A Mathematical Model to Treat for a Cancer Using Chemotherapy and
Immunotherapy under Mass Action Kinetics for Immunotherapy
A.M.D. CLOTILDA, G.V.R.K. VITHANAGE, D.D. LAKSHIKA
Department of Mathematical Sciences
Wayamba University of Sri Lanka
Kuliyapitiya,
SRI LANKA
Abstract: - Cancer is a burning problem in the modern health field. In this research mainly we focused on how
we can treat for a cancer by using chemotherapy and immune therapy as individual monotherapies and as a
combined therapy. In previously, researchers have constructed mathematical models to analyse the combined
therapy treatment with saturation effects for immune therapy treatment but here we introduce mass action kinetics
for immune therapy and this model reflects as a continues attacking process to tumour cells by immune cells with
the help of chemotherapy drug. We introduce some threshold levels which we can remove cancer completely and
control the cancer in a constant level.
Key-Words: - Mass action, Michaelis-Menton, equilibrium point, tumour, asymptotically stable, chemotherapy
1 Introduction
Cancer causes lots of deaths around the
world. Currently, cancer is the second cause of death
worldwide and is expected to hit 27.1 million people
by 2030 [1],[2].
Food patterns, lack of exercise, using
tobacco, radiation mainly caused cancer. Cancer is a
disease that the cells of the body grow abnormally.
This unusual growth in the body can happen
anywhere in the body and can affect people from any
age group. When we consider about cancer treatment
methods there are lots of strategies that has developed
in this era. Chemotherapy, hormone therapy,
hyperthermia, immunotherapy, radiation therapy,
surgery and targeted therapy are some of prominent
treatment methods in the modern world [3]. Among
these treatments, chemotherapy, and immunotherapy
both are prominent treatment methods in the modern
world. When we consider chemotherapy, there are
many different types of chemotherapy drugs that can
work differently to kill cancer cells. Some may
interfere with the growth and division of these cells.
Others can help cause cancer cells to self-destruct.
Chemotherapy is used for plenty of purposes, it can
disturb the growth of cancer cells, sometimes it can
stop spreading of cancer cells in the body.
Chemotherapy can give to the body as injections,
injecting to the muscle, spinal code, skin and
sometimes it can give as tablets. This therapy has
some side effects. Among them these are the main
side effects bone marrow suppression, neuropathies,
gastrointestinal disorders, integrating patients’
perceptions regarding side effects into decision
making process during cancer treatment is always
important [4]. The immune system is the body’s basic
defence against infection and cancer. It is made up of
a complex network of cells, molecules, organs and
lymph tissues attempting together to defend the body
against microorganisms such as bacteria, viruses and
fungi, as well as against cancer cells. The immune
system plays a crucial role in preventing cancer. It
acts in a cascade manner to counter the pathogenic
response both by the innate and adaptive immune
systems [5]. Immune cells can find cancer cells and
kill them and completely eliminate cancer cells but
sometimes it can control the growth of cancer.
Immune checkpoint inhibitors, adoptive cell transfer,
monoclonal antibodies, T-cell therapy, vaccines are
some treatment methods [6]. Immunotherapy has
some side effects, they are paining, swelling,
soreness, redness and rash. In this research, first we
consider chemotherapy and Immunotherapy
individually and then we consider both of them as a
combined therapy [5],[6].
2 Model
Here we construct the mathematical model to
represent the interactions between cancer cell
population, immune cell population and
chemotherapy agents in blood.
Received: April 13, 2024. Revised: September 7, 2024. Accepted: November 11, 2024. Published: December 19, 2024.
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Fig. 1: The interrelationship among x, y and z. In this
diagram arrow represents the activation and blunt
arrow represents inhibition.
(1)
Consider the parameters of the model.
r = rate of tumour growth.
b = Inverse carrying capacity of tumour cells.
a = parameter of cancer cleans up.
kT = killing rate of tumour cells by chemotherapy.
