# Modeling and Analysis of Breast Cancer with Adverse Reactions of Chemotherapy Treatment through Fractional Derivative

Authors: Tao-Qian Tang; Zahir Shah; Ebenezer Bonyah; Rashid Jan; Meshal Shutaywi; Nasser Alreshidi
Journal: Computational and Mathematical Methods in Medicine
 (2022)

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## Abstract

The abnormal growth of cells in the breast is called malignancy or breast cancer; it is a life-threatening and dangerous cancer in women around the world. In the treatment of cancer, the doctors apply different techniques to stop cancer cell development, remove cancer cells through surgery, or kill cancer cells. In chemotherapy treatment, powerful drugs are used to kill abnormal cells; however, it has adverse reactions on the patient heart which is called cardiotoxicity. In this paper, we formulate the dynamics of cancer in the breast with adverse reactions of chemotherapy treatment on the heart of a patient in the fractional framework to visualize its dynamical behaviour. We listed the fundamental results of the fractional calculus for the analysis of our model. The model is then analyzed for the basic properties, and the existence and uniqueness of the proposed breast cancer system are investigated through fixed point theory. Furthermore, the Adams-Bashforth numerical technique is presented for the solution of fractional-order system to illustrate the time series of breast cancer model. The dynamical behaviour of different stages of breast cancer is then highlighted numerically to show the effect of fractional-orderϑ and to visualize the role of input parameter on the dynamics of breast cancer.


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## Body

## 1. Introduction

Medical experts reported that breast cancer is the abnormal growth of cells in the breast which is a life-threatening disease and is mostly found in women. It is reported in [1] that breast cancer has the highest incidence rate as compared to the other cancer types. It destroys breast tissue and breast cells to grow out of control and to change the breast to abnormal shape. After lung cancer, breast cancer is declared to be the largest cancer in the globe, and every woman may be infected by this infection. It is stated by the WHO that about 8 to 9 percent of the women are infected by breast cancer in the world; moreover, the root cause is not yet explored by the medical experts. However, several risk factors are predicted which increase the risk of breast cancer in women which include dietary arrangements, drinking alcohol, smoking, being a woman, dense breast, lack of exercise, pregnancy history, genetic, breastfeeding history, race, menstrual history, life history, weight, certain breast changes, personal history of breast cancer, and age. The main symptoms of breast cancer are swollen lymph node nipple discharge, pulling in of the nipple, breast or nipple pain, flaky skin on the breast or nipple, irritation of the skin, dimpled skin red, a change in the size or shape of a breast, thickening of part of a breast, and full or partial swelling. Figure 1 is the representation of breast or milk making factory for the newly baby child with cancer cells which further grow and damage the body of infected individuals.Figure 1
Illustration of milk making (lobules) and shuttling (ducts) glandular epithelial cells anchored by connective tissue.Medical authorities assess cancer condition of a patient by using stages, and these stages are tumor, node, and metastasis which determine the chances of recovery from cancer. The early the stage, the greater the chances of recovery. Numerous treatment techniques have been developed for prevention of cancer which includes surgery, gene therapy, bisphosphonates, immunotherapy, targeted cancer drugs, hormone therapy, bone marrow transplants and stem cell, cancer drugs, radiotherapy, and complementary and alternative therapies. The most common treatment of the above is chemotherapy which involves the use of drugs to kill cancer cells. In chemotherapy, the drugs are either injected to the patient or are used orally which have some effectiveness but may hurt the heart. This bad side effect is called cardiotoxicity and affects children and adults [2]. The failure of patient heart is reported and observed during oncological treatment of anthracyclines and trastuzumab [3]. It is still a challenge for cancer expert and cardiologists to prevent cardiotoxicity experience during chemotherapy treatment. In chemotherapy treatment, anthracycline drugs are used which affect the heart of the patient and lead to cardiotoxicity illustrated in Figure 2.Figure 2
Effect of anthracyclines on the heart of a patient during cancer chemotherapy.Mathematical frameworks are used to conceptualize the intricate dynamics of diseases and to provide accurate results for the control and prevention of these infections [4, 5]. In modeling of cancer, the journey starts from 1954 [6] to explain cancer, and then, the researcher studies different aspects of cancer and tumor growth [7, 8]. A mathematical model of chemotherapy treatment for cancer has been developed by Dixit et al. [9]; the authors represent the treatment procedure for tumor cancer. The dynamics of cancer represent the interactions of tumor cell energy and tumor cell density and the effect of chemotherapy drugs. Recently, a mathematical model has been formulated for low-dose chemotherapy with minimal parameters; they studied angiogenic signals between vasculature and tumors [10]. In [11], a compartmental model has been developed by Jordao and Tavares; they consider cancerous and healthy cells to analyze the proposed model of cancer. The role of time delay on the dynamic of tumor system has been investigated by S. Khajanchi and Nieto [12]. Another important model was developed by Mahlbacher et al. [13] to conceptualize the interactions between immune and tumor and predict better suggestions about cancer. In the literature, several mathematical models have been developed and formulated to study, conceptualize, and visualize the transmission phenomena of cancer [14–17].Recent advancement of fractional calculus showed that the results of fractional operators are more accurate, precious, and reliable as compared to the system of classical derivatives [18, 19]. Novel fractional operators are developed which modeled real-world problem in mathematics, biology, engineering, economics, physics, and other areas of science and technology [20–23]. In fractional calculus, a variety of fractional operators are introduced for the study of real-world issues. These operators, on the other hand, have a power law kernel and can only simulate physical problems to a limited extent. To solve these challenges and limitations, Caputo and Fabrizio presented a new fractional operator with an exponential decay kernel. Because of its nonsingular kernel, this unique operator is a revolutionary fractional derivative operator that has piqued the interest of many scholars. The results of this novel operator are more suitable and have many applications [24, 25]. To be more specific, the transmission phenomena of cancer with treatment and unknown parameters have been successfully represented through CF derivative [26, 27]. To get more realistic findings, we choose to depict the transmission mechanism of breast cancer with side effects on patient heart during chemotherapy through CF fractional derivative.These accurate results and outcomes of fractional calculus motivate us to inspect and interrogate the dynamics of breast cancer with adverse reactions of chemotherapy treatment on patient heart through Caputo-Fabrizio (CF) fractional operator. In Section1 of the article, we represent the fundamental idea of fractional calculus for the analysis of our system. In Section 2, a fractional model is formulated for breast cancer with the adverse reactions of chemotherapy treatment on the heart of a patient in fractional framework. The proposed model of breast cancer is then investigated through mathematical skills. The existence and uniqueness of the solution of the formulated FO model of breast cancer patients through the fixed point theorem are presented in Section 3 of the article. The dynamics of proposed cancer model is then analyzed with the variation of different input parameters numerically in Section 4. Finally, concluding remarks and suggestions are presented in the last section.


