# Dynamics of a Breast Cancer Model for Neutropenia Case due to Chemotherapy Effects

Authors: M. Ivan Ariful Fathoni; Fajar Adi-Kusumo; Gunardi Gunardi; Susanna Hilda Hutajulu
Journal: International Journal of Differential Equations
 (2021)

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## Abstract

Breast cancer is a type of carcinoma with a high prevalence. The treatment of breast cancer through chemotherapy can cause a risk to healthy cells throughout the body. The neutrophil is one of the cells that is influenced by chemotherapy drugs. Chemotherapy-induced neutropenia is one of the most common toxic effects experienced by patients and often threatens chemotherapy to use efficiency. In this paper, we introduce an interaction model between blood components, i.e., neutrophil, lymphocytes, and albumin, with chemotherapy drugs. The model is important to understand the neutropenia effect due to chemotherapy in mathematical perspective and to calculate breast cancer patients’ survival level. Our model is a four-dimensional system of the first-order ODE with 13-dimensional parameter space. We focus our study for understanding the steady-state conditions and the bifurcations when the parameter values are varied. Here, we also study the role of albumin for reducing the neutropenia effects for breast cancer patients mathematically, where the results can be used as an alternative solution for treating neutropenia in a breast cancer case.


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## Body

## 1. Introduction

Breast cancer is a type of a high prevalence carcinoma. It can occur in both men and women, where the higher prevalence is in women. Data from WHO in 2018 showed that the number of new breast cancer cases was 2,088,849 or 11.6% of the world’s total number of cancer cases. This number is only different from 5,027 cases of lung cancer, which occupies the first position. Based on this number, 626,679 of them were the death cases. Asia is in the top rank of the number of breast cancer incidence and mortality in the world [1]. The disease is treatable if it is detected earlier and is still in its earler stages. The higher the stage implies, the smaller the life expectancy is. There is no cancer that has a 100% mortality rate; it means that the treatment can be done to get a cure, especially if treatment is started earlier [2].There are several studies being carried out in cancer modeling and treatment. Schättler et al. [3] developed a mathematical model with minimal parameters for low-dose metronomic chemotherapy. The model takes into account the angiogenic signaling between the tumor and the vasculature and the tumor inhibitory effects of tumor-immune system interactions. Liu and Yang [4] developed a cancer treatment model with radiotherapy followed by chemotherapy. Jordão and Tavares [5] studied the dynamics of healthy and diseased cells. The model for healthy cells is based on the p53 gene network, cell cycle control mechanism, the MAP kinase pathway, and the PI3K or Akt pathway, a cancer cell model embodied by deregulation in a healthy cell model. Solis-Perez et al. in [6] created a mathematical model that considers the population dynamics among cancer stem cells, tumor cells, healthy cells, the effects of excess estrogen, and the body’s natural immune response to cell populations.Several mathematical models that are more specific to breast cancer have been introduced before. Enderling et al. in [7] modeled the breast cancer development, the local treatment, and the recurrence. Furthermore, Prameswari and Adi-Kusumo [8] constructed a model of the phase transition regulation on cell cycle to characterize breast cancer cells’ growth in the earlier stage, which is indicated by the anomalies of cell proliferation. The model knows the behavior changes in cell concentration used as an early indicator of cancer growth. In the same year, Kusuma and Adi-Kusumo [9] introduced a mathematical model of DNA damage as the estrogen response involving the G1/S transition phase in the cell cycle. The model is a foundation to understand the initial behavior of breast cancer.Treatment of breast cancer through chemotherapy can pose a risk to healthy cells throughout the body. The neutrophil is white blood cells that have a short life span (6–10 hours). They provide the first line of defense part of the body’s immune system. Neutrophils are the most common types of leukocytes found in human peripheral blood. The neutrophil concentration is about 60% of white blood cells in human blood. They are major cellular contributors to inflammation, mediating the initial phases of the inflammatory response. They can eliminate microbes through various mechanisms and are essential for the defense of the human body. Acquired neutrophil disorders are usually caused by cytotoxic chemotherapy or immunosuppressive agents. Primary neutrophil disorders usually result from the deactivation of individual gene mutations that result in impaired neutrophil number or function [10].Neutropenia is defined as a decreasing number of neutrophils in the circulating blood. Chemotherapy-induced neutropenia is one of the most common toxic effects experienced by patients and often threatens the efficiency of chemotherapy use. Neutropenia usually occurs 7 to 10 days after chemotherapy. It can increase the risk of a complication of infection and fever known as febrile neutropenia (FN) [11]. In several papers, they consider the prediction of the occurrence of neutropenia in breast cancer patients. A mathematical model for optimizing neutropenia minimization was introduced in the study by Craig et al. [12]. A model that was developed is based on the characterization of neutrophil production in the bone marrow. The model was integrated with a pharmacokinetic and pharmacodynamic (PKPD) model of a chemotherapy agent. In another case with the model proposed by Craig et al. [12], we will also look at a model of neutrophil interaction with variables that, according to the data, affect the survival of breast cancer patients, namely, lymphocytes, chemotherapy drugs, and albumin.Lymphocytes are white blood cells that play a role in maintaining the immune system. Lymphocytes are one of the main components of the immune system and are indispensable in the immune response to cancer. Lymphocytes are divided into two types, namely, T lymphocytes and B lymphocytes. T lymphocytes kill bacteria and viruses, while B lymphocytes function to make antibodies that fight antigens. The neutrophil-lymphocyte ratio (NLR) has been evaluated in various types of cancer. It is proposed as a marker of inflammation to predict outcomes. Increased NLR in the peripheral blood of patients with cancer could be a sign of a manifestation of systemic inflammation induced by cancer cells [14]. Hartono’s study [15] discussed the assessment of the neutrophil-lymphocyte and albumin ratio before and after chemotherapy. The results of the study demonstrated a reduction in NLR and an increase in albumin after chemotherapy.Albumin is the most substantial protein in blood plasma. It is maintaining oncotic pressure, transporting fatty acids to the liver, transporting drugs, shortening the half-life of the drug, transporting bilirubin, bindingCa2 + ions a buffer solution, transporting thyroid hormones, and transporting other hormones, especially those that can dissolve in fat. Albumin also functions as a transport for various substances, including bilirubin, fatty acids, metals, ions, hormones, and drugs. It plays a vital role in the regeneration and repair process of cells, as an agent that forms cell bonds. Albumin also helps to give signal for the immune system about signs of damage to body cells.Chemotherapy causes hematopoietic progenitor cells in the bone marrow to experience depletion and acute myelosuppression [16]. In conditions of severe hemopoiesis stress, stem cells can repair themselves, proliferate, and differentiate into various blood cell series in the periphery. However, suppose the self-renewal ability of hemopoiesis stem cells is impaired. In that case, there will be permanent damage to the hematopoietic system resulting in bone marrow failure. It can occur when the bone marrow is exposed to high doses of chemotherapy agents or radiation [17]. The effect of neutropenia appears to be higher than that of cytopenia of the erythroid and platelet series because the composition of neutrophil precursors occupies the most significant number in the bone marrow, which is more than 50% [18].In this paper, we consider a new mathematical model on breast cancer therapy that includes the neutropenia effects due to the chemotherapy. Our model is a 4-dimensional system of first-order ODE based on the interaction between the concentrations of neutrophils (N), chemotherapy drugs (C), lymphocytes (L), and albumin (A). We focus our analysis to understand the role of parameters to the steady-state conditions of the system. In this case, we employ bifurcation analysis to study the existence and stability of the equilibria. It is important for calculating the survival level of the breast cancer patients.


