This article was submitted to Computational Physiology and Medicine, a section of the journal Frontiers in Physiology
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Roughly 75% of normal myocardial tissue volume is comprised of myocytes, however, fibroblasts by number are the most predominant cells in cardiac tissue. Previous studies have shown distinctive differences in cellular electrophysiology and excitability between myocytes and fibroblasts. However, it is still unclear how the electrical coupling between the two and the increased population of fibroblasts affects the electromechanical dynamics of cardiac tissue. This paper focuses on investigating effects of fibroblast-myocyte electrical coupling (FMEC) and fibroblast population on atrial electrical conduction and mechanical contractility by using a two-dimensional Discrete Element Method (DEM) model of cardiac tissue that is different to finite element method (FEM). In the model, the electro-mechanics of atrial cells are modelled by a biophysically detailed model for atrial electrical action potentials and myofilament kinetics, and the atrial fibroblasts are modelled by an active model that considers four active membrane ionic channel currents. Our simulation results show that the FMEC impairs myocytes’ electrical action potential and mechanical contractibility, manifested by reduced upstroke velocity, amplitude and duration of action potentials, as well as cell length shortening. At the tissue level, the FMEC slows down the conduction of excitation waves, and reduces strain of the tissue produced during a contraction course. These findings provide new insights into understandings of how FMEC impairs cardiac electrical and mechanical dynamics of the heart.
Roughly 75% of normal myocardial tissue volume is comprised of myocytes (
In various conditions, the population of fibroblasts increases with the progression of cardiac diseases. For example, fibrosis resulting from a large number of cardiac fibroblasts increases with aging (
There is growing evidence suggesting an increased incidence of atrial fibrillation with increased fibrosis (
Computer modelling provides an alternative platform to experimental studies for investigating the functional impacts of fibroblasts on cardiac dynamics. In previous studies, computer models incorporating fibroblast-myocyte electrical coupling (FMEC) at various scales have been developed. These models include cell-pair coupling of myocytes and fibroblasts (
There is considerable debate in the cardiac-related literature regarding whether fibroblasts contribute to cardiac electrophysiology in a passive or active manner. Traditionally fibroblasts were viewed as only passive obstacles or insulators (
So far, it is unclear how the FMEC affects cardiac electromechanical dynamics, especially with the myocytes being coupled to active fibroblasts instead of passive ones used in previous studies (
In the present study, the active fibroblast model developed in (
The myocyte model used in this paper was introduced in (
For the electrical action potential, we used the model of
For the mechanical model, we used the
For the fibroblast electrophysiology, we used the “active 2” model developed by
We used DEM to model the mechanical behaviour of the tissue, following the method introduced in (
As introduced in (
We choose a myocyte dimensions to be 125 μm in length and 25 μm in width, whilst fibroblasts have length 25 μm and width 25 μm similar to the study of
Clumps are non-deformable, and so a method was introduced in (
The length scaling of each myocyte is computed at each timestep of the simulation, using the sarcomere length output of the electromechanical single-cell model (
Fibroblasts in the current study are represented not by a clump, but by either a single particle or a group of non-clumped particles (see
DEM contact bonds were formed between particles in the model. Following (
To model the interaction of fibroblasts and myocytes in cardiac tissue, previous studies have considered various methods of coupling the different cell types (
Previous studies (
Exploiting the versatility of the DEM method, we attempted to construct a coupling approach to represent descriptions and images of fibroblast-myocyte coupling
With regards to fibroblast dimensions, freshly-isolated fibroblasts are rounded cells with initial diameters of 7–9 μm, and considering their near-spherical shape, a surface membrane area of 150–250 μm2 (
In this study, we attempted to distribute fibroblast and myocyte particles to form a tissue model which is in accordance with the above descriptions and images. The process is somewhat imprecise, as there is no data on the cell size distribution of fibroblasts
An example of cardiac tissue imaging in
We assume that all cells in the model, whether myocyte or fibroblast, are electrically coupled with other cells that they are physically connected to. Assuming electrical coupling, thus, electrical excitation waves may propagate throughout the system, despite cell heterogeneity. Handling of electrical wave propagation in the DEM distribution was introduced in (
We introduce a subscript
Here,
The DEM parameters used in the model are presented in
Default DEM parameter values used in the model.
Parameter | Description | Value |
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Density of particles |
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Acceleration due to gravity |
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Normal contact stiffness |
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Shear contact stiffness |
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Normal critical dashpot damping ratio |
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Shear critical dashpot damping ratio |
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Contact tensile strength |
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Contact shear strength |
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The computational cycle at each time step is as follows: • For each cell, loop over neighbouring cells and calculate their contribution to that cell’s electrophysiology, • For each cell, solve the single-cell equations defining electrical and mechanical behaviour of that cell, using the explicit Euler method; • For each cell, update the particle radii and particle positions, such that the cell length matches the output of the single-cell equations; • For each contact between particles, solve the force-displacement law, updating the contact forces based on relative particle motion and constitutive contact model; and • For each particle, solve the law of motion, updating the particle position and velocity due to contact forces.
