(Ebook) An Introduction to Modeling of Transport Processes Applications to Biomedical Systems 1st Edition by Ashim Datta, Vineet Rakesh ISBN 9780521119245 0521119243

I Essential steps

1 Problem formulation: From reality to realistic computer representation

1.1 Context: biomedical transport processes

1.1.1 Heat transfer and thermal therapy

1.1.2 Mass transfer and drug delivery

1.1.3 Quantification of goals in a biomedical process

1.2 What is problem formulation?

1.3 Steps in problem formulation

1.4 Defining goals for problem formulation

1.5 Simplify, simplify, simplify

1.6 Geometry: setting the computational domain

1.6.1 What regions need to be included?

Connectivity between the regions

1.6.2 How much of a very large region should be included?

1.6.3 How many dimensions are needed?

1.6.4 How can we consider symmetry to reduce the domain?

How to implement a 1D problem in 2D

1.7 Governing equations

1.7.1 Which governing equations?

Conservation equation for total mass (continuity equation)

Momentum conservation equations (fluid flow equations)

Energy conservation equation (heat transfer equation)

Mass species conservation equation (mass transfer equation)

1.7.2 What terms remain in the governing equation?

Transient

Convection

Diffusion

Generation (or source)

Deciding on 2D/3D/axisymmetric

1.8 Boundary and initial conditions

1.8.1 How many boundary conditions are needed?

1.8.2 What kind of boundary condition?

1.8.3 Boundary conditions changing with time

1.8.4 Boundary condition at an infinite region

1.8.5 Boundary condition at an interface between two materials

1.8.6 Flux boundary condition versus volumetric heat generation

1.8.7 Initial condition

1.9 Material properties

1.9.1 Questions to ask

1.9.2 Simplification is generally needed

1.9.3 When accurate data are not available

1.10 Other input parameters

1.11 Summary

References

1.12 Problems

1.12.1 Short questions

1.12.2 Choice of domain size: heating tumors with ferromagnetic materials

1.12.3 Choice of domain size: drug delivery in the brain

Questions on problem formulation: instructions

1.12.4 Drug delivery in the brain

1.12.5 Heat generation from tooth drilling

1.12.6 Drug delivery from a stent

1.12.7 Laser-interstitial thermal therapy (LITT)

1.12.8 Nitrogen elimination in alveoli

1.12.9 Drug release from a nicotine patch

1.12.10 Therapeutic heating

1.12.11 Radiofrequency ablation

1.12.12 Radiofrequency heating to destroy a tumor

1.12.13 Cryosurgery

1.12.14 Drug release from a therapeutic contact lens

1.12.15 Oxygen transport in alveoli

1.12.16 Refractive laser vision surgery

1.12.17 Conditioning of air in the nose

2 Software implementation 1: What to solve (preprocessing)

2.1 Choosing a software

2.2 Software is not to be used as a blackbox

2.3 Organization of a typical CAE software: preprocessing,processing and postprocessing

2.4 Some general guidelines to preprocessing

2.5 Introduction to preprocessing in a computational software (COMSOL)

2.6 Geometry and analysis type

2.6.1 Geometry

2.6.2 Analysis

2.7 Geometry creation

2.7.1 Drawing geometry in COMSOL

Geometry in 1D

Geometry in 2D

Geometry in 3D

2.7.2 Obtaining exact geometry from a CAD program

2.8 Governing equations

2.8.1 Convection, transient and source terms

2.8.2 Example of heat source: implementing blood flow term in the bioheat equation

2.8.3 Example of mass source: implementing reactions

2.9 Boundary conditions

2.9.1 General implementation

2.9.2 Special cases: time-varying boundary conditions

2.9.3 Example: specifying a parabolic inlet velocity profile

2.10 Initial conditions

2.11 Material properties

2.11.1 Variable properties

2.11.2 Zero diffusivity

2.12 Miscellaneous implementation aspects

2.12.1 Solving ordinary differential equations

2.12.2 Using logical expressions

2.12.3 Modeling in COMSOL Script

References

2.13 Problems

2.13.1 Short questions

2.13.2 Questions on software implementation

3 Software implementation 2: How to solve (preprocessing)

3.1 Which numerical method to use

3.2 Items needed in specifying the solution methodology

3.3 How to discretize the domain: mesh

3.3.1 Elements and mesh

3.3.2 Structured (mapped) versus unstructured (free) mesh

3.3.3 Meshing in COMSOL: 1D geometry

3.3.4 Meshing in COMSOL: 2D geometry

Structured (mapped) mesh

Unstructured (free) mesh

3.3.5 Meshing in COMSOL: 3D geometry

Structured mesh

Unstructured mesh

Combination of structured and unstructured mesh

3.3.6 Deciding on a mesh

3.4 How to choose a time step

3.4.1 Implementation of time step in COMSOL

3.4.2 Specifying end times and saving data in COMSOL

3.5 How to choose a solver to solve the system of linear equations

3.5.1 Solver selection in COMSOL

3.5.2 Setting tolerances for a time-dependent problem

3.6 Problems

3.6.1 Guidelines for mesh and time step

3.6.2 Need for non-uniform mesh

3.6.3 Choice of mesh

3.6.4 Choice of mesh: Case studies VI, VIII, IX and X

