CARDIAC MODELLING

Slide 1:CARDIAC MODELLING

Andy, Hugo, Kim, Liya, Steph and Zanna Nicolas Jeannequin

Slide 2:Why do we want to model the heart? How the Heart Works Cardiac Gating Mechanism Different Cell Models Numerics 1D and 2D Simulations Conclusions

Slide 3:Why do we want to model the heart?

Heart disease is the leading cause of death in America today. Heart attacks may cause immediate cardiac arrest or may progress to cardiac arrest. Defibrillation shocks the heart to help re-establish a heart beat. Want better understanding of all processes involved in the heart. Cardiac arrest means a complete loss of the mechanical function of the heart. Defibrillation is a process in which an electronic device gives an electric shock to the heart. This helps re-establish normal contraction rhythms in a heart having dangerous arrhythmia or in cardiac arrest. Want better understanding of all processes involved in the heart. Cardiac arrest means a complete loss of the mechanical function of the heart. Defibrillation is a process in which an electronic device gives an electric shock to the heart. This helps re-establish normal contraction rhythms in a heart having dangerous arrhythmia or in cardiac arrest.

Slide 4:How can our models help?

If we can: come up with effective models for different types of cardiac tissue run simulations with different initial conditions, or stimuli Analyse our results for useful information We can then make informed comments on the best strategies for regaining a normal heartbeat Researching electrical activity in heart may be beneficial to prevention/treatment of heart disease Replace experimental data using computers For example – where best to stimulate (tissue type) and what type of stimulus (how big, how long?)Researching electrical activity in heart may be beneficial to prevention/treatment of heart disease Replace experimental data using computers For example – where best to stimulate (tissue type) and what type of stimulus (how big, how long?)

Slide 5:How the Heart Works

Slide 6:The Heart

Chambers contract to pump blood to body and lungs Sequence of contractions must be synchronized Electro-chemical wave (Action Potential) signals heart cells to contract waste /gaseous exchange between body and external environment transmitted from the sinoatrial node in the right atrium propagate throughout the tissues of the heart to stimulate contraction waste /gaseous exchange between body and external environment transmitted from the sinoatrial node in the right atrium propagate throughout the tissues of the heart to stimulate contraction

Slide 7:Sinoatrial Node

Electrochemical oscillator Produces a spherical periodic wavefront originating from a point Wave passes through 4 different cell types Pacemaker of heart Frequency of oscillatory wave can change with adrenaline (or other neurotransmitters) Pacemaker of heart Frequency of oscillatory wave can change with adrenaline (or other neurotransmitters)

Slide 8:Propagation through Different Cell-types

ATRIAL TISSUE - Fast - Cells contract - Blood -> ventricles ATRIO-VENT. NODE - Slow - Creates a pause PURKINJE FIBERS - VERY fast - Highly conductive - Signal to ventricles VENTRICULAR TISSUE - Fast - Cells contract - Blood -> body Atrio-ventricular node: from atrium to ventricle non-excitable cells separate atria and ventricles, so signal must pass through node (0.05m/s) Purkinje Fibers: necessary for signal to reach all endocardial tissue at once to produce simultaneous contraction (5m/s). Spreads action potential simultaneously over all the tissue of the ventricles Atrio-ventricular node: from atrium to ventricle non-excitable cells separate atria and ventricles, so signal must pass through node (0.05m/s) Purkinje Fibers: necessary for signal to reach all endocardial tissue at once to produce simultaneous contraction (5m/s). Spreads action potential simultaneously over all the tissue of the ventricles

Slide 9:The Action Potential

Depolarization - wavefront causes the membrane potential to increase Repolarization - slower than the depolarisation Recovery Period – cell is non-excitable, cannot be stimulated to contract again for a certain period of time basic property of cardiac cells is that they are excitable, so depolarisation is much more rapid than repolarization basic property of cardiac cells is that they are excitable, so depolarisation is much more rapid than repolarization

Slide 10:Action Potential Waves for Different Cell Types

Waves are travelling in a 1D or 2D domain. We are modelling the propagation of these travelling waves. Sinoatrial Node Atrial Tissue Purkinje Fibers Ventricular Tissue A-V Node

Slide 11:Cardiac Gating Mechanism and cell modelling

Slide 12:Cell Membrane Transport

CELL CELL MEMBRANE Calcium Sodium Potassium Gates Control the rate at which ions can pass into and out of cells A cell of the heart, is similar to other cells in that they … The cell depicted here, is actually one of The rate at which ions pass into and out of the cell is controlled by the cell’s protein gatesA cell of the heart, is similar to other cells in that they … The cell depicted here, is actually one of The rate at which ions pass into and out of the cell is controlled by the cell’s protein gates

Slide 13:Voltage Dependent Gating

Concentrations differences at the cell membrane cause electric fields V = Vint – Vext There is a different protein gate for each substrate. Experimental evidence shows that ion transport rates through gates depends on V.