## 2. Model Formulation

The interaction between the neutrophils (N), chemotherapy drugs (C), lymphocytes (L), and albumin (A) is shown in Figure1. The interaction between those substances is created by the results of some medical papers (see [10, 15–18]), and the correlation analysis of the data from National Health and Nutrition Examination Survey (see [14]).Figure 1
The transfer diagram of substances. The production, degradation, and interaction are denoted as arrow headed line. The inhibition is denoted as blunt arrow.Our model is formulated as follows:(1)dNdt=α1N−σNL1−θN−β1CN+δ1AN,dCdt=α2C−γCN+L,dLdt=α3L−σNL1−θL−β2CL+δ2AL,dAdt=α4A1−Ak+β3CA.By the transfer diagram shown in Figure1, we construct a mathematical model that shows the interactions between the neutrophil, lymphocyte, and albumin concentrations due to the appearance of chemotherapy drugs concentration (see system (1)). System (1) is a four- dimensional system of the first-order ODE in indimensional parameter space. The parameters used in the model are assumed to be positive, with the information in Table 1.Table 1
Parameters description.
ParameterDescriptionUnitα1Neutrophil production rate1/dayα2Chemotherapy drugs absorbing rate1/dayα3Lymphocyte production rate1/dayα4Albumin production rate1/dayβ1Neutrophil reaction rate due to chemotherapy regimens1/%/dayβ2Lymphocyte reaction rate due to chemotherapy regimens1/%/dayβ3Albumin reaction rate due to chemotherapy regimen1/%/dayδ1Albumin and neutrophils reaction rate1/%/dayδ2Albumin and lymphocytes reaction rate1/%/dayσNeutrophils and lymphocytes reaction rate%/dayγDecay of chemotherapy regimens%/dayθLeucocyte saturation parameter%kMaximum capacity of albumin in blood plasma%


