APPM 2360 Project 3 Mathematical Investigation of Cardiac Dynamics

Transcription

1 APPM 2360 Project 3 Mathematical Investigation of Cardiac Dynamics Due: Thursday, December 6, 2018 by 4:59 p.m. Submit as a PDF to Assignments on Canvas 1 Introduction Cardiac Arrhythmia, or irregular heart beat, is a name for a large family of cardiac behaviors that show abnormalities in the electrical behavior of the heart. For instance, a heartbeat that is too fast ( tachycardia ) or too slow ( bradycardia ) can cause irregular activity and even death. Some arrhythmias such as heart palpitations can exhibit little danger while others such as stroke or embolism can lead to sudden death. Cardiac dynamics are typically modeled with a set of nonlinear ordinary differential equations (ODEs). These systems can be very large and each equation can be very complicated. However, to capture certain behaviors of the system, we can often work with simpler models. In our case, we will use a two-variable system to study the behavior of the voltage flux across cardiac cell membranes. This model is based on the famous FitzHugh-Nagumo model of neural excitation and propagation that results from ion-flow. 1 An understanding of how the voltage flux works can give insight to how or why arrhythmias form. In general, the functions involved in these cardiac models even our simplified, two-variable model cannot be solved analytically. Therefore, numerical solvers are useful for studying these systems. 2 Background The sinoatrial node, the heart s natural pacemaker, is a small mass of specialized cells that produces spontaneous electrical impulses in the heart. These impulses spread to nearby cardiac cells and signal the cells to contract in a coordinated manner, producing a natural and regular heartbeat. Cells are polarized, meaning that there is an electrical voltage (charge difference) across the membrane. When a cardiac cell is at rest, the charge on the outside of the cell is more positive than on the inside of the cell (due to a large concentration of calcium and sodium ions outside the cell and a large concentration of potassium ions inside the cell). The electrical impulse generated by the pacemaker cells is the cardiac Figure 1: Voltage gated ion channels in pacemaker cells action potentials, a brief change in electric charge across the cell membrane. Small ion channels in the cell membrane open and allow ions to pass through the membrane to the other side of the cell. These ion 1 The FitzHugh-Nagumo (FHN) model is itself a 2D simplification of the famous Hodgkin-Huxley model for conductance-based action potentials in squid neurons. 1

2 channels are called voltage gated channels 2 because they open for some values of membrane potential and they close for other values. 3 As positive charge moves into the cell, the voltage flux depolarizes the cell (loses its charge difference, becoming more positive inside) and the cardiac cell contracts. The cell remains contracted until repolarization occurs (gains its charge difference, becoming more negative inside). In order for the pacemaker cells to stimulate other (contractile) cells in the heart, the electric impulse generated in the pacemaker cells must propagate to nearby cells. The pacemaker cells are connected to the contractile cells through small bridges between the cells. Once the pacemaker cells have undergone an action potential, the ions flow through these openings into the nearby cells and signal them to contract. These contractile cells will only depolarize if stimulated. 3 Mathematical Model 3.1 The Basic Model The model under consideration is a type of excitation/recovery model that has the form: dv = kv(v a)(v 1) vh + S(t) dt ( dh dt = ε 0 + µ ) 1h ( h kv(v a 1)) (1) v + µ 2 The excitation in this system comes from the variable v which represents the voltage across the cell membrane. Similarly, recovery of the system is controlled by the gating variable h. The gating variable takes on non-negative values and can be thought of as a physical gate blocking or allowing voltage to pass across the cell membrane. If h = 0, then the gate is open and voltage can pass freely if the cell is stimulated. Conversely, if h > 0 then the gate reduces the voltage passing into the cell. For a large enough value of h, the gate is closed and no voltage is allowed to pass through. The function S(t) is a stimulus parameter that represents the electrical impulse generated by the sinoatrial node. This stimulus is crucial for maintaining a regular heartbeat. The parameters a, k, ε 0, µ 1, µ 2 are positive, unitless constants. Several of these parameters, and specific terms in (1), have clear physical significance: - a represents the threshold excitation in the system - k controls the magnitude of the electric current across the cell membrane. - vh describes the repolarization current in the recovery process ( ) - ε 0 + µ 1h v+µ 2 sets the time-scale for the cell s recovery/ gives a relationship between the excitation and recovery time scales. 3.1 Questions All plots in the (v, h)-phase space should be plotted with v on the horizontal axis and h on the vertical axis. No points will be given for plots that switch the axes. 1. (a) Consider the system defined by (1) where the function S(t) = 0 and the other parameters are constant. Find the nullclines of the system analytically. (Do not use exact values for any of the parameters.) 2 See YouTube video linked on the APPM 2360 website for an animation of this. 3 Fact: the special channels that cause pacemaker cells to depolarize are known as funny channels. 2

