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Lynnique Johnson, Department of Mathematics, Winthrop University, USA
2020 Alumni Drive, Box 6617
Rock Hill, SC 29733
Telephone: (803) 818-1624
[email protected]
Dr. Zachary Abernathy, Department of Mathematics, Winthrop University, USA
Rock Hill, SC 29733
Telephone: (803) 323-4605
[email protected]
USING LINEAR CANCER NETWORKS TO MODEL PHOTODYNAMIC THERAPY
Abstract
Photodynamic therapy (PDT) is a treatment used in the early stages of certain cancers, which
involves a localized injection of a photosensitizer followed by exposure to a light source that
releases a cytotoxic agent, singlet oxygen, into surrounding cells. We create a set of eight
ordinary differential equations that model photodynamic therapy (PDT) within a linear cancer
network. A cancer network is a developmental control network in which cancer stem cells either
divide into stem cells or differentiate into tumor cells. This model illustrates three possible
outcomes: a stable state of coexistence between cancerous and healthy cells, an extended
remission with eventual recurrence of the cancer, or permanent remission. Stability and
sensitivity analyses are performed to support numerical simulations.
Keywords. photodynamic therapy (PDT), cancer stem cell, cancer networks, recurrence,
remission
Introduction
Photodynamic therapy (PDT) is a cancer treatment using a drug called a photosensitiser
(PS), usually porfirmer sodium or temoporfin, and a light source, usually a laser, to kill small
malignant tumors (Gubarev et al., 2007). The PS is intravenously injected into the patient’s
body and absorbed by both the cancerous cells and healthy cells. In order to ensure
minimal healthy cells are killed in the process, the patient must wait 24-72 hours after the PS is
injected so that there is enough time for healthy cells to dispose of the drug. After the allotted
time, a laser is inserted into the affected area and it activates the light-sensitive PS. Once
activated, the drug releases singlet oxygen, a cytotoxic agent, into the tumor which eventually
kills the cancerous cells (Brown et al., 2004).
Although PDT is effective in treating local tumors, it is not useful for advanced or
metastasised cancers because it is difficult to deliver light throughout a large span of the body
and the lasers are not designed to be inserted deep into the body. However, it can be effective
enough to lengthen a patient’s survival by removing a considerable amount of a tumor in one
treatment session.
Eric Werner has recently proposed a theory of cancer networks, which distinguishes itself
from the traditional gene-centered theory of cancer development. The networks are similar to a
set of instructions which cells follow to divide and differentiate; a mutation in such a network can
cause cells to grow uncontrollably. Various types of cancer networks, such as linear, geometric,
and exponential networks, describe the outcome of a cancer stem cell’s division into two
daughter cells (Werner, 2011). Deducing which type of cancer network a certain cancer has
may help determine the most effective treatment to use. Our present goal is to create a
mathematical model which illustrates the possible effects of using PDT within a linear cancer
network.
Description of Model
Previous Models. Gubarev, Makhanek, and Shul’man modeled the interactions among the
ground and excited states of a PS and oxygen molecules, including heat and mass transfer of
the oxygen molecules during therapy (2007: 83-84). From their model we will incorporate the
following into our model:
•
monitoring the concentration of singlet oxygen within the cancer system because different
levels of concentrations has different effects on cells,
•
the intensity of the light emitted from the laser to influence the effectiveness of the drug,
hence releasing more singlet oxygen into the body, because the intensity can be controlled
during an actual PDT treatment and it will be beneficial to monitor and
•
the concentration of the PS within the cells because this will determine how effectively the
treatment is within the cells.
For the purposes of our model, however, we do not take into account the difference between the
ground and excited states of the molecules nor the heat/mass transfer of oxygen. Instead, our
goal is to describe the impact of treatment with PDT within a linear cancer network.
Defining Linear Cancer Networks. A linear cancer network contains cancer stem cells, denoted
by 𝐴, and differentiated tumor cells, denoted by 𝐵, that create a tumor of only 𝐴 and 𝐵 cells.
Each 𝐴 cell divides into two cells, another 𝐴 cell and a 𝐵 cell, to sustain the growth of the tumor.
