## 2 Models  
### Classical Mathematical Models of Tumor Growth  
As said above, the purpose of this work is to analyze the predictive capabilities of various classical mathematical models that have been employed to describe MTS (and tumor) growth. In all the classical mathematical models of the tumor growth kinetics considered in the present work, the variable described is the total volume of the tumor, \(V\), as a function of time, \(t\). Furthermore, this total volume is supposed to be proportional to the total number of cells in the tumor. To simplify the labeling, E, G, L, P, and B have been used to refer to the exponential, Gompertz, logistic, potential, and Bertalanffy models that are described in what follows. The exponential model assumes that all cells within the MTS have a cell cycle with a constant duration [54]. The same growth behavior occurs if it is supposed that the cell proliferation is due to a constant fraction of the MTS volume [25] or if the duration of the cell cycle is a random variable following an exponential distribution. The mathematical expression describing this model is:  
\[V(t)\,=\,V_{0}\,\exp(E\,t)\,, \tag{1}\]  
where \(V_{0}\), the volume at \(t=0\), and \(E\) are the model parameters. The observation of non-constant doubling times in some tumors led to considering more elaborated models. One of the most widespread and accepted is the Gompertz model, which has been used to describe population growth in many different branches of knowledge [25; 35; 37]. The Gompertz model shows a sigmoid shape, i.e., a rising curve with an inflection point that converges asymptotically to a maximum volume. This qualitatively reproduces the growth slowdown observed experimentally [37; 55] and is consistent with general growth patterns of organs and organisms. The essential characteristic of the Gompertz model is that it shows a relative growth rate that reduces exponentially when volume increases. One of the main criticisms of the Gompertz model is that the growth relative rate may become arbitrarily large (or, equivalently, that the volume doubling time may reach arbitrarily small values) for small tumor volumes. In the present work, the following expression has been used for this model:  
\[V(t)\,=\,V_{0}\,\exp\{A\,[1\,-\,\exp(-\,a\,t)]\}\,. \tag{2}\]  
In this model, \(V_{0}\), which gives the MTS volume at \(t=0\), and \(A\) and \(a\) are the model parameters. The logistic model also shows a sigmoid shape [23; 25]. It is based on the equilibrium of metabolic processes and is defined by assuming that the relative growth rate reduces linearly with the volume. In this work, the following analytical expression has been considered:  
\[V(t)\,=\,\frac{V_{0}\,B}{V_{0}\,+\,(B\,-\,V_{0})\exp(-\,b\,t)}\,. \tag{3}\]  
In this case, the model parameters are \(V_{0}\equiv V(t=0)\), \(B\), and \(b\). [PERSON] [56] and others [57] proposed deriving the general laws governing the organic growth from basic principles involving the energy. They established that the net growth rate should be the result of the balance of synthesis and destruction, noting that metabolic rates often follow the law of alometry (i.e., they scale with a power of the total size of the system analyzed), while catabolic rates are proportional to that total volume. The Bertalanffy model has already been successfully applied to describe tumor growth [58; 59]. The mathematical function corresponding to this model is:  
\[V(t)\,=\,\frac{V_{0}}{(1\,-\,C)^{3}}(1\,-\,C\exp(-\,c\,t))^{3}\,. \tag{4}\]  
Here, the model parameters are the volume at \(t=0\), \(V_{0}\), \(C\), and \(c\). The potential model was derived from the Bertalanffy model [56] and does not exhibit a clear saturation phase. Tumor growth is proportional to the number of proliferating cells, under the assumption of a constant cell cycle duration. It is a model that produces a tumor growth with a decreasing growth fraction and, therefore, a decreasing relative growth rate, which also provides a description in terms of a geometric characteristic of the proliferative tissue.