3 (b) There is only one non-negative equilibrium solution (v 0, h 0 ) in this system. Find this equilibrium solution analytically. (Do not use exact values for any of the parameters. It might be helpful to try problem 2(a) first.) (c) Recall for a system of ordinary differential equations the Jacobian matrix 4 is defined as v = f(v, h) h = g(v, h) (2) J(v, h) = [ ] fv f h where the subscripts denote partial derivatives. Find the eigenvalues of the Jacobian evaluated at the non-negative equilibrium point (v 0, h 0 ). (Do not use exact values for any of the parameters.) (d) Use the eigenvalues of J(v 0, h 0 ) to classify the stability 5 of the non-negative equilibrium solution (v 0, h 0 ). Give a biological explanation of your result (a) Let a = 0.15, k = 8, ε 0 = 0.002, µ 1 = 0.2, and µ 2 = 0.3. Plot the nullclines of (1) together on the domain v = [ 0.05, 1.2], h = [ 0.05, 2.75], making the voltage nullclines blue and the gating nullclines red. On this same plot, mark the equilibrium solution you found in Question 1(b) in black. (Be sure to put v on the horizontal axis and h on the vertical axis and include a legend.) (b) Use arrows to indicate the direction of movement in each separate region of the plot from Question 2(a) (i.e. is the flow north-west, north-east, south-west, or south-east in each region?). You may draw these arrows by hand. 3. Assign the parameters in (1) the values a = 0.15, k = 8, ε 0 = 0.002, µ 1 = 0.2, µ 2 = 0.3, and S(t) = 0. Note: The following problems depend on a Matlab script called flow.m that generates the vector field for an EXAMPLE problem (this script has been added to the Files section of Canvas). Modify flow.m so that it finds the vector field for the given problem, (1). (a) Modify the script flow.m in Matlab and plot the vector field of (1) on the domain v = [ 0.05, 1.2], h = [ 0.05, 2.75]. Starting from (v 0, h 0 ) = (0.5, 0.2), use ode45 7 in Matlab to draw a sample solution curve in the vector field and mark the starting point (v 0, h 0 ). (b) Modify the script flow.m in Matlab and plot the vector field of (1) on the domain v = [ 0.05, 1.2], h = [ 0.05, 2.75]. Starting from (v 0, h 0 ) = (0.1, 0.2), use ode45 in Matlab to draw a sample solution curve in the vector field and mark the starting point (v 0, h 0 ). (c) Compare and contrast the sample solution curves in Questions 3(a) and 3(b). How does the solution s trajectory depend on the initial condition? Given the nullclines and the vector fields that you found earlier, do the solutions behave as you would expect? Why or why not? (d) The behavior of the solutions heavily depends on the relationship between the system s excitation threshold a and the initial value of the excitation variable v 0. For each of the vector fields and solution curves found in Questions 3(a) and 3(b), give a relationship between a and v 0 and describe how the relationship between these variables affects the behavior of the samples solution. Describe why this relationship might make sense physically. 4 See page 434 in the textbook for more information about the Jacobian matrix. 5 See page 434 in the textbook for more information about stability of equilibrium solutions. 6 Hint: Biologically, (v 0, h 0) corresponds to a resting state in which the cell is polarized and not contracted. 7 The APPM 2460 lecture notes may be helpful if you have not used ode45 before. See Canvas. g v g h (3) 3