𝐵 cells are terminal - they do not divide into any other cells on their own (Werner, 2011). Also
contained within a cancer network are healthy cells, denoted by 𝐻, which do not interact with 𝐴
or 𝐵 cells. We will include healthy cells within our model because it will be beneficial to monitor
the effects of PDT on these cells since they could potentially be harmed from the treatment
(Brown et al., 2007). We will take the concept of the linear cancer network into consideration
when creating the mathematical representation of PDT treatment. In order to incorporate Eric
Werner's theory, we need to reflect a realistic environment of cancer growth. We derive the
following equations to model the respective change in the number of healthy cells (𝐻), stem
cells (𝐴) and terminal tumor cells (𝐵) over time:
𝑑𝐻
𝐻
= 𝑟1 𝐻 (1 − )
𝑑𝑡
𝐾1
𝑑𝐴
𝐴
= 𝑟2 𝐴 (1 − )
𝑑𝑡
𝐾2
𝑑𝐵
𝐴
𝐵
= 𝑟2 𝐴 ( ) (1 − ) − 𝑑𝐵
𝑑𝑡
𝐾2
𝐾3
Where 𝑟𝑖 is the rate at which the cells are dividing, 𝐾𝑖 is the respective carrying capacity and
d is the rate at which 𝐵 cells are dying. We include this death rate because unlike 𝐴 and 𝐻
cells' logistic growth terms that include cell formation and death, 𝐵′𝑠 growth is dependent on 𝐴,
𝐴
but 𝐵 cell death is not accounted for. The (𝐾 ) term controls the rate at which 𝐴 cells are dividing
2
into 𝐵 cells; 𝐴 cells will want to sustain their population as effectively as possible and depending
on their population, stem cells may not produce tumor cells as quickly. For instance, if stem
cells are at their carrying capacity, they will produce tumor cells normally; if they are above their
carrying capacity, they will differentiate into tumor cells at a faster rate; and if stem cells are
below their carrying capacity, they will divide into tumor cells at a much slower rate (Li and Xie,
2005).
We assume the cells grow logistically instead of linearly or exponentially to keep the cells
contained within the region they are growing. Then, to complete our model, we add treatment to
this system.
Our Model. In order to incorporate the PDT treatment over time, we need a way to monitor
the drug, the singlet oxygen that is being released into the cells, the light source that will
have a certain intensity, and the concentration of the drug in the cells. We do this by adding
separate equations for each of the aspects of the treatment along with interaction terms within
the cells’ equations.
For our model, we made the following assumptions:
(1)
the cancerous cells would not metastasise
(2)
for simplicity, healthy and cancerous cells would not compete for resources,
(3)
the B cells do not divide,
(4)
singlet oxygen will enter the system after introduction of light
Below we refine the previous set of equations to include the change in the amount of drug (𝑁),
the amount of singlet oxygen (𝑆), the concentration of drug within healthy cells, stem cells, and
terminal tumor cells (𝐿𝐻 , 𝐿𝐴 , 𝐿𝐵 respectively); we also introduce a separate function, 𝐼(𝑡), to
represent the intensity of the light administered during treatment.
(1)
𝑑𝐻
𝑑𝑡
= 𝑟1 𝐻 (1 − 𝐾 ) − 𝑏1 𝐿𝐻 𝑆
𝐻
(2)
𝑑𝐴
𝑑𝑡
= 𝑟2 𝐴 (1 −
(3)
𝑑𝐵
𝑑𝑡
= 𝑟2 𝐴 (𝐾 ) (1 − 𝐾 ) − 𝑑𝐵 − 𝑏3 𝐿𝐵 𝑆
(4)
𝑑𝑁
𝑑𝑡
= −𝑎1 𝑁
(5)
𝑑𝑆
𝑑𝑡
= −𝑎2 𝑆 + 𝑗𝑁𝐼(𝑡)
(6)
𝑑𝐿𝐻
𝑑𝑡
= −𝑎3 𝐿𝐻 + 𝑐1 𝑁
(7)
𝑑𝐿𝐴
𝑑𝑡
= −𝑎4 𝐿𝐴 + 𝑐2 𝑁
(8)
𝑑𝐿𝐵
𝑑𝑡
= −𝑎5 𝐿𝐵 + 𝑐3 𝑁
1
𝐴
)
𝐾2
− 𝑏2 𝐿𝐴 𝑆
𝐴
𝐵
2
3
I (t) represents the following piecewise function that controls when the light source is
introduced (𝑇1 ) during treatment and then turned off again (𝑇2 ):
0,
𝐼(𝑡) = {𝐼,
0,
0 ≤ 𝑡 < 𝑇1
𝑇1 < 𝑡 < 𝑇2
𝑡 ≥ 𝑇2
By incorporating treatment into our model we have to slightly change the first set of
healthy cell, stem cell, and tumor cell equations. The −𝑏𝑖 𝐿𝑆 term represents singlet oxygen
killing cells based on the concentration of the drug within each respective cell type. The
chemicals and concentrations will naturally decay within the system (Gkigkitzis, 2012) and 𝑎𝑖
represents the rate at which this occurs. The equation for singlet oxygen has another
term, 𝑗𝑁𝐼(𝑡), which controls how it enters the system; once the light portion of therapy is
introduced to the system, more singlet oxygen is released.