Slide 14:Squid Giant Nerve Cell

Accurately simulates action potential of cardiac tissue Large enough for accurate measurements to be made Two main types of gating proteins m: fraction of open “m” gates h: fraction of open “h” gates

Slide 15:Suggested Mechanism

Slide 16:Action Potential Models

Hodgkin – Huxley Model (1952) Constructed to fit experimental results squid axon Excitability behaviour analogous to cardiac cells

Slide 17:The Hodgkin-Huxley model

Presentation of the model: The modeling:

Slide 18:The Hodgkin-Huxley model

The equations:

Slide 19:The Hodgkin-Huxley model

The results: evolution of V with time

Slide 20:The Hodgkin-Huxley model

The results: evolution of other variables

Slide 21:The Fitzhugh Nagumo Model

The equations:

Slide 22:The Fitzhugh Nagumo Model

The numerical experiment:

Slide 23:Phase Plane Analysis

FitzHugh Nagumo model Simplified H-H model for phase plane analysis Produces threshold phenomenon and limit cycles

Slide 24:Numerics

Kimiya Minoukadeh

Slide 25:MATLAB ODE solversfor simple cell models

Non stiff regions Compare two ODE solvers: ode45 (non-stiff solver) ode15s (stiff solver, more intelligent adaptive time stepping) Fitzhugh Nagumo To demonstrate different methods for solving ODEs we will consider a simple cell model based on the Fitzhugh Nagumo model, which as described before is a simplification to the Hodgkin-Huxley model. For v, we have chosen the initial condition 0.2 which is above the threshold for the action potential to take place. The graph to the top right shows the result. To demonstrate different methods for solving ODEs we will consider a simple cell model based on the Fitzhugh Nagumo model, which as described before is a simplification to the Hodgkin-Huxley model. For v, we have chosen the initial condition 0.2 which is above the threshold for the action potential to take place. The graph to the top right shows the result.

Slide 26:Discretizing diffusion in 1D

Fitzhugh Nagumo Discretizing… x0 x2 ……..… x1 xJ j = 1, 2 ,… , J-1 I will now talk about how we implemented the diffusion term into the equation. I will now talk about how we implemented the diffusion term into the equation.

Slide 27:MATLAB code for 1D model

function f = fitzmolfun(t, y, A, epsilon) n = length(y); num = n/2; f = zeros(n,1); v = y(1:num); n = y(num+1:end); f(1:num) = epsilon*A*v + 100*(v.*(v-0.1).*(1-v)-n); f(num+1:end) = v - 0.5*n; non-steady state initial condition Pass this matrix A as a parameter to the function… I will now talk about how we model diffusion in the 1D case I will now talk about how we model diffusion in the 1D case

Slide 28:Adding a stimulus

function f = fitzmolfun(t, y, A, epsilon, x) … f(1:num) = epsilon*A*v + 100*(v.*(v-0.1).*(1-v)-n)… +100*0.3*(exp(-(0.5-x).^2/0.002)>0.7)*(t<0.1); f(num+1:end) = v - 0.5*n; steady state initial condition stimulus I will now talk about how we model diffusion in the 1D case I will now talk about how we model diffusion in the 1D case

Slide 35:Self-sustaining Waves

Under what circumstances are waves self-sustaining? Two types: Anatomical re-entry: waves travel around an obstacle Functional re-entry: rotating pattern Self sustaining action potential activty underlies many of the most dangerous and lethal cardiac arrythmias.Self sustaining action potential activty underlies many of the most dangerous and lethal cardiac arrythmias.

Slide 36:Spiral Waves

Spiral Waves are easiest to produceSpiral Waves are easiest to produce

Slide 37:Conclusions

Gained understanding of the heart and the underlying electrical processes Developed cell models using experimental data Extended cell models to 1D and then 2D by introducing diffusion Implemented numerical codes and ran simulations

Slide 38:A big thank you…

To Nicolas This would make a nice self contained course for studying an important mathematical model, implementing MatLab codes and analysing results. THE END