3.6.5 Choice of time steps: Case studies I, II, III and IV

4 Software implementation 3: Visualizing and manipulating solution(postprocessing)

4.1 Useful information in a biomedical context

4.2 Obtaining data at a particular location

4.3 Plotting transient data at one or more points, line or surface as a function of time

4.4 Obtaining surface/contour plots (in 2D problems) for observing variation within a region

4.4.1 Contour plot

4.4.2 Surface plot

4.4.3 Additional options for surface and contour plots

4.5 Obtaining a surface plot in a 3D problem

4.5.1 For the entire geometry

4.5.2 For different sections of the geometry

4.6 Obtaining average values at a particular time or as a function of time

4.6.1 Average values at any particular time

4.6.2 Average values as a function of time

4.7 Obtaining arbitrary functions of computed variables

4.8 Creating animations

4.9 Dedicated plotting and postprocessing software

4.10 Relating to the goals of the simulation: guidelines for postprocessing

4.11 Analysis of data obtained from postprocessing

4.12 Presenting the simulation results to others

4.13 Problems

4.13.1 Postprocessing: Case study I

4.13.2 Postprocessing: Case study II

4.13.3 Postprocessing: Case study III

4.13.4 Postprocessing: Case study IV

4.13.5 Postprocessing: Case study VII

4.13.6 Postprocessing: Case study X

4.14 Appendix

5 Validation, sensitivity analysis, optimization and debugging

5.1 Types of errors and error reduction

5.1.1 Defining uncertainty and error

Uncertainty

Error

Acknowledged errors

Unacknowledged errors

5.1.2 Classification of errors

Physical approximation error

Computer round-off error

Iterative convergence error

Discretization errors

Computer programming errors

Usage errors

5.2 Estimating error: validation of the model

5.2.1 What level of agreement is desirable?

5.2.2 Use your eyes and common sense (qualitative checks)

5.2.3 Compare with alternative solutions

5.2.4 Compare with experimental data

5.2.5 Compare over a large parameter range

5.3 Reducing discretization error: mesh convergence

5.3.1 Performing mesh convergence manually

5.3.2 Performing automated mesh convergence: adaptive meshing

5.4 Estimating uncertainty and relating to design: performing sensitivity analysis

5.4.1 How to perform sensitivity or uncertainty analysis

Obtaining range

Obtaining distributions

5.5 Objective functions: simple optimization

5.6 When things don't work: debugging

5.6.1 Is the computer solving the problem you want it to solve?

5.6.2 Intermediate steps where things may be going wrong

5.6.3 Can you help the computer?

5.6.4 What problem would be the easiest to solve (for you and the computer)?

5.6.5 Adding the complexities

5.6.6 Make sure it is really working before you stop

References

5.7 Problems

5.7.1 Short questions

5.7.2 Qualitative checks

5.7.3 Qualitative checks

5.7.4 Qualitative checks

5.7.5 Physical approximation error

5.7.6 Qualitative checks: laser heating to treat glaucoma

5.7.7 Optimization: laser heating to treat glaucoma (2)

5.7.8 Qualitative checks: laser hair removal

5.7.9 Mesh convergence: freezing of tongue on metal pole

5.7.10 Optimization: radiofrequency heating to destroy a tumor

5.7.11 Optimization: radiofrequency heating to destroy a tumor (2)

5.7.12 Experimental validation: cryosurgical procedure

5.7.13 Inverse problem: transdermal drug delivery with penetration enhancer

5.7.14 Inverse problem: detection of a breast tumor

5.7.15 Inverse problem: convection-enhanced interstitial diffusion

5.7.16 Objective function: radiofrequency ablation

II Case studies

6 Case studies

6.1 Introduction

6.2 How to use the case studies

6.3 Additional case studies from the work of students at Cornell University

Thermal therapy

Thermal comfort

Drug delivery

Transport of blood, oxygen and other materials inside the body

I: Thermal ablation of hepatic tumors

Problem formulation

Governing equations

Boundary conditions

Input parameters

Reference

II: Cryosurgery of a wart

Problem formulation

Governing equations

Boundary conditions

Input parameters

Reference

Implementation in COMSOL

Step 1: specifying the problem type

Step 2: setting the grid and creating the geometry

Step 3: meshing

Step 4: defining material properties and initial conditions

Step 5: defining boundary conditions

Step 6: specify solver parameters

Step 7: postprocessing

Step 8: save and exit

Optimization

Step 1: open file

Step 2: define objective function

Step 3: solve

Step 4: plot objective function versus time

III: Drug delivery from a patch

Problem formulation

Governing equations

Boundary conditions

Input parameters

Reference

IV: Drug delivery in therapeutic contact lenses

Problem formulation

Governing equation

Boundary conditions

Input parameters

References

Implementation in COMSOL

Case Study V: Elimination of nitrogen from the blood stream during deep sea diving

