A Prediction Model of a Lung Tumor Growth and Chemotherapy Treatment: Equilibrium Points and Stability Abdellatif Bettayeb 1* , Nadir Teyar 2 and Boutheina Fellahi 2 1 Department of General Studies, Jubail Industrial College, Kingdom of Saudi Arabia 2 Department of Mathematics, Mentouri University of Constantine, Algeria * Corresponding author: Abdellatif Bettayeb, Department of General Studies, Jubail Industrial College, Kingdom of Saudi Arabia, Tel: 00966560755963; E-mail: abdellatif.bettayeb@gmail.com, bettayeb_a@jic.edu.sa Rec date: December 11, 2019; Acc date: February 20, 2020; Pub date: March 02, 2020 Copyright: © 2020 Bettayeb A, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Abstract The purpose of this paper is to study the prediction model of tumor growth non-vascularized lung cancer (first stage of cancer) before and during chemotherapeutic treatment. This model will be represented by a differential equations system, which will describe, among other things, tumor volume and proliferating cells. Finally, all the numerical results presented in this paper have been implemented in Scilab. Keywords: Tumor; Temporal model; Lung; Chemotherapy; Equilibrium; Stability Introduction Proliferation cells: Tese cancerous cells participate in the growth of the tumor by their incessant division, and they are able to use glucose from medium to ensure their energy far more easily than others cells. Quiescent cells: Tey are old proliferating cells which sufer from a lack of nutrients. Tey are waiting to have enough energy at their disposal to become proliferating again. Necrotic cells: Tey are quiescent cells that have died because of a lack of nutrients. Hypoxia: It is the absence of oxygen in the environment. Scanner: Te scanner is a medical imaging method that measures the absorption of X-rays by the tissues. Te apparatus consists of a ring in which the patient is placed. Te scanner makes it possible to have 2D and 3D images with precision. It gives information on the geometry of the tumor and its size. Te duration of the exam takes less than an hour. Methods Te laws of tumor growth Te exponential law: To model tumor growth, the most appropriate way is to consider the growth rate of the tumor. In the frst half of the 20th century, the analysis of observations experimental animal and human population data led to consider an exponential growth of the tumor. Te evolution of the tumor is therefore given by the following dynamics [1,2]: dC/dt=λ C C(t) ……….(1) Where C(t) represents the law of evolution of the quantity of cancerous cells; C 0 =C(t=0) is the initial quantity; and λ C ≥ 0 denotes the growth rate of the tumor. Te solution of the equation (1) is given by: C(t)=C 0 e λ C t Using the data in Table 1, we obtain (Figure 1): Model Parameter Unit Value Exponential C 0 mm 3 13.2 λ C day -1 0.257 Logistic λ C day -1 0.502 C ∞ mm 3 1297 Gompertz λ C day -1 0.742 k day -1 0.0792 Table 1: Parameter values estimated from lung data adjustments. Figure 1: Te evolution of the quantity of cancer cells according to the exponential law. Tumor growth in this model is considered not limited by any factor. But the continuation of tumor growth is linked by mechanical and environmental constraints (problem of oxygen distribution, nutrients), so unlimited proliferation is impossible. Tese constraints are taken into account in the following models: J ou r n a l o f C o m p u t e r S c i e n c e & S y s t e m s B i o l o g y ISSN: 0974-7230 Journal of Computer Science & Systems Biology Bettayeb et al., J Comput Sci Syst Biol 2020, 13:1 Research Article Open Access J Comput Sci Syst Biol, an open access journal ISSN: 0974-7230 Volume 13 • Issue 1 • 1000306