calculating time intervals in sequential drug treatment of cancerous tumors

8/13/2019 Calculating Time Intervals in Sequential Drug Treatment of Cancerous Tumors

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Calculating Time Intervals in Sequential Drug Treatment

of Cancerous Tumors

Mochamad Adrian Prananda & Nathan Hyunsoo Lee

November 27, 2013

Abstract

Treatment of cancerous tumors has evolved from crude surgery to modern techniques such as resection,

radiology, and chemotherapy. In this project, we present a model that uses two drugs to destroy a tumor.

We determine the growth of the normal cell subpopulation, as well as the subpopulations of mutated cells

that are resistant to either of the drugs. We also establish how these subpopulations are reduced in

response to the two drugs. Using techniques such as separation of variables, graph analysis and

programming, we form a developed model that calculates the ideal time intervals at which the two drugs

are sequentially administered.


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Page 2 of 20 Prananda & Lee

1 Introduction of the Problem & Our Motivations

A defining hallmark in oncogenesisthe induction of canceris uncontrolled cell division. These

proliferated cells form a mass called a tumor, and once the cells within the tumor are able to spread

throughout the body via the bloodstream or lymphatic vessels, the tumor becomes malignant and

cancerous[1]. Although surgery may appear to be the most viable option for the removal of the tumor, it

becomes almost useless once the cancerous cells spread, or metastasize, because numerous tumors may

arise even decades after the primary tumor has been removed[2]. If the tumor can be treated and

eradicated before or even during metastasis, a higher chance of survival is possible for the patient. By this

point, invasive procedures then become impractical. Therefore, scientists have turned to more non-

invasive methods in order to treat malignant tumors.

One of such methods consists of the administration of chemotherapeutic drugs. This may appear

to be an effective method to treat these cancerous tumors, but there are many complications in both the

administration of the drugs and the drugs themselves. Listed are few of many issues that arise during

chemotherapeutic drug treatment:

Chemotherapeutic drugs used for tumor treatment are toxic to normal tissues [3]; Resistance from genetic mutations both independent of and dependent from the drug occurs

naturally[4];

Interaction effects and that prevent drugs from being administered simultaneously are common[3], as well as their additive toxicity.

The complications are too many to number, but progress is being made through a tremendous amount of

research in technology.

As engineers that work at the University of Washington Medical Center and Fred Hutchinson

Cancer Research Center, we study the field of cancer from a radiological and a pancreatic perspective.

We recognize that oncologythe study of canceris crucial to discovering cures and the best treatments

for the different types of cancer, as well as learning about how cancer functions and grows. Therefore, we


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have purposed this project to further investigate how to fight cancer through a very practical and

mathematical means. Even though previous scientists have created more sophisticated models addressing

drug therapy of tumors, we have realized that those models are near impossible to understand without a

strong knowledge of oncology. With this in mind, we began our journey in creating a mathematical model

that would be understood without much cancer-specific expertise and jargon and representative of basic

oncology. These are the questions we plan to consider:

How does a tumor grow and proliferate over time? How do mutant cells factor into the model? How can the time intervals between administering a drug be determined, given the toxicity of a

specific drug?

2 Simplifications & Variables

2.1 Simplifications

Drug delivery is not accounted for.Because drug delivery requires more extensive knowledge about vasculature of the tumor,

pressures of different tissues, and the location of where a drug should be administered, we have

chosen to omit the subject entirely. We assume that the drug has been properly administered and

the full concentration of the dose is inserted into the tumor; this also allows us to make the

assumption that the drug effects are active and felt immediately the moment at which the drug is

administered. We believe that the time in which the chemotherapeutic drug will reach the tumor

is insignificant relative to the time intervals between the administrations of the drug.

Instead of randomly occurring, mutations that are resistant to the drug appear in astraightforward fashion that can be observed mathematically.

Mutations occur randomly, and there is neither an accurate equation nor a constant method in

which mutations develop. Also, mutation rates tend to increase as a drug takes its course and


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diffuses into the body system. By simplifying the way mutations occur, we may not be able to

fully capture the effect of resistant mutations in our mathematical model; however, it must be

understood that modeling resistance is quite controversial to this day[4]. Therefore, we believe

this is a valid simplification to make in our model.

