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Advanced simulation in medicine and biology: Opportunities for multiphysics modeling
Steven Conrad, MD PhD Louisiana State University Health Sciences Center
Shreveport, LA
COMSOL Conference 2010 Boston Presented at the
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‘Traditional’ modeling approaches in medicine and biology
ODE-based lumped compartmental models • Mass transport in artificial organs / body systems /
intracellular • Chemical reactions involving fluids and electrolytes • Heat transport in the body • Electrostatic discharge from the body • …
Limitations • Many underlying assumptions • Constrained to specific operating conditions
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Arteriovenous carbon dioxide removal model schematic
Fig. 1
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Gasin
Gasout
TissueComp
PeriphCapillary
PulmonaryCapillary
GasPhase
BloodPhase
AlveolarVentilation
PulmonaryAlveolarComp
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Mathematical model description
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Carbon dioxide removal simulations
0.00 0.05 0.10 0.15 0.20Shunt Flow (fraction of cardiac output)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
CO2
rem
oval
(fra
ctio
n of
VCO
2)
A
Membrane DLCO2 = 0.3Gas:Blood = 3
0.00 0.15 0.30 0.45 0.60
Membrane DLCO2 (ml/min/torr)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
CO2
rem
oval
(fra
ctio
n of
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B
Shunt Fraction = 0.15Gas:Blood = 3
0 3 6 9 12Gas:Blood ratio
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0.2
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oval
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20 torr40 torr80 torr160 torr
Wt = 5 kg
CO = 500 ml/minVCO2 = 4 ml/kg/min
Shunt Fraction = 0.15
C
Baseline PaCO2:
Membrane DLCO2 = 0.3
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Recent modeling approaches in medicine and biology
Single physics finite element analysis • CFD of blood flow in blood
vessels and pumps • CFD of gas transport in the
lungs • Heat transport in organs • Fluid-structure interaction in
the brain • Musculoskeletal stress and
strain Highly focused studies
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Hemodiafilter Design
HF: 240×10-6 m inner diameter 0.1 to 0.2 m in length
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COMSOL example – separation through dialysis
COMSOL model library Axisymmetric hollow
fiber geometry Solute transport
through the membrane by diffusion
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Diffusive vs. convective transport
Hemodialysis (traditional) • Focus on diffusive transport • Lower porosity membranes • Removal of small, readily diffusible solutes thought
solely responsible for toxicity of renal failure Hemofiltration
• Focus on convective transport • Higher porosity membranes • Removal also of larger solutes (‘middle molecules’)
now known to contribute to toxicity of renal failure Combined therapies now common
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Governing principles for modeling
Small channel blood flow (r ≈ 120 µm) • Applied flows/pressures to blood side • Non-Newtonian properties of blood
Blood viscosity varies
• Hemoconcentration Axial alterations in blood density and viscosity
• Blood cell skimming behavior (Fahraeus-Lindqvist) Radial alterations in blood density and viscosity
Presence of proteins in blood phase • Influence convection from osmotic pressure
• Additional axial alterations in density, viscosity and osmotic pressure
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Governing principles for modeling
Solute characteristics • Partitioning between plasma and red blood cell
• Partial solute rejection at membrane surface
Membrane factors • Interaction of convective and diffusive transport
• Concentration polarization of blood cells, protein and partially reflected smaller molecules
• Interaction of hydraulic pressure and osmotic effects to produce forward filtration and backfiltration
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Governing principles for modeling
Dialysate factors • Applied fluid flows and pressures
• Concentration of solutes in dialysate
Dialysate has cooler temperatures than blood • Heat loss to environment
• Temperature dependence of blood density and viscosity
Ionic charge on proteins and some solutes • Protein rejection at membrane
• Impairment of solute transport across membrane
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Previous mathematical models
Single compartment models Multiple compartment models One-dimensional computational models Two/three-dimensional computational model
• Finite element analysis
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Single compartment analytical model
Single blood and dialysate mass balance compartments
Used a length-averaged mass transfer coefficient for solutes • Dependent upon operating
conditions Overestimated filtration rate vs.
