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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

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Action Potential (AP): Net changes of transmembrane potentials

Research Background

Na+

Cardiac Contraction Ventricular cell

APSchematic diagram of a cardiac cell

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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

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Nav

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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

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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

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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

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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

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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

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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

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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

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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∗ )

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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

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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

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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

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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

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Results – State Transitions

Action Potential

Na+ Channel Current - INa

State Occupancy

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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

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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)

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Thank you!

Questions?