=rate of decrement in concentration of
chemotherapy.
= supply rate of chemotherapy drug.
= supply rate of immune cells.
= death rate of immune cells.
= proliferation rate of immune cells.
=killing rate of immune cells by
chemotherapy.
=half saturation constant of immune
proliferation.
These are the state variables.
x = Cancer cell population.
y =Amount of chemotherapeutic agent in the
blood stream.
z = Immune cell population.
m can be negative or positive. Here all the other
parameters are positive.
The first equation states the rate of change in
the cancer cell population, basically logistic growth
model has used to represent the growth of cancer cell
population [11]. Here we consider how
immunotherapy and chemotherapy attack to cancer
cells, represent the attack of immune cells to
cancer cells and on the other hand represent the
attack of chemotherapy drug to tumor cells [12], [13],
[14],[15]. As this phenomenon is a biochemical
reaction, we assume that this reaction happens
according to the mass action law [14]. Previous
researchers have used the attack of immune cells with
saturation effects but in our research work we
construct our model as these immune cells attack
cancer cells without any saturation effect and this
reaction happens according to the Mass action law
[14]. If the immunotherapeutic drug can attack cancer
cells without decaying its strength, the given model
can represent this scenario clearly [9].
The second equation states the rate of change
in the amount of chemotherapeutic agent in the
bloodstream with time. Here it supplies drug in a
constant rate, it is represented by .
Chemotherapy drug concentration is decreasing with
time, specialty in [9] introduced model, the drug
concentration is decreasing with a constant rate.
The third equation states the rate of change
of the immune cell population with time, here
immune cells are injected to body from outside. On
the other hand, when time is increasing these immune
cells die, it can be represented by using the natural
death rate of immune cells. Activation of immune
cells works according to Michaelis-Menten
mechanism [12]. Immune cells can be killed by the
chemotherapy drug, that incident is represented in the
model by using the killing rate of immune cells by
chemotherapy [13]. In the model of [8] they have
considered activation and inactivation of immune
cells separately but in this model according to the
sign of , it can be decided the activation of immune
cells or resistance for immune cells [9,12]. In our
research we consider immune cells proliferate
without any resistance, so . In [8] the model
has used with Viral Therapy and Immune Therapy
but here we modified that model replacing Viral
Therapy form Chemotherapy by taking concepts
form the model [9].
Theorem 1. Solution of (1) exist, remain
nonnegative, and are bounded on [0, ∞).
Proof : Let F(x,y,v,z) be the right hand side of (1).
Since F is locally Lipschitz in
, there exist a
unique solution on [0,) for the initial value problem
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(1), where > 0 may depend on the initial condition.
As and solution
of (1) remain nonnegative on [0,) by [7].
We let . Then,
where. Consider
with X(0) = Since the solution
of are defined on [0, and on
[0, can be extended to Therefore
solutions of (1) are bounded, noticing
implies
.
Consider the second equation in (1).
implies
. When m>0, let suppose
then choose such
that
for all then
then
from this we can obtain
When
< (
) then from this
we can obtain
Hence, we can
conclude that solutions of (1) are bounded.
2 Chemotherapy
In this section we are going to consider only
about chemotherapy treatment. We reduced our
model to the following sub-system.
(2)
Using Dulac Criteria with
. We
consider on the region
(3)
+
=
-
< 0 for
Proposition 2.1. Consequently (2) has no periodic
solutions on .
Biologically this gives an important idea.
There are no periodic orbits means cancer can’t occur
again and again under this chemotherapy.
Finding equilibrium points for the system (2).
Then we can obtain cancer free equilibrium
as (0,
) and further we can find a positive
equilibrium point as
, here
. Then we are going to discuss about the existence
and the stability of these two equilibrium points.
Consider the x isocline,
and consider
the isocline
where is strictly
decreasing with
and
is a
constant function. The range of is 0
. If
then there are no intersections between isoclines.
