In this paper, we introduce a new class of nonlinear discrete fractional equations to model tumor growth rates in mice. For the data fitting purpose, we develop a new method which can be considered as an improved version of the partial sum method for parameter estimations. We demonstrate the goodne ss of fit by comparing the models with three statistical measures.

Background Tumor growth, a relationship between tumor size and time, is of special interest since growth estimation is very critical in a clinical practice. There are some mathematical models which describe tumor growth and have prediction capabilities. Typically, there are three ways to model non-complex growth behavior: exponential, logistic and sigmoidal. In 1825, Benjamin Gompertz introduced the Gompertz function, a sigmoid function, which is found to be applicable to various growth phenomena, in particular tumor growth. Besides the Gompertz model which includes three parameters, the Weibull and Richards models with four parameters are known as sigmoidal models. The aim of this project is to introduce continuous fractional and discrete fractional models of the tumor growth and also estimate parameters of these models in order to have better data fitting.

Let P = {P1, P2, ..., Pn} be set of participants and Γ = {Bi|Bi⊂ P, 1 ≤ i ≤ k} be access structure. Vector space secret sharing realizing access structure Γ requires existence of function φ : P → (Zp)d, where p is a prime number and d ≥ 2 is an integer, satisfying the following condition (1, 0, 0, ..., 0) =i) : Pi∈ B >;⇔ B ∈ Γ = {B1, B2,..., Bk}. There is no known algorithm to construct such a function φ in general. Constructions are mainly done by trial and error. In this paper, we developed a polynomial algorithm to construct a φ function for certain type of access structures. Some examples are given to illustrate the algorithms.