Quiescence as an explanation of Gompertzian tumor growth revisited E.O. Alzahrani a , Asim Asiri a , M.M. El-Dessoky a,b , Y. Kuang c,⇑ a Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia b Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt c School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA article info Article history: Received 19 January 2014 Received in revised form 6 June 2014 Accepted 12 June 2014 Available online 23 June 2014 Keywords: Tumor model Quiescence Proliferation Steady state Stability abstract Gompertz’s empirical equation remains the most popular one in describing cancer cell population growth in a wide spectrum of bio-medical situations due to its good fit to data and simplicity. Many efforts were documented in the literature aimed at understanding the mechanisms that may support Gompertz’s ele- gant model equation. One of the most convincing efforts was carried out by Gyllenberg and Webb. They divide the cancer cell population into the proliferative cells and the quiescent cells. In their two dimen- sional model, the dead cells are assumed to be removed from the tumor instantly. In this paper, we mod- ify their model by keeping track of the dead cells remaining in the tumor. We perform mathematical and computational studies on this three dimensional model and compare the model dynamics to that of the model of Gyllenberg and Webb. Our mathematical findings suggest that if an avascular tumor grows according to our three-compartment model, then as the death rate of quiescent cells decreases to zero, the percentage of proliferative cells also approaches to zero. Moreover, a slow dying quiescent population will increase the size of the tumor. On the other hand, while the tumor size does not depend on the dead cell removal rate, its early and intermediate growth stages are very sensitive to it. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction Tumor growth models have their historical roots in the work of Gompertz [8]. The Gompertz model was first employed in the paper of Laird [12] to model real tumor growth. Ever since, Gompertz’s empirical equation remains the most popular one in describing cancer cell population growth in a wide spectrum of bio-medical situations due to its good fit to data and simplicity [15]. Many efforts were documented in the literature aimed at understanding the mechanisms that may support Gompertz’s elegant model equation [7,9,13,14,11,18]. The key aspect of the approach of these existing efforts is to divide the cancer cell population into the proliferating cells and the quiescent cells, or the proliferating cell and dispersing cells. In the beginning, a tumor often grows in approximately a spher- ical form. If the tumor fails to produce enough signaling proteins such as vascular endothelial growth factor (VEGF) for angiogenesis, then the tumor can only grow to a certain size with available nutri- ent supplies. Indeed, most tumors exhibit a sigmoid growth curve in the early stage. For this reason, many modelers simply employ the well-known logistic equation dN=dt ¼ rNð1  N=KÞ¼ rN  rN 2 =K ð1:1Þ as an initial model for tumor growth. Here N is the size of the tumor, usually measured as a number of cells or as a volume. r is the growth rate while rN=K can be interpreted as the density dependent death rate. The tumor size is an increasing function that tends to the carrying capacity K. Generalizing the logistic model, von Bertalanffy [1] introduced the equation dN=dt ¼ f ðNÞ¼ aN k  bN l ; k < l: ð1:2Þ to represent tumor growth. This is often referred as the (general- ized) von Bertalanffy tumor model. The tumor size is an increasing function that tends to the carrying capacity ða=bÞ 1=ðlkÞ . Tumors tend to approach a steady state size in the nutrient-limited growth phase when nutrient is supplied only by diffusion. A particular case of the von Bertalanffy equation is the surface rule model [1], which states that growth is proportional to surface area (k ¼ 2=3) since nutrients have to enter through the surface, while death is propor- tional to the size (l ¼ 1). In this special case, b is the death rate. Notice that, the birth rate of logistic model and the death rate of the von Bertalanffy model (when l ¼ 1) are constant. Gompertz model is arguably also the most important and prac- tical tumor model. Many researchers reported that Gompertz model provided surprisingly good fit to their experimental data on various tumor growths. The key assumption embodied in the Gompertz model is that the cell growth rate decreases exponen- tially as a function of time. http://dx.doi.org/10.1016/j.mbs.2014.06.009 0025-5564/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. Tel.: +1 480 965 6915; fax: +1 480 965 8119. E-mail address: kuang@asu.edu (Y. Kuang). Mathematical Biosciences 254 (2014) 76–82 Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs