Introduction Past Present Future Summary A comparison of discrete and continuum models of cardiac electrophysiology Doug Bruce Computational Biology Group Department of Computer Science University of Oxford 28th November 2012 Doug Bruce CompBio Group Talk
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A comparison of discrete and continuum models of cardiac electrophysiology · models of cardiac electrophysiology Doug Bruce Computational Biology Group Department of Computer Science
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IntroductionPast
PresentFuture
Summary
A comparison of discrete and continuummodels of cardiac electrophysiology
Doug Bruce
Computational Biology GroupDepartment of Computer Science
University of Oxford
28th November 2012
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Outline1 Introduction
Biological BackgroundResearch Questions
2 PastPhysiology of gap junctionsAdapting models to include gap junctionsResults of simulationsConclusions
3 PresentThe homogenised conductivity tensors
4 FutureHybrid Modelling
5 Summary
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Biological BackgroundResearch Questions
Outline1 Introduction
Biological BackgroundResearch Questions
2 PastPhysiology of gap junctionsAdapting models to include gap junctionsResults of simulationsConclusions
3 PresentThe homogenised conductivity tensors
4 FutureHybrid Modelling
5 Summary
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Biological BackgroundResearch Questions
Discrete Modelling
Many phenomena in biology are discrete:For example, biological tissue consists of discrete cellsThese cells lie in an extracellular matrix
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Biological BackgroundResearch Questions
Discrete Modelling
Different sets of equations usually apply in intracellular andextracellular regions
Thus, we must model each cell individually
This is not practical at organ or tissue level
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Biological BackgroundResearch Questions
Continuum Modelling
To overcome this, we often model the tissue as a continuum:
‘Average’ the quantities we are solving for
Retain their small-scale behaviours
Process known as homogenisation
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Biological BackgroundResearch Questions
Modelling Cardiac Tissue
An example of cardiac cells:
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Biological BackgroundResearch Questions
Modelling Cardiac Tissue
From discrete: To continuum:
What assumptions are made in this process?
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Biological BackgroundResearch Questions
Assumptions in the Homogenisation Process
Problem naturally defined on two scales:Macroscale — Tissue-levelMicroscale — Cell-level
Parameter introduced — ratio of the two scalesIt is assumed very small
Physiology of gap junctionsIncorporation of gap junctionsResults of simulationsConclusions
Effect of changing sodium conductance
Steepness of upstrokerelated to ionic modelparameter gNa
Increase by a factor of 10and look and results ofsimluations
No gap junctions: modelsstill match up
Gap junctions: largerdiscrepancy withincreased upstrokevelocity
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−100
−80
−60
−40
−20
0
20
40
60
80
Distance (mm)
Mem
bran
e Po
tent
ial (
mV)
Comparison of solutions without gap junctions
Continuum, g
Na = 0.04
Continuum, gNa
= 0.4
Discrete, gNa
= 0.04
Discrete, gNa
= 0.4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−100
−80
−60
−40
−20
0
20
40
60
80
100
Distance (mm)
Mem
bran
e Po
tent
ial (
mV)
Comparison of solutions with gap junctions
Continuum, g
Na = 0.04
Continuum, gNa
= 0.4
Discrete, gNa
= 0.04
Discrete, gNa
= 0.4
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Physiology of gap junctionsIncorporation of gap junctionsResults of simulationsConclusions
Outline1 Introduction
Biological BackgroundResearch Questions
2 PastPhysiology of gap junctionsAdapting models to include gap junctionsResults of simulationsConclusions
3 PresentThe homogenised conductivity tensors
4 FutureHybrid Modelling
5 Summary
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Physiology of gap junctionsIncorporation of gap junctionsResults of simulationsConclusions
Conclusions
Gap junctions cause discrepancy in propagation speed &characteristics between discrete and continuum modelsMain area is around upstroke of action potentialCapacitance and ionic current of junction has no effect ondiscrepancyReduction in gap junction conductivity increasesdiscrepancyIncrease in upstroke velocity increases discrepancy
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
The homogenised conductivity tensors
Outline1 Introduction
Biological BackgroundResearch Questions
2 PastPhysiology of gap junctionsAdapting models to include gap junctionsResults of simulationsConclusions
3 PresentThe homogenised conductivity tensors
4 FutureHybrid Modelling
5 Summary
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
The homogenised conductivity tensors
What affects the homogenised conductivity tensors?