So, it follows that (2) has no positive equilibrium
points on the other hand when the opposite of above
inequality happens, when
there is only one
intersection in these two isoclines, it gives that there
is a unique positive equilibrium point for the system
(2). Then let’s consider local stability of the cancer
free equilibrium point by using the Jacobian.
Then we can conclude that,
is locally
asymptotically stable if (both eigenvalues
are negative) and otherwise it becomes a saddle point
if (one eigenvalue is negative and the other
one is positive). Then let’s consider the positive
equilibrium point.
By using x isocline and we
can reduce the above jacobian to following form.
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We are considering on x ≥ 0, y ≥ 0. In this
Jacobian matrix trace is negative and determinant is
positive. Then we can obtain that positive
equilibrium point is locally asymptotically stable.
By using Poincare Bendixen Theorem [7]
and the above results we can obtain the following
theorem.
(4)
Theorem 2. The following statements hold for (2).
a) If then
is globally
asymptotically stable in
b) If then there is a unique positive
equilibrium point which is globally
asymptotically stable in .
When the tumour is not aggressive, so that
the immune system is not dysfunctional, the tumour
can be eradicated completely if the tumour killing
rate is large. On the other hand, if the tumour killing
rate is not large, then the tumour cells will be
stabilized at a positive level that is smaller than
carrying capacity. Further it has an interesting
biological phenomenon, when the basic reproduction
number of the supply rate of chemotherapy is greater
than the basic reproduction number of the supply rate
of tumour cells cancer can eliminate. So, we need a
strong chemotherapy treatment to cure the cancer
completely. On the other hand, if the basic
reproduction number of tumour supply rate is greater
than the basic reproduction number of the supply rate
of chemotherapy, tumour level has to stabilize in a
fixed level. Finally, we can conclude that this gives a
threshold level that under what condition we can
eliminate the tumour completely with the support of
only chemotherapy. Then let’s move towards the
combined therapy and how it works.
Table 1. Parameters values and sources
Parameter
Value
Unit
Reference
0.2773-
0.3466
day-1
estimated
1.02
cell-1
[2]
cell-1day-
1
[2]
0.01-0.7
cell-1day-
1
[3]
0.01-0.9
day-1
[3][4]
0.003-0.6
mg day-1
[3]
5000
cell day-1
[2]
2
day-1
[2]
-1-1.5
cell-1day-
1
[2]
0.06
cell-1day-
1
[3]
40-
cell
[2]
(a) (b)
Fig. 2: Here Green Curve Represents the Amount of
Chemotherapy Drug and Blue Curve Represents the
Cancer Cell Population = 0.3, = 1.02, =
0.05, = 0.01, Initial Tumor Cell Population= 6.7 ×
106 cells, Initial Chemotherapy Drug Amount=
19.95units, (a) = 0.55 (b) = 0.005.
When we decrease the initial supply
amount of chemotherapy, we can observe that it takes
a long-time treatment period to eliminate the tumour
completely. It reveals that, with a strong initial
amount of chemotherapy, we can remove can within
a short time period.
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(a)
Fig. 3: Here blue, red, cyan, yellow, magenta curves
respectively represent the behaviour of tumour cell
population according to each initial amount of
chemotherapy drug respectively 19.95, 10, 5, 2, 1,
= 0.3, = 1.02, = 0.05, = 0.01, initial
tumor cell population= 6.7 × 106cells, = 0.55.
3 Immune Therapy
Now the model reduces to the following form.
(5)
First, we are going to Dulac Criteria to
analyse about periodic solutions. Using Dulac
Criteria, we can obtain that, there are no periodic
orbits in this system.
Using Dulac Criteria with
.
Proposition 3.1. Consequently (5) has no periodic
solutions.
This implies that under immune therapy
cancer can’t occur again and again because there are
no periodic orbits. We proceed to discuss the
existence of positive equilibria. The nontrivial x-
isocline and the z-isocline of (5) are given
respectively by
(6)
Where f is strictly decreasing with f(0) = r/a and
f(1/b) = 0. The slope of the graph of g is
clearly, we can see the slope depends
on the sign of m.
3.1 m
For m , it needed to be
to become g(x)>0, when , g(x)
becomes positive but when m > d we need x <
,
if m > d then g(x) is positive only on [0,
. On the
other hand, if 0 is defined and
positive on [0, . When m > 0, g(x) is increasing
and when x arrives to infinity, curve approach to a
constant level
. If
then it occurs a
positive equilibrium point. Consider the determinant
of the jacobian at that point ().
According to the in (5),
,
then we can see and so we can
conclude that this equilibrium point is locally
asymptotically stable. Then consider about the cancer
free equilibrium point (0, s/d).
According to the eigenvalues we can see this
equilibrium point is unstable if
and on the
other hand If
cancer free equilibrium point
become locally asymptotically stable, in this
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situation, there doesn’t exist a positive equilibrium
point. When the case become trivial. By using
Poincar´e-Bendixson Theroem[7] and above details
we can obtain bellow theorem.
Theorem 3. The following statements hold for (5).
a) If
then (0, s/d) is globally asymptotically
stable in .
b) If
then there is a unique positive
equilibrium point (x1,y1) which is globally
asymptotically stable in
Here also happen a similar situation like
chemotherapy in section 3. We can observe when the
cancer clean-up rate is greater than the rate of tumour
growth, tumour can eliminate completely. On the
other hand, when the tumour growth rate is greater
than the rate of cancer cleanup we have to stabilize
the tumour size in some fixed level. Here it provides
a threshold level to eradicate the cancer completely,
further we can say, it needs a strong immune therapy
treatment to remove the cancer when immune cells
proliferate without the resistance of cancer cells.
4.2 Numerical Simulations
(a) (b)
Fig. 4: Here green curve represents the amount of
immunotherapy and blue curve represents the cancer
cell population r = 0.3, b = 1.02 10-9, s = 5000, d =
2, m = 110-9, g = 2000 initial tumour cell population
= 6.7104 cells, initial immune cell amount = 300,
(a) = 110-4, (b) = 110-3.
When we consider about the Theorem 3,
according to the details of figure 3(a) we can
calculate that
= 2.5 and r = 0.3 then
< r. It is
clear that cancer level arrivers to a fixed level. On the
other hand, according to figure 3(b)
> r = 0.25 and
r =0.3 there
< r and cancer free equilibrium point
becomes globally asymptotically stable and we can
remove the cancer completely. It clearly represents
that when the rate of cancer cleanup by immune cells
is in a strong level, we can eradicate the cancer by
immunotherapy completely. Here we can calculate
that if we need to remove the cancer what is the
needed cancer cleanup rate. Another important result
is if we can identify cancer in early, we can remove
it completely otherwise we have to fix the cancer in
a stable level.
Fig. 5: Here blue, red, cyan, curves respectively
represent the behaviour of tumour cell population
according to each initial amount of tumour cell
population = 6.3104 cells, = 6.3105 cells, =
6.3106 cells.
It is clear that when the tumour cell
population is high, it approaches to a constant level
more rapidly.
4.3 m < 0
When m < 0, g(x) is defined on (0, ] and
g(x) is gradually decreasing and approaches to a
constant level
. is strictly decreasing with
and
Here is in the range of
0 < < 1/ and is in the range of 0 < z < s/d. We
are going to consider about the existence of the
positive equilibrium points and their stability and
then supposed to describe about the biological
phenomena in each case. Setting = g() and
(a)
(b)
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simplifying, the component of a positive
equilibrium is a positive root of
(7)
The vertex of H1(x) is
at
(8)
If ,then has a unique positive
equilibrium point , where < 1/b and <
s/d. Notice T r( , < 0 and consider about
,
)) =
+
+ (9)
( ( =
+
]]
If
> 0 then ,
> 0, from here we can obtain a sufficient condition
that , then , becomes locally
asymptotically stable. As there are no any other
stable equilibrium points in , , becomes
globally asymptotically stable in that region.