## 2. Formulation of the Model

In the formulation of the model, we consider the population of breast cancer patients in a hospital where we categorized the total population of breast cancer patient into stage 1, stage 2, stage 3, and stage 4 subpopulations during the first medical report. It is assumed that all the patient are treated with chemotherapy treatment in the hospital where the patients are passing with different stages during treatment, some patients experience cardiotoxicity, and some patients experience recovery while some getting worse condition of the disease during chemotherapy process. A compartmental model of five subgroups is formulated having subcompartments of stages 1 and 2 (CA), stage 3 (CB), stage 4 (CC), disease-free state (CD), and cardiotoxic (CE) subgroups where the number of cancer patients in stages 1 and 2 is smaller than the other stages; therefore, they are placed in one subgroup.New patients identified to suffer in stages 1 and 2 cancer are assumed to beη1 while for stage 3 and stage 4 are assumed to be η2 and η3, respectively. The subgroup CA having chemotherapy may either move worse subgroup CB with δAB or recover with a rate δAD. The patients of the subgroup CB who are first treated in the hospital are recruited with a rate η2. This subgroup is more intensive chemotherapy as compares to CA, where the patient die from cancer with a rate γ2, move to the recover subgroup with a rate δBD, become more worse with a rate δBC, and at a rate δBE become cardiotoxicity. The patients of cancer are recruited to subgroup CC with a rate η3 during the treatment. In this case, the rate of recovery δCD is smaller than the first two and rate δCE towards cardiotoxicity is greater than the rate of the subgroup CB due to intensive chemotherapy effect. We assume γ3 to be the death rate of cancer patient in this subgroup. In the forth subgroup, the population increased from the first three subgroups and lose recovery at δDB,δDC, and δDE to the subgroups CB, CC, and CE, respectively. The patient in (CE) comes from (CB), (CC) and (CD) and taste cardiac death with rateγ1. Then, the dynamics of breast cancer with chemotherapy treatment with the above assumptions is given by the following system of ODEs:
(1)dCAdt=η1−δADCA−δABCA,dCBdt=η2+δABCA+δDBCD−δBDCB−δBCCB−δBECB−γ2CB,dCCdt=η3+δBCCB+δDCCD−δCDCC−δCECC−γ3CC,dCDdt=δADCA+δBDCB+δCDCC−δDBCD−δDCCD−δDECD,dCEdt=δDECD+δCECC+δBECB−γ1CE,with appropriate initial condition for vector
(2)CA0≥0,CB0≥0,CC0≥0,CD0≥0,CE0≥0.It is well-know that fractional system provides more accurate results of the dynamics of a system developed from natural phenomena. There are several fractional operators in the literature of fractional calculus with power law kernel and have limitations to mimic real-world problems. Therefore, we applied Caputo-Fabrizio operator to our problem which represents the dynamics of mathematical model through exponential decay kernel to overcome these challenges and limitations. The dynamics of breast cancer through CF fractional derivative can be expressed as follows:(3)D0CFtϑht=Uϑ1−ϑ∫ath′xexp−ϑt−x1−ϑdx.A detailed discussion of this operator has been presented in the upcoming section of the article. We represent the system (1) of breast cancer with the help of the above definition of CF derivative as
(4)D0CFtϑCA=η1−δADCA−δABCA,D0CFtϑCB=η2+δABCA+δDBCD−δBDCB−δBCCB−δBECB−γ2CB,D0CFtϑCC=η3+δBCCB+δDCCD−δCDCC−δCECC−γ3CC,D0CFtϑCD=δADCA+δBDCB+δCDCC−δDBCD−δDCCD−δDECD,D0CFtϑCE=δDECD+δCECC+δBECB−γ1CE,whereϑ is the order of CF fractional derivative such that 0<ϑ≤1 and the unit of the above fractional system is ϑ−1. In the next subsection, we will list some basic definitions and statements related to CF fractional derivative for further analysis of the model.