The following table provides a brief summary of each parameter we use in our model.
Parameter
𝑟𝑖 , 𝐾𝑖 , 𝑑
𝑗
𝑏𝑖
Description
same as above
rate at which singlet oxygen is released when drug is exposed
to light
rate at which cells are dying due to the presence of singlet
𝑎𝑖
oxygen
rate at which drug/concentrations are decaying
𝑐𝑖
rate at which the drug is being absorbed
𝐼
𝑇1
𝑇2
intensity of light source
time light therapy begins
time light therapy ends
Results
The following parameters either come directly from the literature (de Pillis et al., 2005) or are
estimated based on the desired behavior of the model. With the parameters, we use the
‘NDSolve’ command in Mathematica to help graph our model to determine its behavior visually
(code used during calculations are available from author, upon request).
Parameter Baseline Value
0.00035
𝑟1
0.000457
𝑟2
Source
(de Pillis et al., 2005)
(de Pillis et al., 2005)
𝑏1
𝑏2
𝑏3
𝑎1
0.00025
0.00025
0.00025
0.00045
(de Pillis et al., 2005)
(de Pillis et al., 2005)
(de Pillis et al., 2005)
Estimated
𝑎2
𝑎3
𝑎4
𝑎5
𝐾1
0.00045
0.009
0.0045
0.0045
1000
Estimated
Estimated
Estimated
Estimated
Estimated
𝐾2
𝐾3
𝑐1
500
2000
0.0005
Estimated
Estimated
Estimated
𝑐2
0.002
Estimated
𝑐3
𝑇1
𝑇2
𝑗
𝑑
𝐼
0.01
5
10
0.05
0.0000139
50-250
Estimated
Estimated
Estimated
Estimated
(de Pillis et al., 2005)
Estimated
By introducing the following realistic parameters to our system and then manipulating the
value of 𝐼, the graphical representation of our system reveals the following behaviors (one may
note manipulating values for 𝑇1 and 𝑇2 causes similar behavior, but in this article, we will just be
discussing the effects of changing 𝐼):
•
stem cells and tumor cells experience some decrease in population but do not fully die
out (Figure 1)
•
tumor cells die but the stem cells do not, causing the cancer to go into temporary
remission (Figure 2)

both stem and tumor cells die (Figure 3)
Figure 1: 𝐼 = 50
Figure 2: 𝐼 = 200
Figure 3: 𝐼 = 250
From the three cases that we are able to produce by varying the value of 𝐼, we perform a
stability analysis to classify possible end behaviors of our system.
Analysis of Model
In order to determine the long-term behavior of our model, we next perform a stability
analysis on a dimensionless form of our equations. Note: When setting the equations equal to
zero, equation (4) would force 𝑁 to be zero, and will cause the respective variables in
equations (5)-(8) to be zero as well, making those equations inadmissible for the stability
analysis. The techniques that we use to perform our stability and sensitivity analyses can be
referenced in the textbook A First Course in Mathematical Modeling (Giordano, 2005).
Stability. We use the following variables in our dimensionless system to represent the original
dependent and independent variables from our original model:
𝑥=
𝐻
𝐴
𝐵
, 𝑦=
, 𝑧=
, 𝜏 = 𝑟2 𝑡
𝐾1
𝐾2
𝐾3
The new parameters we introduce for our dimensionless system are as follows:
𝜌=
𝑟1
𝑑
𝐾2
, 𝛿= , 𝛼=
𝑟2
𝑟2
𝐾3
We nondimensionalize our original system (Giordano, 2005) and obtain the following equations:
𝑑𝑥
= 𝜌𝑥(1 − 𝑥)
𝑑𝜏
𝑑𝑦
= 𝑦(1 − 𝑦)
𝑑𝜏
𝑑𝑧
= 𝛼𝑦 2 (1 − 𝑧) − 𝛿𝑧
𝑑𝜏
From our dimensionless system, we calculate the following equilibria, in the form (𝑥, 𝑦, 𝑧), by
setting the above equations equal to zero and solving for each corresponding variable.

