biomedical engineering minicourse

COMSOLMultiphysics ®

V E R S I O N 3 . 5

Biomedical Engineering Minicourse

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Biomedical Engineering Minicourse© COPYRIGHT 1994–2008 by COMSOL AB. All rights reserved

Patent pending

The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from COMSOL AB.

COMSOL, COMSOL Multiphysics, COMSOL Script, COMSOL Reaction Engineering Lab, and FEMLAB are registered trademarks of COMSOL AB.

Other product or brand names are trademarks or registered trademarks of their respective holders.

Version: September 2008 COMSOL 3.5

C O N T E N T S | i

C O N T E N T S

Introduction 2

Hip Replacement 3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3

Model Definition . . . . . . . . . . . . . . . . . . . . . . . 3

Results. . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Modeling Using the Graphical User Interface . . . . . . . . . . . . 5

Drug Release from a Biomaterial 9

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 9


Results and Discussion. . . . . . . . . . . . . . . . . . . . . 13

Reference . . . . . . . . . . . . . . . . . . . . . . . . . 15

Modeling Using COMSOL Reaction Engineering Lab . . . . . . . . . 16

Modeling Using COMSOL Multiphysics . . . . . . . . . . . . . . 22

Tumor Removal 26

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 26


Results and Discussion. . . . . . . . . . . . . . . . . . . . . 28

Reference . . . . . . . . . . . . . . . . . . . . . . . . . 30


Creating the Geometry from Scratch . . . . . . . . . . . . . . . 36

Biomedical Stent 41

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 41


Results. . . . . . . . . . . . . . . . . . . . . . . . . . . 43


ii | C O N T E N T S

| 1

B i o m e d i c a l E n g i n e e r i n gM i n i c o u r s e

2 | B I O M E D I C A L E N G I N E E R I N G M I N I C O U R S E

I n t r o du c t i o n

Join this class to get insight how to set up biomedical models such as tissue heating and drug targeting. The training exercises are from many different engineering areas and you will use a multitude of different products and application modes of COMSOL Multiphysics. The following examples will be discussed and built during the short-course:

• Structural analysis of a hip joint replacement prosthesis

• Drug Release from a Biomaterial

• Tumor removal by electrode treatment

• Treatment of atherosclerosis with stenting

Enjoy your modeling!

H I P R E P L A C E M E N T | 3

H i p R ep l a c emen t

This example gives an overview of basic modeling: It shows how to import a 3D geometry from a COMSOL native geometry format, how to create the mesh, set boundary conditions and material properties, and how to solve the model.

Introduction

The most common reason that people have hip-replacement surgery is the wearing down of the hip joint that results from osteoarthritis. Other conditions, such as rheumatoid arthritis (a chronic inflammatory disease that causes joint pain, stiffness, and swelling), avascular necrosis (loss of bone caused by insufficient blood supply), injury, and bone tumors can also lead to the breakdown of the hip joint and the need for a hip replacement.

Model Definition

This exercise is inspired by work of Professor Richard Hart, Ohio State University.

G E O M E T R Y

The hip joint is called a ball-and-socket joint because the spherical head of the thighbone (femur) moves inside a cup-shaped hollow socket (acetabulum) in the pelvis. To duplicate this action, a complete hip-replacement implant has three parts: the stem, which fits into the femur and provides stability; the ball, which replaces the spherical head of the femur; and the cup, which replaces the worn-out hip socket (not shown in the nearby figure).

Figure 1: Geometry of the hip replacement.


P H Y S I C S

This example models the replacement hip using the Solid, Stress-Strain application mode.

M A T E R I A L P R O P E R T I E S A N D S U B D O M A I N S E T T I N G S

A metal with FDA approval for orthopedic implants is an alloy made of cobalt, chromium, and molybdenum. A reasonable value for the Young’s modulus is 220 GPa, and 0.33 for Poisson’s ratio.

The Young’s modulus for the thighbone is set to 2 GPa.

B O U N D A R Y C O N D I T I O N S

Although in reality the femoral hip component would be well supported by the surrounding bone, this example applies boundary conditions that idealize a drastically lowered stiffness of the surrounding bone (which would require revision surgery).

Specifically, this example uses boundary-load expressions that approximate the stiffness of the supporting thighbone with the bone’s stiffness drastically lowered (to 10% of the original value) at some height (0.05 m from the bottom of the stem).

It then applies a normal load to the femoral head’s top surface. The load is assumed to vary linearly from zero at the horizontal edge of the top surface to its maximal value at the top of the head. This maximal value, in turn, is adjusted to result in a specified value of the total downward force on the hip. In extreme cases (such as when climbing stairs), the force on the hip can be 3.5 times body weight. Assume that the body weight is 100 kg.

Results

Figure 2 illustrates the von Mises stresses and the deformed shape. The figure reveals moderate stress levels in the stem.


The yield stress levels are approximately in the range 430 MPa to 1028 MPa for cobalt-based alloys, and the von Mises stress levels are thus well below these levels in Figure 2.

Figure 2: The deformed shape and the von Mises stress in the hip replacement.

Modeling Using the Graphical User Interface

M O D E L N A V I G A T O R

1 In the Model Navigator go to the New page and select 3D from the Space dimension list.

2 In the list of application modes select Structural Mechanics Module>Solid, Stress-Strain>Static analysis.

3 Click OK to close the Model Navigator.

G E O M E T R Y M O D E L I N G

Importing the Geometry from a Binary File1 From the File menu, select Import>CAD Data From File.


2 In the Import CAD Data From File dialog box, make sure that either COMSOL

Multiphysics file or All 3D CAD files is selected in the Files of type list.

3 Locate the file hip_replacement_geometry.mphbin, then click Import.

O P T I O N S A N D S E T T I N G S

1 Select Options>Constants, then click the Import Variables From File icon. Browse through the course CD and import the file hip_replacement_constants.txt.

Note that the Young’s modulus defined in the constants file is that for the thighbone; you enter the properties of the alloy material manually.

2 Click OK.

3 Select Options>Expressions>Scalar Expressions, then click the Import Variables From

File icon. Browse through the course CD and import the file hip_replacement_expressions.txt.

These expressions define the boundary loads representing the thighbone in accordance with the assumptions discussed in the “Boundary Conditions” section.

4 from the Options menu, select Expressions>Boundary Expressions.


5 Select Boundaries 13, 14, 38, and 39 by pressing the Ctrl key while clicking on the boundary numbers in the Boundary selection list.

6 In the Name edit field type Fn, and in the Expression edit field type 1.5*sy_mean*(y-0.14[m])/0.02[m].

This defines an expression for the normal load on the femur head’s top surface that depends linearly on the spatial coordinate in the y direction in such a way that it is zero at y = 0.14 m and 1.5 times the average load in the y direction at the top of the spherical head (at y = 0.16 m). As already mentioned, the normalization is chosen to give a total downward load on the femur head that equals the total force on the hip as specified in the Constants dialog box.

P H Y S I C S S E T T I N G S

Subdomain Settings1 From the Physics menu, select Subdomain Settings.

2 From the Subdomain selection list, select Subdomains 1 and 2.

3 Set the Young’s Modulus to 220e9 and Poisson’s ratio to 0.33, as discussed in the section “Model Definition”.

4 Click OK.

Boundary Conditions1 From the Physics menu, open the Boundary Settings dialog box.

2 From the Boundary selection list, select Boundaries 1–6, and 23.

3 On the Load page, enter the expressions you imported in the Scalar Expressions dialog box in the Fx, Fy, and Fz edit fields:

4 From the Boundary selection list, select Boundaries 13, 14, 38, and 39 (the femur head top surface).

5 From the Coordinate system list, select Tangent and normal coord. sys. (t1, t2, n).

6 In the Fn edit field, type Fn.

7 Click OK.

NAME EXPRESSION

Fx stiffness_thighbone_x

Fy stiffness_thighbone_y

Fz stiffness_thighbone_z


M E S H G E N E R A T I O N

Because of the sharp features in the geometry, you need a slightly finer mesh than the standard one to obtain a reasonable mesh-element quality.

1 From the Mesh menu, select Free Mesh Parameters.

2 On the Global page, select Fine from the Predefined mesh sizes list.

3 Click Remesh. When the mesher has finished, click OK.

The resulting mesh should look like that in the following figure.

C O M P U T I N G T H E S O L U T I O N

Click the Solve button on the Main toolbar.

P O S T P R O C E S S I N G A N D V I S U A L I Z A T I O N

1 Click the Plot Parameters button on the Main toolbar.

2 On the General page, clear the Slice check box and select the Subdomain and Deformed

shape check boxes.

3 On the Subdomain page, select MPa from the Unit list.

4 Click OK to close the Plot Parameters dialog box and generate the plot shown in Figure 2 on page 5.

D R U G R E L E A S E F R O M A B I O M A T E R I A L | 9

Drug R e l e a s e f r om a B i oma t e r i a l

Introduction

Biomaterial matrices for drug release are useful for in vivo tissue regeneration. The following example describes the release of a drug from a biomaterial matrix to damaged cell tissue. Specifically, a nerve guide delivers a regenerating drug to damaged nerve ends. The model examines detailed drug-release kinetics with rate expressions handling drug dissociation/association reactions as well as matrix degradation by enzyme catalysis. The enzyme reaction is described by Michaelis-Menten kinetics. With this simulation it is easy to investigate design parameters governing the rate of drug release such as drug-to-biomaterial affinity, biomaterial degradation, drug loading, and of course the influence of geometry and composition of the biomaterial matrix. The model illustrates the use of the Reaction Engineering Lab and the Chemical Engineering Module as powerful modeling tools for bioengineering applications.

Model Definition

In this model a drug is released into a region containing damaged nerve ends. The biomaterial holding the drug has a cylindrical shape and serves a dual purpose: It acts as a guide to help regenerating nerve cells connect, and it also stimulates the healing process through targeted drug release.

The model simulates transient chemical reactions and species transport in a 2D geometry with axial symmetry.

Figure 3 shows the full 3D geometry as well as the 2D modeling domain, reduced by axial symmetry and a mirror plane. The subdomains are:

• The nerve-cell tissue (Ω1)


• The biomaterial matrix (Ω2)

• The surrounding medium (Ω3)

Figure 3: The full 3D geometry (left) and the equivalent modeling domain reduced to 2D by axial symmetry (right). The subdomains are: the nerve-cell tissue (Ω1), the biomaterial matrix (Ω2), and the surrounding medium (Ω3).

In the biomaterial matrix, a drug molecule, d, binds to a peptide, p, which in turn is anchored to the matrix, m. Matrix-bound species are labeled mpd and mp, respectively, the latter referring to a species where no drug is bound to the peptide. Species mpd and mp are active only in subdomain Ω2.

Two mechanisms release the drug from the matrix. First, the drug can simply dissociate from the matrix site mp. Second, matrix degradation by an enzyme, e, originating from the cell-tissue domain, leads to release of the drug-peptide species, pd, from which the drug subsequently dissociates. The unbound species p, d, pd, and e are active in the entire model domain. Figure 4 illustrates the complete reaction scheme.

Ω1

Ω2

Ω3


Figure 4: Reaction scheme describing drug dissociation/association reactions (horizontal) and matrix-degradation reactions (vertical).

The simulation makes use of two diffusion application modes in the Chemical Engineering Module. The time-dependent mass balance per species is described by

(1)

where Dik (m2/s) is the diffusion coefficient for species i in the medium of subdomain Ωk. Further, Rik (mol/(m3·s)) is the rate expression for species i in the medium of

m

p

m

pd +

m

p

m

m

pE

m

pEE

k1r

k1f

k2r k2

f

k3f

E

d

d

p

d

d +

k2r k2

f

k1r

k1f

k3f

Dissociation

Degradation

ci∂t∂------- ∇+ D– ik∇ci( )⋅ Rik in Ωk=


subdomain Ωk. In the matrix subdomain, all the reactions described in Figure 4 are possible, leading to the following rate expressions:

(2)

(3)

(4)

(5)

(6)

The rate terms RMMmp and RMMmpd refer to the Michaelis-Menten kinetics describing the enzyme catalyzed degradation of the matrix:

with

RMMmp describes the disappearance of mp sites and the production of p species. RMMmpd describes the disappearance of mpd sites and the production of pd species. In the cell region (Ω1) and in the surrounding medium (Ω3) only dissociation/association reactions occur, leading to the rate expressions

Rd2 k– 1f cd cmp cp+( ) k1

r cmpd cpd+( )+=

Rp2 k– 1f cd cp k1

r cpd RMMmp+ +=

Rpd2 k1f cd cp k1

r cpd– RMMmpd+=

Rmp2 k– 1f cd cmp k1

r cmpd RMMmp–+=

Rmpd2 k1f cd cmp k1

r cmpd– RMMmpd–=

RMMmpVmax cmpKM cmp+------------------------=

RMMmpdVmax cmpdKM cmpd+---------------------------=

Vmax k3f ce=

KMk3

f k+ 2r

k2f

------------------=

Rd1 Rd3= Rp1 Rp3 k– 1f cd cp k1

r cpd+= = =

Rpd1 Rpd3= k1f cd cp k1

r cpd–=


The boundary condition is axial symmetry along the rotational axis and insulation/symmetry elsewhere. Values for diffusion coefficients and rate constants come from the literature (Ref. 1).

Results and Discussion

Figure 5 shows the concentration transients of the reacting species in a perfectly mixed (space-independent) system.

Figure 5: Concentrations of all reacting species (mol/ m3) as functions of time (s).