The tumor is homogeneous in cell type.Specifically, in our model, homogeneity is simply defined by having the same natural birth rate

and death rate. By targeting a tumor that contains only one cell type, our mathematical model can

produce more accurate results. However, this is a grand simplification because almost all tumors

have a mass of cancer cells that are proliferating, inactive, or obsolete[5]. To address all of these

types of cells, we would have to create three different models that would connect in order to

properly address a fully developed tumor. Thus, an appropriately complex model for this class

would not require us to address multiple types of cells.

The population growth model applies to tumor cell proliferation.Cell proliferation can be modeled using two constants: the proliferation rate and the carrying

capacity at the location of the tumor[5]. Using these constants, we can use the population growth

model to describe tumor cell proliferation. We believe this assumption is justified because a

population of cells is no different than a population of any other species; the cell-specific

constants alter the population growth model to better fit the proposed problem.

Because the drugs are toxic, the follow equation [3] is always fulfilled:!! ! ! !! ! ! !!"#$%!!! ! !!!

!!

!!

Our model specifically suggests the timing of drug administration by looking at the growth of

resistant mutant cells and optimizing the effect of the drugs. We believe that the optimal

concentration can be found through clinical trials and analysis of data, both of which are

unavailable to us. Thus, we have chosen to leave out the concentration aspect from our model and

accept this simplification.


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Mutant cells can only be resistant to a single drug.Although the above statement is not necessarily factual, the capability of a cell being resistant to

both administered drugs would require another complicated factor in our model. With this

simplification, we can better control the growth and death of the mutant cell population. Ideally,

we want to destroy mutant cells with the administration of any two drugs; thus, our model does

not completely satisfy real-life situations and, instead, presents the results for an ideal situation.

Drug-specific constants can be determined by experimental data pertaining to that drugand used as inputs in the final mathematical model.

It is quite difficult to find pertinent data of specific drugs, and without data it is impossible

determine viable constants to be used in our mathematical model. Thus, we assume that

professional scientists and oncologists can acquire the correct constants through methods that are

beyond our knowledge and feasibility. This also makes our model more versatile and relevant to

many types of drugs, drug combinations, and tumors, which is one of our goals in this project.

The natural birth rate is always greater than the natural death rate (L>D).The proliferation of tumor cells requires a growing behavior; therefore, there will be a natural

positive difference between the production and the termination of cells.

The death rate due to drug toxicity is always greater than the growth rate of cells.This is an obvious assumption because the drugs must have a destructive effect over the tumor

cells. Therefore, the growth rate of normal tumor cells and mutant tumor cells is negative in the

presence of a drug, portrayed through the cell populations decreasing values.

2.2 Variables

Natural development of normal tumor cells

t Time (days)

N(t) Normal tumor cell population as a function of time (number of cells)


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L Birth rate of tumor cells7(percent of cell population in decimal form)

D Death rate of tumor cells not from drugs (percent of cell population in decimal form)

K Carrying capacity of cancer models (number of cells)

Drug administration

A Death rate ofN(t) induced by drug A (percent of cell population in decimal form)

B Death rate ofN(t) induced by drug B (percent of cell population in decimal form)

RA(t) Mutant tumor cell population resistant to drug A as a function of time (number of cells)

RB(t) Mutant tumor cell population resistant to drug B as a function of time (number of cells)

RA-B(t) RAreduced by drug B as a function of time (number of cells)

RB-A(t) RBreduced by drug A as a function of time (number of cells)

DAM Death rate ofRBinduced by drug A (percent of cell population in decimal form)

DBM Death rate ofRAinduced by drug B (percent of cell population in decimal form)

Modeling of time intervals

T(o) Time intervals of drug administration as a function of the order of administration (days)

o Order of administration (1st, 2

nd, 3

rd,)

r Parameter to determine the slope of time intervals over order of intervals (days/o2)

3 Mathematical Models

3.1 Simplest Model of Tumor Growth (Prior to Drug Administration)

The purpose of this section is to show how normal tumor cells multiply and when drugs should be

administered to begin reducing the cell population. In the real world, tumors behave in a very

unpredictable way. Since the biology of human body is very complex, it is difficult to play directly with

numerous parameters at once. However, we first want to understand how a normal tumor grows. In early

stages, a tumor grows rapidly and then slows until a maximum cell population is reached6. The closest

equation that we can use is the logistic population growth model. We assume that the normal tumor cell


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population N(t) grows exponentially, accounting for the parameters L, D and K. From these three

parameters, we can model cancer as a logistic growth equation:

!" !