DEM calculations are performed using Itasca’s PFC version 5. Other calculations are performed using a custom C++ plugin interfacing with PFC. Each step of the computational cycle above must be performed before proceeding to the next step. However, each step is accelerated by parallelising the computation onto the eight threads of an Intel Xeon 3.6 GHz CPU. The DEM equations are solved explicitly by a centred finite-difference scheme. Relatively small timesteps are required owing to the relatively stiff contact springs, and calculation of a minimum timestep is calculated based on spring stiffness and particle radii (
Using the method described in
In this section, we analysed the effect of fibroblast insertion in-series to tissue (see
An example of the electro-mechanical activity simulated in a 2D human atrial tissue model including fibroblasts. Snapshots of electrical conduction and tissue contraction were shown at different timings.
In simulations, the excitation waves were initiated by a series of external stimuli, which were applied to the upper-most 10% of the tissue at
Snapshots of electrical wave conduction and mechanical contraction/relaxation in the tissue model are presented in
In order to understand potential impacts of the electrical coupling between myocyte and fibroblasts on atrial electro-mechanical dynamics, further investigations were conducted with varying gap junction conductances between the two cell types and F-M ratio, while the dimension of the tissue remains the same as that in
Effects of variations in fibroblast-myocytes coupling strength and F-M ratio on electrical activity of model. For all figures, blue indicates myocytes, and red indicates fibroblasts. Dots indicate a data point.
Firstly, we investigate the effects of varying the F-M ratio, while keeping the fibroblast gap junction conductance fixed at
Increase in the F-M ratio also reduces the overshoot of the action potential as shown in
However, the curves for max.
Effect of varying the fibroblast conductance
Similarly, myocyte APO (
Finally, the conduction velocity is measured in the tissue along the fibre direction, and the results are shown in
Further analyses are performed to quantitatively evaluate the mechanical contraction produced by the tissue, using our electro-mechanical DEM model. The average strain-rate tensor of the tissue is estimated using a best-fit procedure which minimises the error between the predicted and measured velocities of all particles in the tissue. Strain increments can then be accumulated at each time-step throughout the computational cycle, giving curves for the approximate average strains in the tissue. The equations to calculate the strain-rate tensor are described in (Itasca Consulting Group Inc., particle flow code version five user manual, Minneapolis, MN 2014).
Firstly, we analysed the average strain while varying the fibroblast gap junction conductance
We see that for an F-M ratio of 0.108:1, for the
Using the DEM model, it is possible to increase the strength of DEM contacts involving a fibroblast, in order to simulate the mechanical stiffening often associated with fibrosis. We multiply the normal and shear contact strengths,
We see that there is only a slight difference in maximum strain for each case. This is somewhat expected, as the majority of the strain in the system is generated by the myocyte contraction, which is unaffected by the changes in DEM contact stiffness of fibroblasts. However, there is still a 2.7% variation in maximum strain component
The analysis of
Effects of varying F-M ratio and electrical coupling strength between fibroblast-myocytes on the characteristics of action potential and excitation wave conduction in the interstitial model. For all figures, blue indicates myocytes, and red indicates fibroblasts. Dots indicate a data point.
When analysing the effect of varying
The analysis of the strain in the tissue is repeated as for the in-series model (
In this section, we compare the impact of the two modelling approaches on the plane wave propagation past a central fibrous region. A tissue of size 2.32 mm wide and 6.27 mm high is constructed for each coupling method, with a central rectangular region containing ∼930 fibroblasts in each case. A stimulus is applied from the top of the tissue, and the results are shown in
Plane wave propagation past a central fibrous region. Cells are coloured by their membrane potential as shown in the colour bar. Inset: spatial configurations of fibroblasts in the corresponding in-series and interstitial models.
