GPU-Enabled Spatiotemporal Model of Stochastic Cardiac Calcium Dynamics and Arrhythmias

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1 GPU-Enabled Spatiotemporal Model of Stochastic Cardiac Calcium Dynamics and Arrhythmias M. Saleet Jafri Hoang Trong Minh Tuan Department of Bioinformatics and Computational Biology George Mason University Institute of Computational Medicine, The Johns Hopkins University Department of Biomedical Engineering and Technology The University of Maryland Baltimore

2 Understanding Ca 2+ -Dependent Cardiac Arrhythmias Heart disease is the leading cause of death in developed nations and an increasing problem in the developing world. The contraction of heart muscle pumps blood throughout the body to supply oxygen and nutrients to and remove carbon dioxide and waste products from the body tissues. Death often occurs by cardiac arrhythmias that prevent this normal function of the heart.

3 Basic Components of Cardiac E-C Coupling Membrane Currents Calcium Handling Force Generation Actio n Potential Calcium Transient Force Transient

4 Understanding Ca 2+ -Dependent Cardiac Arrhythmias As cardiac myocyte calcium dynamics are essential for the contraction of the heart, the dysfunction of normal calcium dynamics is often a major factor in cardiac arrhythmias. Computational models are an essential tool to understand the complex dynamcs of cardiac myocyte calcium signaling as the explain mechanisms by integrating the known information about this system.

5 Ca 2+ Transient and Ca 2+ Sparks in Ventricular Myocytes [Ca 2+ ] "sparks" are the elementary release events. They are synchronized by the electrical signal of the cell to produce the elevation of [Ca 2+ ].

6 What is a Ca 2+ spark? seconds cell images at 0.5 sec per image sparks line-scan image at 2 ms per line location time (from Cheng, Lederer & Cannell (1993), Science 262:740) spark

7 (From L. Fernando Santana, unpublished) Heart Cell

8 Sarcomere Geometry Many diads Diads separated NSR connections All sarcomeres shorten uniformly

9 T-Tubules and SR Apposition Z-line SR M RyRs T-tubule TT-SR junction Modified from J. Frank (1990)

10 Ryanodine Receptor Organization from Baddeley, et al., micron

11 RYR'S RYR'S RyR Channel Properties Three properties of RyR gating need to be included in the model. 1. Large number of RyRs (Franzini-Armstrong et al., 1998) 2. SR lumenal [Ca 2+ ] 3. Coupled gating of RyRs [Ca 2+ ] lumen and RyR gating Coupled gating of RyRs control trans [Ca 2+ ]=20 mm trans [Ca 2+ ]=5 mm FK506 from Gyorke & Gyorke (1998) Biophys J. 75:280 Skeletal Muscle RyRs: Marx et al., (1998) Science 281:818. Heart RyRs: Gaburjakova et al. (2001) Biophys. J. 80:380A.

12 Approach Contraction of the heart is caused by the summation of calcium sparks. Build a model of contraction in the heart starting with these stochastic events. Use this model to answer fundamental questions about the mechanisms of arrhythmia that could not be answered with previous modeling efforts.

13 Questions What is the molecular basis of the calcium leak from the SR? How does the leak play a role in arrhythmogenisis?

14 Computational Challenges Stochastic Simulation is very computationally expensive. Various approaches have been suggested to address these. Reduction methods Reduction methods make assumptions about the system to make the reduction possible. These might not be valid under all relevant conditions. Sometimes the reduction require simplification of the system by reducing model complexity. This limits the details that can be included. Monte Carlo Simulation of simpler system Other attempts have simplified the dynamics of ryanodine receptor gating or the system. Using slower kinetics, fewer channels, release units, omitting physiological/biophysical detail reduced veracity of the model. We have developed the Ultrafast Monte Carlo Method that makes the computation possible

15 New Compartment Model Benchmarks Ultra-fast Ultra-fast Method Monte Carlo Monte Carlo Speedup On CPU On GPU No I/O 69 min 3:40 min 19 x I/O 70:52 min 4:26 min 16 x 20,000 release units 49 RyRs 6 DHPR 1 second physiological simulation time NOTE: Colleagues running MPI spatial cardiac code use at

16 Ultrafast Monte Carlo Method CONDITION: Row sums at each matrix are zero

17 Ultrafast Monte Carlo Method (cont.) Single channel state: [x] with x=1..n, N = 5 Cluster state: [y1,y2,,yn] with y = number of i channels in state i-th Heterogeneous cluster state: [y1z1,y1z2,,y1zn,,ynz1,ynz2,,ynzn] with y i = number of channels in state i-th, z j = number of channels in state j-th

