A generalization of Gompertz law compatible with the Gyllenberg–Webb theory for tumour growth Alberto d’Onofrio a,⇑ , Antonio Fasano b , Bernardo Monechi c a Department of Experimental Oncology, European Institute of Oncology, Via Ripamonti 435, Milano, Italy b Department of Mathematics ‘‘Ulisse Dini’’, University of Florence, Florence, Italy c Department of Physics, University of Florence, Florence, Italy article info Article history: Received 9 April 2010 Received in revised form 2 January 2011 Accepted 4 January 2011 Available online 11 January 2011 Keywords: Tumor Laws of growth Quiescent cells Special functions Population-based models abstract We present a new extension of Gompertz law for tumour growth and anti-tumour therapy. After discuss- ing its qualitative and analytical properties, we show, in the spirit of [16], that, like the standard Gom- pertz model, it is fully compatible with the two-population model of Gyllenberg and Webb, formulated in [14] in order to provide a theoretical basis to Gompertz law. Compatibility with the model proposed in [17] is also investigated. Comparisons with some experimental data confirm the practical applicability of the model. Numerical simulations about the method performance are presented. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction In 1825 Gompertz [13] formulated his model for the mortality rate of a population, which later became one of the most frequently used laws to describe tumour growth (it is currently applied in other contexts, both in biology and in economics). Gompertz differ- ential law can be written for instance in the form N 0 ¼ Nða  bLnNÞ; a > 0; b > 0; ð1Þ where N(t) represents either the number of individuals in the pop- ulation or any quantity associated with its size (for instance the vol- ume). The requirement a > b must be fulfilled, if it has to express growth. Clearly a has to be interpreted as proliferation rate, while b summarizes mutual inhibitions between cells and the competi- tion for nutrients, and it is sometimes called the growth retardation factor. Of course the physical meaning of the parameters has to be adapted to the one assumed for N. Adopting the normalization N(0) = 1, the integral of (1) is NðtÞ¼ exp a b 1  e bt     ; ð2Þ which has a typical double exponential structure. The normalized carrying capacity is N 1 = e a/b , and the coefficient b determines the rate of convergence to it. Gompertz law belongs to the large class of phenomenological growth models based on the competition of two terms, one repre- senting production and the other associated with death. The num- ber of such models is amazingly large, including the ubiquitous logistic law: (with N replacing Ln(N) in (1)) and the generalized lo- gistic (the same term being now a power of N), the von Bertalanffy law [31] with its generalizations, etc. A survey of many classical models, supplemented with an interesting comparative analysis, is due to Marusic [23] (see also [20,21]). The more recent paper by one of us [8], on modelling tumours and immune system inter- action, also describes several growth models developed in recent years. Collecting a complete list of growth models seems however to be a hopeless task. In the paper [22] fourteen models are tested on two specific sets of experimental data. In [24] for logistic and Gompertz laws and in [9] for the general family N 0 = f(N)N it has been showed how phenomenological models may be linked to in- ter-cellular inhibitory interactions. Among the models reviewed in [23] one can find the so-called hyper-Gompertz law: N 0 ¼ Nða  bLnNÞ 1þp ; p > 1; ð3Þ proposed in [30] as a limit case of a general class of three-parameter power-law models, which includes also the so-called hyper-logistic law: N 0 ¼ Nða  bNÞ 1þp ; p > 1: ð4Þ 0025-5564/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2011.01.001 ⇑ Corresponding author. Tel.: +39 0257489819; fax: +39 0257489922. E-mail address: alberto.donofrio@ifom-ieo-campus.it (A. d’Onofrio). Mathematical Biosciences 230 (2011) 45–54 Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs