Cancer is the cause of more than 13% of deaths in the world and the first cause of death in developing countries. Glioblastoma is the most common malignant brain tumor in humans, which is categorized as the fourth type of glioma tumor by the World Health Organization. The main characteristic is the formation of a network of blood vessels and a necrotic nucleus. In this type of tumor, the interval between the identification stage and the time of death is 12 to 18 months. For this reason, the use of methods for modeling, simulating and personalizing tumor behavior can facilitate the process of cancer diagnosis and treatment. In this study, tumor growth simulation was first performed using five PDE equations. According to the results obtained from the simulation and its correlation with the theory of tumor growth in different parts, it can be seen that this model is the most suitable model for predicting the behavior of the brain tumor and its use in choosing treatment methods, because determining the different parts of a tumor can lead to the correct choice of treatment. According to these results, the main factor for increasing the size of the tumor is the increase in the number of normoxic cells and it is possible to determine the tumor boundary by determining the concentration of the normoxic cells. Also, by this model, it is possible to determine the range of hypoxic, necrotic, and blood vessels cells that are important for treatment methods such as radiotherapy and chemotherapy. In tumor growth models, estimating the model's coefficients for each patient is described as a patient-specific problem. To estimate the coefficients of the model, two stages of the image of a patient are used in a given time period. The first stage image will be used as the initial function of the model and the second stage image will be used to compare the simulation result with real conditions. The purpose of a patient-specific problem is to find the coefficients of the model that the results of simulation has the least error with actual conditions of the tumor. In the next step, due to the lack of access to a complete set of medical images of a cancer patient, images of rat mice and a simpler model including one PDE equation, have been used. To estimate unknown model parameters and enable patient-specific simulations, we formulate and solve a PDE-constrained optimization problem by adjoint method. To solve the equations, LiveLink for MATLAB interface with COMSOL has been used, which is the first time in related research in this field. In this research, three stages of experimental mice imaging were used to estimate the model's coefficients. The parameter estimation with the first and second stage images has an error of less than 10% and the validation of the method with the third stage image also led to an error of less than 15%. Our study shows that, the presented patient-specific model is independent from the initial guess for model parameters, and its inappropriate selection only prolongs the time of solving, and no significant error comes to the result. Keywords : Glioblastoma tumor, Tumor growth model, Patient-specific patient-specific mathematical model, Optimization, Adjoint method.