Advanced simulation in medicine and biology: Opportunities for multiphysics modeling

Transcription

1 Presented at the COMSOL Conference 2010 Boston Advanced simulation in medicine and biology: Opportunities for multiphysics modeling Steven Conrad, MD PhD Louisiana State University Health Sciences Center Shreveport, LA

2 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 2

3 Arteriovenous carbon dioxide removal model schematic Fig. 1 Pulmonary Alveolar Comp Pulmonary Capillary Tissue Comp Periph Capillary Gas Phase Blood Phase Gas in Gas out Alveolar Ventilation Qalv Qshunt 3

4 Mathematical model description t PCO2 Qalv DLCO2 CalvCO2 VE dt x 0 Q pul PCO2 C CO C CO F D CO dt t i pul 2i 1 pul 2i puli L 2 x 0 Q tis t 0 VCO 2 D CO T 2 PCO x 2 dt Q cap t 0 C art CO 2 C ven CO 2 F cap D CO T 2 PCO x 2 dt Q C F n PCO x t devi, j DDCO2 2 bld artco CdevCO i, j 2 2 i, j b nb n 0 g i, j dt 4 Q C F n PCO x t gasi, j DDCO2 2i gas inco CoutCO i, j 2 2 i, j g nb n 0 g, j dt

5 Carbon dioxide removal simulations 0.6 A 0.7 B CO2 removal (fraction of VCO2) Membrane DLCO2 = 0.3 Gas:Blood = 3 CO2 removal (fraction of VCO2) Shunt Fraction = 0.15 Gas:Blood = Shunt Flow (fraction of cardiac output) Membrane DLCO2 (ml/min/torr) 0.5 C Wt = 5 kg CO2 removal (fraction of VCO2) Shunt Fraction = 0.15 Membrane DLCO2 = 0.3 Baseline PaCO2: CO = 500 ml/min VCO2 = 4 ml/kg/min 20 torr 40 torr 80 torr 160 torr Gas:Blood ratio 5

6 6 Clinical application

7 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 7

8 8 Artificial kidney

9 9 Dialysis membrane technologies

10 Hemodiafilter Design HF: m inner diameter 0.1 to 0.2 m in length 10

11 COMSOL example separation through dialysis COMSOL model library Axisymmetric hollow fiber geometry Solute transport through the membrane by diffusion 11

12 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 13

13 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 14

14 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 15

15 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 16

16 Previous mathematical models Single compartment models Multiple compartment models One-dimensional computational models Two/three-dimensional computational model Finite element analysis 17

17 Single compartment analytical model Pallone TL: Kidney Int 1988;33: Pallone TL: Kidney Int 1989;35: 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 18

18 Multiple compartment analytical model Fiore GB: Contrib Nephrol 2005;149:27-34 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 19

19 One-dimensional computational models Legallis C: J Membr Sci 2000;168:3-15 Raff M: J Membr Sci 2003;216:1-11 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 20

20 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 21

21 Hollow Fiber Geometry Modeling Blood HF Dialysate Hospal M60 AN69S membrane Blood Membrane Dialysate 120 µm 170 µm 225 µm Circular approximation Axial symmetry 23

22 COMSOL Application Modes Protein: Convection- Diffusion Solute: Convection- Diffusion Heat: Heat transport Blood: Non-Newtonian Dialysate: Navier-Stokes 24 Membrane: Brinkman

23 Non-Newtonian Blood Flow Carreau equation: 25

24 Fahraeus-Lindqvist Effect H () r n( n 1)( n 1) H 2 n n1 n2 r 2r r 2 n n 1 n 2 n( n 1)( n 2) n where H

25 Colloid Osmotic Pressure F u C pro P osm P 1.2 C 0.16 C C 2 3 osm p p p 27 C pro in

26 Solute reflection Convective flux C dia in us = um (1-σ) 28 C bld in C dia out

27 29 Protein concentration in blood phase

28 Ultrafiltration rate validation 50 Manufacturer s data: Gambro M60 hemofilter Qb 180 ml/min Ultrafiltration rate (ml/min) Solid lines: experimental measurements Symbols: model predictions Qb 100 ml/min Transmembrane pressure (torr)

29 31 Hemofiltration vs dialysis

30 Solute clearance during dialysis Historical assumption: Vbldin Vdiain Vbldout Vdiaout Vdiain = Vdiaout zero net ultrafiltration diffusive clearance only Recent recognition: Pressure drop Additional convective clearance 32 Internal filtration Backfiltration

31 33 Internal filtration / backfiltration

32 Sensitivity of fluid flux to the protein diffusion coefficient 55 Ultrafiltration rate (ml/min) Experimental UF rate Predicted UF rate Gambro M60 TMP 300 torr Qb 180 ml.min Cpro 60 g/l Diffusion coefficient (10 10 m 2 /s)

33 Expanding the application modes Solute charge: Electrostatics Protein charge: Electrostatics Protein: Convection- Diffusion Membrane charge: Electrostatics Solute: Convection- Diffusion Heat: Heat transport 35 Blood: Non-Newtonian Membrane: Brinkman Dialysate: Navier-Stokes

34 Modeling the circadian rhythm existing ordinary differential equation models Cell shape Cytoplasmic streaming Intracellular organelles DNA binding kinetics 36 Momentum transport Diffusive transport Convective transport Chemical reactions

35 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 37