4 3.2 Model Improvement: Periodic stimulation As noted in the introduction, cardiac cells need to be stimulated by the heart s pacemaker to function properly. To maintain a steady heart beat this stimulation must come periodically. The period of stimulation in our model will be denoted by the parameter T, where the value of T defines the number of time units that pass between each stimulation. For example, if the heart is stimulated once every 100 time units, then we should set T = 100. In order to simulate realistic cardiac behavior, we must incorporate the stimulation period. The following code snippet will incorporate this periodic stimulation into (1) by giving the system a push every T time units of the simulation. It does this by setting S(t) = 0.25 if 100 time units have elapsed since the last stimulation occurred and S(t) = 0 otherwise: if ( mod(t,t) >= 10.0) && ( mod(t,t) <= 13.0 ) S = 0.25; else S = 0.0; end For any time between stimulation times, the cell will not be stimulated. 3.2 Questions Assign the parameters in (1) the values a = 0.15, k = 8, ε 0 = 0.002, µ 1 = 0.2, µ 2 = 0.3, and S(t) = Suppose that the system is initially at (v 0, h 0 ) = (0, 0) and a large, positive stimulus β is added to the voltage. Based on the plots made in Section 3.1, what direction will the trajectory move initially? What is the maximum value that v will attain? 2. Plot the nullclines of (1) again on the domain v = [ 0.05, 1.2], h = [ 0.05, 2.75]. In this same plot, use ode45 in Matlab to simulate solutions to (1) starting at (v, h) = (0, 0) given an initial large, positive stimulus β = 0.25 of the voltage. ( ) Clarification: To obtain this sample solution curve, use ode45 8 to solve (1) over the time interval t = [0, 500]. Start from the initial point (v 0, h 0 ) = (0 + β, 0) = (0.25, 0). (This time interval will give the full solution curve in phase space and the initial condition will mimic what happens when the system, starting from rest, receives a single large, positive stimulus β.) Is the flow clockwise or counter-clockwise? If no more stimuli are given after the initial push where will the system go as t? Does this agree with your reasoning in Question 1 above? 3. (a) Using ode45 in Matlab, simulate solutions to (1) starting from the initial condition (v 0, h 0 ) = (β, 0) (where β = 0.25 is a positive, initial stimulus as in Problems 1 and 2 above) over the time interval t [0, 500] using a time-step of t = 0.2. (The solution curves for v(t) and h(t) should be plotted against t in the same figure. Include a legend.) (b) Using ode45 in Matlab, simulate solutions to (1) starting from the initial condition (v 0, h 0 ) = (0, 0) over the time interval t [0, 500] using a time-step of t = 0.2, a stimulation period T = 100, and stimulus of S(t) = (The solution curves for v(t) and h(t) should be plotted against t in the same figure. Include a legend.) (c) Compare and contrast the solutions to the system with and without constant stimulus. 8 The ode45 call should be set-up as follows: [t,x] = time interval, initial conditions ) 9 Include the exact code snippet from this section in your script 4

5 4 Studying the Cardiac Action Potential One quantity of great interest and importance in cardiac modeling is the Action Potential Duration (APD). Formally, the APD is the duration from the time a cell is stimulated (and depolarizes) to the time it repolarizes. Heuristically, this measures the time duration that a heart cell is contracted. Mathematically, we calculate the APD as the time difference Figure 2: One Action Potential Duration AP D beat = t down t up (4) where t up is the time at which the voltage v passes a constant critical voltage v c on the way up (as the cardiac cells depolarize) and t down is the time at which the voltage v passes v c on the way down (after the cardiac cells repolarize). 4.0 Questions 1. Let the critical voltage be v c = 0.1. (a) Calculating the APD for the last full beat of the v(t) solution curve to (1) corresponds to the steady state APD and is denoted APD 0. Using the initial condition (v 0, h 0 ) = (0, 0) over the time interval t [0, 1000] with a time-step of t = 0.2 and stimulus S(t) = 0.25, simulate solutions to (1) and find the APD 0 for T 1 = 100, T 2 = 90, T 3 = 80, T 4 = 70, T 5 = 60, and T 6 = 50. Plot APD 0 versus stimulation period T. (b) Referring to the plot in part (b), does APD 0 increase or decrease with T? Interpret this from a biological viewpoint. That is, what is happening in the heart cell as it is stimulated more frequently? One important feature of cardiac tissue is that the APD needs to be long enough, especially for large animals. Give a reason for why this might be. 2. (a) The minimum value of h between two beats is denoted h (this is a local minimum in the h versus t plot) and is essentially a measure of how much the heart cell has been allowed to relax before the next stimulation (smaller h = more relaxed). The value of the last h between the last two beats corresponds to the steady-state h. Using the same initial conditions, step size t, time interval t, and stimulus S(t) as in Question 1(a), find the steady-state h for T 1 = 100, T 2 = 90, T 3 = 80, T 4 = 70, T 5 = 60, and T 6 = 50. Plot the steady-state h versus stimulation period T. 5