Problem formulation

Governing equation

Boundary conditions

Input parameters

References

Implementation in COMSOL

VI: Flow in human carotid artery bifurcation

Problem formulation

Governing equation

Boundary conditions

Input parameters

References

VII: Radioimmunotherapy of metastatic melanoma

Problem formulation

Governing equations

Boundary conditions

Input parameters

References

Implementation in COMSOL

Problem type specification

Meshing

Governing equations, source terms, I.C., B.C. – diffusion

Solver settings

Solution

Postprocessing

VIII: Burn injury in blood-perfused skin

Problem formulation

Governing equations

Boundary conditions

Input parameters

References

IX: Radiofrequency cardiac ablation

Problem formulation

Governing equations

Boundary conditions

Input parameters

Reference

Implementation in COMSOL

Problem type specification

Geometry creation

Meshing

Governing equations, source terms, I.C., B.C. – conduction

Subdomain settings

Boundary settings

Governing equations, source terms, I.C., B.C. – Conductive media DC

Subdomain settings

Boundary settings

Define expressions

Solver settings

Solution

Postprocessing

Alternative method

Problem type specification

Geometry creation

Meshing

Governing equations, source terms, I.C., B.C. – conduction

Subdomain settings

Boundary settings

Governing equations, source terms, I.C., B.C. – diffusion

Subdomain settings

Boundary settings

Define expressions

Solver settings

Solution

Postprocessing

Adaptive mesh refinement

Open file

Modify mesh

Change solver settings

Solution

Postprocessing

Change solver settings for adaptive mesh refinement

Solution

Postprocessing

X: Laser irradiation of human breast tumor

Problem formulation

Governing equations

References

Implementation in COMSOL

Problem type specification

Geometry creation

Meshing

Governing equations, source terms, I.C., B.C. – conduction

Subdomain settings

Boundary settings

Define expressions

Define postprocessing variables

Solver settings

Solution

Postprocessing

Calculations for subsequent analysis

Problem type specification

Geometry creation

Meshing

Governing equations, source terms, I.C., B.C. : Moving mesh

Subdomain settings

Boundary settings

Governing equations, source terms, I.C., B.C.:Incompressible Navier–Stokes

Subdomain settings

Boundary settings

Governing equations, source terms, I.C., B.C. : Convection and diffusion

Subdomain settings

Boundary settings

Define function

Solver settings and solution

Postprocessing

6.4 Problems

Supplementary case study questions

6.4.1 Case study I

6.4.2 Case study II

6.4.3 Case study III

6.4.4 Case study IV

6.4.5 Case study V

6.4.6 Case study VI

6.4.7 Case study VII

6.4.8 Case study VIII

6.4.9 Case study IX

6.4.10 Case study X

6.4.11 Case study X (2)

Complete problem formulations

6.4.12 Laser surgery

6.4.13 Thermal balloon endometrial ablation

Implementation in COMSOL

6.4.14 Transdermal scopolamine patch

Problem formulation

Governing equation

Boundary and initial conditions

Input parameters

Questions

6.4.15 Cold therapy

Problem formulation

Governing equations

Boundary conditions

Input parameters

Questions

References

III Background theory

7 Governing equations and boundary conditions

7.1 Conservation of mass: the continuity equation

7.2 Conservation of momentum: governing equation for fluid flow

7.3 Conservation of thermal energy: governing equation for heat transfer

7.3.1 Lumped parameter analysis: special case of the governing equation for heat transfer

7.4 Governing equation for heat conduction with change of phase

7.4.1 Freezing/melting

7.4.2 Evaporation

7.5 The bioheat transfer equation for mammalian tissue

7.6 Conservation of a mass species: governing equation for mass transfer

7.6.1 Multiple species in the same problem and their coupling

7.7 Non-dimensionalization of the governing equations

7.7.1 Heat transfer

7.7.2 Species mass transfer

7.7.3 Momentum equation

7.8 Coupling of governing equations

7.9 Summary: governing equations

7.10 Boundary conditions: general comments

7.11 Boundary conditions: fluid mechanics

7.11.1 Velocity at the boundary is specified

No slip on a solid boundary

Velocity continuity at a fluid boundary

Symmetry condition

Velocity at an inlet is specified

7.11.2 Pressure at a boundary is specified

7.12 Boundary conditions for heat transfer

7.12.1 Surface temperature is specified

7.12.2 Surface heat flux is specified

Special case: insulated condition

Special case: symmetry condition

7.12.3 Convection at the surface

7.12.4 Evaporation at surface

7.12.5 Radiation at surface

7.12.6 Combined convection, radiation and evaporation

7.13 Boundary conditions for mass transfer

7.13.1 Interphase equilibrium as it relates to boundary conditions

Gas and liquid

Liquid and solid

Solid and gas

7.13.2 Surface concentration is specified

7.13.3 Surface mass flux is specified

Special case: impermeable condition

Special case: symmetry condition

7.13.4 Convection at the surface

7.13.5 Multiple materials and discontinuities at the boundary

7.13.6 Summary of boundary conditions

7.14 Governing equations in various coordinate systems

7.14.1 The equation of continuity in rectangular, cylindricaland spherical coordinate systems

Rectangular coordinates (x,y,z):

Cylindrical coordinates (r,, z):

Spherical coordinates (r, , ):

7.14.2 The governing equations for fluid flow in rectangular coordinates (x,y,z): in terms of shear

7.14.3 The governing equations for fluid flow in rectangular coordinates (x,y,z): in terms of veloci

7.14.4 The governing equations for fluid flow in cylindrical coordinates (r,0,z): in terms of shear

7.14.5 The governing equations for fluid flow in cylindrical coordinates (r,0,z): in terms of veloci

7.14.6 Governing equation for heat transfer (assuming Newtonian fluids of constant p and k) in recta

7.14.7 Governing equation for heat transfer (assuming Newtonian fluids of constant and k) in cylind

7.14.8 Governing equation for heat transfer (assuming Newtonian fluids of constant p and k) in spher

7.14.9 Governing equation for species mass transfer in rectangular coordinates for constant p and DA

7.14.10 Governing equation for species mass transfer in cylindrical coordinates for constant p and D

7.14.11 Governing equation for species mass transfer in spherical coordinates for constant p and DAB

References

7.15 Problems

7.15.1 Heat transfer with larger vessels

7.15.2 Governing equation for a freezing problem

7.15.3 Difficulties with modeling freezing using a sharp interface formulation

8 Source terms

Heat source term

Mass source term

8.1 Heat source terms due to metabolism and blood flow

8.2 A generic form for the heat source term

8.3 Heat source term for electromagnetic heating

8.3.1 Governing equations and the heat source term

8.4 Microwave heating and its heat source term

8.4.1 Mechanism of heating

8.4.2 Dependence of electromagnetic properties on frequency, temperature, composition and other fact

8.4.3 Clinical applicator example

8.4.4 Simplified heat generation term

8.4.5 Complexities

8.5 Radiofrequency heating and its heat source term

8.5.1 Mechanism of heating

8.5.2 Clinical applicator example

8.5.3 Simplified heat generation term

8.5.4 Dependence of properties on temperature, composition and other factors

8.5.5 Complexities: when a simplified heat generation term is not enough

8.6 Ferromagnetic heating and its heat source term

8.7 Infrared heating and its heat source term

8.7.1 Mechanism of heating

8.7.2 Clinical applicator example

8.7.3 Simplified heat generation term

8.8 Laser heating and its heat source term

8.8.1 Mechanism of heating

8.8.2 Clinical applicator example

8.8.3 Simplified heat generation term

8.8.4 Dependence of laser absorption on various factors

8.8.5 Complexities: when a simplified heat generation term is not enough

Heat source term computed from light diffusion

Very short duration laser heating

8.9 Ultrasonic heating

8.9.1 Mechanism of heating

8.9.2 Clinical applicator example

8.9.3 Simplified heat generation term

8.9.4 Dependence of properties on temperature, composition and other factors

8.9.5 Complexities: when a simplified heat generation term is not enough

8.10 Mass source terms

8.10.1 Zero-order reactions

8.10.2 First-order reactions

8.10.3 Various other reactions

8.10.4 Modeling of various reactions

Tissue oxygen consumption

Thermal injury of tissue

8.11 Summary

References

General

Laser

Microwave

Radiofrequency

Ferromagnetic

Ultrasonics

Mass transfer

8.12 Problems

8.12.1 Penetration depth in the heat source term

8.12.2 Variable penetration depth

8.12.3 Zero penetration depth

8.12.4 Thermal dosimetry

8.12.5 Heat source term for ferromagnetic heating

8.12.6 Heat source term for laser heating

8.12.7 Coupling of equations: effect of temperature dependence of properties

8.12.8 Example of coupling: radiofrequency heating to destroy a tumor

8.12.9 Implementation of any rate process as a species equation

9 Material properties and other input parameters

9.1 What material property data and input parameters do we need?

9.2 Where do we get data?

9.2.1 Measurement

Own measurement

Outsourcing

9.2.2 Literature

Databases

Handbooks and research publications

9.3 How accurate should the data be?

9.4 What to do when accurate data is not available

Start with a reasonable guess

Perform sensitivity analysis

9.5 Anatomical and physiological parameters

9.5.1 Standard anatomical data

9.5.2 Body surface area

9.5.3 Blood perfusion

9.5.4 Metabolic heat generation

9.5.5 Growth rate and metabolic heat generation in a tumor

9.5.6 Tissue shrinkage or swelling

9.5.7 Temperatures for tissue destruction

9.6 Rheological properties

9.7 Thermal conductivity

9.8 Specific heat

9.8.1 Specific heat during phase change of biomaterials: apparent specific heat

9.8.2 Specific heat during phase change of pure materials:adaptation of apparent specific heat

9.9 Density

9.10 Thermal diffusivity

9.11 Thermal properties of related materials

Thermal Properties of Cryoprotectants

9.12 Latent heat of fusion and evaporation

9.13 Radiative properties

9.14 Equilibrium vapor pressure

9.15 Properties of an air--water vapor mixture

9.16 Mass diffusivity

9.16.1 Diffusivity in gases

9.16.2 Diffusivity in liquid

9.16.3 Diffusion through solids (gels and tissues)

9.17 Partition coefficient

9.18 Diffusive permeability and transmissibility

9.19 Reaction rate constants

9.20 Other parameters

9.20.1 Convective heat and mass transfer coefficients

9.20.2 Radiative heat transfer coefficient

9.20.3 Combined heat transfer coefficient

9.21 Summary

References

9.22 Problems (short questions)

10 Solving the equations: numerical methods

10.1 Flexibility of numerical methods

10.1.1 An example of a numerical solution

10.2 Finite difference method (FDM)

10.2.1 FDM: A few common formulations

forward difference

backward difference

central difference

10.3 FDM: converting the 1D heat equation to algebraic equations

Governing equation

Boundary conditions

Initial condition

10.3.1 Discretization of the spatial term

10.3.2 Discretization of the transient term

10.3.3 Complete finite difference formulation of the 1D heat equation

10.4 FDM: stability (limitations in choosing step sizes)

10.5 FDM: summary

10.6 Finite element method (FEM)

10.7 FEM: converting the 1D heat equation to algebraic equations

Governing equation

Boundary conditions

Initial condition

10.7.1 Discretization of the spatial term

Element 1

Element 2

Element 1

Element 2

10.7.2 Discretizing the transient term

Explicit (also called forward) method

Implicit (also called backward) method

10.7.3 Inclusion of boundary conditions

Temperature specified at the boundary

Heat flux specified at the boundary

Convective boundary condition

10.8 FEM: solving the linear system of algebraic equations

10.8.1 Direct methods

LU decomposition

10.8.2 Iterative methods

10.9 FEM: choice between linear solvers

10.10 FEM: linearization of non-linear equations

10.11 FEM: error in the finite element method and its reduction

10.12 FEM: convergence of the numerical solution as the mesh is refined

Improving convergence

10.13 FEM: stability of the numerical solution

10.14 FEM: generalization of methodology to more complex situations

10.15 FEM: summary

10.16 Problems

10.16.1 Short questions

10.16.2 Finite difference formulation of the variable property heat transfer equation

10.16.3 Finite difference formulation of the bioheat equation

10.16.4 Finite difference formulation of the heat equation in 2D

10.16.5 Checking the accuracy of a finite difference formulationby comparing with an analytical solu

10.16.6 Programming the finite difference equationin a spreadsheet program

10.16.7 Finite element formulation of the bioheat equation

10.16.8 Finite element method for boundary temperature specified

10.16.9 Finite element formulation for unequal element size

10.16.10 Error in finite element computation and its reduction

10.16.11 Limitations of analytical solution

10.16.12 Convergence of finite element computations

Index