The effect of enzyme degradation is clearly visible, with matrix-bound peptide species (mp and mpd) decreasing and free peptide species (p and pd) increasing with time. The matrix is completely degraded after approximately 5000 seconds. As the drug and peptide species have the same association/dissociation kinetics, no matter whether the peptide is free or matrix-bound, the steady-state concentration of drug is constant during the degradation process.


Solving the space-dependent mass balances of Equation 1 results in concentration distributions of all participating species as functions of time. Figure 6 shows the concentration of drug across the modeling domain.

Figure 6: Drug concentration at the end of the simulation (7200 s).

You can visualize the effect of matrix degradation on the release of the drug by plotting the total drug concentration over a cross section of the matrix as a function of time.

Figure 7: Concentration profiles describing the total drug concentration (cd + cpd) across the modeling domain. Profiles were collected at times up to 3600 s.

As noted earlier, the enzyme originates from the nerve-cell subdomain. From Figure 7 it is clear that matrix degradation has a directing effect on the drug release toward the damaged cell region.


To visualize the degradation front passing through the biomaterial geometry, plot the total matrix site concentration (cmp + cmpd) across the matrix subdomain (Figure 8).

Figure 8: Concentration profiles describing the total matrix site concentration (cmp + cmpd) with profiles collected at times up to 7200 s.

The detailed reaction/transport description in this model allows for the investigation of many design parameters relevant to bioengineering. This case presents the effect of matrix degradation on drug release as a function of time and geometry. Furthermore, it is straightforward to study the influence of the drug/peptide affinity by varying the rate constants kf

1 and kr1, or the influence of drug loading by varying the cmp:cmpd

ratio. Of course, the ability to examine alternative geometries and mixed biomaterial domains gives even more design flexibility.

Reference

1. D.J. Maxwell, B.C. Hicks, S. Parsons, and S.E. Sakiyama-Elbert, Acta Biomat., vol. 1, p. 101, 2005.

Note: The complete model requires the Chemical Engineering Module and COMSOL Multiphysics.

The following path shows the location of the COMSOL Multiphysics model:


Modeling Using COMSOL Reaction Engineering Lab

To set up this model, enter the chemistry described in Figure 4 into the Reaction Engineering Lab in two steps. First describe the association/dissociation chemistry by entering the corresponding chemical reaction formulas. Second, for the degradation of the biomaterial matrix due to enzyme catalysis, directly enter the well-known expressions for the Michaelis-Menten rates.


Start the Reaction Engineering Lab. In the Model Navigator, click the New button.


1 Click the Model Settings button on the Main toolbar.

2 In the Reacting fluid list, select Liquid.

3 Select the Calculate thermodynamic properties check box.

4 Select the Calculate species transport properties check box.

5 In the T edit field, type 310.

6 Click Close.

7 Choose Model>Constants, then define the following names and expressions. Note that Km is the Michaelis-Menten constant.

8 Click OK.

9 Choose Model>Expressions, then define the Michaelis-Menten expressions by entering the following settings:

10 Click OK.

R E A C T I O N S I N T E R F A C E

1 Choose Model>Reaction Settings.

NAME EXPRESSION

kf_mm3 7.336e-3

Km 0.01

NAME EXPRESSION

Vmax kf_mm3*c_e

R_mm_mp Vmax*c_mp/(Km+c_mp)

R_mm_mpd Vmax*c_mpd/(Km+c_mpd)


2 On the Reactions page, create two entries in the Reaction selection list by clicking the New button twice.

3 Enter the following reaction formulas by first selecting the appropriate row in the Reaction selection list and then typing the text in the Formula edit field:

4 On the Kinetics page, enter the following forward and reverse rate constants. To do so, first select the appropriate row in the Reaction selection list, and then enter the numbers in the appropriate edit fields. Note how you set the rate constants of Reaction 2 equal to those of Reaction 1 by invoking the variable names corresponding to the Rate constant edit fields, kf_1 and kr_1.

S P E C I E S I N T E R F A C E

1 Click the Species tab.

2 Look at the Species selection list and note the five species that result automatically when you enter the reaction formulas as described earlier.

3 Create a new species entry by clicking New, then in the Formula edit field enter e. This entry corresponds to the enzyme.

4 Click the General tab. For each entry in the Species selection list, type the following initial concentrations in the c0 edit field:

REACTION ID # REACTION FORMULA

1 d+mp<=>mpd

2 d+p<=>pd

REACTION ID # kf kr

1 1.5e4 13.5

2 kf_1 kr_1

SPECIES NAME c0

d 0

mp 1e-2

mpd 1e-2

p 0

pd 0

e 2e-3


To include the effects of the enzyme degradation into the kinetic description, you must modify the predefined reaction expressions with the Michaelis-Menten rates, which you defined earlier in the scalar expressions.

5 While still on the General page, select the appropriate entry in the Species selection

list and modify the rate expressions in the corresponding R edit field.

6 Click the Transport tab.

7 For each species in the list below, select the corresponding entry in the Species

selection list. Then select the Specify diffusivity check box and type the appropriate diffusivity value in the D0 edit field.

8 Click Close.

Start by investigating the kinetics in a perfectly mixed system in the Reaction Engineering Lab.


Click the Solve Problem button (=) on the Main toolbar.

P O S T P R O C E S S I N G

To reproduce results of Figure 5, click the x log button on the Main toolbar.

To reproduce results of Figure 6, follow the steps listed below.

SPECIES NAME R

mp -r_1-R_mm_mp

mpd r_1-R_mm_mpd

p -r_2+R_mm_mp

pd r_2+R_mm_mpd

SPECIES NAME D0

d 8.93e-11

mp 0

mpd 0

p 1.58e-10

pd 8.3e-11

e 5e-11



1 Choose Model>Reaction Settings.


3 Go to the General tab.

4 Choose the species e from the Species selection list and type 0 in the c0 edit field.

You change the value of the initial concentration because the first export of the reaction model is to a geometry domain where the enzyme concentration is zero.

5 Click Close.

E X P O R T I N G T H E M A S S B A L A N C E

1 Choose File>Export>Model to COMSOL Multiphysics.

2 Choose 2D in the Space dimension list, then click OK.

This step launches COMSOL Multiphysics and opens the Export to COMSOL Multiphysics dialog box.

3 On the Settings page in the Export mass balance area, select Diffusion: New from the Application mode list.

4 In the Group name edit field on the same page, type biomaterial.

5 Clear the check box at the top of the Export energy balance area.

6 Clear the check box at the top of the Export momentum balance area.

7 Click Export to close the Export to COMSOL Multiphysics dialog box.

8 In the Reaction Engineering Lab, choose File>Save As and save the current reaction model as drug_release_1.rxn.

Now create a new reaction model that describes the reaction occurring outside the biomaterial matrix. The reaction to include is the association/dissociation reaction of the unbound peptide with the drug. When entering the reaction formulas, use new species names. You later couple the concentration variables for peptide, drug, peptide-drug complex, and enzyme into COMSOL Multiphysics.


Create a new reaction model by choosing File>New.


1 Click the Model Settings button on the Main toolbar.

2 In the Reacting fluid list, select Liquid.


3 Select the Calculate thermodynamic properties check box.

4 Select the Calculate species transport properties check box.

5 In the T edit field, type 310.

6 Click Close.

R E A C T I O N S I N T E R F A C E

1 Click the Reaction Settings button on the Main toolbar.

2 On the Reactions page, create an entry in the Reaction selection list by clicking the New button.

3 Type the following data in the Formula edit field:

4 Go to the Kinetics page and enter the following forward and reverse rate constants:



In the Species selection list, notice the three species that resulted automatically when you entered the reaction formula.

2 Create a new species entry by clicking New, then type E in the Formula edit field. This entry corresponds to the enzyme.


4 Select the appropriate entry in the Species selection list, select the Specify diffusivity check box, and enter the following diffusivities in the D0 edit field:

5 Click Close.

REACTION ID # REACTION FORMULA

1 D+P<=>PD

REACTION ID # kf kr

1 1.5e4 13.5

SPECIES NAME D0

D 8.3e-11

P 1.68e-10

PD 9.31e-11

E 5.5e-11


Now export these settings to a second application mode in COMSOL Multiphysics. You want to create two separate groups: one corresponding to the conditions in the fluid surrounding the biomaterial and nerve, and another corresponding to the nerve-cell domain.

E X P O R T I N G T H E M A S S B A L A N C E

1 Choose File>Export>Model to COMSOL Multiphysics.

2 On the Settings page in the Export mass balance area, select Diffusion: New from the Application mode list.

3 In the Group name edit field on the same page, type fluid.

4 Clear the check box at the top of the Export energy balance area.

5 Clear the check box at the top of the Export momentum balance area.

6 Click Export to close the Export to COMSOL Multiphysics dialog box.

Now modify the settings of the reaction model to reflect the situation in the nerve-cell region.


1 Click the Reaction Settings button on the Main toolbar.


3 Click the General tab, then enter the following initial concentrations, one by one, in the c0 edit field after first having selecting the appropriate entry in the Species

selection list:


5 For each species in the table below, select the entry in the Species selection list and type the corresponding diffusivity value in the D0 edit field:

SPECIES NAME c0

D 0

P 0

PD 0

E 2e-3

SPECIES NAME D0

D 8.93e-11

P 1.58e-10


6 Click Close.

E X P O R T S E T T I N G S

From the File menu, choose Export>Model to COMSOL Multiphysics.

Export the Mass Balance1 In the Export mass balance area, select Diffusion (chdi2) from the Application mode list.

2 In the Group name edit field, type nerve.

3 Go to the Export energy balance area and clear the check box at the top.

4 Go to the Export momentum balance area and clear the check box.

5 Click Export.

6 Choose File>Save as and save the current reaction model as drug_release_2.rxn.

Modeling Using COMSOL Multiphysics

1 Switch to COMSOL Multiphysics.

2 Choose Multiphysics>Model Navigator.

3 Verify that the model contains two Diffusion application modes as a result of the exports from the Reaction Engineering Lab.

4 Click OK.


1 While holding down the Shift key, click the Rectangle/Square button on the Draw toolbar. Enter the values for the rectangle R1 from the following table, then repeat the procedure for the rectangles R2 and R3.

2 Click the Zoom Extents button on the Main toolbar.

PD 8.3e-11

E 5e-11

EDIT FIELD R1 R2 R3

Width 6e-3 1e-3 2e-3

Height 9e-3 9e-3 6e-3

Base Corner Corner Corner

x 0 0 1e-3

y 0 0 0

SPECIES NAME D0



Subdomain Settings1 Choose Multiphysics>1 Diffusion (chdi).

2 Choose Physics>Subdomain Settings.

3 From the Subdomain selection list, select 1, then clear the Active in this domain check box. Repeat this procedure for Subdomain 3.

4 From the Subdomain selection list, select 2.

5 From the Group list, select biomaterial. COMSOL Multiphysics created this group when you performed the export from the Reaction Engineering Lab. Note how all the relevant edit fields of the subdomain setting are automatically filled in.

6 Click OK.

7 Choose Multiphysics>2 Diffusion (chdi2).

8 Choose Physics>Subdomain Settings.

9 From the Subdomain selection list, select 2, and then clear the Active in this domain check box.


11 From the Group list, select nerve.


13 From the Group list, select fluid.

14 Click OK.

Boundary Conditions1 Choose Multiphysics>1 Diffusion (chdi).

2 Choose Physics>Boundary Settings.

3 Go to the Boundary Selection list and select 4, 7, and 9 by Ctrl-clicking the entries.

4 Couple the species concentration variables of the biomaterial to the surroundings. For each species in the table, find the Boundary condition list and select Concentration. Then type the entries from the following table:

5 Leave the remaining boundary conditions at their default setting Insulation/

Symmetry.

6 Click OK.

BOUNDARY c_d c_p c_pd c_e

4, 7, 9 c_D c_P c_PD c_E


7 Choose Multiphysics>2 Diffusion (chdi2).

8 Choose Physics>Boundary Settings.

9 Couple the species-concentration variables of the surroundings to the biomaterial. For each species in the table, find the Boundary condition list and select Concentration. Then type the entries from the following table:

10 Leave all the other boundary conditions at their default setting Insulation/symmetry.

11 Click OK.


1 Define the following expressions by choosing Options>Expressions>Subdomain

Expressions and entering the settings in the table:

2 Click OK.


1 Choose Mesh>Free Mesh Parameters.

2 Click the Boundary tab, go to the Boundary selection list, and select 4, 6, 7, and 9. Type 1e-4 in the Maximum element size edit field.

3 Click OK.


1 Click the Solver Parameters button on the Main toolbar.

2 In the Times edit field, type 0:600:7200.

3 In the Relative tolerance edit field, type 1e-5.

4 In the Absolute tolerance edit field, type 1e-6.

5 Click OK.

6 Click the Solve button.

BOUNDARY c_D c_P c_PD c_E

4, 7, 9 c_d c_p c_pd c_e

NAME SUBDOMAINS 1, 3 SUBDOMAIN 2

c_peptide c_P c_p

c_drug c_D c_d

c_peptidedrug c_PD c_pd

c_enzyme c_E c_e



1 From the Postprocessing menu open the Plot Parameters dialog box.

2 On the Surface Data page, find the Expression edit field and type c_drug.

3 Click OK.

4 To create Figure 7, first go to the Postprocessing menu and open theCross-Section Plot Parameters dialog box.

5 Go to the General page. In the Solutions to use area, select output times between 0 and 3600 by clicking 0 and then Shift-clicking 3600.

6 Click the Line/Extrusion tab.

7 In the y-axis data area, find the Expression edit field and type c_drug+c_peptidedrug.

8 Go to the Cross-section line data area and enter the following values:

9 Click Apply.

10 To create Figure 8, return to the Cross-section Plot Parameters dialog box.

11 On the General page, go to the Solutions to use area and select all the entries by pressing Ctrl+A and clicking the output list.