### 2.1. Rudimentary Knowledge

In this subsection of the article, the fundamental results and definitions of fractional Caputo-Fabrizio (CF) is presented for the analysis of our breast cancer model with chemotherapy treatment. The basic definitions and results are given below:Definition 1.
Let us supposeh∈H1a,b, where b is greater than a; then, the CF derivative [28] of order ϑ is given by
(5)Dtϑht=Uϑ1−ϑ∫ath′xexp−ϑt−x1−ϑdx,
whereϑ∈0,1 and Uτ denotes normality with U0=U1=1 [28]. In the case, when h∉H1a,b, then the following fractional derivative is obtained:
(6)Dtϑht=ϑUϑ1−ϑ∫atht−hxexp−ϑt−x1−ϑdx.Remark 2.
Let us takeα=1−ϑ/ϑ∈0,∞ and ϑ=1/1+α∈0,1; then, equation (6) can be written in the following form:
(7)Dtϑht=Mαα∫ath′xe−t−xαdx,M0=M∞=1.
Furthermore,(8)limα⟶01αexp−t−xα=δx−t.In [29], the authors introduced the concept of fractional integral which is defined as follows:Definition 3.
Leth be a given function; then, the fractional integral is defined in the following manner:
(9)Itϑht=21−ϑ2−ϑUϑht+2ϑ2−ϑUϑ∫0thudu,t≥0,
where0<ϑ<1 is the order of the above fractional integral.Remark 4.
From the above Definition3, we can conclude that
(10)21−ϑ2−ϑUϑ+2ϑ2−ϑUϑ=1,
which givesUϑ=2/2−ϑ, 0<ϑ<1. A new Caputo derivative of order ϑ was introduced by Losada and Nieto in [29] by using equation (10) and is given by
(11)Dtϑht=11−ϑ∫0th′xexpϑt−x1−ϑdx,0<ϑ<1.For the equilibrium point of breast cancer model (4), we set all the fractional derivative of model (4) to zero and obtain the equilibrium point given by Ee=CA∗,CB∗,CC∗,CD∗,CE∗. The equilibrium point exists and is
(12)CA∗=η1k1,CB∗=αk1λ,CC∗=βk1λ,CD∗=ξk1λ,CE∗=vk1λγ1,where,
(13)α=k3δBD+δDE+δDCγ3+δDE+δDCδCE+δDEδCDη2δAB+k3η1+δCDη3δDB+η1δDE+δDCγ3+δDE+δDCδCE+δDEδCDδAB+δADk3δDB+δDE+δDCγ3+δDE+δDCδCE+δDEδCDη2+k3η1+δCDη3δDB,β=k2δDC+δDB+δDEδBC+γ2+δBD+δBEδDE+δDBγ2+δBEk1η3+η1+η2δBC+γ2+δBD+δBEη1+δBDη2δDC+η2δDE+δDBη1+η2δBCδAD+δBC+δBDδDC+δDB+δDEδBCη1+η2δAB,ξ=k2η1+η1+η3δBD+δBCη2+η3γ2+δBC+δBEδCD+k2η1+δBDη2γ3+δCEδAD+δABδBC+δBDη1+η2+η3δBD+δBCη2+η3γ2+δBC+δBEδCD+δBDη1+η2γ3+δCE,v=η1+η2+η3δBE+γ2+δBC+δBDη1+δBC+δBDη2+η3γ2+δBC+δBDδDEδADδCE+δDB+δDCη1+η2+η3δBE+δDB+δDCδBC+δDCγ2+δBDη1+δBD+δDCδBC+δBDδDCη2+δDB+δDCδBC+γ2δDB+δDCγ2+δBDη3δADδCE+η+η2+η3δBE+δBC+δBDη1+δBC+δBDη2+η3γ2+δBC+δBDδDEδABδCE+δDB+δDCη1+η2+η3δBE+δDB+δDCδBC+δBDδDCη1δABδCE+δDB+δDCδBC+δBDδDCη2+δDB+δDCδBC+γ2δDB+δDCγ2+δBDη3δABδCE+δDEδADγ3+δCDη1+γ3+δCDη2+η3δCDδBE+γ3+δCDγ2+δBC+δBDη1+δBC+δBDδDC+γ3δBDη2+η3δCDγ2+δBC+δBDδDEδAD+δBEγ3+δCDδDBη1+δDBδCD+γ3δDB+δDCη2+η3δCDδDBδAD+δABδDEγ3+δCDη1+γ3+δCDη2+η3δCDδBEδBC+δBDδCD+γ3δBDη1+δABδDEδBC+δBDδCD+γ3δBDη2+η3δCDγ2+δBC+δBD+δBEδABδBDδCD+γ3δDB+δDCη1+δDBδCD+γ3δBD+δDCη2+η3δCDδDB,λ=k1k2δDE+δDB+δDCγ2+δDB+δDCδBE+δBCδDB+δBC+δBDδDCγ3+δDB+δDCγ2+δDB+δDCδBE+δBCδDB+δBC+δBDδDCδCE+δCDδDBγ2+δBE,in whichk1=δAD+δAB,k2=δBD+δBC+δBE+γ2,andk3=δCD+δCE+γ3. Here, for equilibrium of our fractional-order breast cancer model, we have the following conclusion. The disease-free equilibrium point can be easily determined by taking the steady state of our system without infection. These equilibrium points are important for the analysis of the proposed fractional model of cancer with chemotherapy treatment and can predict sufficient condition for the control and spread of the infection. We have the following result based on the above investigation:Theorem 5.
There exists an equilibrium of the proposed fractional model (4) of breast cancer without any condition.