experimental data • Could not account for
concentration polarization or membrane rejection
Unable to account for processes such as backfiltration Pallone TL: Kidney Int 1988;33:685-98
Pallone TL: Kidney Int 1989;35:125-133
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Multiple compartment analytical model
Accounted for local fluid balance changes (axially only) • Pressure drops • Viscosity • Osmotic effects
Addressed fluid fluxes only • Did not include convective or
diffusive solute transport Described backfiltration not
possible in simpler models Does not account for radial
effects • Concentration polarization
Fiore GB: Contrib Nephrol 2005;149:27-34
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One-dimensional computational models
Ordinary differential equations solved numerically along the axis • Included local viscosity,
osmotic pressure, etc. Include solute transport
with convection-diffusion interaction • Formulas of Zydney
(extension of Villarroel) • Flat membrane
Included approximations of boundary layer effect on mass transfer coefficient • Able to account for
concentration polarization
Legallis C: J Membr Sci 2000;168:3-15 Raff M: J Membr Sci 2003;216:1-11
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Limitations of previous models
Simplifying assumptions • Overall mass transfer coefficients • Approximations to boundary conditions
Geometrically naïve • Single compartments naïve to axis and radius • Multiple compartments naïve to radius, and approximate axial
conditions • One dimensional computational models approximate radial
conditions • Membranes modeled as boundaries only
Biased estimates of convection-diffusion interaction • Ignored in previous models • Based on older flat membrane analysis in advanced models
Model geometry based on hollow fiber membranes
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Hollow Fiber Geometry Modeling
Hospal M60 AN69S membrane
Axial symmetry
Blood
Membrane
Dialysate
120 µm
170 µm 225 µm
Circular approximation
Blood
HF
Dialysate
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Dialysate: Navier-Stokes
Membrane: Brinkman
Blood: Non-Newtonian
Heat: Heat
transport
COMSOL Application Modes
Protein: Convection-
Diffusion
Solute: Convection-
Diffusion
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Fahraeus-Lindqvist Effect
1 2
( 1)( 1)( )
2
2 2
1 2 ( 1)( 2)
where
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.206
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n n nH r H
r r r
n n n n n n
nH
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Colloid Osmotic Pressure
Cproin
Cpro → Posm
Fu
2 31.2 0.16 0.009osm p p pP C C C
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Ultrafiltration rate validation
0 100 200 300 400Transmembrane pressure (torr)
-10
10
30
50
Ultr
afilt
ratio
n ra
te (m
l/min
)
Qb 100 ml/min
Solid lines: experimental measurementsSymbols: model predictions
Qb 180 ml/minManufacturer’s data: Gambro M60 hemofilter
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Solute clearance during dialysis
Vbldin Vbldout
Historical assumption:
Vdiaout Vdiain
Vdiain = Vdiaout → zero net ultrafiltration → diffusive clearance only
Pressure drop
Recent recognition:
Internal filtration Backfiltration
Additional convective clearance
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Sensitivity of fluid flux to the protein diffusion coefficient
0.5 1.0 1.5 2.0 2.5 3.0
Diffusion coefficient (1010 m2/s)
25
30
35
40
45
50
55
Ultr
afilt
ratio
n ra
te (m
l/min
)
Gambro M60TMP 300 torrQb 180 ml.minCpro 60 g/L
Experimental UF rate
Predicted UF rate
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Dialysate: Navier-Stokes
Membrane: Brinkman
Blood: Non-Newtonian
Heat: Heat
transport
Expanding the application modes
Protein: Convection-
Diffusion
Protein charge:
Electrostatics
Solute: Convection-
Diffusion
Solute charge: Electrostatics
Membrane charge: Electrostatics
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Modeling the circadian rhythm – existing ordinary differential equation models
• Cell shape • Cytoplasmic streaming • Intracellular organelles • DNA binding kinetics
• Momentum transport • Diffusive transport • Convective transport • Chemical reactions
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In closing
Multiphysics modeling has not been widely exploited in medicine and biology
Most modeling to date has been in the application of technology to medical treatments, such as artificial organs and tissue heating
Little multiphysics finite element analysis has been applied to study of underlying physiological and cellular processes