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1
Statistical Metamodeling and Computer
Experiments of Large-Scale Cardiac Models
University of South Florida
Dongping Du, Hui YangIndustrial and Management Systems Engineering
Andrew R. Ednie, Eric S. BennettMolecular Pharmacology and Physiology
2
Action Potential (AP): Net changes of transmembrane potentials
Research Background
Na+
Cardiac Contraction Ventricular cell
APSchematic diagram of a cardiac cell
3
Background
Glycosylation:
The enzymatic process that attaches glycans to proteins, lipids, or
other organic molecules
4
Background
Voltage-gated sodium channel (Nav) – Initiation of cellular excitation
Altered Glycosylation Nav channel Cardiac function
4
Nav
5
Motivations
Congenital Disorders of Glycosylation (CDG):
High infant mortality rate
Multi-system effects, e.g., diabetes, cardiac diseases
Prevalence of cardiac involvement
Unknown etiology of cardiomyopathy among young CDG patients
Goals
Understand the pathology of cardiac disease among CDG patients
Develop pertinent therapeutic solutions
6
Gaps
Little is known on how reduced glycosylation affects cardiac electrical signaling
Limitations
Study the changes at molecular levels
Connect changes at one organizational level, e.g., ion channels, to another
organizational level, cardiac cells.
Objectives:
Couple in-silico studies with the wealth of data from our electrophysiological
experiments to model, mechanistically, how reduced glycosylation affects Na+
activity and cardiac electrical signaling.
Gaps
Na+
K+
Ca+
myocyte
7
Challenges
Nonlinear model characteristics
High dimensionality of design space: 25 parameters
Markov Model of Na+ channel
𝑑𝑷(𝑡)
𝑑𝑡= 𝒇(𝑡, 𝑉, 𝑨, 𝑷 𝑡 )
where 𝐏 = [𝑃𝐼𝐶3, 𝑃𝐼𝐶2, 𝑃𝐼𝐹 , 𝑃𝐼1, 𝑃𝐼2, 𝑃𝐶3, 𝑃𝐶2, 𝑃𝐶1, 𝑃𝑂]T, 𝐀 is the 9 × 9 transition rate
matrix, e.g. 𝐴1,1 = − 𝛼31 + 𝛼111 , 𝛼31 =1
2.5exp𝜽1+V
7.0+8.0exp
𝜽2+V
7.0
𝜶𝟐
𝜶𝟏𝟏
𝜷𝟏𝟐𝜷𝟏𝟏
𝜷𝟐𝜷𝟑𝟑
𝜷𝟑𝟏
𝜷𝟓𝜷𝟏𝟏𝟏
𝜷𝟏𝟑
𝜶𝟏𝟑
C3
𝜶𝟏𝟏𝟏 𝜶𝟒 𝜶𝟓
𝜶𝟑𝟏
𝜷𝟒
𝜶𝟏𝟏𝟐
𝜶𝟑𝟐 𝜷𝟑𝟐
C1C2O
IF I1 I2
𝜶𝟑𝟑
IC2IC3𝜷𝟏𝟏𝟐
𝜶𝟏𝟐
8
Challenges
3 different cardiac functional responses from pulse protocols
High computational cost
𝑓(𝜽, 𝒕, 𝑽)𝜽
patch clamp protocols
𝑰 𝒚
normalize peak currents
ion currents
𝑰𝒑𝒆𝒂𝒌
model parameter modeled
outputs
-80 -60 -40 -20
0.2
0.4
0.6
0.8
20 40 60 80 100120-25
-20
-15
-10
-5
0 50 100-100
-50
0
(a) protocol(b) ion currents
(c) normalized peak
current
9
Research Methodology
Design
variablesMachine parameters,
Enviromental factors,
Human Factors ...
Response
variablesQuality, Integrity,
Performance Metrics, ...
Simulation Model
Costly
Design
variables
Response
variables
Economical
Statistical MetamodelingGaussian Process, Kriging, Neural Network,...