## 3. Existence of Equilibrium Solutions

System (1) has eight equilibrium points, but we consider the analysis of the first seven equilibrium points. Calculating the last equilibrium point is very complicated, so we left it as an open problem in this paper. The existences of those equilibrium points are presented in Theorem 1 and Theorem 2.Theorem 1 (the existences of equilibrium pointsT1 to T6 of System (1)).
(i)
The equilibrium pointT1=α3−θα1+σ/−θ2α1α3+σ2,0,α1−θα3+σ/−θ2α1α3+σ2,0 exists if σ>θα1 and σ>θα3(ii)
The equilibrium pointsT2=0,α3/β2,γ/α2,0 and T3=γ/α2,α1/β1,0,0 exist for all parameter values(iii)
The equilibrium pointT4=kδ1+α1θ−σkδ2+α3/kδ2+α3kδ1+α1θ2−σ2,0,kδ1+α1kδ2+α3θ−σ/kδ2+α3kδ1+α1θ2−σ2,k exists if it satisfies the conditions in Table 2(iv)
The equilibrium pointT5=0,−kδ2+α3α4/kβ3δ2−α4β2,γ/α2,−kα3β3+α4β2/kβ3δ2−α4β2 exists if kβ3δ2<α4β2(v)
The equilibrium pointT6=γ/α2,−kδ1+α1α4/kβ3δ1−α4β1,0,−kα1β3+α4β1/kβ3δ1−α4β1 exists if kβ3δ1<α4β1Table 2
Terms of existencesT4.
No.Terms of existences1α1<−δ1θk−σ/θ,α3<−δ2θk−σ/θ,δ1<σ/kθ,δ2<σ/kθ2α3<−δ1δ2θ2k2−σ2/kθ2δ1,δ1<σ/kθ,δ2<σ/kθ,−δ1θk−σ/θ<α1,−δ2θk−σ/θ<α33α3=−δ1δ2θ2k2−σ2/kθ2δ1,δ1<σ/kθ,δ2<σ/kθ,−δ1θk−σ/θ<α14δ1<σ/kθ,δ2<σ/kθ,−δ1δ2θ2k2−σ2/kθ2δ1<α3,−δ1θk−σ/θ<α15δ1=σ/kθ,δ2<σ/kθ,−δ2θk−σ/θ<α36δ1<σ2/δ2θ2k2,δ2<σ/kθ,σ/kθ<δ1,−δ2θk−σ/θ<α37δ1=σ2/δ2θ2k2,δ2<σ/kθ,−δ2θk−σ/θ<α38δ2<σ/kθ,σ2/δ2θ2k2<δ1,−δ2θk−σ/θ<α39δ2=σ/kθ,α3<−σδ1θk−σ/δ1θ2k,δ1<σ/kθ,−δ1θk−σ/θ<α110α3=−σδ1θk−σ/δ1θ2k,δ2=σ/kθ,δ1<σ/kθ,−δ1θk−σ/θ<α111δ2=σ/kθ,δ1<σ/kθ,−δ1θk−σ/θ<α1,−σδ1θk−σ/δ1θ2k<α312δ1=σ/kθ,δ2=σ/kθ13δ2=σ/kθ,σ/kθ<δ114α3<−δ1δ2θ2k2−σ2/kθ2δ1,δ1<σ2/δ2θ2k2,σ/kθ<δ2,−δ1θk−σ/θ<α115α3=−δ1δ2θ2k2−σ2/kθ2δ1,δ1<σ2/δ2θ2k2,σ/kθ<δ2,−δ1θk−σ/θ<α116δ1<σ2/δ2θ2k2,σ/kθ<δ2,−δ1θk−σ/θ<α1,−δ1δ2θ2k2−σ2/kθ2δ1<α317δ1=σ2/δ2θ2k2,σ/kθ<δ2,σδ2θk−σ/δ2θ2k<α118δ1<σ/kθ,σ/kθ<δ2,σ2/δ2θ2k2<δ1,−δ1θk−σ/θ<α119δ1=σ/kθ,σ/kθ<δ220σ/kθ<δ1,σ/kθ<δ2Theorem 2 (the existences of equilibrium pointsT7 of System (1)).
The equilibrium pointT7N7,C7,L7,A7 with C7=±wθα1−σ+2α2x+yα1−2xγσθ−σy/β1θ2α2x±w+y and N7=−±w+y/2α2θx,L7=2γθx±w+y/2α2θx,A7=0 where w=γ2θ4z2−4γα2θ3z2+θ24α22z2+2γσα2β1−β2z+4γ2σ2β1β2−8α2σθα2β1−β2z/2+γσβ1β2+σ2α22β1+β22, x=θα1β2−θα3β1+σβ1−σβ2, y=−γθ2α1β2+γθ2α3β1+2γσθβ2−σα2β1−σα2β2, and z=−α1β2+α3β1 has two possible existences due to value ±w. If w value is negative, then condition of existence is −σ2γθx−w+y/θ−2α2x+w−y<α1<−σ2γθx−w+y/θ−2α2x+w−y,w>0,x>0,y>0. If w value is positive, then condition of existence is 2γθx+w+yσ/θ2α2x+w+y<α1<2γθx+w+yσ/θ2α2x+w+y,w>0,x>0,y>0.
Then, the results of Theorem1 are summarized in Table 3 in the appendix. Based on the results of the theorem, we show the area in parameter spaces that two or more equilibria coexist in Table 4.Table 3
The existences of equilibrium points.
Equilibrium pointsTnNn,Cn,Ln,AnTerms of existencesT1=α3−θα1+σ/−θ2α1α3+σ2,0,α1−θα3+σ/−θ2α1α3+σ2,0σ>θα1 and σ>θα3.T2=0,α3/β2,γ/α2,0Always existT3=γ/α2,α1/β1,0,0Always existT4=kδ1+α1θ−σkδ2+α3/kδ2+α3kδ1+α1θ2−σ2,0,kδ1+α1kδ2+α3θ−σ/kδ2+α3kδ1+α1θ2−σ2,kTable2T5=0,−kδ2+α3α4/kβ3δ2−α4β2,γ/α2,−kα3β3+α4β2/kβ3δ2−α4β2kβ3δ2<α4β2T6=γ/α2,−kδ1+α1α4/kβ3δ1−α4β1,0,−kα1β3+α4β1/kβ3δ1−α4β1kβ3δ1<α4β1Table 4
Coexistence of the equilibrium pointsT1 to T6 of System (1).
The area of coexistenceEquilibriumσ|δ1kθ≥σ>θα1∧δ2kθ≥σ>θα3,δ1<α4β1/kβ3,δ2<α4β2/kβ3,σ∈ℛT1,T2,T3,T4,T5,T6σ|δ1kθ≥σ>θα1∧δ2kθ≥σ>θα3,δ1≥α4β1/kβ3,δ2<α4β2/kβ3,σ∈ℛT1,T2,T3,T4,T5σ|δ1kθ≥σ>θα1∧δ2kθ≥σ>θα3,δ1<α4β1/kβ3,δ2≥α4β2/kβ3,σ∈ℛT1,T2,T3,T4,T6σ|δ1kθ≥σ>θα1∧δ2kθ≥σ>θα3,σ∈ℛT1,T2,T3,T4σ|σ>θα1∧σ>θα3,σ∈ℛT1,T2,T3σ|σ∈ℛT1,T2∗ means that the condition for the existence of T4 is number 20 in Table 2.