12 In the Expression edit field, on the Line/Extrusion page, type c_mp+c_mpd.

13 Click OK.

EDIT FIELD ENTRY

x0 0

y0 3e-3

x1 5e-3

y1 3e-3


T umo r R emova l

Introduction

One method for removing cancerous tumors from healthy tissue is to heat the malignant tissue to a critical temperature that kills the cancer cells. This example accomplishes the localized heating by inserting a four-armed electric probe through which an electric current runs. Equations for the electric field for this case appear in the Conductive Media DC application mode, and this example couples them to the bioheat equation, which models the temperature field in the tissue. The heat source resulting from the electric field is also known as resistive heating or Joule heating. The original model comes from S. Tungjitkusolmun and others (Ref. 1), but we have made some simplifications. For instance, while the original uses RF heating (with AC currents), the COMSOL Multiphysics model approximates the energy with DC currents.

This medical procedure removes the tumorous tissue by heating it above 45 °C to 50 °C. Doing so requires a local heat source, which physicians create by inserting a small electric probe. The probe is made of a trocar (the main rod) and four electrode arms as shown in Figure 9. The trocar is electrically insulated except near the electrode arms.

An electric current through the probe creates an electric field in the tissue. The field is strongest in the immediate vicinity of the probe and generates resistive heating, which dominates around the probe’s electrode arms because of the strong electric field.

Figure 9: Cylindrical modeling domain with the four-armed electric probe in the middle, which is located next to a large blood vessel.

electrodes

trocar tip

blood vessel

trocar base (coated)

liver tissue

TU M O R R E M O V A L | 27

Model Definition

This model uses the Bioheat Equation and the Conductive Media DC application modes to implement a transient analysis.

The standard temperature unit in COMSOL Multiphysics is kelvin (K). This model uses the Celsius temperature scale, which is more convenient for models involving the Bioheat Equation.

The model approximates the body tissue with a large cylinder and assumes that its boundary temperature remains at 37 °C during the entire procedure. The tumor is located near the center of the cylinder and has the same thermal properties as the surrounding tissue. The model locates the probe along the cylinder’s center line such that its electrodes span the region where the tumor is located. The geometry also includes a large blood vessel.

The bioheat equation governs heat transfer in the tissue

where δts is a time-scaling coefficient; ρ is the tissue density (kg/m3); C is the tissue’s specific heat (J/(kg·K)); and k is its thermal conductivity (W/(m·K)). On the right side of the equality, ρb gives the blood’s density (kg/m3); Cb is the blood’s specific heat (J/(kg·K)); ωb is its perfusion rate (1/s); Tb is the arterial blood temperature (°C); while Qmet and Qext are the heat sources from metabolism and spatial heating, respectively (W/m3).

In this model, the bioheat equation also models heat transfer in various parts of the probe with the appropriate values for the specific heat, C (J/(kg·K)), and thermal conductivity, k (W/(m·K)). For these parts, all terms on the right-hand side are zero.

The model next sets the boundary conditions at the outer boundaries of the cylinder and at the walls of the blood vessel to a temperature of 37 °C. Assume heat flux continuity on all other boundaries.

The initial temperature equals 37 °C in all domains.

The governing equation for the Conductive Media DC application mode is

where V is the potential (V), σ the electric conductivity (S/m), Je an externally generated current density (A/m2), Qj the current source (A/m3).

δtsρCt∂

∂T ∇ k T∇–( )⋅+ ρb Cbωb Tb T–( ) Qmet Qext+ +=

∇– σ V∇ Je–( )⋅ Qj=


In this model both Je and Qj are zero. The governing equation therefore simplifies into:

.

The boundary conditions at the cylinder’s outer boundaries is ground (0 V potential). At the electrode boundaries the potential equals 22 V. Assume continuity for all other boundaries.

The boundary conditions for the Conductive Media DC application mode are:

The boundary conditions for the bioheat equation are:

The model solves the above equations with the given boundary conditions to obtain the temperature field as a function time.

Results and Discussion

The model shows how the temperature increases with time in the tissue around the electrode.

∇– σ V∇( )⋅ 0=

V 0= on the cylinder wallV V0= on the electrode surfaces

n J1 J2–( )⋅ 0= on all other boundaries

T Tb= on the cylinder wall and blood-vessel wall

n k1 T1∇ k2 T2∇–( )⋅ 0= on all interior boundaries


The slice plot in Figure 10 illustrates the temperature field 60 seconds after starting the procedure.

Figure 10: Temperature field at time = 60 seconds.

Figure 11 shows the temperature at the tip of one of the electrode arms. The temperature rises quickly until it reaches a steady-state temperature of about 90 °C.

Figure 11: Temperature versus time at the tip of one of the electrode arms.


It is also interesting to visualize the region where cancer cells die, that is, where the temperature has reached at least 50 °C. You can visualize this area with an isosurface for that temperature; Figure 12 shows one after 8 minutes.

Figure 12: Visualization of the region that has reached 50 °C after 8 minutes.

Reference

1. S. Tungjitkusolmun, S. Tyler Staelin, D. Haemmerich, J.Z. Tsai, H. Cao, J.G. Webster, F.T. Lee, Jr., D.M. Mahvi, and V.R. Vorperian, “Three-Dimensional Finite Element Analyses for Radio-Frequency Hepatic Tumor Ablation,” IEEE Transactions on Biomedical Engineering, vol. 49, no. 1, 2002.



1 Open the Model Navigator. On the New page, select 3D in the Space dimension list.

2 Go to the Heat Transfer Module menu and select Bioheat Equation>Transient analysis.

3 Click the Multiphysics button and add the application mode to the model by clicking the Add button.

4 Similarly add the Conductive Media DC application mode, from theCOMSOL Multiphysics>Electromagnetics menu.


5 Click OK.


1 From the Options menu, choose Constants.

2 Define the following constants (the descriptions are optional). The constants can also be loaded from the course CD. To do this click on the Folder button in the lower left corner of the Constants dialog box and brows to the file tumor_ablation_constants.txt.

