ANALYSING THE EFFECT OF TRASTUZUMAB TREATMENT ON BREAST CANCER STAGES AND CARDIAC FUNCTION: A MATHEMATICAL MODELING AND NUMERICAL SIMULATION

. Breast cancer is the second most common cause of death among women worldwide. Trastuzumab is the ﬁrst humanised monoclonal antibody against HER2-positive metastatic breast cancer. However, the most serious side effect of trastuzumab is cardiotoxicity, which has become a limiting factor in the drug’s safe use. In this study, we investigated the effect of trastuzumab treatment on breast cancer stages and cardiac function. Therefore, we constructed a mathematical model based on breast cancer patients. The model was created using a differential equation system, and equilibrium point and stability analysis were employed to study the associated temporal dynamics. The stability of the equilibrium point was analysed using the Routh-Hurwitz criteria, which identiﬁed an asymptotically stable equilibrium point. To valudate these ﬁndings, numerical simulations were performed, which demonstrated that the equilibrium point is always stable regardless of the initial conditions. Finally, our results suggest that the ﬁve sub-populations of patients will reach a stable state upon reaching the equilibrium point.


INTRODUCTION
Breast cancer is the most common malignant disease and among the most frequent causes of cancer mortality in females worldwide [1]. Breast cancer has the highest occurrence rate when compared to other types of cancer. It occurs when breast tissue and cells grow uncontrollably, resulting in an abnormal breast shape. Breast cancer is the world's second most common type of cancer, after lung cancer, and it can affect any woman. According to The World Health Organization (WHO) [2], there were 2.3 million women diagnosed with breast cancer and 685 000 deaths globally. As of the end of 2020, there were 7.8 million women alive who were diagnosed with breast cancer in the past 5 years, making it the worlds most prevalent cancer. World health experts are still unsure what causes breast cancer. Only a few risk factors that influence a woman's likelihood of developing breast cancer can be identified by medical professionals.
Some cancer risk factors, such as age and race, cannot be changed, whereas others, particularly those related to environment and behavior, such as smoking, alcohol consumption, and dietary habits, can change over time.
The stages of cancer determine the severity of the disease. Doctors use the TNM system (Tumor, Node, Metastasis) to describe the stage of cancer. This system employs three criteria to determine the cancer stage: tumor size, spread to lymph nodes, and spread to other organs (metastasis). If breast cancer is detected at an early stage, the healing process is relatively simple. However, the chances of recovery decrease as the cancer advances to higher stages.There are various treatment techniques available for cancer prevention, including surgery, gene therapy, bisphosphonates, immunotherapy, targeted therapy drugs, hormone therapy, bone marrow transplants and stem cell therapy, radiotherapy, and complementary and alternative therapies.
The chances of successful treatment for breast cancer are higher when it is detected and diagnosed early, as is the case with most types of cancer.
There are several types of breast cancer, and any of them can metastasise. About 15% to 20% of breast tumors have higher levels of a protein known as HER2. These cancers are called HER2-positive breast cancers [3] and it's the type we are discussing in this paper. Cancers that are characterised as 'HER-2 positive' can be effectively treated using targeted biological therapies, like trastuzumab. These biological therapies are very effective but also very expensive, because they are antibodies rather than chemicals. When targeted biological therapies are given, they are combined with chemotherapy to make them effective at killing cancer cells [2].