!"! ! ! ! ! ! ! !

!

!! ! (1)

3.2 More Sophisticated Tumor Models (Subsequent to Drug Administration)

Upon understanding the mechanism of normal tumor cell growth, we recognize another aspect of tumor

growth: the mutant tumor cell population R(t). These mutant cells correlate closely with the drugs we

administer. However, the mutant cells behave more rigorously than the tumor cells, and, in real-life

situations, different kinds of mutant cells exist even in a single tumor. In our model, mutant cells can be

resistant to one of the administered drugs. For the sake of simplicity, let us consider two types of drugs:

drug A and B. Both drugs kill normal tumor cells N(t) whenN(t) reaches a steady state. In addition, let us

consider mutant cellsRA(t) that are resistant to drug A and mutant cells RB(t) that are resistant to drug B.

The changes in these populations can be modeled with exponential growth and decay. Because mutant

cancer cells can metastasize[6], mutant cancer can increase exponentially; this behavior is what makes

cancer cells so deadly and uncontrollable. From these facts, we derive the following equations:

Treatments with Drug A

The development of mutant cells resistant to drug A!!!

!"! ! ! ! !! (2)

The treatment of normal tumor cells with drug A!"

!"! !

!!!

! ! (3)

The treatment of mutant cells resistant to drug B with drug A!!!!!

!"! ! ! ! ! !!" !!!! (4)


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Treatments with Drug B

The development of mutant cells resistant to drug B

!!!

!"! ! ! ! !! (5)

The treatment of normal tumor cells with drug B

!"

!"! ! ! ! ! ! ! (6)

The treatment of mutant cells resistant to drug A with drug B

!!!!!

!"! ! ! ! ! !!" !!!! (7)

DAMandDBMare greater thanAandB, respectively, because mutant cells are more sensitive to the effects

of the drug they are not resistant to. Cells that harbor mutations that do not affect resistance to a drug are

more susceptible to that specific drug-induced damage [6].

3.2.1 Improvisation

Since exponential models are simple by this point, we wish to determine when we should administer

drugs A and B. Upon further research, we find that sequential administration of drugs A and B could

effectively reduce the total population. The reason behind this idea is growth of mutant cells resistant to

either drug A or B. We must destroy the optimal amount of normal tumor cells while accounting for the

growth of mutant cells resistant to the drug being used, then we must begin to use the other drug to

destroy the mass of mutant cells that emerged while destroying normal tumor cells as well. This produces

a sequential schedule of administration.

We do not want the mutant cells to grow to 80% of the normal tumor cell population; this

situation may allow the mutant cells to metastasize. Therefore, we can reasonably say that a drug should

be administered when the mutant cells grow to 50% of the normal tumor cell population at a given time.

When drug A is administered, the mutant cells resistant to drug B can be killed while killing numerous

normal tumor cells. Then, when the population of mutant cells resistant to drug A increases, we can


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immediately kill that subpopulation with drug B. This method is repeated until the mutant cells

exponential growths become very dismal.

3.3 Time Interval Modeling

We wish to model how time intervals T(o) (in days) will behave over the order of drug administration (1st

interval, 2nd

interval,). This model will determine how often we have to administer drugs to the system

so thatR(t) andN(t) cells can be killed as effectively as possible. Our sequential schedule states that the

1stadministration uses drug A, the 2

ndadministration uses drug B, the 3

rdadministration uses drug A, and

so on. Thus, T(1) is the time interval between the first administrations of drug A and drug B.

We hypothesized that the time intervals could be modeled exponentially, where the parameterris

a decay rate because we believe that administration of the drugs would occur more often as the order of

drug administration increased. Thus, the exponential equation can be modeled by:

!!!!!!

!"! !! (8)

4 Solutions

4.1 Analytical Solutions

Using Mathematica, we used the function DSolveto solve eq. (1):

!" !

!"! ! ! ! ! ! ! !

!

!! !

! ! !!!