## 2.1. Rudimentary Knowledge

In this subsection of the article, the fundamental results and definitions of fractional Caputo-Fabrizio (CF) is presented for the analysis of our breast cancer model with chemotherapy treatment. The basic definitions and results are given below:Definition 1.
Let us supposeh∈H1a,b, where b is greater than a; then, the CF derivative [28] of order ϑ is given by
(5)Dtϑht=Uϑ1−ϑ∫ath′xexp−ϑt−x1−ϑdx,
whereϑ∈0,1 and Uτ denotes normality with U0=U1=1 [28]. In the case, when h∉H1a,b, then the following fractional derivative is obtained:
(6)Dtϑht=ϑUϑ1−ϑ∫atht−hxexp−ϑt−x1−ϑdx.Remark 2.
Let us takeα=1−ϑ/ϑ∈0,∞ and ϑ=1/1+α∈0,1; then, equation (6) can be written in the following form:
(7)Dtϑht=Mαα∫ath′xe−t−xαdx,M0=M∞=1.
Furthermore,(8)limα⟶01αexp−t−xα=δx−t.In [29], the authors introduced the concept of fractional integral which is defined as follows:Definition 3.
Leth be a given function; then, the fractional integral is defined in the following manner:
(9)Itϑht=21−ϑ2−ϑUϑht+2ϑ2−ϑUϑ∫0thudu,t≥0,
where0<ϑ<1 is the order of the above fractional integral.Remark 4.
From the above Definition3, we can conclude that
(10)21−ϑ2−ϑUϑ+2ϑ2−ϑUϑ=1,
which givesUϑ=2/2−ϑ, 0<ϑ<1. A new Caputo derivative of order ϑ was introduced by Losada and Nieto in [29] by using equation (10) and is given by
(11)Dtϑht=11−ϑ∫0th′xexpϑt−x1−ϑdx,0<ϑ<1.For the equilibrium point of breast cancer model (4), we set all the fractional derivative of model (4) to zero and obtain the equilibrium point given by Ee=CA∗,CB∗,CC∗,CD∗,CE∗. The equilibrium point exists and is
(12)CA∗=η1k1,CB∗=αk1λ,CC∗=βk1λ,CD∗=ξk1λ,CE∗=vk1λγ1,where,
(13)α=k3δBD+δDE+δDCγ3+δDE+δDCδCE+δDEδCDη2δAB+k3η1+δCDη3δDB+η1δDE+δDCγ3+δDE+δDCδCE+δDEδCDδAB+δADk3δDB+δDE+δDCγ3+δDE+δDCδCE+δDEδCDη2+k3η1+δCDη3δDB,β=k2δDC+δDB+δDEδBC+γ2+δBD+δBEδDE+δDBγ2+δBEk1η3+η1+η2δBC+γ2+δBD+δBEη1+δBDη2δDC+η2δDE+δDBη1+η2δBCδAD+δBC+δBDδDC+δDB+δDEδBCη1+η2δAB,ξ=k2η1+η1+η3δBD+δBCη2+η3γ2+δBC+δBEδCD+k2η1+δBDη2γ3+δCEδAD+δABδBC+δBDη1+η2+η3δBD+δBCη2+η3γ2+δBC+δBEδCD+δBDη1+η2γ3+δCE,v=η1+η2+η3δBE+γ2+δBC+δBDη1+δBC+δBDη2+η3γ2+δBC+δBDδDEδADδCE+δDB+δDCη1+η2+η3δBE+δDB+δDCδBC+δDCγ2+δBDη1+δBD+δDCδBC+δBDδDCη2+δDB+δDCδBC+γ2δDB+δDCγ2+δBDη3δADδCE+η+η2+η3δBE+δBC+δBDη1+δBC+δBDη2+η3γ2+δBC+δBDδDEδABδCE+δDB+δDCη1+η2+η3δBE+δDB+δDCδBC+δBDδDCη1δABδCE+δDB+δDCδBC+δBDδDCη2+δDB+δDCδBC+γ2δDB+δDCγ2+δBDη3δABδCE+δDEδADγ3+δCDη1+γ3+δCDη2+η3δCDδBE+γ3+δCDγ2+δBC+δBDη1+δBC+δBDδDC+γ3δBDη2+η3δCDγ2+δBC+δBDδDEδAD+δBEγ3+δCDδDBη1+δDBδCD+γ3δDB+δDCη2+η3δCDδDBδAD+δABδDEγ3+δCDη1+γ3+δCDη2+η3δCDδBEδBC+δBDδCD+γ3δBDη1+δABδDEδBC+δBDδCD+γ3δBDη2+η3δCDγ2+δBC+δBD+δBEδABδBDδCD+γ3δDB+δDCη1+δDBδCD+γ3δBD+δDCη2+η3δCDδDB,λ=k1k2δDE+δDB+δDCγ2+δDB+δDCδBE+δBCδDB+δBC+δBDδDCγ3+δDB+δDCγ2+δDB+δDCδBE+δBCδDB+δBC+δBDδDCδCE+δCDδDBγ2+δBE,in whichk1=δAD+δAB,k2=δBD+δBC+δBE+γ2,andk3=δCD+δCE+γ3. Here, for equilibrium of our fractional-order breast cancer model, we have the following conclusion. The disease-free equilibrium point can be easily determined by taking the steady state of our system without infection. These equilibrium points are important for the analysis of the proposed fractional model of cancer with chemotherapy treatment and can predict sufficient condition for the control and spread of the infection. We have the following result based on the above investigation:Theorem 5.
There exists an equilibrium of the proposed fractional model (4) of breast cancer without any condition.


