Vol.:(0123456789) 1 3 Iranian Journal of Science and Technology, Transactions A: Science https://doi.org/10.1007/s40995-019-00681-w REVIEW PAPER Various Mathematical Models of Tumor Growth with Reference to Cancer Stem Cells: A Review Azim Rivaz 1  · Mahdieh Azizian 2  · Madjid Soltani 3,4 Received: 10 September 2018 / Accepted: 18 January 2019 © Shiraz University 2019 Abstract Using mathematical models to simulate biological systems has a long history. An increasing number of such models have been applied to various aspects of tumor growth, with the ultimate goal of controlling cancer. Nevertheless, very little has been done in the feld of cancer stem cells. Herein, we have reviewed some mathematical models of tumor growth and their specifc properties. Considering the importance of the role cancer stem cells play in the production, progression and recurrence of cancer, we have also examined a mathematical growth model describing the dynamics of tumor growth in the presence of cancer stem cells. Keywords Mathematical modeling · Tumor growth · Cancer stem cells 1 Introduction The challenge the oncology community faces at a global scale is understanding the complex nature of cancer and the mysteries underlying its development. It requires laboratory and clinical experiments to collect useful data which could lead to promising and successful therapy fnally. Efective treatment protocols can be developed by identifying the mechanisms which afect the tumor growth (Byrne 1999). So, if we understand the mechanism of the disease progres- sion, we can identify key components of disease control and its treatment. The achievement of this goal can be speeded up through the application of mathematical modeling, con- trol theory and optimization to describe diferent aspects of tumor growth in the absence or presence of anticancer agents (Alfonso and Flanagan 2018). Mathematical modeling is a versatile tool to test hypoth- eses, confrm experiments and simulate the dynamics of complex systems in a relatively short time without the enor- mous costs of laboratory experiments. By understanding a system and developing a valid model, the system process can be predicted and controlled. A mathematical model is a summary of the process of a system. It concludes model equations and parameters. Usually, available experimental data are used for estimating the model parameters and for validating its prediction ability. Then, parametric analysis (sensitivity analysis with respect to parameters) of the model is performed to understand the domain and variations of the system behavior with the variation in the parameters (Rod- rigues and Minceva 2005). Various uses can be made from cancer mathematical modeling ; for example, the tumor growth can be predicted and the main parameters responsible for it can be under- stood better. Another application is that these models can be combined with pharmacokinetic models of the therapeutic agents to study their impact on cancer progress (Quaranta et al. 2005). Moreover, modeling and in silico experiments can provide new insights and ofer several possibilities to understand cancer process and consequently its treatment. * Mahdieh Azizian m_azizian@kmu.ac.ir Azim Rivaz arivaz@uk.ac.ir Madjid Soltani cbb@uwaterloo.ca; madsoltani@gmail.com 1 Department of Applied Mathematics, Faculty of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran 2 Department of General Educations, Afzalipour School of Medicine, Kerman University of Medical Sciences, Kerman, Iran 3 Department of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran 4 Center for Biotechnology and Bioengineering (CBB), University of Waterloo, Waterloo, ON, Canada