!"!!"

!!"!!"

!!!"

! ! !!!!!

!"

!!!!"!!

!", where !! ! !!"

(1)

To model natural growth, we must assume that L D must be greater than zero; thus, when L > D, then

N(t)!"as t!", which models exponential growth.


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In this class, we have studied differential equations extensively. We have learned about the

appropriate use of constants and how to best formulate differential equations from a given situation. All

of equations above are first-order ordinary differential equations (ODEs), and we have applied the

techniques we learned in this class to solve them. Furthermore, we also learned to assess the behavior of a

solution in the long run; this concept is represented by evaluating the solutions as time goes to infinity

with different values of the constants.

4.2 Numerical Solutions

The numerical solutions for the three different models are derived in Matlab based on the analytical

solutions. The parameters are within reasonable values [7].

4.2.1 Proliferation of Normal Tumor Cells

For our first trial, we setL= 0.5,D= 0.3 andK= 500000, with the initial conditionN(0) = 1. Therefore,

L is the increase in population at the rate of 50% of current population over time, D is the decrease in

population at the rate of 30% of current population over time, and K is the limiting tumor population (that

is, 500,000 cells). The initial

population is one cell because

cells can divide extremely fast

even from a single cell[5]. Using

ode45 and the function

logistic.m containing eq. (1), we

obtained the following trajectory:

Figure 1.Normal tumor cell growth model with L = 0.5, D = 0.1, K =

500000


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Lets try other parameter values.

If we setL= 0.9,D= 0.1, andK=

500,000, with the initial condition

N(0) = 1, we will acquire the

following trajectory behavior:

4.2.2 Response to Drug Administration

When the normal cells reach the carrying capacity, we must administer drugs immediately[5].

However, mutant cells respond differently to the drugs. To model the sequential treatment schedule, we

set L= 0.5,D= 0.1,A= 0.42,B= 0.42,DAM= 0.421,DBM= 0.422, with initial conditions of RB(0) = 1,

RA(0) = 1,N(0) =K= 900,000. The value of every rate is less than one [6]. We used four functions in our

code: wA.m, which governs RA(t) (eq. (1)) and N(t) (eq. (2)) when drug A is administered; wA_drug.m,

which governsRB(t) (eq. (3)) when drug A is administered; wB.m,which governsRB(t) (eq. (4)) andN(t)

(eq. (5)) when drug B is administered; and wB_drug.m,which governs RA(t) (eq. (6)) when drug B is

administered. Also, we used the main functionpopul_intern.mto govern the functions.

To obtain solutions that describe how mutant and normal cells behave during drug administration,

we can set a logical loop expressed by the pseudo-code below to attain the graph using Matlab:

Set parameters L, D, K, A, B, D_AM, D_BM, and initial

conditions

Set time interval recorder

Figure 2.Normal tumor cell growth model with L = 0.9, D = 0.1, K =

500000


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% one cycle = administration of one drug

for every 2 cycles

% when drug A is administered

solve R_A(t), N(t) due to drug Acheck when R_A(t) = 0.5*N(t)

plot both R_A(t) and N(t) with shifted time

affected by drug B

% for first cycle, R_B still not mutant due to

% drug A

% for first cycle, time is not shifted yet

solve using ode45 for R_B(t) decrease due to A

shift time to when R_A(t) = 0.5*N(t)

record time interval (from shifted time)

reset initial condition for N and R_A(t)

depending on last known condition

% when drug B is administered

solve using ode45 R_B(t) and N(t) due to drug B

check when R_B = 0.5*N(t)

plot both R_B(t) and N(t) with shifted time

affected by drug A

solve for R_A(t) decrease to B

shift time to when R_B(t) = 0.5*N(t)

record time interval (from shifted time)

reset initial condition for R_A, R_B, and N

depending on last known condition

end loop

Unfortunately, we were not able to make the loop work because of its high complexity. However, we

were able to code three cycles forRA(t) and two forRB(t), a total of five cycles forN(t), which means that

five drug administrations were issued in the given amount of time. When we ran the file, we found

sometimes possible values of RA(t) andRB(t) that is close to 50% of N(t). And, we take the median of

these values to look for the closest value to 50% of N(t). However, sometimes the median value is not in

RA(t) and RB(t) array values. To solve this problem, we used another file closest.m to help finding the

closest value ofRA(t) andRB(t) to the median.