3 Click OK.

NAME EXPRESSION DESCRIPTION

rho_e 6450[kg/m^3] Density, electrodes

rho_t 21500[kg/m^3] Density, trocar tip

rho_l 1060[kg/m^3] Density, liver tissue

rho_b 1000[kg/m^3] Density, blood

rho_c 70[kg/m^3] Density, trocar base

c_e 840[J/(kg*K)] Heat capacity, electrodes

c_t 132[J/(kg*K)] Heat capacity, trocar tip

c_l 3600[J/(kg*K)] Heat capacity, liver tissue

c_b 4180[J/(kg*K)] Heat capacity, blood

c_c 1045[J/(kg*K)] Heat capacity, trocar base

k_e 18[W/(m*K)] Thermal conductivity, electrodes

k_t 71[W/(m*K)] Thermal conductivity, trocar tip

k_l 0.512[W/(m*K)] Thermal conductivity, liver tissue

k_b 0.543[W/(m*K)] Thermal conductivity, blood

k_c 0.026[W/(m*K)] Thermal conductivity, trocar base

sigma_e 1e8[S/m] Electric conductivity, electrodes

sigma_t 4e6[S/m] Electric conductivity, trocar tip

sigma_l 0.333[S/m] Electric conductivity, liver tissue

sigma_b 0.667[S/m] Electric conductivity, blood

sigma_c 1e-5[S/m] Electric conductivity, trocar base

omega_b 6.4e-3[1/s] Blood perfusion rate

T_b 37[degC] Arterial blood temperature

T0 37[degC] Initial and boundary temperature

V0 22[V] Electric voltage



To save time, you can import the geometry from file. The complete step-by-step instruction of how to build the geometry can be found in the section “Creating the Geometry from Scratch” on page 36.

1 From the File menu, choose Import>CAD Data From File.

2 In the Import CAD Data From File dialog box, make sure that the COMSOL Multiphysics

file or All 3D CAD files is selected in the Files of type list.

3 Locate the tumor_ablation.mphbin file on the course CD, and click Import.


Subdomain Settings—Bioheat Equation1 From the Multiphysics menu, select 1 Bioheat Equation (htbh).

2 From the Physics menu, select Subdomain Settings.

3 Select Subdomain 8 and clear the Active in this domain check box.

4 Enter the properties for the remaining subdomains as follows:

5 On the Init page, select all active subdomains and in the T(t0) edit field type T0.

6 To reduce the size of the computation problem, select a lower element order by first clicking the Element tab and then selecting Lagrange - Linear from the Predefined

elements list for all active subdomains.

7 Click OK.

Boundary Conditions—Bioheat Equation8 From the Physics menu, select Boundary Settings.

SETTINGS SUBDOMAIN 1 SUBDOMAINS 2, 5-7 SUBDOMAIN 3 SUBDOMAIN 4

k k_l k_e k_t k_c

ρ rho_l rho_e rho_t rho_c

C c_l c_e c_t c_c

ρb rho_b

Cb c_b

ωb omega_b

Tb T_b

Qmet 0

Qext Q_dc


9 Enter boundary coefficients as follows:

10 Click OK.

Subdomain Settings—Conductive Media DC1 From the Multiphysics menu, select 2 Conductive Media DC (dc).

2 From the Physics menu, select Subdomain Settings.

3 Enter the subdomain properties as in the following table:

4 Reduce the element order also for this application mode: On the Element page, select Lagrange - Linear from the Predefined elements list for all subdomains.

5 Click OK.

Boundary Conditions—Conductive Media DC1 From the Physics menu, open the Boundary Settings dialog box.

2 Select the Interior boundaries check box and enter boundary coefficients as in the table below. You only need to set the boundary condition on the boundaries that use Electric potential. The remaining boundaries already have the correct conditions set by default, that is, Ground for outer boundaries and Continuity for interior boundaries.

3 Click OK.


1 From the Mesh menu, select Free Mesh Parameters.

2 From the Predefined mesh sizes list, select Fine.

SETTINGS BOUNDARIES 1-4, 34 47, 58, 59, 62, 63

BOUNDARY 20

Boundary condition Temperature Thermal insulation

T0 T_b

SETTINGS SUBDOMAIN 1 SUBDOMAINS 2, 5–7 SUBDOMAIN 3 SUBDOMAIN 4 SUBDOMAIN 8

σ sigma_l sigma_e sigma_t sigma_c sigma_b

SETTINGS BOUNDARIES 5-16, 21-33, 35-39, 41-43, 45, 46, 48-57

BOUNDARIES 1-4, 20, 34, 47, 60, 61

BOUNDARIES 17-19, 40, 44, 58, 59, 62, 63

Boundary condition

Electric potential Ground Continuity

V V0


3 Click the Custom mesh size button, then modify the Element growth rate from the default value to 1.7.

4 Click Remesh, then click OK.


1 Click the Solver Parameters button on the Main toolbar.

2 From the Solver list, select Time dependent.

3 On the General page, type 0 600 in the Times field.

4 From the Linear system solver list, select Direct (PARDISO).

5 On the Time Stepping page, select Time steps from solver from the Times to store in

output list.

6 Select the Manual tuning of step size check box. Type 0.01 in the Initial time step edit field and 50 in the Maximum time step edit field.

7 Click OK.

8 Click the Solve button on the Main toolbar.


The default plot is a slice plot of the temperature at time = 600 seconds. In order to generate Figure 10, follow these steps:


2 On the General page, select Interpolated from the Solution at time list.

3 In the Time edit field, type 60.

4 Click the Slice tab and locate the Slice data area. From the Predefined quantities list select Temperature and from the Unit list select oC.

5 In the Slice positioning area set the number of x levels, y levels, and z levels to 1, 1, and 0, respectively.

6 Click Apply.

The following steps describe how to generate the plot in Figure 12, which shows the isosurface for the temperature 50 °C after 8 minutes:

1 Return to the General page, then clear the Slice check box and select the Isosurface check box.

2 In the Time edit field type 480.

3 Click the Isosurface tab. On the Isosurface Data page, select Temperature from the list of Predefined quantities and select oC from the Unit list.


4 In the Isosurface levels area, type 50 in the edit field for Vector with isolevels.

5 Click OK.

6 To finish the plot, click the Scene Light button on the Plot toolbar, then click the Increase Transparency button 5–6 times.

Clicking the Decrease Transparency button repeatedly returns the transparency settings to the original state.

To create Figure 11, which shows temperature versus time, do the following steps:

1 From the Postprocessing menu, select Domain Plot Parameters.

2 On the General page, click the Point plot option button under Plot type.

3 On the Point page, select Point 43. From the Unit list select oC, then click OK.

The postprocessing image that is shown when you open the tumor ablation model from the model library is created with the following steps:


2 On the General page, clear the Isosurface check box and select the Streamline and Slice check boxes.

3 On the Streamline page, select Heat flux from the Predefined quantities list.

4 On the Start Points page, select Specify start point coordinates, then specify the following settings in the x, y, and z edit fields:

5 On the Line Color page, click the Use expression option button, then click the Color

Expression button. In the dialog box that appears, clear the Color scale check box (leave the default settings in the Streamline color data area). Click OK.

6 Set the Line type to Tube, then click the Tube Radius button. In the dialog box that appears, clear the Auto check box for the Radius scale factor, then type 0.1 in the edit field. Click OK.

7 Click the Slice tab. In the Slice positioning area set the number of x levels, y levels, and z levels to 1, 0, and 0, respectively.

8 Click OK.

COORD. VALUE

x linspace(0.02,0.02,30)

y linspace(-0.04,0.04,10) linspace(-0.04,0.04,10) linspace(-0.04,0.04,10)

z linspace(0.05,0.05,10) linspace(0.08,0.08,10) linspace(0.02,0.02,10)


Creating the Geometry from Scratch

1 Go to the Draw menu and select Work-Plane Settings.

2 Create an x-y plane at the z coordinate 0.06.

3 Click OK.

4 Press the Shift key and click the Ellipse/Circle (Centered) button.

5 In the dialog box that appears, enter the following circle properties:

6 Click OK.


8 Repeat Steps 4–6 to create five additional circles, for smaller ones and a large one. The properties of each circle appear in the following five tables:

OBJECT DIMENSIONS EXPRESSION

Radius 9.144e-4

Base Center

x 0

y 0


Radius 2.667e-4

Base Center

x -5e-4

y 0


Radius 2.667e-4

Base Center

x 5e-4

y 0


Radius 2.667e-4

Base Center

x 0

y -5e-4



This completes the geometry needed in the 2D work plane, and the result appears in Figure 13.

Figure 13: 2D working plane geometry.

The following steps describe how to create the 3D geometry by extruding and revolving the 2D geometry:

1 To create the conducting part of the trocar, go to the Draw menu and select Extrude.

2 In the dialog box that appears, select C1 in the Objects to extrude list and type 10e-3 in the Distance edit field.

3 Click OK.


Radius 2.667e-4

Base Center

x 0

y 5e-4


Radius 5e-3

Base Center

x 2.6e-2

y 0


4 To create the insulated part of the trocar, return to the Geom2 work plane, then go to the Draw menu and select Extrude.

5 Select C1 in the Objects to extrude list and type 50e-3 in the Distance edit field.

6 Click OK.

7 Select the EXT2 geometry object.

8 Click the Move button on the Draw toolbar.

9 Enter the following displacement values; when finished, click OK.

To create the electrode arms, proceed as follows:

10 Return to the Geom2 work plane.

11 From the Draw menu select Revolve.

12 From the Objects to revolve list select C2.

13 Enter the following values in the dialog box; when finished, click OK.

PROPERTY VALUE

x 0

y 0

z 10e-3

PROPERTY VALUE

α1 0

α2 180

x (point on axis) -8e-3

y (point on axis) 0

x (second point) -8e-3

y (second point) 1


14 Revolve the circles C3, C4, and C5 in the same manner using the following values:

- Revolve parameters for the circle C3:



15 To create the blood vessel, return to the Geom2 work plane and select C6.

16 From the Draw menu, select Extrude.

17 In the Distance field, type 120e-3.

18 Click OK.

19 Select the EXT3 geometry object, go to the Draw menu, and select Move under Modify.

PROPERTY VALUE

α1 0

α2 -180

x (point on axis) 8e-3

y (point on axis) 0

x (second point) 8e-3

y (second point) 1

PROPERTY VALUE

α1 -180

α2 0

x (point on axis) 0

y (point on axis) -8e-3

x (second point) 1

y (second point) -8e-3

PROPERTY VALUE

α1 180

α2 0

x (point on axis) 0

y (point on axis) 8e-3

x (second point) 1

y (second point) 8e-3


20 Enter the following displacement values; when finished, click OK.

21 To create the large cylinder, press the Shift key and then click the Cylinder button.

22 Enter the following cylinder properties; when finished, click OK.


This concludes the drawing stage. To get a better view of the geometry you have created, do as follows:

1 To hide the coordinate axes, double-click the AXIS button on the status bar at the bottom of the user interface.

2 From the Options menu, select Visualization/Selection Settings.

3 Clear the Geometry labels check box, then click OK.

4 Choose Options>Suppress>Suppress Boundaries.

5 Select Boundaries 34, 47, 58, 59, 62, and 63, then click OK.

6 Click the Perspective Projection button on the Camera toolbar.

7 Finally, after rotating the geometry in the drawing area upside down you should have a view similar to that in Figure 9 on page 26.

PROPERTY VALUE

x 0

y 0

z -60e-3

PROPERTY VALUE

Radius 0.05

Height 0.12

B I O M E D I C A L S T E N T | 41

B i omed i c a l S t e n t

Introduction

Percutaneous transluminal angioplasty with stenting is a widely spread method for the treatment of atherosclerosis. During this procedure a stent, which is a small tube like structure, is deployed in the blood vessel by using a balloon as an expander. Once the balloon-stent package is in place in the artery, the balloon is inflated to expand the stent. The balloon is then removed, but the expanded stent remains to act as a scaffold that keeps the blood vessel open.

Stent design is of significance for the procedure, since damage can be inflicted on the artery during the expansion process. One way this may happen is by the non-uniform deformation of the stent, where the ends expand more than the mid parts, which is also called dogboning. Foreshortening of the stent can also be damaging to the artery, and it can make the positioning difficult.

The dogboning is defined according to

where rend and rmid are the radii at the end and middle of the stent, respectively.

The foreshortening is defined as

where Lorig is the original length of the stent and Lload is the deformed length of the stent.

To check the viability of a stent design, you can study the deformation process under the influence of a radial pressure which expands the stent. With a model you can easily monitor both the dogboning and foreshortening and draw conclusions on how to change the geometry design parameters for optimum performance.

dogboningrend rmid–

rmid---------------------------=

foreshorteningLorig Lload–

Lorig-------------------------------=


Model Definition

Due to the circumferential and longitudinal symmetry of the stent, it is possible to model only one twenty-fourth of the geometry. But, for easier visualization of the deformation you can use one quarter of the geometry in the model.

Figure 14: One quarter of the stent geometry.

M A T E R I A L

Assume that the stent is made of stainless steel with material parameters according to the following table.

MATERIAL PROPERTY VALUE

Young’s modulus 193 GPa

Poisson’s ratio 0.3

Yield strength 300 MPa

Isotropic hardening modulus 2GPa

Free end

Symmetry boundary condition

Symmetry boundarycondition

Fixed point


C O N S T R A I N T S

To prevent rigid body translation and rotation, apply the following constraints, which are also shown in Figure 14.

• Symmetry boundary conditions (normal displacements zero) for the boundaries lying in xy-plane and for the boundaries in the plane parallel to zx-plane going through the middle of the stent.

• Fix the rigid body translation in the remaining direction by constraining a point in the x direction.

L O A D S

Apply a radially outward pressure on the inner surface of the stent. During loading, increase the pressure with the parametric solver to a maximum value of pmax = 0.3 MPa. Follow by decreasing the load to zero to obtain the final shape of the deformed stent.

T H E F I N I T E E L E M E N T M E S H

Use the predefined fine mesh size to mesh the geometry with the free mesher. The mesh created this way consists of approximately 7300 tetrahedral elements.

Results

The stent is expanded from an original diameter of 1.2 mm to a diameter of 3.6 mm in the mid section after unloading. The dogboning is 61% and the foreshortening is


-18%. These values are quite large and may require a change of the geometry parameters to improve the stent design.

Figure 15: Deformed shape plot of the total displacement of the stent after unloading.

In the figure below you can see the diameter of the stent versus the load parameter. Parameter values above one signify the unloading phase. You can follow the deformation, which is initially linear and gets larger as the material is loaded beyond


the yield stress. The unloading is purely elastic for this elasto-plastic material model with isotropic hardening.

Figure 16: Diameter increase in the middle of the stent versus the load parameter.

You can also study the dogboning as the stent is expanded. In the following figure you can notice that the free end of the stent reaches the yield point sooner than the mid parts. As a result of this the dogboning parameter increases until the sections further in also start to deform plastically.

Figure 17: Dogboning versus the load parameter.


Note: Due to the nonlinearity of this model, a fairly dense mesh is used. the solution time for the model does permit solving during class. You can solve the firs few steps and then cancel and open the prepared file in the course CD.



1 Click the New button or start COMSOL Multiphysics to open the Model Navigator.

2 Select 3D from the Space dimension list.

3 Select Structural Mechanics Module>Solid, Stress-Strain>Static analysis elasto-plastic

material.

4 Click OK.

I M P O R T O F C A D G E O M E T R Y

1 On the File menu select Import>CAD Data From File.

2 In the Files of type list select COMSOL Multiphysics file (*.mphtxt; *.mphbin; ...).

3 Browse to the directory models/Structural_Mechanics_Module/Bioengineering located in the COMSOL installation directory and select the file biomedical_stent.mphbin.

4 Click Import.


Constants1 From the Options menu select Constants.

2 Enter the constants from the following table. The constants can also be loaded from the course CD. To do this click on the Folder button in the lower left corner of the Constants dialog box and brows to the file biomedical_stent_constants.txt.


Load_max 3e5 Maximum applied load to the stent

E 1.93e11 Young’s modulus

nu 0.3 Poisson’s ratio


3 Click OK.

Extrusion Coupling VariablesYou can set up extrusion coupling variables to gain access to coordinates and displacements of points on the entire geometry.

1 Select Options>Extrusion Coupling Variables>Point Variables.

2 Use the following table to enter variables for selected points on the Source page. Click the General transformation option button for each variable, after entering both the name and expression.

3 Click the Destination tab.

4 Select first x_mid from the Variable list, then from the Level list select Subdomain and select the check box next to Subdomain 1 in the Subdomain selection list.

5 Repeat the previous step for all the variables you have entered under step 2.

6 Click OK.

Global ExpressionsBy defining the quantities you are interested in as global expressions you can access them for postprocessing or during the solution process.

1 From the Options menu select Expressions>Global Expressions.

S_yield 3e8 Yield stress

E_tan 2e9 Isotropic tangent modulus

POINTS NAME EXPRESSION

6 x_mid x

6 y_mid y

6 u_mid u

272 y_end1 y

272 z_end1 z

272 u_end1 u

272 v_end1 v

272 w_end1 w

12 x_end2 x

12 u_end2 u

12 w_end2 w



2 Enter expressions according to the following table. The expressions can also be loaded from the course CD. To do this click on the Folder button in the lower left corner of the Global Expression dialog box and brows to the file biomedical_stent_expressions.txt.

3 Click OK.


Application Mode Properties1 Select Physics>Properties to open the Application Mode Properties dialog box.

2 In the Large deformation list box select On.

3 Click OK.

Subdomain Settings1 Select Physics>Subdomain Settings to open the Subdomain Settings dialog box.

2 Select Subdomain 1.

3 On the Material page, select Elasto-plastic from the Material model list.

4 In the E edit field enter E.

5 In the ν edit field enter nu.

6 Click the Elasto-Plastic Material Settings button.

7 In the σys edit field enter S_yield.

8 In the ETiso edit field enter E_tan.

9 Click OK.

10 Click OK.

NAME EXPRESSION

foreshortening (l_orig-l_load)/l_orig

l_orig abs(y_mid-y_end1)

l_load l_orig+v_end1

dogboning (r_end-r_mid)/r_mid

r_mid r_orig+abs(u_mid)/2

r_orig abs(x_mid)

r_end sqrt((z_end1+w_end1)^2+(dist12/2)^2)

dist12 sqrt((w_end2-w_end1)^2+(u_end2-u_end1+2*x_end2)^2)


Boundary ConditionsBy applying the symmetry boundary conditions you can prevent rigid body translation in the y and z direction and rotation around all coordinate axes.

1 Select Physics>Boundary Settings.

2 From the Boundary selection list select Boundaries 2, 5, 8, 10, 42, 86, 130, 144, 145, and 146.

3 On the Constraint page select Symmetry plane from the Constraint condition list.

4 Select Boundary 28.

5 On the Load page, from the Coordinate system list box select Tangent and normal

coord. sys. (t1, t2, n).

6 In the Fn edit field enter -Load_max*((para<=1)*para+(para>1)*(2-para)).

7 Click OK.

Point SettingsTo fix the rigid body translation in the remaining x direction constrain a point in this direction.

8 Select Physics>Point Settings.

9 From the Point selection list select Point 288.

10 On the Constraint page make sure that the Standard notation button is selected.

11 Select the Rx check box and check that the value in the corresponding edit field is zero.

12 Click OK.


1 From the Mesh menu, choose Free Mesh Parameters.

2 Select Fine from the Predefined mesh sizes list.

3 Click the Remesh button, then click OK.


Solver Parameters1 Click the Solver Parameters button on the Main toolbar.

2 On the General page, in the Parameter name edit field enter para.

3 In the Parameter values edit field enter 0 0.3 0.35 0.4 0.45 0.5 0.54 0.58 0.6 0.62 0.64:0.04:1 1.05 1.5 2.


4 Click OK.

Probe PlotTo monitor the deformation of the stent during the solution process, you can create a probe plot. You can see this diagram, updated for each parameter step, on the Plot page of the Progress dialog box, that COMSOL Multiphysics displays while the solver is running.

1 Select Postprocessing>Probe Plot Parameters.

2 In the Probe Plot Parameters dialog box click the New button.

3 In the dialog box that opens select Global from the Plot type list box.

4 In the Plot name edit field enter Stent diameter plot.

5 Click OK.

6 In the Expression edit field type 2*r_mid.

7 Click OK.

8 Click the Solve button on the Main toolbar to solve the problem.


When the solution is done the plot of the stent diameter versus the solver parameter is automatically displayed in a separate figure window. You can also visualize the deformation of the stent after unloading.

1 Switch back to the COMSOL Multiphysics window.


3 On the General page deselect the Slice check box under the Plot type area.

4 Deselect the Geometry edges check box.

5 Select the Boundary check box.

6 Select the Deformed shape check box.

7 Click the Deform tab, and then clear the Auto check box and type 1 in Scale factor edit field.

8 Click OK to display the plot in Figure 15 on page 44.

You can also display the foreshortening and dogboning of the stent.

9 Select Postprocessing>Data Display>Global.

10 In the Expression edit field type dogboning.

11 Click Apply to display the value in the message log.

12 In the Expression edit field type foreshortening and click OK.

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