# A Mathematical Model for the Treatment of Melanoma with the BRAF/MEK Inhibitor and Anti-PD-1
## 1 Introduction  
Melanoma is a rapid growth of melanocyte cells in the skin. This cancerous growth usually starts in the skin or moles. Rarely, it may appear in organs other than the skin, such as the eyeball (extensive information on this subject is provided in a separate chapter below), the oral mucosa, or under the nails. Sometimes, this tumor spreads to other organs in the body through the lymph ducts or the bloodstream. Cancerous growth of the melanoma may appear in the skin and all over the body; however, in women, the most common area for the appearance of melanoma is the skin of calves, and in men, melanoma is more common in the skin of the back [1].  
Types of malignant melanoma These include superficially spreading melanoma, nodular melanoma, lentigo maligna melanoma, and acral melanoma. In advanced melanoma, the cancer cells spread from their primary site to other parts of the body, causing another cancerous growth, called a secondary or metastatic cancer. Melanoma spreads to other distant skin areas, distant lymph nodes, lungs, liver, bones, and the brain. Sometimes, advanced melanoma is diagnosed as melanoma that has returned to another part of the body, even many years after the primary melanoma is removed, or when metastatic disease is diagnosed after additional tests are performed after the initial lesion is removed [2].  
Symptoms of the disease: Most types of malignant melanoma begin with a change in the skin, such as a new or existing mole that has changed its shape.  
Geometry: A mole with an asymmetrical shape.  
Border: A mole whose border is not rounded, clear, or defined.  
Shade (Color): A mole whose color is not uniform, and consists of several shades, such as brown, black, pink, and more.  
Size: A mole exceeding 5 mm in diameter.  
Height: Change in height or any change in the previous four sections [3].  
Treatment of melanoma is very broad and depends on the stage of the disease. In this section, we will not review all the possibilities of treating the disease but will present those relevant to the current study.  
MEK Inhibitors Combined with BRAF Inhibitors for Metastatic Melanoma. These treatments are intended for melanoma at an advanced stage (stage 4, metastatic disease). Patients with melanoma (approximately 40-50 percent) have a mutation in the BRAF gene, which causes uncontrolled cell division and the development of a cancerous tumor. A group of drugs targeting mutations in the BRAF gene neutralizes the mutation and leads to regulated cellular division. These drugs are effective in patients with the BRAF V600E mutation (accounting for 70-80 percent of the BRAF mutation). The MEK gene is closely related to the BRAF gene, and MEK-targeted drugs can treat melanoma with BRAF mutations. It is reported that a combination of a BRAF inhibitor with an MEK inhibitor is more effective in the treatment of melanoma than either type alone [4]. In patients with the BRAF V600K mutation (which accounts for 20-30 percent of BRAF mutations), immunotherapy treatments were more effective. The most common immunotherapeutic drugs to treat melanoma are checkpoint inhibitors that work by releasing T cells--immune cells that can destroy tumors. For example, pembrolizumab is a monoclonal antibody that blocks the activity of the PD-1 protein by binding to its receptor, allowing the body's immune system to attack melanoma cells [5; 6; 7; 8; 9; 10; 11].  
In this study, we present a new personalized treatment protocol for melanoma that combines biological therapy with immunotherapy using a mathematical model. The mathematical model contains analytical functions that depend on parameters and variables that can be changed to control the time of drug administration and dosage.  
The purpose of the current study is to propose a treatment protocol for melanoma that is personalized for each patient and differs from the existing standard protocol. The treatment protocol considers changes in the treatment time intervals and dose given to each patient.