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From the provided parameter values, equations and functions, we present a numerical graph for our

model:

In this specific model, we only presented one set of parameters because we want to use this specific set of

parameters to analyze the time intervals over the order of drug administrations. Since this model is very

related to the following model, we want to keep everything consistent.

4.2.3 Time Interval Modeling

Using the previous model, we save the time intervals between each drug issuance as shown in the pseudo-

code while we run the loop logic. Then, we pair each interval to the order of administration (1st, 2

nd, 3

rd

interval). From the same code and parameter values of the previous model, we acquire the following data:

Order of Interval Time Interval

1st 29.3547 days

2nd 27.8955 days

3rd 5.3968 days

4th 2.1587 days

5th 1.0545 days

Table 1. The values of time intervals between drug administrations

Figure 3.Sequential treatment schedule for about 55 days, with arrows

showing three of the five drug administrations

!"#$ &!"#$ ' !"#$ &


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We can now use eq. (8) and the data above to find both rand C8; take, for example, the values of the 1st

and 5thintervals: 29.3547 days and 1.0545 days, respectively.

To find C8:

!"!

!"#$ ! !!!! !

!!

!! ! !"!!"#$!"#$To find r:

!!!"#" ! !"!!"#$!!!!!!!

!"!!!"#"

!"!!"#$ ! !!

! ! !!!!"#!"#$!!

!"#$"!"

!"#$%$&'(!'$)%!

!

Thus, the complete equation is as follows:

! ! ! !"!!"#$!!!!!"#!!!!!

The fitted exponential curve mainly follows most of the data from the table except for the 2nd

interval, and

this discrepancy is discussed in the following section.

Figure 4. Comparison of the actual data versus the exponential fit of

the time interval data


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5 Results

5.1 Natural Development of Normal Tumor Cells

In Fig. 1 and Fig. 2, the two solutions differ slightly. Here, we can see that if the birth rate of cells is

much greater than the death rate, the cell population can increase quickly at dramatic pace, and the

population stays at a steady state of 500,000 cellsa smaller tumorwhen time goes to infinity. When

we set L = 0.5 and D = 0.3, it takes about 80 days for the cells to reach the steady state; the difference

between L and D is much smaller than the second solution in Fig. 2. When we set L = 0.9 and D = 0.1, it

takes only 10 days for the cells to reach the steady state. This model answers our question regarding how

normal tumor cells proliferate. We also know that no matter what the initial condition is, the normal cell

population will always reach K in the ideal situation. The faster the growth is, the more dangerous the

cells are, because this provides room for mutant cells to develop [8]. In a real-life situation, we must

administer a drug as soon as we can before the tumor reaches carrying capacity [4]. Our values are

reasonable in comparison to the values used in previous research papers (104to 10

6cells in a tumor) [7].

5.2 Drug Administration Sessions

When the normal tumor cells population has already reached its carrying capacity, and mutant cells

proliferate naturally and in response to drugs, we can administer the drugs in a sequential manner that can

eradicate the cancerous cells. When we administer the drugs sequentially, we notice a decrease in

amplitude for both RA(t) (from 250,000 to about 100,000) and RB(t) (from 150,000 to 100,000), and a

decreasing value of N(t) (from 900,000 to 300,000); these are all desired results. We can say that our

model calculates the death of both mutant cells and normal cells to a certain point, answering one of our

questions from the introduction, because the values ofRA(t) andRB(t) seemed to reach a steady state value

of 100,000 cells as seen from the graph. However, we do know what the model would look like in the

long run because we were only able to program up to five cycles.


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5.3 Time Interval Modeling

Here, we used two points, the 1stand 5

thintervals, to find the fitted exponential curve for the model. After

obtaining the data, we saw that the time intervals decrease as the order of administration increases. Also,

the fitted curve actually suits the model very well. The fitted curve in Fig. 4 fits almost all of the points,

except on the 2nd

interval where our model is slightly off. We believe this discrepancy occurs because the

time interval between the first administrations of drug A and drug B varies greatly due to different growth

constants varied by 0.1% (looking at DAM and DBM). However, the near-perfect fitting of most of the

points shows that our model is potentially applicable to real life. This model answers our question of how

often should drugs should be administered. Compared to previous papers, the time intervals of

administration in our model are reasonable (about every three days to one month) [3,4].