If where the tumor free equilibrium
point is locally asymptotically stable. If
It is clear that has
no positive real roots since for all
and thus
is globally asymptotically stable by
the Poincar´e-Bendixson Theorem. Let
and then there is no positive
equilibrium point in the system (5) because there are
no any intersections with the quadratic curve with the
axis. If quadratic curve touches the axis
at one point and it means (5) has one positive
equilibrium point and as implies the
equilibrium point become non hyperbolic. If
and , there
exist two positive equilibrium points
where We can simplify
Theorem 4. If m<0, then the following statements
hold for (5).
a) If sa < rd there is a unique positive equilibrium
point which is globally asymptotically stable
in
b) If sa>rd and then
there is no positive equilibrium points and
is
globally asymptotically stable in
c) If sa>rd and
then there is no positive equilibrium
points for (5) and
is globally asymptotically
stable in
(5) has a unique equilibrium
point which is non –hyperbolic if
d) If sa>rd,
, there exist two positive equilibrium
points
5 Combined Therapy
In this section we are going to consider the
effects of combined therapy. When we are giving
Immune therapy and Chemotherapy together, we
need to observe how this dynamical system behaves.
Here we are considering the full system on
(10)
(11)
First let’s considered the isoclines of the
system.
(13)
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From (6) second equation, we can obtain the
value of y component at equilibrium state and we can
observe it’s a constant value such as
. After
substituting for y in first and second equations in (6)
we can obtain a two-dimensional equation system.
We can obtain x-isocline and the z-isocline
as,
Suppose
Where P is strictly decreasing with
and
The slope of the graph
of is
.
Clearly, we can see that the sign of the slope depends
on the sign of m.
5.1
When
to become
, but when
we need
. If
then is positive only on
On the other hand, if
is defined and positive on When
m > 0, is increasing when x arrives to infinity,
curve approaches to a constant level
When there are no
intersections of two isoclines in the first quadrant.
Then system contains a cancer free equilibrium point.
After calculating we can obtain it as
consider the Jacobian at
that point.
This is an upper triangular matrix, if all the
diagonal elements become negative then this system
becomes locally asymptotically stable at . We can
obtain a condition that if then
(11) becomes locally asymptotically stable at
This (11) is asymptotically autonomous with (5), in
that (5) subsystem cancer free equilibrium point is
globally asymptotically stable on hence we can
say is globally asymptotically stable on .
Theorem 5. The following statement hold for (11). If
then is globally asymptotically
stable on
From the above theorem we can obtain, when
the tumour killing rate of chemotherapy and immune
therapy (combined therapy) is greater than the
tumour growth rate cancer can eliminate completely.
Here it provides a threshold level to eliminate cancer
completely. Then let’s consider . Here
these two isoclines have only one intersection in Γ .
In this situation there exist a unique positive
equilibrium point in the system and here
. This (11) is asymptotically autonomous with
(5), according to the stability of (5) subsystem we can
obtain that this positive equilibrium point is globally
asymptotically stable on
Theorem 6. The following statement hold for (11). If
, then there exists a unique positive
equilibrium point and it is globally asymptotically
stable on
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From this theorem we can obtain that if the
tumour growth rate is greater than the tumour killing
rate of combined therapy, we have to fix the cancer
in some stable level but we can’t eradicate the cancer
completely.
5.2 Numerical Simulations
(a)
(b) (c)
Fig. 6: (a) γ = 0.4 (b) γ = 0.6 (c) γ = 0.9. Here green,
blue and red curves represent the logarithm of cancer
cell population, concentration of chemotherapy and
number of immune cells against number of days.