## 4. Stability Analysis

We apply the linearization method to determine the stability of the equilibrium point of System (1), and we have the Jacobian matrix as follows:(2)JT=j11−β1NσNθN−1δ1NγCL+N2j22γCL+N20σLLθ−1−β2Lj33δ2L0β3A0j44,where j11=α1+σL/θN−1−σNLθ/θN−12−β1C+δ1A, j22=α2−γ/L+N, j33=α3+σN/Lθ−1−σNLθ/Lθ−12−β2C+δ2A, and j44=β3C+α4k−2α4A/k.The eigenvalues can be obtained from the characteristic equationJT−λI=0 evaluated at each point of equilibrium. The stability analysis discussed in this study is only at the equilibrium point T1 to T6. The stability analysis of the equilibrium point will be carried out by considering the numerical simulation in special cases. The stability analysis of the equilibrium point T4 will be carried out by considering the numerical simulation in special cases.


### 4.1. Equilibrium PointT1

Linearization of System (1) near the equilibrium T1 produces the characteristic equation as follows:(3)ab−α4+λ/α1+α3σ−2α3α1θ−θα3+σ−θα1+σσ2=0,where a=γσ2+α1+α3λ−α2σ−θα1α3γθ+2λ−2α2 and b=λ2−α1α3σ4−−α3α12+λ2−2λα3−α32α1+λ2α3θσ3+θ2λα1α3λ−2α1−2α3σ2+θ3α1λ−α3α12−α32α1+λα32α3σ+θ4α13α33.From (3), it can be seen that there is a positive eigenvalue, i.e., λ1=α4. It can be concluded that the equilibrium point T1 is unstable.


### 4.2. Equilibrium PointT2

The characteristic equation of the linearization of System (1) near the equilibrium point T2 is −λ−α1α2+γσβ2+β1α3α2λ2+α2α3α4−λβ2+α3β3/α2β22=0, so that its eigenvalues are λ1=−α2α3,λ2=−−α2α3,λ3=α3β3+α4β2/β2, and λ4=−γσβ2−α1β2α2+β1α3α2/α2β2. In this case, we have a positive definite eigenvalue, so the equilibrium point of T2 is unstable.


### 4.3. Equilibrium PointT3

The characteristic equation of the linearization of System (1) near the equilibrium point J3 is −λ2+α1α2α4−λβ1+α1β3λ−α3α2+γσβ1+α1β2α2/α2β12=0 where the eigenvalues are λ1=−α1α2,λ2=−−α1α2,λ3=α1β3+α4β1/β1, and λ4=λ=−γσβ1+α1β2α2−β1α3α2/α2β1. The positive eigenvalues indicate that the equilibrium point of T3 is unstable.


### 4.4. Equilibrium PointT4

The Jacobian matrix obtained from the linearization of System (1) near the equilibrium point T4 is(4)J4=−def1+eσ−β1ede+fσ+ef−defδ1ede+fσ+ef0−γe+fσ+ef−σd+c+2de+2σd+α200−cfe−β2cfe+fσ+ef−dfe1+eσcδ2fe+fσ+ef0β3k0−α4,where c=kδ1+α1,d=kδ2+α3, e=θc−σ,andf=θd−σ. It is assumed that the values of c,d,e, and f are positive. The characteristic equation formed is(5)ghα4+λσ2fσc−2de−σde=0,where g=−cλσ+cσα2+2deλ−2deα2+dλσ−dσα2+efγ+eγσ+fγσ and h=cdσ2fe−d2e3f−2d2e2fσ−d2efσ2−de3λσ−e2dλσ2−def2λσ−df2λσ2−efλ2σ2which has the eigenvaluesλ1=−α4,λ2,3=−de3+de2σ+def2+df2σ±m/efσ where m=4cde2f2σ2+d2e6+2d2e5σ−2d2e4f2+d2e4σ2−4d2e3f2σ+d2e2f4−2d2e2f2σ2+2d2ef4σ+d2f4σ2 and λ4=α2+efγ+eγσ+fγσ/σc−2de−σd. Based on the equations of e,f, and λ4, the stability conditions for the equilibrium point of T4 are obtained, namely, θc>σ,θd>σ, and α2<efγ+eγσ+fγσ/σc−2de−σd.


