1

Name:__________________________

Group #:__________________________

BEE/MAE 4530

Prelim 2

Notes:

1) Governing Equations are provided at the end

2) Multiple parts per question

3) Only write/keep the necessary terms in an equation

Question 1 2 3 4

Points 25 22 38 15

2

Problem 1

Iontophoresis works by applying a voltage gradient between an anode and cathode that

increases mass transport from a reservoir, with concentration cs, into the body. A drug, cA, that

is a cation will be driven from the positive anode (V=V0) in an arc to the cathode (V=0) in such

a way to increase transport into the blood vessel and into the blood stream. The goals of the

model are to simulate the transport from the reservoir into the blood stream and accumulation

of the drug in the blood stream and body.

Once the drug leaves the domain shown in Figure 1 it enters the body, cbody, where it is

degraded in the body with a second order reaction and rate constant, k. The volume of blood in

the body is Vb. The blood vessels around the patch have a cross sectional area Ab.

Figure 1 shows an incomplete schematic, governing equations (governing equations are

defined here as any differential equation, i.e. PDE or ODE), and boundary conditions. The GEs

are for the drug (the cation), the anion in tissue, and for charge conservation. The anion

accumulation in the body can be ignored.

Figure 1: Iontophoresis schematic and boundary conditions and governing equation at the bottom. The no flux boundary applies for voltage, charge, and mass transport.

3

(12 points) Write the 4 missing governing equations and for each equation state whether they

are in the skin, blood vessel, body, or a combination of all 3. These are the governing equations

needed to simulate the transport from the reservoir into the blood stream and accumulation of

the drug in the blood stream and body.

(4 points) What is the velocity, w, in the skin and the blood for the governing equations shown

in Figure 1? Make sure to explain what the variables are and where they come from.

4

(7 points) Write the expressions for any missing boundary conditions and where they occur.

You can write them below or on the figure. Please note the figure caption.

(2 points) Write all the missing initial conditions below.

5

Problem 2

(5 points) For laser heating (light diffusion), write the 1D finite difference equation for the

fluence rate, tt

x

. Use a 1st order accurate explicit form for time and a 2nd order accurate,

central difference form for space. The time step is Δt and the nodal spacing is Δx. Your

equation should be a function of t

x ,

t

xx ,

t

xx , and appropriate transport properties.

(6 points) Simplify the 1D, 2 element, 1st order finite element method (FEM) transient heat

conduction formulation (shown below) to the steady state form only keeping the necessary

terms.

3

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2

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(3 points) Write the set of equations (in matrix form) for a 1D, 3 element, 1st order FEM

transient heat conduction problem where h=h1=h2=h3.

6

(8 points) Using the matrix equation in the previous question, apply the boundary conditions

for nodes 1 and 4. Where λ is the latent heat of evaporation of water and nw,s is the constant

surface mass flux of water.

Node 1: sw

tt

tt

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Tk

,1

1

Node 4: S

ttt

TTT

44

Problem 3

7

(2 points) COMSOL uses ______________________ for the direct numerical solution of

Ax = b instead of Gaussian elimination.

(2 points) If a model has 100 elements and 200 degrees of freedom (DOF), how many

equations is COMSOL solving?

(3 points) For a 1D FEM problem, write the 3 shape functions where the nodes are located at z

= 0, 0.5, and 1.

p1 =

p2 =

p3 =

(2 points) Four methods of reducing discretization error were discussed in class, list 2 of those:

1) _____________________

2) _____________________

(2 points) Four methods of obtaining computational model parameters were discussed in class,

list all 4:

1) _____________________

2) _____________________

3) _____________________

4) _____________________

(1 point) When accurate data is unavailable, we can perform a ________________________ to

determine its relative importance.

(1 point) Modeling microwave heating with Maxwell’s equations and the bioheat equation is

one/two way coupled? Circle the correct answer.

8

(2 points) A user is trying to implement a binary laser pulse by using modular arithmetic and

the rectangle function in COMSOL. The laser pulses for 10 µs and is off for 100 µs. The user

created the rectangle function shown below. Fill in the 3 blanks for the expression below:

rect1(mod( ____ [ ______ ], ______)

(5 points) Under some simplifying assumptions, a heat source (such as laser, microwave, etc.),

can be reduced from a heat generation term, Q=Q0e-x/δ where Q is in W/m3, to a heat flux, F.

How would you obtain the heat flux, F, in W/m2.

(2 points) A key assumption of this conversion is that the penetration depth is

_____________________ than the domain characteristic length.

9

(3 points) We discussed the bioheat equation in class and property values for different tissues

and organs, for a frozen liver:

1) The units of blood

V are:

2) The numerical value of bloodbloodpblood

Vc ,

3) The numerical value of met

Q

(2 points) For transport in a porous media, a transdermal patch uses ______________ diffusion

to deliver a drug through the skin. While soaking an organ in a nutrient solution to rehydrate it

uses ______________ diffusion.

Note: writing the same answer in both will result in a 0/2.

(2 points) When the rate of water diffusion is much faster than degradation rate,

________________ erosion occurs. When the degradation rate is much faster than water

diffusion, ____________________ erosion occurs.

(2 points) In estimating diffusion and degradation time scales during tablet erosion, the time it

takes for water to diffuse into a tablet of radius, R, can be estimated by (in symbolic form):

tdiff ≈

(7 points) The following is a 3 part question below.

10

When modeling tumor hyperthermia during ultrasound heating (see figure below), we use the

wave equation to model propagation of the ultrasound waves, the bioheat equation to model

temperature, the t43 equation to model cell death.

Figure 2: Left pressure waves and right is temperature in the healthy tissue and tumor

Part 1/3: Is the equations for cell death due to hyperthermia and temperature one or two way

coupled?

Part 2/3: Explain why.

Part 3/3: Write an equation for the objective function for the process.

Problem 4

tumor

Healthy tissue

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Please only answer the 3 questions for your group below.

Group 1

(5 points) On the boundary where the laser is hitting the eye, is the mass flux equal everywhere or different?

Explain

(5 points) The ablation rate is proportional to the evaporation rate of water in your current model. But the eye is

not 100% water, and so why is this an over simplification and how would the equation for the ablation rate

change?

(5 points) Is your laser source term empirical (based on experimentation and curve fitting) or mechanistic

(based on theory)? What would be a more mechanistic way to model it?

Group 2

(5 points) For the fully embedded stent, what assumption about the model geometry is obviously unphysical?

(5 points) Why is there no convection term in your GE if there is flood flow in the vein (not the wall)?

(5 points) Is your diffusivity modeled with a series or parallel formulation? Explain

Group 3

(5 points) Does the 3D brain you were given have rotational or planar symmetry? Explain.

12

(5 points) If the brain has a characteristic length of 1 cm, a drug diffusivity of 1×10-9 m2/s, and a velocity

magnitude of 10 nm/s, does convection matter? Use a dimensionless constant to back up your claim.

(5 points) Are you modeling flow in the brain as compressible or incompressible? What is inconsistent between

these equations?

Group 5

(5 points) What term in the blood perfusion heat source term goes to 0 when the tissue is frozen?

(5 points) The spatial derivative of the temperature on the outer boundary of the healthy tissue (the large

cylinder), should equal what value? And why.

(5 points) Explain how the tissue (the tumor, prostate, etc.) densities are being changed to account for different

numbers and sizes of probes.

Group 6

(5 points) Does convection matter for heat and mass transfer in the aqueous humor? Explain how you

determined this.

(5 points) If the person was standing instead of lying down (as you are assuming now), could you still use axial

symmetry for the model? Explain.

13

(5 points) What governing equation are you missing? Here is your heat transfer and Navier-Stokes equation

(convection-diffusion equation is not shown).

Group 7

(5 points) Below is your governing equation for fluid flow in the vein. What equation are you missing

pertaining to the vein?

(5 points) The model assumes the patch (and needles) volume is equal to the height times width times depth,

why is this overestimating the patch volume?

(5 points) If you had more computational power and time, what would be a more accurate way to model drug

delivery from the patch? Right now you are using a flux/concentration out of the patch. What would be more

realistic at the expense of time and computing resources?

Group 8

14

(5 points) Is your laser source term empirical (based on experimentation and curve fitting) or mechanistic

(based on theory)? What would be a more mechanistic way to model it?

(5 points) In your report, you do not list the initial conditions for pressure and velocity in the vein. What would

they be or how would you obtain them?

(5 points) Below is your governing equation for fluid flow in the vein. What equation are you missing

pertaining to the vein?

Group 9

(5 points) Does your model assume planar or rotational symmetry? Explain

(5 points) Based on your model’s symmetry, what is missing in the equation below?

(5 points) Are your governing equations for uterine copper concentration and body copper concentration one-

way or two-way coupled? Explain.

Group 10

15

(5 points) How many planes of symmetry does your 3D model have and what would be the heat transfer

boundary condition there?

(5 points) How are you modeling the location of the suit in the 3D model? What equation determines where is

tissue and where is fabric? How does the solution of this equation affect the heat source terms (i.e. how is it

coupled to the heat equation)?

(5 points) What is the heat transfer boundary condition at the neck, wrists, and ankles? Could this be modeled

another way? Explain.

Group 11

(5 points) Does your model have any planes of symmetry? If yes, where are they (you can describe them or

draw it)? Explain.

(5 points) Below is your governing equation for fluid flow in the vein. What equation are you missing

pertaining to the vein?

(5 points) What is unphysical about your equation for fibrin formation? What is a more accurate equation?

Equations

16

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