Trastuzumab is a monoclonal antibody used to treat HER2-positive breast cancer. However, one of the major side effects of trastuzumab treatment is cardiotoxicity, which can lead to heart failure. Mathematical modeling has been used to understand the mechanisms underlying trastuzumab-induced cardiotoxicity and to optimise treatment strategies to minimise cardiotoxicity. Over time, modeling breast cancer has become an invaluable tool for understanding the dynamic behavior of tumor growth during the treatment process. The role of mathematical modeling in combination therapy for tumors was explored by Onofrio et al. [4]. Studies conducted by Kermack and McKendrick [6,5], as well as other investigations [7,8], have demonstrated the usefulness of mathematical modeling of biological phenomena in solving epidemiological problems. Most mathematical models of cancer are based on differential equations. Ordinary differential equations with two or three cell populations containing tumor cells and effector cells have been used to investigate essential patterns of tumor growth and decay [9,10]. Several mathematical models have been proposed to study the effects of trastuzumab on breast cancer cells and the development of cardiotoxicity. For example, Jarrett et al. [11], develop an integrated, mathematical-experimental approach for understanding the interactions between the immune system and the effects of trastuzumab on breast cancer that overexpresses the human epidermal growth factor receptor 2 (HER2+). Kitani et al. [12] aimed to develop a humaninduced pluripotent stem cell (hiPSC) model of trastuzumab-induced cardiac dysfunction in breast cancer patients. The researchers compared hiPSCs derived from patients with breast cancer who had received trastuzumab treatment to hiPSCs derived from healthy individuals. The hiPSCs were then differentiated into cardiomyocytes and tested for response to trastuzumab treatment. Maadi and Wang [13] aimed to investigate the mechanism underlying the inhibitory effects of trastuzumab on the growth of HER2-positive breast cancer cells. The researchers used several HER2-positive breast cancer cell lines and treated them with trastuzumab to evaluate the effects on cell proliferation, cell cycle progression, and apoptosis. They also performed gene expression analysis to identify potential signaling pathways involved in the response to trastuzumab. Therefore, based on the studies that we have discussed so far, none of them appear to have focused on a mathematical model of breast cancer at the patient population level.
for that, this study explores a mathematical model that examines the stages of breast cancer and the potential side effects on the heart in patients receiving trastuzumab treatment.
Therefore, this work is organised as follows: In the first section, we construct a mathematical model of breast cancer with side effects on the heart in trastuzumab patients. The five compartments are modeled by creating a system of differential equations. We utilise this differential equation system to construct the model and analyse the associated temporal dynamics by studying the equilibrium point and its stability. To analyse the stability of the equilibrium point, we use the Routh-Hurwitz criteria, which allows us to identify an asymptotically stable equilibrium point. We then confirm these results through numerical simulations, which demonstrate that the equilibrium point is always stable regardless of the initial conditions, without any additional conditions. Ultimately, our findings suggest that the five sub-populations of patients will reach a stable state upon reaching the equilibrium point.

FORMULATION OF THE MODEL
Motivated by the empirical observation that compartmental epidemiological models have played a significant role in advancing our understanding of epidemic transmission mechanisms and the various preventive strategies used against it, we propose a deterministic mathematical model that utilises ordinary differential equations.
We assumed that no patients were healthy upon their first arrival at the hospital. Upon their first medical record, patients were classified into sub-populations of stage I, stage II, stage III, or stage IV. During their treatment process at the hospital, all patients were presumed to receive Trastuzumab treatment. Some patients experienced recovery, while others experienced worsening conditions of the disease during Trastuzumab treatment, and some experienced cardiotoxic effects during the treatment process.
The model consists of five compartments that represent sub-populations of breast cancer patients. Each sub-population is denoted by variables A, B, C, D, and E. Subpopulation A represents patients with stage I and II cancer. Subpopulation B represents patients with stage III cancer. Subpopulation C represents patients with stage IV cancer. Subpopulation D represents ANALYSING THE EFFECT OF TRASTUZUMAB TREATMENT ON BREAST CANCER STAGES   5 patients with a disease-free condition after Trastuzumab treatment. Subpopulation E represents cancer patients who experience cardiotoxicity. The compartment diagram is illustrated in Figure   (1).

FIGURE 1. Compartment Diagram
Patients affected by breast cancer are assumed to occur when the patient comes for the first time to the hospital.
The patients whose tumor size exceeds 2cm but not more than 5cm are diagnosed with stage I and II cancer, which are referred to as early breast cancer. Invasive breast cancer occurs when cancer cells within the milk duct or lobules break out into nearby tissue. Early breast cancer is invasive breast cancer that is contained within the breast and may or may not have spread to the lymph nodes in the breast or armpit. Some cancer cells may have spread outside the breast and armpit area but cannot be detected. Because the number of patients diagnosed with stage I and II cancer is smaller than the other stages, they are placed in a sub-population denoted as A. The majority of cancer patients who have already been treated have advanced cancer. Therefore, patients diagnosed with stage I and II cancer are included in sub-population A with a rate of ν 1 . Patients in this sub-population who undergo Trastuzumab treatment have two possibilities: they may either recover (become disease-free) with a rate of δ AD or worsen with a rate of δ AB .
Patients diagnosed with stage III cancer upon their first visit to the hospital are included in sub-population B with rate ν 2 . At this stage, the axillary lymph node is typically enlarged (greater than 5 cm), and the cancer becomes inflammatory. In sub-population B, patients may transition from sub-population A (stages I and II) with rate δ AB or from the disease-free subpopulation D with rate δ DB . This sub-population receives more intense Trastuzumab treatment than sub-population A, with patients either dying from cancer with rate γ 1 , transitioning to the recovery sub-population D with rate δ BD , or worsening with rate δ BC . Additionally, subpopulation B is more prone to experiencing cardiotoxicity due to the intensive Trastuzumab treatment, with a rate of δ BE .
Patients who are treated for the first time can also enter into sub-population C because the cancer has metastasised or spread to distant parts of the body, typically the bones, lungs, or liver. This is an advanced stage of cancer, called metastatic breast cancer or Stage IV breast cancer. During this stage, targeted treatment is unlikely to cure cancer, so the rate towards recovery is low with rate δ CD . This sub-population can grow by moving from stage III with a rate δ BC and from disease-free with a rate δ DC . On the other hand, this sub-population may also be reduced with a cancer death rate of γ 2 . It is assumed that the rate of cardiotoxicity δ CE is high because patients are receiving very intensive Trastuzumab treatment. This sub-population can also experience cancer death with a rate of γ 2 .
Subpopulation D (disease-free) increased from the first tree sub-population A, B and C and lose recovery with the rates δ DB , δ DC and δ DC respectively.
The patients in sub-population E comes from B, C and D and taste cardiac death with a rate γ 3 .
Then the dynamics of breast cancer with Trastuzumab treatement treatment with the above assumptions is given by the following system of EDOs: with appropriate initial condition for vector

DYNAMICAL ANALYSIS
In this section we study the existence and stability of the equilibrium point who is also represents the critical point of the system (1).

Equilibrium Point.
To find the equilibrium point of the system (1), we set all the fractional derivative of (1) to zero and obtain the equilibrium point given by P * e = (A * , B * ,C * , D * , E * ). The equilibrium point exist and it is where, These equilibrium points are important in analysing this breast cancer model with Trastuzumab treatement and can predict sufficient condition for the spread of the infection.
The results of this investigation are as follows: There exist an equilibrium of the system (1) of the breast cancer without any condition.

Stability.
To determine the stability of the equilibium point, the equation is first given in matrix form We will use the Rough-Hurwitz criterion [22] to determine the stability of the system.
The characteristic equation is calculated using det(F − λ I). We have: We get, We can clearly see that this (2) has five eigenvalues −k 1 , −k 5 and the rest of (three) eigenvalues are the roots of the following equations given below: where the coefficients are: To study their nature, we Establish a Routh table for this equation, we have: Therefore, From the Routh Array, the sign of the first column are all positive, meaning that all real eigenvalues are negative [22].
Further, it is easy to show that a 1 > 0, b 1 = a 1 a 2 − a 3 > 0 and a 3 > 0. Thus, it follows from the Routh-Hurwitz criteria that the system (1) is locally asymptotically stable.

NUMERICAL SIMULATION AND DISCUSSIONS
In   For the second and third simulations, we will try to lower the relapse rate and cardiotoxity rate.