## 3. Interrogation of Fractional System

Here, the solution of the proposed breast cancer model will be investigated for existence through fixed point theory. We use the concept of CF fractional derivative on the system (4) and get the following:
(14)CAt−CA0=I0CFtϑη1−δADCA−δABCA,CBt−CB0=I0CFtϑη2+δABCA+δDBCD−δBDCB−δBCCB−δBECB−γ2CB,CCt−CC0=I0CFtϑη3+δBCCB+δDCCD−δCDCC−δCECC−γ3CC,cCDt−CD0=I0CFtϑδADCA+δBDCB+δCDCC−δDBCD−δDCCD−δDECD,CEt−CE0=I0CFtϑδDECD+δCECC+δBECB−γ1CE.By applying the idea presented in [29], we get
(15)CAt−CA0=21−ϑ2−ϑUϑη1−δADCA−δABCA+2ϑ2−ϑUϑ∫0tη1−δADCA−δABCAdy,(16)CBt−CB0=21−ϑ2−ϑUϑη2+δABCA+δDBCD−δBDCB−δBCCB−δBECB−γ2CB+2ϑ2−ϑUϑ∫0tη2+δABCA+δDBCD−δBDCB−δBCCB−δBECB−γ2CBdy,(17)CCt−CC0=21−ϑ2−ϑUϑη3+δBCCB+δDCCD−δCDCC−δCECC−γ3CC+2ϑ2−ϑUϑ∫0tη3+δBCCB+δDCCD−δCDCC−δCECC−γ3CCdy,(18)CDt−CD0=21−ϑ2−ϑUϑδADCA+δBDCB+δCDCC−δDBCD−δDCCD−δDECD+2ϑ2−ϑUϑ∫0tδADCA+δBDCB+δCDCC−δDBCD−δDCCD−δDECDdy,(19)CEt−CE0=21−ϑ2−ϑUϑδDECD+δCECC+δBECB−γ1CE+2ϑ2−ϑUϑ∫0tδDECD+δCECC+δBECB−γ1CE.In the next step, we proceed in the following manner:(20)L1t,CA=η1−δADCA−δABCA,L2t,CB=η2+δABCA+δDBCD−δBDCB−δBCCB−δBECB−γ2CB,L3t,CC=η3+δBCCB+δDCCD−δCDCC−δCECC−γ3CC,L4t,CD=δADCA+δBDCB+δCDCC−δDBCD−δDCCD−δDECD,L5t,CE=δDECD+δCECC+δBECB−γ1CE.Theorem 6.
If the following condition satisfies(21)0≤δAD+δAB<1,then, the kernels L1,L2,L3,L4, and L5 assure the Lipschitz condition.Proof.
For the proof of Theorem7, we first take CA and CA1 and start from L1 in the following manner:
(22)L1t,CA−L1t,CA1=−δADCA+δADCA1−δABCA+δABCA1.
Here, apply norm on Eq. (19) and simplify; we get the following:
(23)L1t,CA−L1t,CA1≤−δADCA+δADCA1+−δABCA+δABCA1≤δADCA−CA1+δABCA−CA1≤δAD+δABCA−CA1.
At this stage, we assumeμ1=δAD+δAB; then, the following result is obtained:
(24)L1t,CA−L1t,CA1≤μ1CAt−CAt1.
Thus, the Lipschitz condition is fulfilled forL1; in addition to this, the condition 0≤δAD+δAB<1 assure that the contraction is also satisfied. In the same way, the Lipschitz conditions for the other cases of our system are determined as
(25)L2t,CB−L2t,CB1≤μ2CBt−CBt1,L3t,CC−L3t,CC1≤μ3CCt−CCt1,L4t,CD−L4t,CD1≤μ4CDt−CDt1,L5t,CE−L5t,CE1≤μ5CEt−CEt1.