18 Ultrafast Monte Carlo Method (cont.) The method works Non-stationary state transition, i.e. functional transition rate Heterogeneous clusters where channels are modeled as Markov-chain Memory efficient using compact form representation of state transition matrix Lazy approach: only calculate probability of next state transition if needed

19 SR Ca 2+ Leak The leak of calcium out of the SR helps maintain calcium homeostasis. It balances the SR Ca 2+ -ATPase flux. It increases when Ca 2+ in the SR increases limiting SR loading. Increases in SR Ca 2+ can lead to larger Ca 2+ release events. In some conditions, such as heart failure, disease, and Ca 2+ overload, the leak has been suggested to increase the generation of cardiac arrhythmias.

20 Ca 2+ Leak Mechanisms Calcium release through the RyRs in the form of calcium sparks has been suggested to account for part of the SR Ca 2+ leak. Calcium spark rate increases with increasing SR load. However, there remains a certain amount of leak, called invisible leak that is not yet measured. Various sources have been suggested: Backflux through the SR Ca 2+ ATP ase Other ion channels IP 3 receptors Non-junctional RyRs

21 Ca 2+ Leak Mechanisms Backflux through the SR Ca2+ ATPase Backflux through the SR Ca 2+ ATPase has not been observed under physiological conditions. Only extreme experimental manipulation can do so. Other ion channels IP3 receptors Other ion channels have not been found. Calcium flux through IP 3 receptors has not been observed in adult myocytes where they are low in number <5% the number of RyRs. Non-junctional RyRs Non-junctional channels are very small in number less than 5% of the total number of RyRs. They see bulk myoplasmic calcium which does not reach the high levels needed to trigger calcium release in the diad. Flux through these is likely small. We propose and alternative mechanism to account for invisible leak

22 Model: Revised Sticky Cluster

23 Model Equations

24 SERCA Models

25 Model Solution RyR open state calculated using our Ultrafast Monte Carlo Method Fluxes calculated to determine derivatives Differential equations solved using a Euler Method Programmed in Fortran 90/CUDA Fortran (Portland Group Compiler) on a HP z800 Linux Workstation with NVIDIA Fermi 2050 GPUs

26 Ca 2+ Dependence of Open Probability

27 Calcium Spark Mechanism

28 Calcium Transient

29 Resting Ca 2+ spark behavior 20,000 CRU - 1% plotted

30 Individual Ca 2+ Spark

31 Spark and Quark Visualization

32 RyR Opening to Spark Transition

33 SR Ca 2+ Leak - Experiment (From Zima et al., 2010)

34 SR Ca 2+ Leak - Simulation

35 Leak Analysis

36 Effects of Phosphorylation

37 New Spatial Model Benchmarks Ultra-fast Method Monte Carlo On GPU No I/O I/O 3:09 hr 3:50 hr 20,000 release units 49 RyRs 6 DHPR > 4,000,000 grid elements 1 second physiological simulation time

38 Whole-cell Modeling The cell of size 100x20x18 mm 3 is modeled with a rectangular grid with a mesh of 0.2 mm At each grid point, it contains calcium in the myoplasm, calcium in the network SR.The T-tubule is assumed to be everywhere.

39 Whole-cell modelling (cont.) Euler method with forward difference in time and central difference in space Neumann boundary condition

40 Resting Myocyte Activity Experimental Spontaneous Calcium Sparks

41 Resting Myocyte Activity Simulated Spontaneous Calcium Sparks

42 Calcium Entrained Arrhythmias Experimental Calcium overload

43 Calcium Entrained Arrhythmias Simulated Calcium Overload

44 Conclusions Our Ultrafast Markov chain Monte Carlo method make stochastic simulation of calcium dynamics possible. Calcium release initiation occurs stochastically with the opening of one RyR that can trigger additional RyRs to open. One a critical number (~6 RyRs) opens, the remaining channels open causing a spark. Calcium release termination occurs through a combination of reduced SR Ca 2+ that results in reduced RyR opening, stochastic closure, and coupled gating. Calcium leak is comprised of spontaneous calcium sparks, and the opening of one or a few RyR channels in the release sites (invisible leak). Propagation between release sites depends upon calcium load, and release site placement.