For the in-series model, the top row
In this study, we developed a 2D DEM-based electro-mechanical model of human atrial tissue by using the discrete element method approach. Using the model, we investigated possible functional impacts of FMEC on the electrical and mechanical activities of the human atrial tissue. Our major findings are: 1) the FMEC impairs the electrical and mechanical activities of the cardiac tissue, which are manifested by abbreviated action potential duration, reduced conduction velocity of excitation waves and tissue strain during contraction, suggesting impaired muscle contractility; 2) the observed effects of FMEC are dependent on their population and the coupling strength. With an increase in the F-M ratio or the coupling strength, the observed APD abbreviation, reduction in the conduction velocity of excitation waves and product of muscle strain were further decreased; and 3) the observed effects of fibroblasts rely on how the fibroblasts are distributed and coupled to myocytes. In the case of in-series coupling, there was a monotonical relationship between the measured changes in the electrical activities (i.e., reduction of action potential duration and conduction velocity) and the increase of F-M ratio or the coupling strength. However, in the case of interstitial coupling, such relationship became complex and multi-phasic as the measured conduction velocity of excitation was either decreased or increased depending on the F-M ratio. In terms of mechanical contraction, the in-series coupling caused more reduction in the measured strain than the interstitials coupling. Collectively, these findings provide new insights into the functional impacts of FMEC on impairing cardiac electrical and mechanical functions.
Due to differences in the membrane of ion channel kinetics and properties, fibroblasts present different resting membrane potential and cellular excitability to myocytes (
The distribution of fibroblast in cardiac tissue is complex, both in-series and interstitial distribution. The developed DEM based 2D cardiac tissue model allowed us to investigate the modulation of fibroblast to cardiac excitation wave conduction in both in-series and interstitial coupling scenarios. Our results showed that while both coupling schemes showed similar actions of fibroblast on cardiac excitation and conduction, there is a subtle difference in the measured conduction velocity when the F-M ratio is altered. In the case of in-series coupling, increase of the F-M ratio or the coupling strength produced a monotonic decrease in the conduction velocity. This is attributable to the fact that the fibroblasts are distributed in the principal conduction pathway along the tissue fibre, and form a lower excitable gap between adjacent myocytes, impeding the conduction of excitation waves. However, in the case of interstitial coupling, though the increase in the FMEC caused a monotonic decrease of conduction velocity, following the similar trend as in the in-series coupling, the increase in the fibroblast population (F-M ratio) produced a complex pattern. In simulations, when the F-M ratio was increased, the measured conduction velocity first decreased when F-M ratio was below 0.5, however, it then increased when F-M ratio was between 0.5–0.9 and then decreased again at F-M ratio greater than 0.9. Such complex multi-phasic changes in the measured conduction velocity may be attributable to the balance of a driver or a load the fibroblasts act on myocytes functionally. As the fibroblasts are distributed at the side of myocytes always of the principal conduction pathway along the cardiac fibres, their actions to modulate cardiac excitation wave conduction rely on how they modulate the maximal upstroke velocity of action potentials and cellular excitability of cardiac cells. When the population of fibroblast is low (<0.5), the fibroblast acted more like an electrotonic load, which reduced the maximal upstroke velocity of the action potential, leading to a reduced conduction velocity. When the F-M ratio was increased, in one way the coupling reduced the maximal upstroke velocity that would lead to reduction of the conduction velocity, and in the other way it acted as a driver, which elevated the resting potential of myocytes that would increase cellular excitability leading to an increased conduction velocity. These two actions compete, and at the range of F-M ratio between 0.5 and 0.9, the driver action of the fibroblast dominated resulting in an increased conduction velocity. When F-M ratio is further increased (>0.9), the load action dominated leading a decrease in the conduction velocity again.
Cardiac diseases such as myocardial infarction with increased fibrosis population are often accompanied by contractile dysfunction of the heart, which may result in cardiovascular mortality (Naccarella et al., 2000). However, few studies have attempted to investigate the functional impacts of myocyte-fibroblast coupling on the mechanical activities of cardiac tissue and most previous studies have focused on the investigation of the electrical consequence of the coupling. In their study,
The present model has several limitations which may be improved in later model development. The single cell electro-mechanical model of the human atrial cell implemented the
Atrial fibrillation is a most common cardiac diseases, impairing the ability of the heart to pump blood
In this study, we developed a 2D tissue model of electromechanical dynamics of atrial tissue using the Discrete Element Method by taking into considerations the discrete nature of myocytes and fibroblast. Our simulation results show that the FMEC impairs tissue’s electrical and mechanical function, which is enhanced by the increased F-M ratio or the coupling strength. Such compromising effects of the FMEC were also dependent on the way how fibroblasts are distributed among cardiac cells, with in-series coupling showing more impacts than the interstitial coupling. These findings provide new insights into understandings of how the increase of fibroblasts impairs atrial electrical conduction and mechanical contractility.
The original contributions presented in the study are included in the article/
HZ and JY conceived and designed this study. PB developed the model and conducted simulations. PB, HZ, and JY wrote the manuscript. All authors have given their approval of the final draft for submission to this Journal.
The authors are grateful for financial support provided to the project by EPSRC (EP/J009482/1; EP/I029826/1).
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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