### 4.5. Equilibrium PointT5

Linearization of System (1) near the equilibrium point T5 obtains the characteristic equation as follows:(6)λ4+λ3−qoα2+nα4o+λ2−nα4qα2o2+β2α4pα2o+λ−α4β2pqo2+α2α4pno−α4pnqo2=0,where n=α3β3+α4β2,o=α4β2−kβ3δ2,andp=kδ2+α3,q=α2kδ1n−β1α4p+oα1−γoσ, all of which are assumed to be positive. The terms of the stability of this point are analyzed using the Routh–Hurwitz criteria, and we obtain(7)0<−α4pnqo2,0<α4nα2−qoα2,0<−nop−α4β2+oα23+nqα2α4−q2α4α2o2α4nα2−q,0<p−α4β2+ono2pα24α4−opqα23α4β2+nq2α4α2−q3op−α4β2+oα23+nqα2α4−q2o2.


### 4.6. Equilibrium PointT6

Linearization of System (1) near the equilibrium T6 produces the characteristic equation as follows:(8)λ4+λ3−vα2s+rα4s+λ2−rα4vα2s2+α4uβ1α2s+λ−α4uβ1vs2+α4urα2s−α4urvs2=0,where r=α1β3+α4β1,s=−kβ3δ1+α4β1,u=kδ1+α1,andv=α2kδ2r−α4β2u+α3s−γsσ, all of which are assumed to be positive. The terms of the stability of this point are analyzed using the Routh–Hurwitz criteria with the result as follows:(9)0<−α4urvs2,0<rα4α2−vα2s,0<−rsu−α4β1+sα23+rvα2α4−v2α4α2s2rα4α2−v,0<u−α4β1+srs2uα24α4−suvα23α4β1+rv2α2α4−v3su−α4β1+sα23+rvα2α4−v2s2.The stability of the equilibrium pointsT2 and T5 is very important for neutropenia. If those equilibria are stable, the neutrophils will decrease and go to zero. This situation triggers the neutropenia case due to the chemotherapy. The other important equilibrium point is T6. The stability of this equilibrium shows the reduction of lymphocyte which is one of the main component in the immune system against the cancer.


## 4.1. Equilibrium PointT1

Linearization of System (1) near the equilibrium T1 produces the characteristic equation as follows:(3)ab−α4+λ/α1+α3σ−2α3α1θ−θα3+σ−θα1+σσ2=0,where a=γσ2+α1+α3λ−α2σ−θα1α3γθ+2λ−2α2 and b=λ2−α1α3σ4−−α3α12+λ2−2λα3−α32α1+λ2α3θσ3+θ2λα1α3λ−2α1−2α3σ2+θ3α1λ−α3α12−α32α1+λα32α3σ+θ4α13α33.From (3), it can be seen that there is a positive eigenvalue, i.e., λ1=α4. It can be concluded that the equilibrium point T1 is unstable.


## 4.2. Equilibrium PointT2

The characteristic equation of the linearization of System (1) near the equilibrium point T2 is −λ−α1α2+γσβ2+β1α3α2λ2+α2α3α4−λβ2+α3β3/α2β22=0, so that its eigenvalues are λ1=−α2α3,λ2=−−α2α3,λ3=α3β3+α4β2/β2, and λ4=−γσβ2−α1β2α2+β1α3α2/α2β2. In this case, we have a positive definite eigenvalue, so the equilibrium point of T2 is unstable.


## 4.3. Equilibrium PointT3

The characteristic equation of the linearization of System (1) near the equilibrium point J3 is −λ2+α1α2α4−λβ1+α1β3λ−α3α2+γσβ1+α1β2α2/α2β12=0 where the eigenvalues are λ1=−α1α2,λ2=−−α1α2,λ3=α1β3+α4β1/β1, and λ4=λ=−γσβ1+α1β2α2−β1α3α2/α2β1. The positive eigenvalues indicate that the equilibrium point of T3 is unstable.


