where \(\mu_{A_{i}^{l}}\left(x_{i}\right)\) is the membership function
of the fuzzy set \(A_{i}^{l}\), \(\mu_{B^{l}}\left(y\right)\) is the
membership function of the fuzzy set \(B^{l}\), and\(\mu_{B^{{}^{\prime}}}\left(y\right)\) is the membership function of the output
fuzzy set.
In this paper, the designed Mamdani fuzzy system has one input-one
output. The age of the patient is considered as a language variable and
a Mamdani fuzzy system is designed for controlling chemo-drug dose. The
input of the fuzzy system is the age of the patient and its output is
the upper limit of the chemo-drug dose. The age of the patient is
divided into four groups: Baby, young, middle-aged, and old. The
chemo-drug dose is considered in three groups: Low, medium and high. The
membership functions and the rule-base are shown in Figure 5 and Table
2, respectively.
Mixed treatment simulation
By considering the parameters’ value in Table 1, the treatment must be
imposed in the whole life of the patient because of relapsing cancer
after stopping the treatment (see Figure 1). Due to the instability of
the tumor-free equilibrium point, the dynamics modification is essential
for finite-duration treatment, which is done by immunotherapy. The
vaccine therapy modifies the dynamics of the system by changing the two
parameters \(\sigma\) and \(\delta\). In other words, the phase plane of
the system changes from Figure 1 to Figure 3. After immunotherapy, the
optimal chemotherapy pushes the trajectory of the system toward the
equilibrium point E1.
The proposed therapeutic approach is simulated by considering the
patient’s age. The simulation results are presented for three cases:
young, middle-age and old patients with no specific disease. In each
case the immunotherapy is first implemented. By immunotherapy the free
system dynamics changes from Figure 1 to Figure 4 and then to Figure 3.
Chemotherapy is then implemented by using the pseudo-spectral method.
Case 1: A young patient
In this case, a 20-year-old patient is considered. The fuzzy system
presented a maximum chemo-drug dose as 8.2. As shown in Figure 6, the
vaccine has modified the system’s dynamics, and other equilibrium points
are removed from the system’s phase plane. The task of chemotherapy is
to push the system’s trajectory toward the tumor free equilibrium point
E1. As shown in this figure, the trajectory of the system is pushed to
the point E1 and the imposed chmoe-drug approaches to zero.
Case 2: A middle-age patient
In this case, a 45-year-old patient is considered. The combined
treatment of vaccine therapy and chemotherapy is applied. The fuzzy
system presented a maximum chemo-drug dose as 4.1. As shown in Figure 7,
the vaccine has modified the system’s dynamics and the system’s
trajectory is guided to the equilibrium point E1. When the trajectory of
the system placed in this area the chemotherapy is ceased. In this
situation, the new modified free system is able to guide the trajectory
of the system toward the equilibrium point E1 without any external
treatment.
Case 3: An old patient
In this section, a 60-year-old patient is considered. The fuzzy system
implemented the maximum dose of the chemotherapy drug as 1.6. The
behavior of the system is shown in Figure 8. After the modification of
the dynamics of the system by immunotherapy, the chemotherapy pushes the
trajectory of the system to the the equilibrium point E1. The chemo-drug
dose doses not converge to zero. However, the treatment can be stopped
due to the stability of the tumor free equilibrium point.
Figure 1 shows that the system’s dynamics correction is necessary in
order to have a finite-duration treatment. Hence, some of the
therapeutic inputs should modify the system’s dynamic. In other words,
some therapeutic inputs should have a permanent effect on the system’s
parameters. Hence, a chemo-immonutherpy is regarded, which despite other
researches, the immunotherapy modifyies the dynamics of the system.
By comparing Figures 6, 7 and 8, the chemo-drug dose is smaller in the
old patient. However, the duration of chemotherapy is longer in the old
patient. Chemotherapy is ceased in a shorter duration in the younger
patient.
Compared with other works such as [9, 12, 14, 28, 29], the dynamics
modification during the treatment is considered as a necessary phase for
finite duration treatment. Also, the special condition of the patient is
regarded by designing a fuzzy system. The simulation results show the
effectiveness of the pseudo-spectral controller and its ease of
implimrntation.
Conclusion
A mathematical model of cancer is ectended by adding chemotherapy and
immunotherapy terms in a new approach. The immunotherapy affects on the
parameters of the system and reinforcing the immune system’s ability.
Chemotherapy pushes the trajectory of the system toward the tumor-free
equilibrium point. The chemotherapy protocol is derived using a
pseudo-spectral method. The continuous –time nonlinear model of the
cancer is converted into an NLP problem and is solved. To restrict the
maximum chemo-drug dose a Mamdani fuzzy system is designed. In this
fuzzy system the maximum chemo-drug dose based on the age of the patient
is calculated. Three cases with different age are regarded to
simulation. In all cases the dynamics of the system is modified, and the
trajectoty of the system is guided toward the desired equilibrium point.
The maximum chemo-drug dose in the young patient is higher than other
patients due to his/her immune system’s ability to recovery. Moreover,
the treatment duration in the young patient is shorter. The simulation
results show the effectiveness of the proposed treatment strategy and
ease of implementation of the presented pseudo-spectral method to design
a stabilizing controller.
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