Here r = 0.3465, a = 10−5, = 0.53, h = 0.3, m = 1.3
× 10−9 and other values are same as in the Table 1.
It is clear that when we increase the rate of
decrement of chemotherapy drug the tumour cell
population will increase. It is Important to keep in
a small level. Then it will be more efficient in the
combined therapy.
6 Local Sensitivity Analysis
Table 2. Comparison of cancer cell population
change: baseline values
Parameter
Old-Value
New-Value
r
0.3465
0.38115
b
a
0.0001
0.00011
k τ
0.53
0.583
γ
0.8
0.88
h
0.3
0.33
s
5000
5500
d
2
2.2
m
kg
0.06
0.066
g
10000
11000
Table 3. Comparison of cancer cell population
change: For 50 days with X-old value 0.0315 and
for 20 days with X-old value 0.0004 for each
parameter
t = 50
t = 20
X-new
Diff.
Per
(%)
X-new
Diff.
Per
(%)
0.1773
0.1458
462.86
0.0007
0.1458
75.00
0.0314
-0.0001
-0.32
0.0004
-0.0001
0.00
0.0314
-0.0001
-0.32
0.0004
-0.0001
0.00
0.0032
-0.0283
-89.84
0.0001
-0.0283
-75.00
0.247
0.2155
684.13
0.0017
0.2155
325.00
0.0119
-0.0196
-62.22
0.0003
-0.0196
-25.00
0.0315
0
0.00
0.0004
0
0.00
0.0315
0
0.00
0.0004
0
0.00
0.0315
0
0.00
0.0004
0
0.00
0.0315
0
0.00
0.0004
0
0.00
0.0315
0
0.00
0.0004
0
0.00
The primary objective of these calculations
is to discern the impact of key parameters on the
eradication of cancer cells [17], particularly focusing
on their influence during both short (t=20) and
extended (t=50) time intervals. The aim is to
investigate how variations in these parameters
contribute to a 10% increase in the efficacy of
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eliminating cancer cells over both brief and
prolonged durations.
Fig. 7: Percentage difference in cancer cell
population change at t=20 days visualization
The depicted figure shows information
gathered over a short time period. In this data, the
factor that has the biggest effect is . What stands out
is that when values go up, the number of cancer
cells in the body also increases. This finding, based
on combined therapy results, highlights that higher
rate of decrement of concentration of chemotherapy
values lead to more cancer cells in the body.
The parameter next most influencing
parameter is r, where an increase in its value
corresponds to a heightened proliferation of cancer
cells within the body. It's clear that when tumours
grow faster, getting rid of cancer becomes more
challenging. Increasing the Rate of tumour growth
makes it harder to eliminate cancer.
The parameters and h play a positive role
in the elimination of cancer cells. Increasing by
10% leads to a remarkable 75% improvement in the
effectiveness of killing cancer cells, while h
contributes a 25% positive impact. Unlike , r, h, and
the other parameters show no notable impact on
the cancer cell population within the 10% of change
of itself.
Fig. 8: Percentage difference in cancer cell
population change at t=50 days visualization
The presented figure encapsulates an
extended duration, revealing key influences. In this
dataset, γ emerges as the most impactful factor, while
r follows closely in significance, particularly in the
short term. This underscores that a higher rate of
chemotherapy concentration reduction and
accelerated tumour growth present challenges in
eliminating cancer cells. Beyond the long-term
influence of and h, parameters a and b also
contribute positively to cancer cell elimination.
However, their impact is notably smaller compared
to and h, amounting 0.32%. It's noteworthy that,
similar to the short-term scenario, other parameters
exhibit no notable impact on the 10% change in the
population of cancer cells.