(0, 0, 0)

(1, 0, 0)

(0, 1,
𝛼
)
𝛼+𝛿

(1, 1,
𝛼
)
𝛼+𝛿
Also, we calculate the following Jacobian matrix, a matrix of partial derivatives, which we use to
determine the stability of our equilibrium points.
𝜌 − 2𝜌𝑥
𝐽=[ 0
0
0
1 − 2𝑦
2𝛼𝑦 − 2𝛼𝑦𝑧
0
0
]
−𝛼𝑦 2 − 𝛿
By evaluating the above Jacobian matrix at each of our four equilibrium points and
analyzing the eigenvalues for each point, we find that (1, 1,
𝛼
)
𝛼+𝛿
is locally stable whereas the
other three are unstable because of the negative eigenvalues (Giordano, 2005). Note that by
distinguishing between the growth and death rates of the tumor cells, the 𝑧 component of the
stable equilibrium point is lower than the dimensionless carrying capacity of one.
By incorporating values for the parameters of our model earlier, we find that although
cases two and three represent a stable equilibrium state, those cases were not reflected in the
stability analysis. The end behavior in case two is predicted, but the temporary remission is not;
full remission in case three is not predicted. Therefore, we examine these special behaviors of
each case more closely by performing a sensitivity analysis.
Sensitivity. Since our model displays two other cases that have different behaviors in their
graphs that our stability analysis does not predict, we perform a sensitivity analysis; we examine
these special behaviors more closely to determine the effects of modifying each parameter. For
each case, our sensitivity analysis reveals which parameters have the largest effect on the
length of remission or eradication time. In addition, we find that sufficient changes in the most
sensitive parameters may cause the model to transition from a state of temporary remission to
permanent cure or vice versa. These changes are summarized below:
Behavior 1: Coexistence of Healthy and Cancerous Cells. In the case where a slight decrease in
tumor and stem cell populations occurs (Figure 1), we analyze the ending population of tumor
cells within the area. From the previous stability analysis, we conclude the tumor cells are
always going to relative be the same ending number since the parameters that directly influence
the tumor cells’ end behavior will not change. The fraction from the 𝑧 component of the
𝛼
equilibrium point ( 𝛼+𝛿 ) converts to the following fraction in nondimensionless form:
𝐾2 𝐾3 𝑟2
).
𝑑𝐾3 +𝐾2 𝑟2
𝐾2 (
The latter formula clearly expresses which parameters will affect the final tumor
size, therefore it is not necessary to perform a sensitivity analysis for this specific case since
there is a concise formula to describe the final tumor size; we do not need to monitor the effects
of modifying all of the parameters.
Behavior 2: Temporary Remission of Cancer. For the case where the cancer goes into
temporary remission with eventual recurrence (Figure 2), we analyze the difference in the
amount of time taken before the tumor grows back as we adjust parameters. In the original
model, the length of remission was 10114.12 minutes (approx. 7.02 days).
Parameter Variation that Causes
Permanent Remission
𝑏2
Increase by 25%
𝑐2
Increase by 25%
𝑗
Increase by 25%
𝑎1
Decrease by 25%
𝑎2
Decrease by 25%
Figure 4. Sensitivity of Time for Recurrence. We show the difference in the amount of time
elapsed from when the tumor cells die to their recurrence. Some parameters cause both tumor
and stem cells to die out (which is explained later on in the article) hence, there was no time for
the tumor to recur and proves these parameters are very sensitive to the behavior of the model.
On the graph, these parameters are not visually represented at the values where cases change
since there is no temporary remission of cancer.
Behavior 3: Permanent Remission of Cancer. Our sensitivity analysis for this case reflects the
changes in the amount of time, in minutes, taken for cancerous cells to completely die out. With
our baseline values, stem cells completely die out in 2085.81 minutes (approx. 1.45 days) and
tumor cells die in 1030 minutes (approx. 0.72 days).
