Chemotherapy: Drug Diffusion
through Solid Tumor
William Chen
Jesse Fallick
Jennifer Hsu
Jason Perlmutter
Comron Saifi
BEE 453: Computer-Aided Engineering
Professor Ashim K. Datta
Spring 2005
122
Table of Contents
[1] Executive Summary…………………………………...3
[2] Introduction……………………………………………3
[3] Results and Discussion………………………………...7
Sensitivity Analysis………………………………...11
[4] Conclusions and Design Recommendations…………14
[5] Appendices…………………………………………...16
Appendix A…………………………………………16
Appendix B…………………………………………19
Appendix C…………………………………………20
Appendix D…………………………………………25
123
[1] Executive summary
Diffusion of the anti-cancer drug Verteporfin in solid tumor tissue was modeled using
computer-aided design software. This was performed to determine the penetration depth
at a minimum effective concentration which was compared to the penetration depth
required for complete treatment of tumor tissue. The penetration depth of the drug was
found to be 54.9 microns, which is less than the required 72.5 microns. This indicates that
the minimum effective concentration was not reached throughout the tumor. Sensitivity
analysis was performed to test the stability of our solution and to discover factors that
might affect penetration depth. The factors that had the most influence on penetration
depth were the diffusivity of the drug and the time allowed for diffusion. Manipulation of
these two factors may allow for increased penetration depth, which would result in better
treatment of solid tumors.
[2] Introduction
Verteporfin (Figure 1) is a photosensitive drug used in the treatment of cancer. The drug
is delivered to the body through intravenous infusion directly into the blood stream. The
drug then diffuses from the capillaries into tumor tissue. Fifteen minutes after infusion is
complete, the drug is activated through irradiation at a wavelength of 689 nm at an
intensity of 600 mW/cm2 for 83 seconds.4,10 After excitation by photons, Verteporfin
produces a highly reactive singlet oxygen. This oxygen reacts within 40 nanoseconds and
results in the destruction of membrane bound organelles within the cell. Calcium ions are
released from organelles such as the mitochondria and the endoplasmic reticulum which
induces cell apoptosis. In the case of the mitochondria the loss of the electro-chemical
gradient plays a significant role in inducing apoptosis.5,10
Figure 1. Molecular structure of Verteporfin13
One characteristic of solid tumors is a lack of vascularization. The delivery of the drug to
the tumor is hampered by the lack of blood flow to the target tissue. A minimum effective
concentration of 0.1375 g/m3 is required in all parts of the tumor for effective treatment.9
Therefore, penetration depth can be defined as the depth from a capillary at which this
minimum concentration is achieved. The mean intercapillary distance is 145 microns in
124
solid mass tumors.6 Therefore it is important that the penetration depth of the drug is at
least half that distance, from each capillary, in order to treat the entire tumor.
Utilizing FIDAP, a computer aided design program, diffusion of Verteporfin into a solid
mass tumor from an adjacent capillary was modeled. Computer aided simulation allowed
for the use of time and space dependent boundary conditions in order to calculate the
concentration of Verteporfin as a function of time and space. Furthermore, factors
influencing the rate of drug diffusion were determined through sensitivity analysis in
order to maximize drug delivery.
Design Objectives:
1. To accurately model the diffusion of Verteporfin through solid tumor tissue
2. To determine the penetration depth of Verteporfin achieved within the solid tumor
tissue at Tlight = 15 min. and compare to the penetration depth required for complete
treatment of the tumor tissue.
3. To determine factors that influence penetration depth to possibly improve the treatment
process.
125
Schematic:
The physical representation of a solid mass tumor is a cylindrical capillary surrounded by
a cylindrical mass of tumor cells. Therefore, we were able to assume an axi-symmetric
geometry. This simplified geometry was used to model a solid mass tumor in order to
reduce the computational processing needed to run the simulation. Figure 2 depicts the
3D representation of the capillary surrounded by tumor, with the blue boxed-out section
representing the region of focus in our model. Figure 3 illustrates the 2D-axisimmetric
geometry, which is split into Region 1 (capillary) and Region 2 (tumor).