When we compare short-term and long-term
treatments, a noticeable trend emerges. The negative
impact from γ and r tends to increase over time,
signifying that higher values of the rate of
chemotherapy reduction γ and tumour growth r
become more challenging for cancer elimination as
time progresses. On the flip side, the positive impact
from and h shows an upward trajectory. This
suggests that, with the passage of time, increasing the
values of and h becomes more effective in
enhancing the process of eliminating cancer cells. In
essence, the dynamics indicate a shift where
challenges posed by higher γ and r values intensify
over time, while the effectiveness of higher and h
values becomes more pronounced in the long-term
treatment approach.
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7 Conclusion
In conclusion, the comprehensive analysis of
cancer treatment dynamics presented in this report
unveils critical insights that bear significant
implications for advancing therapeutic strategies. In
this study, we delve into the intricacies of three
distinct cancer treatment methodologies:
chemotherapy, immunotherapy, and the combined
approach integrating both modalities. Our
exploration is grounded in a robust mathematical
framework, meticulously crafted by amalgamating
insights from existing research papers and leveraging
mathematical concepts such as the Logistic Growth
Model [8],[9],[16], Mass Action Law, and Michaelis-
Menten mechanism.
Our mathematical model reveals two
equilibrium points. One indicating a cancer-free state
and another depicting a situation where cancer
persists at a constant level without further growth.
Focusing on chemotherapy as a subsystem,
cancer free equilibrium point exists when
and proved that positive equilibrium point
exists when .
Similarly, our examination extends to the
immunotherapy subsystem, where in equilibrium
points are identified for scenarios representing (
)
equilibrium point as absence of cancer when
.
And proved that positive equilibrium point exists
when
These findings contribute to obtain
threshold levels for parameters as and
. Leveraging the powerful capabilities of
MATLAB software, we translated our mathematical
findings into insightful visualizations for the two
scenarios involving chemotherapy and
immunotherapy subsystems[8],[9].
The two equilibrium points was found in the
combined approach as existence of cancer free
equilibrium point when
+
and cancer
persists at a constant level equilibrium point when
+
. We generated plots by varying the
γ parameter, exploring its impact on the system for
the values of γ set at 0.3, 0.8, and 0.9. And final
insight was decreasing chemotherapy concentration,
as represented by higher γ values, poses a challenge
for effectively reducing the cancer cell population.
Finally, a sensitivity analysis was conducted
to gauge the influence of parameters [17] on the
eradication of cancer cells, with a specific emphasis
on short (t=20) and extended (t=50) time intervals.
The analysis revealed that γ (Rate of decrement of
concentration of chemotherapy) and r (Rate of
tumour growth) exhibited a negative impact on
cancer elimination, while Killing rate of tumor
cells by chemotherapy) and h (Supply rate of
chemotherapy drug) exerted a substantial positive
influence. Additionally, parameters a (Parameter of
cancer cleanup) and b (Inverse carrying capacity of
tumor cells) were found to contribute a very small
positive impact to the process of cancer elimination.
These findings underscore the nuanced interplay of
different parameters in shaping the effectiveness of
cancer treatment strategies across varying time
frames.
The obtained results indicate the fulfilment
of our research objectives, underscoring the
effectiveness of the undertaken study in addressing
key goals. This achievement not only validates the
research methodology but also contributes valuable
insights to the field, paving the way for innovative
approaches redefine the landscape of hope and
healing.
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G. V. R. K. Vithanage, D. D. Lakshika
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Conceptualization, A.M.D. Clotilda, G.V.R.K.
Vithanage
Formal Analysis, A.M.D. Clotilda, G.V.R.K.
Vithanage, D.D. Lakshika
Numerical Simulations, A.M.D. Clotilda, G.V.R.K.
Vithanage, D.D. Lakshika
Writing, A.M.D. Clotilda, G.V.R.K. Vithanage, D.D.
Lakshika
MOLECULAR SCIENCES AND APPLICATIONS
DOI: 10.37394/232023.2024.4.15
A. M. D. Clotilda,
G. V. R. K. Vithanage, D. D. Lakshika
E-ISSN: 2732-9992
161
Volume 4, 2024
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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