Image-Based Modeling of the Heterogeneity of Propagation of the Cardiac Action Potential. Example of Rat Heart High Resolution MRI

International audienceIn this paper we present a modified bidomain model, derived with homogenization technique from assumption of existence of diffusive inclusions in the cardiac tissue. The diffusive inclusions represent regions without electrically active myocytes, e.g. fat, fibrosis etc. We present the application of this model to a rat heart. Starting from high resolution (HR) MRI, geometry is built and meshed using image processing techniques. We perform a study on the effects of tissue heterogeneities induced with diffusive inclusions on the velocity and shape of the depolarization wavefront. We study several test cases with different geometries for diffusive inclusions, and we find that the velocity might be affected by 5% and up to 37% in some cases. Additionally, the shape of the wavefront is affected

Effects of non-linear GJ channels on the AP propagation : a modelling insight

International audienceBackground: Velocity and pattern of propagation of cardiac AP depends on structural andfunctional properties of the tissue, such as conductivity, dynamics of transmembrane ionicchannels and gap junctions (GJ). Gap junctions are clusters of channels that connect adjacent cells. A gap junction channel(GJC) is made of proteins named ­ connexins. Electrical behavior of GJCs depend on the typeand arrangement of their connexin composition. The dominating connexins in cardiacmyocytes are Cx43, Cx45 and Cx40. Methods and results: In current mathematical models, GJCs are considered to be passive.But, the experimental results, obtained by the dual­voltage clamp technique, show that GJCs display biophysical electrical properties such as voltage gating,i.e. a time and voltage dependence. Here we model Cx43 GJCs. We use the Hodgkin­Huxley formalism to describe GJCsconductance via one gating variable g j = g j (t, V j ). From our experimental results we obtain model parameters: the normalisedsteady state conductance and the time constant to reach the steady state, both voltagedependent. Once we have described the behavior of the single GJC, we write the mathematical model ofthe tissue, where we apply GJ current on specific parts of the cells’ membranes. Numerics and outlook: Some 3D numerical experiments are currently being performed on athin strip of cells, in order to compare the model’s results with the experimental ones. We use a monolayer of 50 × 3 cells, represented by cylinders of 100μm lengthand 10μm radius, with 2μm inter­cellular distance. We model GJCs on the cross sections of the cylinders. Finally, we apply an external stimulus on the border of the domain, and observe the propagation of theAP. Our goal is to make a mathematical model of the heterogeneous GJCs, including Cx45channels, as these have been shown to play a role in arrythmogenesis

Decolorization of azo dye Methyl Orange with crude fungal laccase obtained by growing Ganoderma spp. on cereal mix

In recent years, one of the biggest environmental problems is the pollution of water with colored wastewater which has negative effect on the environment and human health. Wastewaters contain complex structural compounds, such as azo dyes that used today in many industrial fields. Removing of azo dyes from wastewater using traditional methods is an extremely complex and in many cases ineffective process. In recent decades, there is a tendency towards the application of environmentally acceptable methods of removing synthetic dyes from wastewater. Method which has proven to be very effective, is the degradation of synthetic dyes using various fungal enzymes. In this study, the crude fungal laccase (31,42 UmL-1) obtained by growing fungal mycelium Ganoderma spp. on cereal mix was used for decolorization of Methyl Orange. Decolorization procedure was carried out at different temperatures (30-70 ºC) and pH (3-8) in order to determine the optimal conditions for dye decolorization. The incubation time was 180 min and every 15 min during the incubation time, the change in color intensity was monitored spectrophotometrically at 472 nm and decolorization efficiency (DE) was calculated. The optimal pH was 5 with DE of 57 % at 30 ºC, while in the case of other pH values, DE was lower. The lowest DE (1,2 %) was in the case of pH 8, which indicates that laccase activity decreases in the alkaline medium. The optimal temperature of decolorization was 50 ºC with DE of 62 % at pH 5, while the DE was lower at higher and lower temperatures, which is in accordance with the literature data on the laccase activity optimal temperature of the Ganoderma spp. The lowest DE was 35 % at 70 ºC and pH 5. The obtained results show that laccase with good decolorization properties can be obtained using cheap agro-industrial wastes, such as cereal mix. The low cost of laccase production as well as the relatively high DE in a short time may further broaden its application in wastewater treatment

Calibration of stochastic biochemical models using single-cell video-microscopy experiments

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Role and modelling of some heterogeneities for cardiac electrophysiology