Figure 2:
Figure 3:
126
Mesh:
Figure 4 depicts the geometry and mesh created in GAMBIT and modeled after the 2Daxisimmetric geometry shown in Figure 3. The length of region 2 (tumor) was purposely
created longer than needed so that the penetration depth of the drug would be reached
within the boundaries of the mesh. Note that the nodes are densest at the interface
between the capillary region and the tumor region. It is important to accurately monitor
concentration in this area, since this is the region where diffusion occurs most rapidly.
Figure 4:
Flux = 0
Flux =0
Flux = 0
Velocity = 214 -214x2
Concentration = 8.321*t-0.5286
Velocity = 0
(Capillary)
(Capillary surface)
Flux =0 dV/dx = 0
(Capillary Center)
127
[3] Results and Discussion
The simulation was initially run for 15 minutes, as per recommendations in the literature
for incubating time after injection of Verteporfin and before light treatment. All
properties, boundary conditions, initial conditions, governing equations are stated in
Appendix A. Our goal was to observe the general trends in the diffusion of the drug
through the tumor and to determine the exact penetration depth, as defined earlier, of the
drug into the tumor. Figure 5 shows the species contour plot at 15 minutes with a line
indicating the penetration depth that resulted. A straight line could be used to indicate the
penetration depth due to the uniform diffusion of the drug as depicted in the contour plot.
This line was automatically generated by FIDAP and bolded for visual acuity. Also
displayed in the plot is the exact penetration depth in microns. This number was
determined by first finding a node at the minimum effective concentration, then finding
the distance of this node from the Z-axis.
54.9 microns
Figure 5. Species contour plot at T = 15 min. Line represents depth at minimum effective concentration .
The contour plot is as expected, with uniform diffusion of the drug from the highest
concentration region in the capillary into the lower concentration region in the tumor. The
penetration depth, 54.9 microns, is in fact less than the required penetration depth
(intercapillary distance / 2) previously determined to be 72.5 microns. Thus, based on our
previous assumptions, the drug would not diffuse enough in 15 minutes such that the
concentration in all parts of the tumor in the original schematic would reach the minimum
effective concentration.
128
Figure 6 shows several plots of concentration versus time for a vertical line of nodes on
the left side of the tumor. The location of these nodes is outlined in Figure 7. The plots
show an interesting trend in how concentration in the tumor changes with time. In most
of the plots, concentration actually reaches a maximum then begins to decrease and level
off as time progresses. The maximum that is reached is inversely proportional to the
distance of the node from the capillary. This trend is due to the decreasing concentration
of drug that is coming into the capillary over time (see Boundary Conditions in Appendix
A). At first, the diffusion into each node is high enough relative to the diffusion out of the
node to steadily increase the concentration over time. After the concentration maximum
is reached, however, the diffusion out is greater than the diffusion in, and the
concentration begins to decrease. For most of the nodes shown in Figure 6, it can be
predicted that the solution would become steady soon after 15 minutes.
Figure 6. Plot of concentration versus time at a vertical line of nodes on the left side of the tumor region.
Nodes shown in mesh plot of Figure 7.
129
Figure 7. Mesh plot with nodes 123-150 outlined in red. Refers to history plot in Figure 6.
130
In an attempt to increase the penetration depth of the drug as per our design objectives,
the simulation was additionally run for 30 minutes and 60 minutes. In the physical
problem, this translates to an increase in incubation time after the injection of Verteporfin
and before light treatment. The penetration depth for these simulations was obtained and
graphed versus incubation time. This graph is depicted below in Figure 8.
70
30 min.
60 min.
Penetration Depth (microns)
60
15 min.
50
40
30
20
10
0
0
10
20
30
40
50
60
70
Incubation Time (minutes)
Figure 8. Plot of penetration depth vs. duration of simulation
From the graph it is clear that with increasing incubation time before light treatment, the
penetration depth of the drug is greater. However, it should be noted that at all three
times, the penetration depth is still less than the required penetration depth of 72.5
microns. Therefore, even after 60 minutes, which is four times the recommended
incubation time of 15 minutes, the drug has still not diffused far enough to reach all parts
of the tumor. It should also be noted that the penetration depth begins to steady out by 60
minutes, suggesting that increasing the incubation time from 60 minutes will not
necessarily increase the penetration depth far enough to surpass the required 72.5
microns.
131
Sensitivity Analysis
In order to determine the effects of error in our property values and to test the stability of
our solution, the effects of varying blood velocity, Verteporfin diffusivity through the
tumor, and the reaction rate of Verteporfin were tested. It was assumed that the
properties that were found in the literature were off by a maximum of 20%. Therefore,
values 20% above and below the initial property values were tested.
Blood Velocity
Sensitivity analysis was performed on blood velocity because the value that was used in
our simulation is based on assumptions that simplify real life situations. Blood velocity
was set to the average velocity in a capillary that is 9 micrometers in diameter.12
However, in reality, not all capillaries are 9 micrometers thick and blood velocity also
varies from person to person. Therefore, in different cases, the variations mentioned
above can cause the blood velocity to be slightly different than the initial velocity chosen.
From Figure 9, it can be observed that there is a negligible variation between the
concentration plots for the varying blood velocities. At t500, the concentrations at both the
high and low blood velocities vary from the concentration at the original blood velocity
by 1*10-4, which is a 0.11% deviation. Since changing velocity had little effect on the
original solution, it can be concluded that the assumptions made about blood flow are
safe.
0.05
214 (Original)
256.8 (20% higher)
171.2 (20% lower)
0.045
0.04
Non-dimensional
Concentration
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0
50
100
150
200
250
300
350
400
450
500
Non-dimensional Time
Figure 9. Plots of concentration versus time for our original blood velocity, a velocity that is 20% higher,
and a velocity that is 20% lower
132
Reaction Rate of Verteporfin in Tumor
The degradation of Verteporfin follows a biexponential function. In the capillary the
biexponential degradation can be modeled as a time-dependent boundary condition (See
Appendix B). However, it is not possible to model this biexponential elimination in the
tumor region using FIDAP. Thus we approximated the degradation reaction as a first
order reaction, since biexponential functions degrade almost linearly initially and the
duration of our simulation was only 15 minutes. Figure 10 illustrates that there is a
negligible difference between the concentration plots for the different reaction rates. At
t500, the concentration difference between the upper and lower limits is 4*10-4, which is a
0.44% deviation. Since varying reaction rate has virtually no effect on the solution, any
error in its value due to our approximations would still result in the same penetration
depth.
0.05
8.772*10^(-5)
7.31*10^(-5)
0.045
5.848*10^(-5)
0.04
Non-dimensional concentration
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0
50
100
150
200
250
300
350
400
450
500
-0.005
Non-dimensional time
Figure 10. Plots of concentration versus time for our original reaction rate of Verteporfin in the tumor, a
reaction rate that is 20% higher, and a reaction rate that is 20% lower.
Diffusion of Verteporfin through Tumor
Unlike blood velocity and Verteporfin reaction rate, Figure 11 shows a significant
difference in the concentration plots for the different diffusivities of Verteporfin through
the tumor. From Figure 12, it is clear that penetration depth increases with diffusivity
through the tumor. The penetration depth increases by 5.841 micrometers (10.64% of
original penetration depth) after changing the original diffusivity to one that is 20%
higher, and it decreases by 5.184 micrometers (9.44% of original penetration depth) after
changing the original diffusivity to one that is 20% lower. These significant variations in
penetration depth demonstrate the process’s extreme sensitivity to the diffusivity of
133
Verteporfin through the tumor. While varying blood velocity and Verteporfin reaction
rate by 20% hardly affected the original diffusion process, the same change in the
diffusivity of Verteporfin through the tumor resulted in considerable changes in
penetration depth.