Recap:
Scalar conductivities σ(i,e)
Continuum tensors Σ(i,e) obtained via:
Σ(i,e) =1
Vcell
ZΩ(i,e)
σ(i,e)
„I +
∂W(i,e)
∂z
«dVz,
Functions W (i,e)j satisfy:
∇z · (σi∇zW ij ) = −∂σi
∂zj, ∇z · (σi∇zW e
j ) =∂σe
∂zj
With boundary conditions:
∇zW ij · n = −nj , ∇zW e
j · n = nj
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
The homogenised conductivity tensors
What affects the homogenised conductivity tensors?
Thus, tensors affected by:
Scalar conductivity — gap junctions changing intracellularconductivitySize & shape of cell membraneProportions of intra- and extracellular space
For bidomain equations, quantities of interest are Σi/χ and(Σi + Σe)/χ.
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
The homogenised conductivity tensors
Building a unit cell in 2D
CellHeight
δx
δyUnit cell to be homogenised
Curve: Axn + Byn + C = 0
σg
CellSeparation
Side-side gap junctions
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
The homogenised conductivity tensors
Performing calculations
We can calculate tensors for all possible parameter values
Huge parameter space with little physiological meaning
Need to isolate which values vary and by how much
Liaising with experimentalists to determine these
Combine with sensitivity analysis
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
The homogenised conductivity tensors
Limitations to 2D approach
No side-side gap junctions: propagation only in fibredirection
Side-side junctions included: no extracellular propagation
However, between these methods we should be able toextract the relevant answers
3D approach removes the problem, currently working onthis
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
The homogenised conductivity tensors
Initial thoughts/results
Change one variable at a time, keep others at a ‘default’level
Height of cell has large effect on (Σi + Σe)/χ
Gap junction height δx also does
Looking to combine this with experimental data
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
The homogenised conductivity tensors
How does the choice of unit cell affect results?
Length of cells: distribution of lengths in fibre & off-fibredirection
Compare results with those expected using mean values
Again combine with experimental data
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Hybrid Modelling
Outline1 Introduction
Biological BackgroundResearch Questions
2 PastPhysiology of gap junctionsAdapting models to include gap junctionsResults of simulationsConclusions
3 PresentThe homogenised conductivity tensors
4 FutureHybrid Modelling
5 Summary
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Hybrid Modelling
Creation of New Hybrid Models
We know what might case a discrepancy between discrete &continuum systems
Gap junctions (especially if lowered conductivity)Steepness of upstroke (specific to cell type)Cell shape & size? Results of previous section willelucidate
We propose a hybrid solution methodUse continuum model (e.g. bidomain) where possibleUse discrete model if continuum assumptions invalid
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Hybrid Modelling
Creation of New Hybrid Models
First, we must:Find the appropriate discrete parameters (conductivitiesetc.)
Derive the corresponding continuum model
Get a ‘feel’ for under what conditions the continuum modeldoes not replicate the discrete model
i.e. perform simulations using both models individually
Convert this into mathematical criteriaDoug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Hybrid Modelling
Hybrid Modelling
A possible method for implementation:
Solve continuum system everywhereLook for regions where solution varies rapidlyi.e. Vmax >= some thresholdRe-solve, using discrete model in such regions
Motivated by work of Hand et al. (*)
They have 1D (cable/monodomain) model of this formWe wish to extend to 2D & 3DIncorporating more realistic geometries (driven byhomogenised tensor data)
(*) Hand PE, Griffith BE: Adaptive multiscale model for simulating cardiac conduction. Proceedings of the
National Academy of Sciences 2012, 107(33):14603-14608
Doug Bruce CompBio Group Talk
IntroductionPast
PresentFuture
Summary
Summary
Inclusion of gap junctions causes discrepancy betweendiscrete & continuum models
Hybrid model may be appropriate: requires investigationinto suitable criteria
Shape of cell boundary also likely to be relevant —considering homogenised conductivity tensors in moredetail