Space-filling Design
Sequential Design
Global Optimization
Adaptive Modeling
10
Research Methodology
Screening Design:
Identify important control variables from the high-dimensional set of
potential parameters
2𝑉25−15 fractional
factorial design
𝜽𝑆𝑆𝐴 = {𝜽1,𝜽2,𝜽3,𝜽4,𝜽5,𝜽6,𝜽17,𝜽22}
𝜽𝑅𝐸𝐶 = 𝜽11, 𝜽12, 𝜽15, 𝜽18, 𝜽19, 𝜽23, 𝜽25
𝜽
𝒁𝑺𝑺𝑨
𝒁𝑺𝑺𝑰
𝒁𝑹𝑬𝑪
The large set of
voltage-relevant
parameters
Sensitive to model
outputs
Curse of Dimensionality
𝜽𝑆𝑆𝐼 = 𝜽11, 𝜽12, 𝜽13, 𝜽14, 𝜽23, 𝜽24
11
Research Methodology
Space-Filling Design:
Generate design points (samples) over the control variable space (𝜽)
Maximin Latin Hypercube Design (LHD):
𝑑𝑖𝑗 = 𝒙𝑖 − 𝒙𝑗
max𝐗∈ 0,1 𝑑 min𝑖≠𝑗𝑑𝑖𝑗
Maximin LHD on high
dimensional space …
2D 3D
12
Gaussian Process
GP: a collection of random variables, any finite number of
which have joint Gaussian Distribution
𝒇𝒇∗
~𝑵 𝟎,𝑲(𝑿, 𝑿) 𝑲(𝑿, 𝑿∗)𝑲(𝑿∗, 𝑿) 𝑲(𝑿∗, 𝑿∗)
𝒇 training outputs, 𝒇∗ test outputs, 𝑲(, ) covariance function
Posterior distribution
𝒇|𝒇∗~𝑵(𝑲 𝑿∗, 𝑿 𝑲−𝟏𝒇,𝑲 𝑿∗, 𝑿∗ −𝑲 𝑿∗, 𝑿𝑻𝑲−𝟏𝑲 𝑿,𝑿∗ )
13
Research Methodology
Probability of Improvement
ProbI = Φ(𝑇 − 𝐸 𝑓 𝒙∗
𝑠{𝑓(𝒙∗})
where 𝐸 𝑓 𝒙∗ and 𝑠{𝑓(𝒙∗} are mean and standard error of
predictions.𝒩(∙) is the Normal CDF function.
𝑇
𝑓𝑚𝑖𝑛(𝒙∗, 𝛿∗)Φ(
𝑇 − 𝛿∗𝑆(𝛿∗)
)
Probability of improvement
14
Research Methodology
Probability of Improvement
ProbI = Φ(𝑇 − 𝐸 𝑓 𝒙∗
𝑠{𝑓(𝒙∗})
where 𝐸 𝑓 𝒙∗ and 𝑠{𝑓(𝒙∗} are mean and standard error of
predictions.𝒩(∙) is the Normal CDF function.
-1.5 -1 -0.5 0 0.5 1 1.5
-3
-2
-1
0
1
input, x
𝑇
𝑓𝑚𝑖𝑛
(𝒙∗, 𝛿∗)Φ(
𝑇 − 𝛿∗𝑆(𝛿∗)
)
-1.5 -1 -0.5 0 0.5 1 1.5
-0.5
0
0.5
1
1.5
2
input, x
ou
tpu
t, y
1
prediction
data points
Probability of improvement
15
Research Methodology
Probability of Improvement
ProbI = Φ(𝑇 − 𝐸 𝑓 𝒙∗
𝑠{𝑓(𝒙∗})
where 𝐸 𝑓 𝒙∗ and 𝑠{𝑓(𝒙∗} are mean and standard error of
predictions.𝒩(∙) is the Normal CDF function.
𝑇
𝑓𝑚𝑖𝑛
Optimal solution
16
Results
Refractory periods of ST3Gal4-/- and WT cells
Computer experiments: WT: 138.0ms, ST3Gal4-/-: 109.5ms
Physical experiments: WT: 139.8 ± 8.6ms, ST3Gal4-/-: 110.2 ± 10.0ms
0 50 100 150 200
-80
-60
-40
-20
0
Time(ms)
Acti
on
Po
ten
tial
(mV
)
WT
ST3Gal4-/-
Refractory
Period
17
Results – State Transitions
Action Potential
Na+ Channel Current - INa
State Occupancy
18
Conclusions
Challenges
Nonlinear differential equations
High-dimensional design space
Experimental protocols and functional responses
Methodological and Biomedical Merits
Statistical metamodeling and sequential design of experiments
Computer experiments and calibration of large-scale cardiac models
Improve fundamental knowledge about the functional role of
glycosylation in cardiac electrical signaling
New pharmaceutical designs to correct aberrant glycosylation
19
Acknowledges
CMMI-1266331
IIP-1447289
IOS-1146882
Publications
1. D. Du, H. Yang, A. R. Ednie, and E. S. Bennett, “Statistical Metamodeling and Sequential Design
of Computer Experiments to Model Glyco-altered Gating of Sodium Channels in Cardiac
Myocytes” IEEE Journal of Biomedical and Health Informatics, second round review.
2. D. Du, H. Yang, S. Norring, and E. S. Bennett, “In-silico modeling of glycosylation modulation
dynamics in hERG channels and cardiac electrical signaling,” IEEE Journal of Biomedical and
Health Informatics, vol. 18, issue 1, p. 205-214, 2014. (Feature article by IEEE Journal of
Biomedical and Health Informatics)
3. D. Du, H. Yang, S. Norring, and E. Bennett, "Multiscale modeling of glycosylation modulation
dynamics in cardiac electrical signaling," Proceedings of 2011 IEEE Engineering in Medicine and
Biology Society Conference (EMBC), pp.104-107, Aug. 30 2011-Sept. 3 2011, Boston, MA.
(Placed first in IBM Best Student Paper competition)
20
Thank you!
Questions?