0.06
0.088 (Original)
0.0704 (20% lower)
0.1056 (20% higher)
NonDimensional
Concentraion
0.05
0.04
0.03
0.02
0.01
0.00
0.00
100.00
200.00
300.00
400.00
500.00
Non-dimensional Time
Figure 11. Plots of concentration versus time for our original diffusivity of verteporfin in the tumor, one
20% higher, and one 20% lower.
70
1.06E-08
60
8.80E-09
50
Penetration Depth
(micrometers)
7.04E-09
40
30
20
10
0
6.00E-09
6.50E-09
7.00E-09
7.50E-09
8.00E-09
8.50E-09
9.00E-09
9.50E-09
1.00E-08
1.05E-08
1.10E-08
Diffusivity (cm^2/s)
Figure 12. Penetration depth achieved at our original diffusivity of verteporfin in the tumor, one 20%
higher, and one 20% lower.
134
[4] Conclusions and Design Recommendations
Design Recommendations:
As indicated by sensitivity analysis, change in the penetration depth of Verteporfin with a
concentration above 0.1375 g/m3 was influenced primarily by the diffusion coefficient of
the drug in the tumor. Two possible methods are sited for increased diffusivity in order to
obtain a penetration depth of 72.5 microns which is needed to saturate the tumor. A third
recommendation is given with respect to the clinical aspects of drug administration. The
first method would involve modification of the drug. Therefore the mechanism by which
the drug functions, which is discussed in the introduction, becomes of central importance
in the recommendation of possible methods to increase the drug’s diffusion coefficient.
kT
, it is only possible to
6r
increase diffusivity in a constant medium by decreasing the effective radius of the drug.
One possible solution is the modification of the drug’s planar structure to utilize a three
dimensional geometry in order to decrease radius while maintaining the original number
of porphryn rings which would consequently increase density. Since the effective radius
also includes a certain amount of solvation, modification of functional groups to decrease
solvation would also increase diffusivity. If drug specificity for tumor cells could be
increased, higher dosages could be given without unwanted side effects. Recently
addition of polyethylene glycol to photosensitizer/polymer conjugates was found to
increase tumor toxicity by 9-fold and reduce toxicity to normal cells by 4-fold.5
Since the Stokes-Einstein relation is expressed as D 
A second possible method would involve modification of the solid tumor material which
the drug diffuses through. Although direct alteration of the tumor mass is not possible,
combinational therapies involving simultaneous treatment with other drugs could induce
the expansion of pore openings in tumor cells which would increase the diffusivity of the
chemotherapy agent.
Pharmokinetic data of Verteporfin concentration in the human body was only present for
patients with choroidal neovascularization and healthy individuals. Concentration as a
function of time in the plasma of healthy individuals was only available for 6 mg/m2 BSA
during and a ten minute intravenous infusion period. Further studies are needed to
determine the mathematical relationship between infusion period, dosage, and
concentration of the drug in the plasma as a function of time. If this relationship can be
modeled, different combinations could be tested with our computational model to
determine dosage requirements for obtaining a penetration depth of 72.5 microns.
135
Discussion on Realistic Constraints:
Health and Safety
Although the suggestions given within ‘design recommendation’ will help increase the
penetration depth of the drug into the tumor, some of these may not be feasible due to
potential side-effects. It is suggested that significant modifications to the chemical
structure of Verteporfin, to the dosage, or the addition of other drugs be re-evaluated
beginning with stage I preclinical testing to ensure safety.
Manufacturing
Due to bond angle strain and steric hindrances it may not be possible to transform the
drug from a planar into a three dimensional geometry in order to increase diffusivity.
Large scale production of Verteporfin with any chemical modifications may encounter
difficulty during synthesis and purification processes.