Further simplification of Eq. (19) implies
(26)CAt=CAt0+21−ϑ2−ϑUϑL1t,CAt+2ϑ2−ϑUϑ∫0tL1y,CAtdy,CBt=CBt0+21−ϑ2−ϑUϑL2t,CBt+2ϑ2−ϑUϑ∫0tL2y,CBtdy,CCt=CCt0+21−ϑ2−ϑUϑL3t,CBt+2ϑ2−ϑUϑ∫0tL3y,CCtdy,CDt=CDt0+21−ϑ2−ϑUϑL4t,CDt+2ϑ2−ϑUϑ∫0tL4y,CDtdy,CEt=CEt0+21−ϑ2−ϑUϑL5t,CEt+2ϑ2−ϑUϑ∫0tL5y,CEtdy.Its recursive form is given by
(27)CAnt=21−ϑ2−ϑUϑL1t,CAn−1+2ϑ2−ϑUϑ∫0tL1y,CAn−1dy,CBnt=21−ϑ2−ϑUϑL2t,CBn−1+2ϑ2−ϑUϑ∫0tL2y,CBn−1dy,CCnt=21−ϑ2−ϑUϑL3t,CCn−1+2ϑ2−ϑUϑ∫0tL3y,CCn−1dy,CDnt=21−ϑ2−ϑUϑL4t,CDn−1+2ϑ2−ϑUϑ∫0tL4y,CDn−1dy,CEnt=21−ϑ2−ϑUϑL5t,CEn−1+2ϑ2−ϑUϑ∫0tL5y,CEn−1dy,with proper initial conditions
(28)CA0t=CA0,CB0t=CB0,CC0t=CC0,CD0t=CD0,CE0t=CE0.
Here, the succeeding difference form is(29)ℏ1nt=CAnt−CAn−1t=21−ϑ2−ϑUϑL1t,CAn−1−L1t,CAn−2+2ϑ2−ϑUϑ∫0tL1y,CAn−1−L1y,CAn−2dy,ℏ2nt=CBnt−CBn−1t=21−ϑ2−ϑUϑL2t,CBn−1−L2t,CBn−2+2ϑ2−ϑUϑ∫0tL2y,CBn−1−L2y,CBn−2dy,ℏ3nt=CCnt−CCn−1t=21−ϑ2−ϑUϑL3t,CCn−1−L3t,CCn−2+2ϑ2−ϑUϑ∫0tL3y,CCn−1−L3y,CCn−2dy,ℏ4nt=CDnt−CDn−1t=21−ϑ2−ϑUϑL4t,CDn−1−L4t,CDn−2+2ϑ2−ϑUϑ∫0tL4y,CDn−1−L4y,CDn−2dy,ℏ5nt=CEnt−CEn−1t=21−ϑ2−ϑUϑL5t,CEn−1−L5t,CEn−2+2ϑ2−ϑUϑ∫0tL5y,CEn−1−L5y,CEn−2dy.
Here, we observe the following:(30)CAnt=∑i=1nℏ1it,CBnt=∑i=1nℏ2it,CCnt=∑i=1nℏ3it,CDnt=∑i=1nℏ4it,CEnt=∑i=1nℏ5it.
Following the same way, we have(31)ℏ1nt=CAnt−CAn−1t=1−ϑ2−ϑUϑL1t,CAn−1−L1t,CAn−2+2ϑ2−ϑUϑ∫0tL1y,CAn−1−L1y,CAn−2dy
By triangular inequality, Eq. (31) becomes
(32)CAnt−CAn−1t≤21−ϑ2−ϑUϑL1t,CAn−1−L1t,CAn−2+2ϑ2−ϑUϑ∫0tL1y,CAn−1−L1y,CAn−2dy.
Lipschitz condition leads us to(33)CAnt−CAn−1t≤21−ϑ2−ϑUϑμ1CAn−1−CAn−2+2ϑ2−ϑUϑμ1×∫0tCAn−1−CAn−2dy.
Next, we have(34)ℏ1nt≤21−ϑ2−ϑUϑμ1ℏ1n−1t+2ϑ2−ϑUϑμ1∫0tℏ1n−1ydy.
Taking the same steps, we get(35)ℏ2nt≤21−ϑ2−ϑUϑμ2ℏ2n−1t+2ϑ2−ϑUϑμ2∫0tℏ2n−1ydy,(36)ℏ3nt≤21−ϑ2−ϑUϑμ3ℏ3n−1t+2ϑ2−ϑUϑμ1∫0tℏ3n−1ydy,(37)ℏ4nt≤21−ϑ2−ϑUϑμ4ℏ4n−1t+2ϑ2−ϑUϑμ4∫0tℏ4n−1ydy,(38)ℏ5nt≤21−ϑ2−ϑUϑμ5ℏ5n−1t+2ϑ2−ϑUϑμ5∫0tℏ5n−1ydy.Theorem 7.
Exact coupled solutions of the proposed breast cancer model (4) exists if the below mentioned condition satisfies. That is, one can find t0 in a way that
(39)21−ϑ2−ϑUϑμ1+2ϑ2−ϑUϑμ1t0<1.Proof.