Figure 5. Sensitivity for Stem (A) and Tumor (B) Cells. As we vary each parameter, we
examined the changes in the amount of time taken for stem cells and cancer cells to die
separately. Similar to the sensitivity of our model to recurrence, there are certain parameters
that cause the time taken for the stem and tumor cells to die to vary significantly; 𝑗 however,
was the only parameter that causes the cancer to go into temporary remission when it
decreases by 25%.
Discussion
From our stability and sensitivity analyses, we determine the specific role of the
parameters within our model. During the stability analysis, we did not explicitly present the
effects of the drug in any of the equations since we account for the behavior of the therapy after
the drug is already injected into the patient. Also, certain manipulations of parameters cause our
model to behave differently during the sensitivity analysis.
For the second case, the entire model was sensitive to any change in the parameters
but there are five specific parameters that causes us to exhibit the behavior of another case.
Increasing 𝑏2 by 25% causes permanent remission of cancer because stem cells are dying
easier from the amount of singlet oxygen present, and as a result, there is not enough
production of tumor cells. Decreasing 𝑎𝑖 by 25% causes permanent remission of cancer
because each variation of the parameter represents a rate at which the drug or oxygen is
decaying in the system; if those variables are decaying at a much slower rate, then there is
more drug/oxygen available to have a negative effect on the cells in the system and causing
them to die. Increasing 𝑐2 by 25% causes complete remission of the cancer because the rate at
which the drug is entering stem cells increases, making the treatment more effective in killing off
the stem cells and as an effect, killing the tumor cells as well. Increasing 𝑗 by 25% causes the
cancer to go into permanent remission because there is more singlet oxygen dispersing into the
system once the light is administered to the affected area.
Similarly, the third case has a parameter that causes the behavior to change to another
case. Decreasing 𝑗 by 25% causes the cancer to go into temporary remission instead of
complete remission because there is not enough singlet oxygen releasing into the site of the
cancer to effectively kill the stem cells.
In previous research conducted by Gubarev, Makhanek, and Shul’man, they used both
ordinary and partial differential equations and found that by changing the radiation intensity and
hyperoxygenation during PDT will increase the success of the treatment. We were able to use
ordinary differential equations to find that varying the intensity of light or the duration of
treatment produces different effects of PDT on cancerous cells explicitly. Even though past
models explored the underlying effects of the chemical reactions within the treatment, they did
not analyze the effects of PDT on cancer cells as a whole.
With our model, we recognise the importance of separating the stem and tumor cells into
separate populations since we encounter instances where the tumor cells grow back after
treatment stops because the stem cells are not effectively eradicated. Treatment is not
successful if the stem cells are not killed along with the tumor cells.
While we were successful in creating a model of PDT for a linear cancer network, we
plan to extend this research by investigating other cancer networks and specific types of
cancers that PDT can be used to treat. Also, since our model predicts that manipulating both the
intensity and duration of treatment can lead to temporary remission or cure, it may be beneficial
to determine an optimal combination of these two parameters which maximize the chance of
curing cancer while minimizing any potentially harmful effects to healthy cells. Furthermore, the
sensitivity of our model suggests that further research should be conducted to determine
whether the parameters we estimated are realistic for an actual PDT treatment.
With the certain variations of parameters we explore in our model, the system can
predict when PDT will cure the cancer, cause the cancer to go into temporary remission, and
have no effect on the cancer. With our model we effectively display three different behaviors of
PDT therapy based on the intensity of light. But, we can make our model more applicable by
developing methods for determining specific parameters for actual cancer patients to
individualize his or her treatment and ensure he or she is receiving treatment that best fits his or
her case.
Acknowledgements
We would like to thank the MAA for sponsoring our research, with funding from the NSA (grant
H98230-13-1-0270) and NSF (grant DMS-1156582), and Dr. Joseph Rusinko for helping
coordinate the NREUP initiative at Winthrop University.
List of Figures
Figure 1: Graph of coexistence between healthy cells, stem cells, and tumor cells.
Figure 2: Graph of temporary remission of cancer with eventual recurrence.
Figure 3: Graph of permanent remission of cancer.
Figure 4: Sensitivity for Recurrence Time
Figure 5: Sensitivity for Permanent Remission Time
List of Tables
Table 1: Parameters
Table 2: Baseline Values
Table 3: Description of Changing from Temporary to Permanent Remission
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