6 Limitations, Improvements & Future Work

Our results show that the time intervals between administration of the drugs decreases at every issuance

of each drug. This does not seem to be the case in reality. Instead, one drug may be given every two

weeks, while another drug is given each day; the time intervals between administrations may remain

uniform even after several months. Therefore, a possible venture in the future may consist of comparing

the two methods and analyzing why one method may be more effective than the other.

More future work may be accomplished through a more sophisticated and viable software than

Matlab. Matlab prevented us from seeing the long-run behavior of our model because we could not code a

successful loop that would sustain our equations. Through lengthy and repetitive coding, we were able to

generate five cycles of drug administration. However, we do not deem this amount of data to be a success,

and we want to find another program that would provide the necessary means to create a viable loop.

With simplification come limitations, and our model is no exception. One of the greatest

limitations of our model consists of its reliability on the homogeneous condition of a tumor. As stated

before, a typical tumor is a spherical mass of cells that are either multiplying, idle or dead5. Our model


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only addresses treatment of the proliferating population of tumor cells, not the cells that are already idle

or dead. We do not necessarily know how the idle and dead cells can be removed or if they have to be

removed, but they should be considered as part of the total population of tumor cells.

Furthermore, our model does not take into account other sources of natural resistance. For

example, compact grouping of exterior tumor cells may reduce penetration of the drug into the interior of

the tumor tissue [9]. Thus, less tumor cells would be killed by the same concentration of a drug. Another

characteristic of tumors that may decrease the efficacy of a chemotherapeutic drug is the vasculature

collapse within the tumor microenvironment [2]. Many, if not all, drugs must travel through blood

vessels, and disorganization of the vascular architecture within a tumor prevents drugs from being potent

throughout the tumor [10]. These are only two of the sources of natural resistance that occur in typical

tumors that we did not address, which accentuates the limitations of our proposed model.

Because of the complexity of cancer and its treatments, many improvements can be made to our

model to yield more accurate results. For example, to model a more authentic manifestation of mutated

cells, we would have to consider the arbitrary characteristic of mutations. Upon further research, we

discovered the concept of stochastic differential equations (SDEs). Replacing the ODEs of the mutant

cells with SDEs may produce more realistic results because mutations occur randomly. By using SDEs,

we could generate time intervals that would better represent the optimal schedule for drug administration.

Another improvement we could make in our model is to provide the optimal concentration of

each drug during a certain time interval. Because timing is only half of the required information for

proper drug administration, our model could be enhanced to calculate the appropriate dosage. We took the

equation !! ! ! !! ! ! !!"#$%!!! ! !!!!!

!!

for granted in our model; however, we can add a

component that addresses how to determine CFixedand how the toxicities of each drug work together. This

addition would validate our model if it produced dosages that were realistic and similar to real-life data.

Deeper knowledge of oncology and human biology, as well as access to drug-specific clinical

trial data, would allow us to complete more future work and improve our model. Having said that,


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acquiring such assets will take long periods of time, further education and much effort on our part. We

acknowledge that our model is relatively basic in comparison to previously made models, but the

potential to further develop our model is apparent and extremely possible.

7 Conclusion

Developing a model to calculate the time intervals of sequential drug administrations proved to be a

difficult problem. We created eight ODEs that represented the growth of normal and mutated

subpopulations of tumor cells, responses to the administrations of a certain drug, and the relationship

between time intervals versus the number of administrations. Using techniques like separation of

variables and behavioral analysis, as well as software programs like Matlab and Mathematica, we were

able to solve these ODEs and combine them to form a graph that would describe when to administer the

drugs over time. As we had originally hypothesized, our results showed that the time intervals decreased

as the number of administrations increased; our model was also able to calculate the exact days at which

the drugs should be administered. However, our model does not take other natural resistances into

account. The lack of functional blood vessels and a tumors compact nature are two of the many

characteristics of tumors that our model does not acknowledge. With improvements and future work, our

model could provide oncologists and doctors with an effective and important tool.


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calculating time intervals in sequential drug treatment of cancerous tumors

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