Introduction: The most used model in the elctrophysiology of the heart,known as the bidomain model, is the system of degenerate parabolic PDEs cou-pled with the non-linear ODE. Even though these equations provide quite ac-curate results, they are based on the fact that active cardiomyocytes are presenteverywhere in the heart, while it is known that non-small regions exist wherefibroblasts and other non-excitable cells or additional extracellular media takeplace. These regions, which play an important role in diseased hearts, are oftentaken into account through ad-hoc rough tuning of the tissue conductivities. Inthis work, we introduce a rigorous way to derive these conductivities from amicroscopic description of the heterogeneities in the tissue.Method: We assume a periodic alternation of the healthy tissue (bidomainmodel) and the fibrotic tissue (diffusive part). Such a microscopic model canbe simulated directly, at the price of a very fine discretization and a high com-putational cost. Instead we derive a homogenized model at the macroscopicscale, following a two-scale method technique. There are two problems risinghere. First one has to deal with the degeneracy of parabolic equations and sec-ond one comes from the non-linearity of the ionic model of the cardiac cells.In order to study the model and illustrate its relevance, we computed numeri-cal simulations of both the microscopic and homogenized models based on anon-physical linear model, and then on the Mitchell-Schaeffer ionic model.Results: Interestingly, we recover a bidomain type model, but with modifiedconductivities, that depend on the volume fraction of the diffusive inclusionsbut also on their geometries. The numerical results confirm the convergenceof the microscopic model to the homogenized equations in the linear case. Weare currently working on the numerical simulations for the non-linear case,where we expect to observe the influence of the diffusive inclusions on thepropagation of action potentials.Conclusion: With the final non-linear model, we shall provide cheap mod-eling tools to account for tissue heterogeneities at intermediate scales, as canbe observed, e.g., in the fibrotic tissue

The Modified Bidomain Model with Diffusive Inclusions

Bidomain equations are the standard way to model the electric potential in cardiac tissue. They are based on the fact that active cardiomyocytes are present everywhere in the heart, while it is known that non-small regions exist where additional extracellular media take place. These regions, which play an important role in diseased hearts, are often taken into account through ad-hoc rough tuning of the tissue conductivities. In this work, we introduce a rigorous way to derive these conductivities from a microscopic description of the heterogeneities in the tissue. We assume a periodic alternation of the healthy tissue and the fibrotic tissue. Such a microscopic model can be simulated directly, at the price of a very high computational cost. Instead we derive a homogenized model at the macroscopic scale, following a standard multiscale technique.We recover a bidomain type model, but with modified conductivities, that depend on the volume fraction of the diffusive inclusions but also on their geometries. The numerical results confirm the convergence of the microscopic model to the homogenized equations. We observe the influence of the diffusive inclusions on the propagation of action potentials. With the final model we shall provide cheap modeling tools to account for tissue heterogeneities at intermediate scales. The diffusive volume ratio, that enters the model, might be available through functional imaging, which enlightens the practical interest of the method

Modified bidomain model with passive periodic heterogeneities

International audienceIn this paper we study how mesoscopic heterogeneities affect electrical signal propagation in cardiac tissue. The standard model used in cardiac electrophysiology is a bidomain model-a system of degenerate parabolic PDEs, coupled with a set of ODEs, representing the ionic behviour of the cardiac cells. We assume that the heterogeneities in the tissue are periodically distributed diffusive regions, that are significantly larger than a cardiac cell. These regions represent the fibrotic tissue, collagen or fat, that is electrically passive. We give a mathematical setting of the model. Using semigroup theory we prove that such model has a uniformly bounded solution. Finally, we use two-scale convergence to find the limit problem that represents the average behviour of the electrical signal in this setting

The Modified Bidomain Model with Periodic Diffusive Inclusions

Bidomain equations are the standard way to model the electric potential in cardiac tissue. They are based on the fact that active cardiomyocytes are present everywhere in the heart, while it is known that non-small regions exist where additional extracellular media take place. These regions, which play an important role in diseased hearts, are often taken into account through ad-hoc rough tuning of the tissue conductivities. In this work, we introduce a rigorous way to derive these conductivities from a microscopic description of the heterogeneities in the tissue. We assume a periodic alternation of the healthy tissue and the fibrotic tissue. Such a microscopic model can be simulated directly, at the price of a very high computational cost. Instead we derive a homogenized model at the macroscopic scale, following a standard multiscale technique. We recover a bidomain type model, but with modified conductivities, that depend on the volume fraction of the diffusive inclusions but also on their geometries. The numerical results confirm the convergence of the microscopic model to the homogenized equations. We observe the influence of the diffusive inclusions on the propagation of action potentials. With the final model we shall provide cheap modeling tools to account for tissue heterogeneities at intermediate scales. The diffusive volume ratio, that enters the model, might be available through functional imaging, which enlightens the practical interest of the method

Influence of periodic diffusive inclusions on the bidomain model

We present a new mathematical model of the electric activity of the heart. In the standard bidomainmodel we can distinguish the intra- and the extracellular space with different conductivities for excitablecells and the fibrotic tissue around them. The main drawback is that it assumes the existence of excitablecells everywhere in the heart, while it is known that there exist non small regions where fibroblasts takeplace. The fibroblasts are equally distributed and since they are non excitable cells, they can beconsidered as a diffusive part. Hence we extend the standard bidomain model as follows: we assumethat we have periodic alternation of the healthy tissue (linear bidomain model) and fibrotic extracellularspace (diffusive part). We use homogenisation techniques to derive our macroscopic partial differentialequations. Interestingly, we obtain again a bidomain type model with modified conductivities thatinvolve the volume fraction of the diffusive domain. Preliminary numerical experiments will concludeon the influence of these diffusive inclusions

The Modified Bidomain Model with Periodic Diffusive Inclusions

Modèles numériques haute résolution de l'électrophysiologie cardiaqu