As the state variablesCAt, CBt, CCt,CDt, and CDt are bounded. Moreover, we have shown that the Lipschitz condition is fulfilled by the kernels; Eqs. (34) and (38) give the following by applying the recursive technique:
(40)ℏ1nt≤CAn021−ϑ2−ϑUϑμ1+2ϑ2−ϑUϑμ1tn,ℏ2ntCBn021−ϑ2−ϑUϑμ2+2ϑ2−ϑUϑμ2tn,ℏ3nt≤CCn021−ϑ2−ϑUϑμ3+2ϑ2−ϑUϑμ3tn,ℏ4nt≤CDn021−ϑ2−ϑUϑμ4+2ϑ2−ϑUϑμ4tn,ℏ5nt≤CEn021−ϑ2−ϑUϑμ5+2ϑ2−ϑUϑμ5tn.
As a result, the existence of solutions of the breast cancer model and its continuity are achieved. In addition to this, we will show that that the above is a solution of system (4) and proceed as follows:
(41)CAt−CA0=CAnt−M1nt,CBt−CB0=CAnt−M2nt,CCt−CC0=CCnt−M3nt,CDt−CD0=CDnt−M4nt,CEt−CE0=CEnt−M5nt.
Thus, we have(42)Hnt=21−ϑ2−ϑUϑL1t,CAn−L1t,CAn−1+2ϑ2−ϑUϑ×∫0tL1y,CAn−L1y,CAn−1dy,≤21−ℓ2−ℓUℓL1t,CAn−L1t,CAn−1+2ϑ2−ϑUϑ×∫0tL1y,CA−L1y,CAn−1dy,≤21−ϑ2−ϑUϑμ1CAn−CAn−1+2ϑ2−ϑUϑμ1CAn−CAn−1t.
Following the technique, we proceed as(43)M1nt≤21−ϑ2−ϑUϑ+2ϑ2−ϑUϑtn+1μ1n+1a.
Then, the following is obtained att0:
(44)M1nt≤21−ϑ2−ϑUϑ+2ϑ2−ϑUℓt0n+1μ1n+1a.
Eq. (44) implies that
(45)M1nt⟶0,n⟶∞.
Following the same procedure, we obtain thatM2nt,M3nt,M4nt,andM5nt approaches to 0 as n tends to ∞.In the next step, we focus on the uniqueness of the solution of system (4); on contrast, we assume that CA1t,CB1t,CC1t,CA1t is another solution of system (4); then, we have
(46)CAt−CA1t=21−ϑ2−ϑUϑL1t,CA−L1t,CA1+2ϑ2−ϑUϑ×∫0tL1y,CA−L1y,CA1dy.Applying the properties of norm, the above (46) converted into the following:
(47)CAt−CA1t≤21−ϑ2−ϑUϑL1t,CA−L1t,CA1+2ϑ2−ϑUϑ×∫0tL1y,CA−L1y,CA1dy.Here, Lipschitz condition of kernel gives the following:(48)CAt−CA1t≤21−ϑ2−ϑUϑμ1CAt−CA1t+2ϑ2−ϑUϑ×∫0tμ1tCAt−CA1tdy,which gives
(49)CAt−CA1t1−21−ϑ2−ϑUϑμ1−2ϑ2−ϑUϑμ1t≤0.Theorem 8.
There exists a unique solution of breast cancer system (4) if
(50)1−21−ϑ2−ϑUϑμ1−2ϑ2−ϑUϑμ1t>0.Proof.
For the required result, we assume that the above condition (50) holds true; then, (49) implies that
(51)CAt−CA1t=0.As a result of this, we have the following:
(52)CAt=CA1t.
Following similar steps, we attain(53)CBt=CB1t,CCt=CC1t,CDt=CD1t,CEt=CE1t.


## 4. Simulations and Discussions

In this section of the article, we perform several simulations to examine the complex dynamics of our proposed breast cancer model with adverse reaction of chemotherapy treatment at population level of patients in a hospital through the Adams-Bashforth two-step method [?]. These simulations are important in order to identify the most significant input parameters that highly disturb the population level of cancer patients. The dynamical behaviour of the fractional breast cancer system is investigated numerically to provide more accurate picture of breast cancer chemotherapy patients. The values of parameters are assumed for simulation purposes in numerical analysis of the system.Here, we perform seven simulations to conceptualize the effect of the input parameters on the dynamical behaviour of the system. Figures3 and 4 illustrate the dynamics of breast cancer with the variation of fractional-order ϑ. We observed that the index of memory ϑ has significant influence on the solution pathway of breast cancer model and the control of ϑ can highly control the dynamics of breast cancer in all the subgroups. The solution pathway of the proposed fractional-order model (4) of breast cancer with the variation of input parameter δAD is shown in Figure 5. We noticed in the second scenario that the increase of parameter δAD decreases the level of CA and CB while increases the level of CC,CD, and CE. In this case, the increase of the level of subgroups CC and CE is critical in the sense to increase the cancer-induced and cardiac death rates in the patients. In the third scenario presented in Figure 6, we illustrate the dynamical behaviour of our proposed fractional-order model (4) of breast cancer with the variation of input parameter δBD. It is clear that the increase of this parameter δBD will decrease the number of patients CB and will increase the number of patients in the subgroup CD which is effective. However, the population level of CB and CE increases slightly. In Figure 7, we represent the solution pathway of the fractional-order model (4) with the variation of input parameter δBE. We observed that the increase of this parameter decreases the level of cancer patients in CB and CC which implies that the death rate of cancer will be decreased. However, the increase of δBE increases the level of patients in CE, and as a result, the cardiac death rate will be increased. We perform numerical comparison of ordinary model (1) and fractional model (4) in Figure 8 with fractional order 0.9 which illustrate that fractional results are better than the ordinary one in the sense to decrease the level of different cancer stages.Figure 3
Time series of our proposed fractional-order model (4) of breast cancer with the variation of fractional-order ϑ, i.e., ϑ=0.6,0.8,1.
(a)(b)(c)(d)(e)Figure 4
Time series of our proposed fractional-order model (4) of breast cancer with the variation of fractional-order ϑ, i.e., ϑ=0.4,0.5,0.7.
(a)(b)(c)(d)(e)Figure 5
Illustration of dynamical behaviour of our proposed fractional-order model (4) of breast cancer with the variation of input parameter δAD, i.e., δAD=0.0351,0.0651,0.0951.
(a)(b)(c)(d)(e)Figure 6
Illustration of dynamical behaviour of our proposed fractional-order model (4) of breast cancer with the variation of input parameter δBD, i.e., δBD=0.242,0.442,0.642.
(a)(b)(c)(d)(e)Figure 7
Time series of our proposed fractional-order model (4) of breast cancer with the variation of fractional-order δBE, i.e., δBE=0.531,0.631,0.731.
(a)(b)(c)(d)(e)Figure 8
Competitive analysis of ordinary model and fractional model where the curve of blue dots illustrates ordinary model while the curve of red dots illustrates the fractional model of breast cancer.
(a)(b)(c)(d)(e)In the fifth scenario presented in Figure9, we demonstrate the time series of breast cancer model with the variation of input parameter δCD. It is noticed that the increase of δCD increases the cardiac death rate while decreases the cancer death rate. In Figures 10 and 11, we visualize the solution pathway of fractional-order breast cancer model with the variation of δCE and δDE, respectively. We observed that similar to the fifth scenario, the level of cancer patient decreases which leads to a decrease in the cancer death rate, while the level of cardiotoxic patient increases which increase cardiac death rate. The roles of parameters are visualized through these simulations, and one can easily understand how to lower cancer and cardiac death rate in the hospital of cancer patient. Furthermore, we observed that the control of fractional order can control the number of cancer patients, and as a result, the cancer and cardiac death rate will be controlled. Therefore, the index of memory ϑ is suggested to the policy makers and medical experts for the control of cancer and cardiac death.Figure 9
Illustration of dynamical behaviour of our proposed fractional-order model (4) of breast cancer with the variation of input parameter δCD, i.e., δCD=0.358,0.458,0.558.
(a)(b)(c)(d)(e)Figure 10
Time series of our proposed fractional-order model (4) of breast cancer with the variation of fractional-order δCE, i.e., δCE=0.296,0.396,0.496.
(a)(b)(c)(d)(e)Figure 11
Time series of our proposed fractional-order model (4) of breast cancer with the variation of fractional-order δDE, i.e., δDE=0.311,0.422,0.533.
(a)(b)(c)(d)(e)


## 5. Conclusion

It is eminent that the treatment and vaccination play a vital contribution to overcome the infectious diseases. However, sometimes the treatment and vaccination are not fully fruitful and have adverse reactions on the patient. Chemotherapy treatment is the most common treatment for breast cancer where the use of drugs affects the heart of a patient which leads to cardiotoxicity. In this article, we formulated a mathematical model for breast cancer with chemotherapy treatment at population level of patients in fractional framework to investigate the adverse reaction of chemotherapy on the heart of a patient. The proposed fractional model of breast cancer is then investigated for the basic properties through fractional calculus. The existence and uniqueness of the proposed breast cancer system are investigated through fixed point theory. Moreover, we highlighted the dynamical behaviour of our fractional system of breast cancer with the help of the Adams-Bashforth method. We have shown the dynamical behaviour of different stages of breast cancer model with variation of fractional-orderϑ. Our findings suggest that the index of memory ϑ is an important input parameter and recommended to the policy makers. Finally, the dynamical behaviour of different stages of breast cancer is highlighted numerically to show the influence of several input parameters on the time series of breast cancer. The impact of input parameters of the breast cancer system has been illustrated the most critical factors highlighted for the control and prevention of breast cancer. In future study, we will extend our model using delay differential equations to investigate the significance of time delay in breast cancer dynamics. We will also incorporate some control measures to lessen the progression of breast cancer to different stages in our future work.



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