45 Co-workers: Modeling Ca 2+ sparks and leak W. Jonathan Lederer BioMET / Johns Hopkins University Aristide Chikando Univ. Maryland Baltimore George (Blair) Williams George Mason U. / Univ. Maryland Baltimore Greg Smith William & Mary College Hoang Trong Minh Tuan George Mason University Eric A. Sobie Mt. Sinai School of Medicine This work was supported by the National Science Foundation, the National Institutes of Health, and the European Union 7 th Framework Program. Thanks to Steve Worley for use of his Pseudo-Random Number Generator GPU code.

46 Traditional Monte Carlo Algorithm (A) State Diagram with states X, Y, Z and transition probabilities p and q (B) A Uniform random number [0, 1] determines state transition.

47 Ultra-Fast Markov Chain Monte Carlo Algorithm Submitted for Publication and Patent Pending Properties: Fast Exact stochastic method Low memory usage Can be used for any Monte Carlo simulation.

48 Q-Matrix of Transition Probabilities Ryanodine Receptor M=2 states minimal model x = f(ca) : Ca-dependent: C O y = g(*) : Ca-independent: O C Single-channel rate-transition matrix Chapman-Kolmogorow equation

49 State Matrix A cluster of RyR State: (c 1, c 2 ) with c 1 = number of Closed RyRs Sˆ ( M N 1)! c 2 = number of Open RyRs N!( M 1)! E.g: N=5 RyRs Cluster rate-transition matrix AR(:,:), BR(:,:) of size 6x6 This reduces the state space from N M states increasing computational efficiency.

50 Transition Matrix for Cluster Exact simulation: In a small time-step, only a SINGLE channel can change state NOTE: rate out + rate in = 0 This allows use of vector-matrix algebra to perform Monte Carlo leveraging CPU/GPU design

51 Adaptive Time Step t min 0.1 P min The time step is chosen so that transitions only occur 10% of the time. P min is the most negative row sum in the transition matrix. Using the adaptive time step decreases simulation time by about 100x

52 Heterogenous Cluster Transition Matrix A heterogeneous cluster: e.g. release site with DHPR + RyR m = # of k-state RyR cluster states n = # of j-state DHPR cluster states

53 Motivation Complexity 50 2-state RyR: cluster of 51 states 7 6-state DHPR: cluster of 924 states Release site: 47,124 states Memory demand (double-precision): 16GB K matrix: Highly sparse Question - How to handle the computation with such sparse matrices in the GPU? Answer - We have developed a novel compact form representation and compact form Kronecker product.

54 Compact Form Matrix Representation Compact form for K: Use two separate matrices: Kcomp(:,:) = keep ONLY non-zero rate transition Kidx(:,:) = keep true column index of Kcomp(i,j) A Acomp Aidx x x

55 Compact Form Matrix Representation A homogeneous cluster: Full matrix Compact form Less number of conditional comparisons Less memory demand

56 Compact Form Matrix Representation A heterogeneous cluster, e.g. RyR + DHPR: Full matrix Compact form

57 Ultra-fast Monte Carlo Method Pros: Based upon matrix representation of the release units and their state space derived in part from the theory of solution of stochastic automata networks. Uses an adaptive time step. Only considers possible transitions from current state of each release units (scales as the number of possible transitions rather than the size of the state space) Reduces number of conditional comparisons. Amenable to parallelization.

58 Ultra-fast Monte Carlo Method Caveats: Reduces number of conditional comparisons by treating channels as identical so the number of channels in a particular configuration is counted. The transition from one state to another should be single agent dependent, e.g. either V or [Ca], but not both. m This does not allow for more than one event in a time step. In our simulations this only occurs ~6% of the time and this fraction can be reduced if needed.

59 Presentation Outline 1. Introduction to Excitation-Contraction Coupling 2. Calcium Sparks Experiment and Model 3. Ultrafast Monte Carlo Algorithm 4. GPU Implementation 5. Calcium-Entrained Arrhythmias 6. Conclusions

60 Computational Model of Resting Ca 2+ Dynamics RyR Minimal model with 2 states Release site: Each site has 50 RyR Whole-cell model 10,000 release sites

61 GPU Issues How GPU fit to our problem & algorithm Highly independent of release site computation Large amount of computation can be done in parallel State space is reused at every computational step Low memory demand makes it fit to the limited device memory (4GB in Tesla 1060, 3GB in Fermi) Need to minimize transfers between CPU and GPU due to Memory access latency Large amounts of I/O