gpu-enabled spatiotemporal model: stochastic cardiac ca+...
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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
Understanding Ca2+-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.
Membrane
Currents
Calcium
Handling
Force
Generation
Basic Components of
Cardiac E-C Coupling
Action
Potential
Calcium
Transient
Force
Transient
Understanding Ca2+-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.
Ca2+ Transient and Ca2+ Sparks in
Ventricular Myocytes
[Ca2+] "sparks" are the elementary release events. They are synchronized by the electrical signal of the cell to produce the elevation of [Ca2+].
What is a Ca2+ spark? 0.0 0.5 1.0 1.5 2.0 2.5 seconds
cell images at 0.5 sec per image
(from Cheng, Lederer & Cannell (1993), Science 262:740)
sparks
line-scan image at 2 ms per line
spark time
location
(From L. Fernando Santana, unpublished)
Heart Cell
• Many diads
• Diads separated
• NSR connections
• All sarcomeres shorten
uniformly
Sarcomere Geometry
M Z-line
Modified from J. Frank (1990)
T-tubule
TT-SR
junction
SR
RyRs
T-Tubules and SR Apposition
39
25
36
61
1 micron
from Baddeley, et al., 2009
Ryanodine Receptor Organization
RyR Channel Properties
[Ca2+]lumen and RyR gating
from Gyorke & Gyorke (1998) Biophys J. 75:280
trans [Ca2+]=20 mM trans [Ca2+]=5 mM
Three properties of RyR gating need to be included in the model.
2. SR lumenal [Ca2+]
3. Coupled gating of RyRs
Coupled gating of RyRs
Skeletal Muscle RyRs: Marx et al., (1998) Science 281:818.
Heart RyRs: Gaburjakova et al. (2001) Biophys. J. 80:380A.
control
+ FK506
0
2 1
RY
R'S
0
2 1
RY
R'S
1. Large number of RyRs (Franzini-Armstrong et al., 1998)
• 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.
Approach
• What is the molecular basis of the calcium leak from the SR?
• How does the leak play a role in arrhythmogenisis?
Questions
Computational Challenges
• 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 …
Stochastic Simulation is very computationally expensive. Various approaches
have been suggested to address these.
New Compartment Model Benchmarks
• 20,000 release units
• 49 RyRs
• 6 DHPR
• 1 second physiological simulation time
NOTE: Colleagues running MPI spatial cardiac code use at most 50 CPUs before losing speedup.
Method
Ultra-fast
Monte Carlo
On CPU
Ultra-fast
Monte Carlo
On GPU
Speedup
No I/O 69 min 3:40 min 19 x
I/O 70:52 min 4:26 min 16 x
Ultrafast Monte Carlo Method
CONDITION: Row sums at each matrix are zero
Ultrafast Monte Carlo Method (cont.)
• Single channel state: [x] with x=1..N, N = 5
• Cluster state: [y1,y2,…,yN] with yi = number of
channels in state i-th
• Heterogeneous cluster state:
– [y1z1,y1z2,…,y1zN,…,yNz1,yNz2,…,yNzN] with yi
= number of channels in state i-th, zj = number of
channels in state j-th
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
SR Ca2+ Leak
• The leak of calcium out of the SR helps maintain calcium
homeostasis. It balances the SR Ca2+-ATPase flux.
• It increases when Ca2+ in the SR increases limiting SR loading.
Increases in SR Ca2+ can lead to larger Ca2+ release events.
• In some conditions, such as heart failure, disease, and Ca2+
overload, the leak has been suggested to increase the generation
of cardiac arrhythmias.
Ca2+ Leak Mechanisms
• Calcium release through the RyRs in the form of calcium sparks has been suggested to account for part of the SR Ca2+ 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 Ca2+ ATP ase
– Other ion channels – IP3 receptors
– Non-junctional RyRs
Ca2+ Leak Mechanisms
• Backflux through the SR Ca2+ ATPase – Backflux through the SR Ca2+ 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 IP3 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 …
Model: “Revised Sticky Cluster”
Model Equations
SERCA Models
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
Ca2+ Dependence of Open Probability
Calcium Spark Mechanism
Calcium Transient
Resting Ca2+ spark behavior
20,000 CRU - 1% plotted
Individual Ca2+ Spark
Spark and Quark Visualization
RyR Opening to Spark Transition
SR Ca2+ Leak - Experiment
(From Zima et al., 2010)
SR Ca2+ Leak - Simulation
Leak Analysis
Effects of Phosphorylation
New Spatial Model Benchmarks
• 20,000 release units
• 49 RyRs
• 6 DHPR
• > 4,000,000 grid elements
• 1 second physiological simulation time
Method
Ultra-fast
Monte Carlo
On GPU
No I/O 3:09 hr
I/O 3:50 hr
Whole-cell Modeling • The cell of size 100x20x18
mm3 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.
Whole-cell modelling (cont.)
• Euler method with forward difference in time and
central difference in space
• Neumann boundary condition
Resting Myocyte Activity Experimental Spontaneous Calcium Sparks
Resting Myocyte Activity Simulated Spontaneous Calcium Sparks
Calcium Entrained Arrhythmias Experimental Calcium overload
Calcium Entrained Arrhythmias Simulated Calcium Overload
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 Ca2+ 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.
Co-workers: Modeling Ca2+ sparks and leak
W. Jonathan Lederer
BioMET / Johns Hopkins University
Hoang Trong Minh Tuan
George Mason University
Aristide Chikando
Univ. Maryland Baltimore
George (Blair) Williams
George Mason U. / Univ. Maryland Baltimore
Eric A. Sobie
Mt. Sinai School of Medicine Greg Smith
William & Mary College
• This work was supported by the National Science Foundation, the National Institutes of Health, and
the European Union 7th Framework Program.
• Thanks to Steve Worley for use of his Pseudo-Random Number Generator GPU code.
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.
Submitted for Publication and Patent Pending
Properties:
Fast
Exact stochastic method
Low memory usage
Can be used for any Monte Carlo simulation.
Ultra-Fast Markov Chain Monte Carlo
Algorithm
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
State Matrix A cluster of RyR
State: (c1, c2) with
c1 = number of Closed RyRs
c2 = number of Open RyRs
E.g: N=5 RyRs
Cluster rate-transition matrix
AR(:,:), BR(:,:) of size 6x6
This reduces the state space from NM states increasing computational efficiency.
)!1(!
)!1(ˆ
MN
NMS
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
Adaptive Time Step
• The time step is chosen so that transitions only occur 10% of the time.
• Pmin is the most negative row sum in the transition matrix.
Using the adaptive time step decreases simulation time by about 100x
min1.0min Pt
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
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.
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)
0 12 0 1
0 0 0 2
... ...
... ...
12 1
2 x
2 2 4
1 4 x
Acomp
Aidx A
Compact Form Matrix Representation
• A homogeneous cluster:
– Full matrix
– Compact form
Less number of conditional comparisons
Less memory demand
Compact Form Matrix Representation
• A heterogeneous cluster, e.g. RyR + DHPR:
– Full matrix
– Compact form
• 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.
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 Vm or [Ca], but not both.
– 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.
Ultra-fast Monte Carlo Method
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
Computational Model of Resting Ca2+ Dynamics
RyR
Minimal model with 2 states
Release site:
Each site has 50 RyR
Whole-cell model
10,000 release sites
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
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