Economy
Currently in the United States the average drug costs $900 million and takes 15 years to
develop.2 This is a significant barrier to the creation of new drugs. For this reason
alterations to the clinical administration of Verteporfin should be analyzed first. By
noting the effect of various dosages and infusion period on the concentration in the
plasma an optimal treatment maybe determined which reaches the necessary penetration
depth within the tumor. This plan would be the most cost effective as it would require
minimal FDA approval compared to the other suggestions and would cost the least since
it does not require further research and development expenses.
136
[5] Appendices
Appendix A
– Mathematical Statement of Problem
Governing Equations
Capillary Region
Momentum Equation:

  1 
vr
p
 v 
rvr   gr
   vr r   
  
t
r
 r 

 r  r r
Species Equation:
 1   c A   2 c A 
c A  c A 
  RA
  vr
  D AB 
r

2 
t  r 
 r r  r  z 
Tumor Region
Species Equation:
 1   c A   2 c A 
c A  c A 
  RA
  vr
  D AB 
r

2 
t  r 
 r r  r  z 
Initial Conditions
Cblood0 = 0
Ctumor0 = 0
Boundary Conditions
Velocity:
Capillary Inlet: u  214  214 x 2
v
0
Capillary Center:
x
Capillary Wall: v  0
137
Species:
Capillary Inlet: C  8.321t 0.5286 (see Appendix B)
c
0
Capillary Center:
x
c
0
Left Tumor, Right Tumor, Top Tumor:
x
Input Parameters:
Variable
Blood
Blood Viscosity3
Blood Velocity8
Value (dimensional)
1060 kg/m3
2.7 x 10-3 Pa*s
4.7x10-4 m/s
Value* (non-dimensional)
1
2.5729x105
2.14x102
Diffusivitytumor11
8.8 x 10-13 m2/s
8.8x10-2
Diffusivityblood7
Reaction Rate constant (tumor)4
Duration of Simulation (tfinal)
Time Step (dt)
10-11 m2/s
3.61x10-5 s-1
9.0x102 s
1s
1
7.31x10-5
4.445x102
0.5 s
Density1
Methods of Non-dimensionalization:
Characteristic Properties:
Characteristic Length12 = 4.5 *10 6 m (radius of capillary)
10 11 ms
u 

 2.2 * 10 6
6
L
4.5 * 10 m
2
Dcap
m
s
Calculations:
Blood density* 
Blood density
1
Blood density
Blood viscosity * 
Blood velocity* 
Blood viscosity
 2.5729x105
Blood density u  L 
Blood velocity
 2.14  10 2
u
Diffusivit y Tumor * 
Diffusivit y Tumor
 8.8 x10 2
Diffusivit y Blood
138
Diffusivit y Blood* 
Diffusivit y Blood
1
Diffusivit y Blood

Reaction rate constant L2 
Reaction rate constant* (tumor) =
 7.31x10 -5
Diffusivit y
Reaction rate constant*(blood) =
Reaction rate constant L2   7.31x10 -5
Diffusivit y
Duration of simulation (tfinal)* =
Concentration* =
C i C
C  Ci
Blood
t final Dblood
2
L
Blood
= 4.445x102
139
Appendix B – Explanation of Capillary Inlet Concentration Time Curve
The kinetics of Verteporfin can be modeled as biexponential elimination, which takes the
form:
C t   A t  B t
Unfortunately, it is nearly impossible to model this type of elimination in FIDAP. A set
of (time, concentration) points was obtained from a paper written on the kinetics of
Verteporfin (see sources), and Microsoft Excel was used to non-dimensionalize and fit a
series of curves to the data. The curve with the highest value of R2 was chosen. The data
and best fit curve are shown in Figure 13.
0.7000
0.6000
Concentration
0.5000
y = 8.321x -0.5286
R2 = 0.9843
0.4000
0.3000
0.2000
0.1000
0.0000
0.0000
200.0000
400.0000
600.0000
800.0000
1000.0000
1200.0000
Tim e
Figure 13. Concentration vs. Time data fitted with a power curve
The equation of best fit curve is displayed in the graph above. Once again, this equation
cannot be modeled in FIDAP. Therefore, this equation was used to define 10 points in the
time range 0-900 seconds (the duration of the simulation). The 1st point was defined as (t
= 0, c = 1), which was required for proper non-dimensionalization even though this point
does not lie on the curve depicted above. The other 9 points were defined using the
equation shown in Figure 13. These 10 points were used in PRESTO to define the timecurve species boundary condition at the capillary inlet.