6 (b) Does the steady-state h increase or decrease with T? Interpret this from a biological point of view. That is, what is happening in the heart cell as it is stimulated more frequently. 3. Plot steady-state APD versus steady-state h. Does steady-state APD increase or decrease with steadystate h? Use this to describe the role of the two variables and how they interact with each other. 5 References 1. Aliev R, Panfilov A. A simple two-variable model of cardiac excitation. Chaos, Solitons, and Fractals. 7(3), (1996). 2. Gani, MO. Stability of periodic traveling waves in the Aliev-Panfilov reaction-diffusion system. Communications in Nonlinear Science and Numerical Simulation (2015). 3. Izhikevich EM, FitzHugh R. FitzHugh-Nagumo model. Scholarpedia, 1(9):1349, (2006). 4. Williams GS, Smith GD, Sobie EA, Jafri MS. Models of cardiac excitation-contraction coupling in ventricular myocytes. Math Biosciences. 226(1), 1-15 (2010). Helpful media sources: 5. Alila Medical (Media). Cardiac Action Potential, Animation. YouTube, YouTube, 24 Jan Draw It to Know It (Media). 6 Report Guidelines Your group will submit your project write-up on Canvas to the appropriate Project Assignment (you can find these under the assignments tab in Canvas). Adhere to the following guidelines: Do not put off finding a group (you must works in groups of 2-3). You should have a group set-up within one week of the project assignment due date. Submit your project in a pdf format and submit ALL code used for your project (.nb files for Mathematica,.m files for Matlab,.py or ipynb for python). Code in Matlab, Mathematica, Maple, R, python or Julia is acceptable (Matlab is recommended). Code in Microsoft Word or Excel (or any other spreadsheet program) is not acceptable. All other languages need instructor permission (please ask as soon as possible). Code may be included in the appendix if you wish. DO NOT submit anything on Canvas as a.zip file. Contents of.zip files will not necessarily be graded. Have only ONE group member submit the project. Having multiple people in your group submit the project to Canvas will result in multiple grades, and we will take the LOWEST one. Include the names and recitation section numbers of all group members working on the project on the cover page of the report. 6

7 When you submit the report to Canvas, please include each group member s information (name, student number, and section number) in the comments. This allows us to quickly search for a student s report. Your report needs to accurately and consistently describe the steps you took to answer the posed questions. This report should have the look and feel of a technical paper. Presentation and clarity are very important. Here are some important items to remember: Remember: you are to submit a complete report for this project. Documents submitted with numbered responses will be severely penalized. Labs must be typed, including all equations (part of your learning experience is to learn how to use an equation editor). An exception can be made for lengthy calculations in the appendix, which may be hand written (as long as they are neat and clear), and minor labels on plots, arrows in the text, and a few subscripts. Write your report in an organized and logical fashion. Section headers such as Introduction, Background, Problem Statement, Calculations, Results, Conclusion,... are not mandatory but are highly recommended. They not only help you write your report, but help the reader navigate your paper. Start with an introduction that describes what you will discuss in the body of the report. A brief summary of important concepts used in your discussion could be helpful here as well. Always introduce relevant equations that will be used or discussed in the report. Always include units in your answers DO include and label any plot that supports your conclusions or gives you insight in your investigations (these should be found in the body of your report). However, DO NOT use screen-shots of your figures. DO NOT include printouts of computer software screens or code snippets. You simply need to state which software you used in each step and what it did for you. DO include the results of any calculations in the main body of your report. DO NOT include lengthy calculations in the main body of your paper (calculations should be included in a labeled appendix and should be referred to in the main body). Your report does not have to be long. You need quality, not quantity, of work. Do not omit any important piece of information, but do not feel obligated to add any extras. Summarize what you have accomplished in the conclusion. No new information or new results should appear in your conclusion. You should only review the highlights of what your wrote about in the body of the report. 7