## 4.4. Equilibrium PointT4

The Jacobian matrix obtained from the linearization of System (1) near the equilibrium point T4 is(4)J4=−def1+eσ−β1ede+fσ+ef−defδ1ede+fσ+ef0−γe+fσ+ef−σd+c+2de+2σd+α200−cfe−β2cfe+fσ+ef−dfe1+eσcδ2fe+fσ+ef0β3k0−α4,where c=kδ1+α1,d=kδ2+α3, e=θc−σ,andf=θd−σ. It is assumed that the values of c,d,e, and f are positive. The characteristic equation formed is(5)ghα4+λσ2fσc−2de−σde=0,where g=−cλσ+cσα2+2deλ−2deα2+dλσ−dσα2+efγ+eγσ+fγσ and h=cdσ2fe−d2e3f−2d2e2fσ−d2efσ2−de3λσ−e2dλσ2−def2λσ−df2λσ2−efλ2σ2which has the eigenvaluesλ1=−α4,λ2,3=−de3+de2σ+def2+df2σ±m/efσ where m=4cde2f2σ2+d2e6+2d2e5σ−2d2e4f2+d2e4σ2−4d2e3f2σ+d2e2f4−2d2e2f2σ2+2d2ef4σ+d2f4σ2 and λ4=α2+efγ+eγσ+fγσ/σc−2de−σd. Based on the equations of e,f, and λ4, the stability conditions for the equilibrium point of T4 are obtained, namely, θc>σ,θd>σ, and α2<efγ+eγσ+fγσ/σc−2de−σd.


## 4.5. Equilibrium PointT5

Linearization of System (1) near the equilibrium point T5 obtains the characteristic equation as follows:(6)λ4+λ3−qoα2+nα4o+λ2−nα4qα2o2+β2α4pα2o+λ−α4β2pqo2+α2α4pno−α4pnqo2=0,where n=α3β3+α4β2,o=α4β2−kβ3δ2,andp=kδ2+α3,q=α2kδ1n−β1α4p+oα1−γoσ, all of which are assumed to be positive. The terms of the stability of this point are analyzed using the Routh–Hurwitz criteria, and we obtain(7)0<−α4pnqo2,0<α4nα2−qoα2,0<−nop−α4β2+oα23+nqα2α4−q2α4α2o2α4nα2−q,0<p−α4β2+ono2pα24α4−opqα23α4β2+nq2α4α2−q3op−α4β2+oα23+nqα2α4−q2o2.


## 4.6. Equilibrium PointT6

Linearization of System (1) near the equilibrium T6 produces the characteristic equation as follows:(8)λ4+λ3−vα2s+rα4s+λ2−rα4vα2s2+α4uβ1α2s+λ−α4uβ1vs2+α4urα2s−α4urvs2=0,where r=α1β3+α4β1,s=−kβ3δ1+α4β1,u=kδ1+α1,andv=α2kδ2r−α4β2u+α3s−γsσ, all of which are assumed to be positive. The terms of the stability of this point are analyzed using the Routh–Hurwitz criteria with the result as follows:(9)0<−α4urvs2,0<rα4α2−vα2s,0<−rsu−α4β1+sα23+rvα2α4−v2α4α2s2rα4α2−v,0<u−α4β1+srs2uα24α4−suvα23α4β1+rv2α2α4−v3su−α4β1+sα23+rvα2α4−v2s2.The stability of the equilibrium pointsT2 and T5 is very important for neutropenia. If those equilibria are stable, the neutrophils will decrease and go to zero. This situation triggers the neutropenia case due to the chemotherapy. The other important equilibrium point is T6. The stability of this equilibrium shows the reduction of lymphocyte which is one of the main component in the immune system against the cancer.