140
Appendix C- Convergence and Input File
Input File:
TITLE
/
/ *** FIPREP Commands ***
/
FIPREP
PROB (AXI-, ISOT, LAMI, TRAN, NONL, FIXE, NEWT, INCO, SPEC = 1.0)
PRES (MIXE = 0.100000000000E-08, DISC)
EXEC (NEWJ)
SOLU (S.S. = 50, VELC = 0.100000000000E-02, RESC = 0.100000000000E-01,
SCHA = 0.000000000000E+00, ACCF = 0.000000000000E+00)
TIME (BACK, FIXE, TSTA = 0.000000000000E+00, TEND = 500.0, DT = 0.5,
NSTE = 1000)
OPTI (SIDE)
DATA (CONT)
PRIN (NONE)
POST (RESU)
TMFU (SET = 1, NPOI = 10)
0.0000000000E+00, 0.1000000000E+01, 0.1000000000E+03, 0.7294000000E+00,
0.1500000000E+03, 0.5887000000E+00, 0.2000000000E+03, 0.5057000000E+00,
0.2500000000E+03, 0.4494000000E+00, 0.3000000000E+03, 0.4081000000E+00,
0.3500000000E+03, 0.3762000000E+00, 0.4000000000E+03, 0.3505000000E+00,
0.4500000000E+03, 0.3294000000E+00, 0.5000000000E+03, 0.3115000000E+00
SCAL (VALU = 1.0)
ENTI (NAME = "Tumor", SOLI, PROP = "mat2", SPEC = 1.0, MDIF = "C1_Tumor",
MREA = "C1_Tumor")
ENTI (NAME = "Capillary", FLUI, PROP = "mat1", SPEC = 1.0,
MDIF = "C1_Capillary", MREA = "C1_Capillary")
ENTI (NAME = "Axis", PLOT)
ENTI (NAME = "In Capillary", PLOT)
ENTI (NAME = "Out Capillary", PLOT)
ENTI (NAME = "Interface", PLOT)
ENTI (NAME = "Right Tumor", PLOT)
ENTI (NAME = "Left Tumor", PLOT)
ENTI (NAME = "Top Tumor", PLOT)
DENS (SET = "mat1", CONS = 1.0)
VISC (SET = "mat1", CONS = 257290.0)
DIFF (SET = "C1_Tumor", CONS = 0.880000000000E-01)
DIFF (SET = "C1_Capillary", CONS = 1.0)
REAC (SET = "C1_Tumor", TERM = 1, KINE)
-0.7310000000E-04, 0.0000000000E+00, 0.0000000000E+00, 0.1000000000E+01,
0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00,
141
0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00,
0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00,
0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00
REAC (SET = "C1_Capillary", TERM = 1, KINE)
0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00, 0.1000000000E+01,
0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00,
0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00,
0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00,
0.0000000000E+00, 0.0000000000E+00, 0.0000000000E+00
BCNO (UZC, POLY = 1, ENTI = "In Capillary")
0.2140000000E+03, -0.2140000000E+03, 0.0000000000E+00, 0.2000000000E+01,
0.0000000000E+00
BCNO (URC, CONS = 0.000000000000E+00, ENTI = "In Capillary")
BCNO (SPEC = 1.0, CURV = 1, CONS = 1.0, FACT = 1.0, ENTI = "In Capillary")
BCFL (SPEC = 1.0, CONS = 0.000000000000E+00, ENTI = "Axis")
BCFL (SPEC = 1.0, CONS = 0.000000000000E+00, ENTI = "Right Tumor")
BCFL (SPEC = 1.0, CONS = 0.000000000000E+00, ENTI = "Left Tumor")
BCFL (SPEC = 1.0, CONS = 0.000000000000E+00, ENTI = "Top Tumor")
ICNO (SPEC = 1.0, CONS = 0.000000000000E+00, ENTI = "Tumor")
ICNO (SPEC = 1.0, CONS = 0.000000000000E+00, ENTI = "Capillary")
EXTR (ON, AFTE = 5, EVER = 5, ORDE = 3, NOKE, NOFR)
END
/ *** of FIPREP Commands
CREATE(FIPREP,DELE)
CREATE(FISOLV)
PARAMETER(LIST)
Problem Statement Keywords:
PROB (AXI-, ISOT, LAMI, TRAN, NONL, FIXE, NEWT, INCO, SPEC = 1.0)
Descriptor
Geometry Type
Value
AXISYMMETRIC
Temperature Dependence
ISOTHERMAL
Flow Type
Simulation Type
Convective Term
Surface Type
Fluid Type
LAMINAR
TRANSIENT
NONLINEAR
FIXED
NEWTONIAN
Flow Regime
Species Dependence
INCOMPRESSIBLE
SPECIES=1
Explanation
System is symmetric about
an axis
Constant temperature
system
Laminar flow
Solution is time dependent
Bulk flow present
Surface is fixed
Fluid can be considered
Newtonian
Fluids are incompressible
Species present
142
Solution Statement Keywords:
SOLU (S.S. = 50, VELC = 0.100000000000E-02, RESC = 0.100000000000E-01,
SCHA = 0.000000000000E+00, ACCF = 0.000000000000E+00)
Descriptor
Solution Method
Velocity Convergence
Value
Successive Substitution =
50
.1e-02
Residual Convergence
.1e-1
Solution Change
0
Relaxation Factor
ACCF = 0
Explanation
Maximum number of
iterations
Velocity convergence
tolerance
Residual vector
convergence tolerance
Default percentage change
in solution magnitude
For acceleration of
convergence
Time Integration Keywords:
TIME (BACK, FIXE, TSTA = 0.000000000000E+00, TEND = 500.0, DT = 0.5,
NSTE = 1000)
Descriptor
Time Integration
Value
BACKWARD
Time Step Algorithm
Start Time
End Time
FIXED
0
500
Time Step
Number of Time Steps
0.5
1000
Explanation
FIDAP uses implicit
method (t + ∆t)
Time step is constant
Our problem starts at t = 0s
Our problem ends at t =
500s
Time increment is 0.5s
There are 1000 fixed time
steps
143
Mesh and Time Step Convergence
Initially, a mesh with 550 nodes and a time step of 0.5 non-dimensional time was used.
A denser mesh and smaller time step were tested to determine if our mesh and time step
were not potential sources of error.
14
0.1
0.5
Penetration Depth (micrometers)
12
10
8
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Tim e Step
Figure 14. Displays the penetration depth achieved at our original time step of 0.5 non-dimensional time
and a smaller one of 0.1 non-dimensional time.
Figure 14 shows that penetration depth remains constant when our time step is reduced
from 0.5 to 0.1 non-dimensional time. Therefore, it can be concluded that our solution
has converged at a time step of 0.5. Furthermore, computing power is saved by using our
original time step of 0.5.
Penetration Depth (micrometers)
144
14
550
1230
12
10
8
6
4
2
0
0
200
400
600
800
1000
1200
1400
Number of Nodes in Mesh
Figure 15. Displays the penetration depth achieved with our original mesh containing 550 nodes and a
denser mesh containing 1230 nodes.
Figure 15 shows that penetration depth remains constant when the number of nodes in
our original mesh is increased significantly. Therefore, it can be concluded that our
solution has converged with a 550 node mesh. This original mesh took a much shorter
time to run that the mesh of 1230 nodes. Thus, computing power can be saved without
affecting the solution by running our simulation on our original mesh.
145
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