…
The field of biomedical imaging has undergone a rapid growth in recent years, mostly due to the implementation of ad-hoc designed experimental setups, theoretical support methods and numerical reconstructions. Especially for biological samples, the high number of scattering events occurring during the photon propagation process limit the penetration depth and the possibility to outperform direct imaging in thicker and not transparent samples. In this thesis, we will examine theoretically and experimentally the scattering process from two opposite points of view, focusing also on the continuous…
Biomedical applications requiring tissue diagnosis, activation, and treatment could be substantially leveraged by optical methods, owing to their unique feature set. However, their widespread application is severely limited by the strong light scattering that occurs in many tissues of interest, which dramatically limits achievable penetration depths. Here we demonstrate a new method to solve this issue by utilizing free-running ultrasound waves, transversal to the light propagation direction, to guide light into deeper tissue regions. We study the formation of the ultrasound-induced refractive…
We present systematic numerical studies of electrical-wave propagation in two-dimensional (2D) and three-dimensional (3D) mathematical models, for human, ventricular tissue with myocyte cells that are attached (a) regularly and (b) randomly to distributed fibroblasts. In both these cases we show that there is a parameter regime in which single rotating spiral- and scroll-wave states (RS) retain their integrity and do not evolve to a state ST that displays spatiotemporal chaos and turbulence. However, in another range of parameters, we observe a transition from ST to RS states in both 2D or 3D …
The van de Hulst formula provides an expression for the effective refractive index or effective propagation constant of a suspension of particles of arbitrary shape, size and refractive index in an optically homogeneous medium. However, its validity for biological matter, which often consists of very dense suspensions of cells, is unclear because existing derivations of the formula or similar results rely on far-field scattering and/or on the suspension in question being dilute. We present a derivation of the van de Hulst formula valid for suspensions of large, tenuous scatterers -- the type b…
We introduce a simple procedure of multivariate signal analysis to uncover the functional connectivity among cells composing a living tissue and describe how to apply it for extracting insight on the effect of drugs in the tissue. The procedure is based on the covariance matrix of time resolved activity signals. By determining the time-lag that maximizes covariance, one derives the weight of the corresponding connection between cells. Introducing simple constraints, it is possible to conclude whether pairs of cells are functionally connected and in which direction. After testing the method aga…
When modelling tissue-level cardiac electrophysiology, continuum approximations to the discrete cell-level equations are used to maintain computational tractability. One of the most commonly used models is represented by the bidomain equations, the derivation of which relies on a homogenisation technique to construct a suitable approximation to the discrete model. This derivation does not explicitly account for the presence of gap junctions connecting one cell to another. It has been seen experimentally [Rohr, Cardiovasc. Res. 2004] that these gap junctions have a marked effect on the propagat…
Ultra-wideband (UWB) electromagnetic pulses of nanosecond duration, or nanopulses, are of considerable interest to the communications industry and are being explored for various applications in biotechnology and medicine. The propagation of a nanopulse through biological matter has been computed in the time domain using the finite difference-time domain method (FDTD). The approach required existing Cole-Cole model-based descriptions of dielectric properties of biological matter to be re-parametrized using the Debye model, but without loss of accuracy. The approach has been applied to several t…
Major topics in medical ultrasound rest on the physics of wave propagation through tissue. These include fundamental treatments of backscatter, speed of sound, attenuation, and speckle formation. Each topic has developed its own rich history, lexicography, and particular treatments. However, there is ample evidence to suggest that power law relations are operating at a fundamental level in all the basic phenomena related to medical ultrasound. This review paper develops, from literature over the past 60 years, the accumulating theoretical basis and experimental evidence that point to power law…
Complex spatiotemporal patterns of action potential duration have been shown to occur in many mammalian hearts due to a period-doubling bifurcation that develops with increasing frequency of stimulation. Here, through high-resolution optical mapping and numerical simulations, we quantify voltage length scales in canine ventricles via spatiotemporal correlation analysis as a function of stimulation frequency and during fibrillation. We show that i) length scales can vary from 40 to 20 cm during one to one responses, ii) a critical decay length for the onset of the period-doubling bifurcation is…
We present numerical simulations for modelling optoacoustic (OA) signals observed in PVA hydrogel tissue phantoms. We review the computational approach to model the underlying mechanisms, i.e. optical absorption of laser energy and acoustic propagation of mechanical stress, geared towards experiments that involve absorbing media only. We apply the numerical procedure to model signals observed in the acoustic near- and farfield in both, forward and backward detection mode, in PVA hydrogel tissue phantoms (i.e. an elastic solid). Further, we illustrate the computational approach by modeling OA s…
Dreaming of Electrical Waves: Generative Modeling of Cardiac Excitation Waves using Diffusion Models
Cardiovascular Research Institute, University of California, San Francisco（加州大学旧金山分校心血管研究所）；University of California, Berkeley（加州大学伯克利分校）；Division of Cardiology, University of California, San Francisco（加州大学旧金山分校心脏病学部）