## 5. Numerical Simulations

Before treatment, all breast cancer patients will perform a complete blood laboratory examination, including examining the concentration of neutrophils, lymphocytes, and albumin. If the patient is indicated that there is breast cancer, then the patient will be treated medically through chemotherapy. However, chemotherapy drugs also consider the side effects of neutropenia. Mathematical models are needed to understand and to anticipate the side effects of neutropenia. Simulations of the changes in neutrophils, chemotherapy drugs, lymphocytes, and albumin concentration during chemotherapy are performed to predict neutropenia.The simulation was carried out based on the assumption that the concentration of neutrophils was 0.8 (80%) and that of lymphocytes was 0.2 (20%) from white blood cells, and that of albumin was 0.7 (70%) from blood plasma. Chemotherapy concentration is assumed to be 0.4 (40%) from the maximum limit capacity of chemotherapy drugs in the body. Medically, neutrophils, albumin, and neutrophils in human blood circulation cannot be zero, so the first case used in this simulation is directed towards the equilibrium point ofT4. By using the parameter values α1=0.3,α2=0.3,α3=0.2,α4=0.33,β1=0.7,β2=0.6,β3=0.75,σ=0.53,θ=0.95,k=0.8,δ1=0.6,δ2=0.6,andγ=0.9, we obtained dynamic changes in the model as shown in the curves of Figures 2 and 3. Numerical simulations are also used to verify the results of the dynamic analysis that has been carried out. Based on the predetermined parameter values, using four different initial conditions, the solution will go to the equilibrium point T4, as shown in Figure 4.Figure 2
The dynamics of change in N, C, L, and A over time untilt = 10 in the first case.Figure 3
The dynamics of change in N, C, L, and A over time untilt = 40 in the first case.Figure 4
Phase portrait of the system solution with four different initial conditions.Figures2 and 3 show the dynamic changes of the four observed variables, where neutrophils at the beginning of chemotherapy increased until the second day and then decreased significantly to equilibrium at 0.725. It is consistent with the medical facts of the side effects of neutropenia after chemotherapy. The chemotherapy concentration at the beginning of chemotherapy is 0.4, gradually decreasing until it reaches zero. Albumin plays a role in forming white blood cells; in this case, the albumin has increased to 0.96 before dropping to 0.8 on day 25. The increase in albumin correlates with an increase in lymphocytes. Lymphocytes that play a role against cancer cells and chemotherapy have increased to 0.456. Based on these dynamic changes, it is known that the neutrophil-lymphocyte ratio has decreased, where at the beginning of chemotherapy, it has a ratio of 4. After chemotherapy, it drops to 1.6.The conditions in Figures2 and 3 show the occurrence of neutropenia, where the neutrophil concentration is significantly reduced. One alternative to prevent neutropenia is to reduce albumin. In the second case, we take the same parameter value as first case for α1,α2,α3,α4,β1,β2,β3,σ,θ,δ1,δ2, and γ, by changing k value. Albumin can be reduced by reducing its maximum capacity (k) to 0.68 so that the albumin concentrates towards the steady-state at this value. In this case, the neutrophils raised at the beginning of chemotherapy will not decrease significantly, and it reaches the steady-state, which is at 0.8. The lymphocytes continued to increase towards the final state at 0.31. The neutrophil-lymphocyte ratio in this second case was reduced to 2.58 after chemotherapy. This second case simulation is shown in the graph in Figure 5.Figure 5
The dynamics of change in N, C, L, and A over time in the second case.In this section, we can find dynamic simulations of the interaction of blood components, represented by neutrophils, lymphocytes, and albumin, with the effects of chemotherapy drugs. Based on a mathematical model simulation, it can be seen that the regulation of albumin concentration in blood has a role in preventing and predicting the side effects of neutropenia due to chemotherapy drugs. So, if the patient’s condition requires chemotherapy treatment, medical experts can reduce the level of albumin concentration in the blood. It is an alternative solution to prevent neutropenia based on a mathematical approach.


## 6. Concluding Remarks

Breast cancer is a dangerous disease with the side effects of treatment. A side effect of chemotherapy for breast cancer that needs to be watched out for is neutropenia, where the neutrophils in the white blood cells are reduced to being abnormal. The dynamics of the three components of blood cells with chemotherapy concentrations have been constructed using a differential equation system (1) based on data and medical facts described previously.Simulations are carried out to determine changes in the number of neutrophils, chemotherapy, lymphocytes, and albumin concentrations from time to time if given the initial conditions. By knowing the dynamics that will occur, we can control changes in the concentration of the three components of blood cells to return to normal after chemotherapy. This study formed a dynamic model of blood components and chemotherapy based on hospital data and medical literature.The mathematical model constructed in this study can describe the dynamics of changes in the concentration of neutrophils, chemotherapy, lymphocytes, and albumin in the patient’s body to predict the possibility of neutropenia. One alternative in preventing neutropenia based on the model in this study is to reduce the albumin concentration. The results of this study can be used as an alternative solution for treating neutropenia in breast cancer patients, in mathematical point of view.



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*Source: 3401639-2021-10-20.xml*