New formulation of the Gompertz equation to describe the kinetics of untreated tumors

Autoři: Antonio Rafael Selva Castañeda aff001;  Erick Ramírez Torres aff003;  Narciso Antonio Villar Goris aff004;  Maraelys Morales González aff007;  Juan Bory Reyes aff008;  Victoriano Gustavo Sierra González aff009;  María Schonbek aff010;  Juan Ignacio Montijano aff001;  Luis Enrique Bergues Cabrales aff001
Působiště autorů: Departamento de Matemática Aplicada, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, Zaragoza, Spain aff001;  Departamento de Telecomunicaciones, Facultad de Ingeniería en Telecomunicaciones Informática y Biomédica, Universidad de Oriente, Santiago de Cuba, Cuba aff002;  Departamento de Biomédica, Facultad de Ingeniería en Telecomunicaciones Informática y Biomédica, Universidad de Oriente, Santiago de Cuba, Cuba aff003;  Universidad Autónoma de Santo Domingo, Santo Domingo, Dominican Republic aff004;  Universidad Católica Tecnológica del CIBAO, Ucateci, La Vega, Dominican Republic aff005;  Departamento de Ciencia e Innovación, Centro Nacional de Electromagnetismo Aplicado, Universidad de Oriente, Santiago de Cuba, Cuba aff006;  Departamento de Farmacia, Facultad de Ciencias Naturales y Exactas, Universidad de Oriente, Santiago de Cuba, Cuba aff007;  ESIME-Zacatenco, Instituto Politécnico Nacional, CD-MX, Mexico aff008;  Grupo de las Industrias Biotecnológica y Farmacéuticas (BioCubaFarma), La Habana, Cuba aff009;  Department of Mathematics, University of California Santa Cruz, Santa Cruz, CA, United States of America aff010
Vyšlo v časopise: PLoS ONE 14(11)
Kategorie: Research Article
doi: 10.1371/journal.pone.0224978



Different equations have been used to describe and understand the growth kinetics of undisturbed malignant solid tumors. The aim of this paper is to propose a new formulation of the Gompertz equation in terms of different parameters of a malignant tumor: the intrinsic growth rate, the deceleration factor, the apoptosis rate, the number of cells corresponding to the tumor latency time, and the fractal dimensions of the tumor and its contour.


Furthermore, different formulations of the Gompertz equation are used to fit experimental data of the Ehrlich and fibrosarcoma Sa-37 tumors that grow in male BALB/c/Cenp mice. The parameters of each equation are obtained from these fittings.


The new formulation of the Gompertz equation reveals that the initial number of cancerous cells in the conventional Gompertz equation is not a constant but a variable that depends nonlinearly on time and the tumor deceleration factor. In turn, this deceleration factor depends on the apoptosis rate of tumor cells and the fractal dimensions of the tumor and its irregular contour.


It is concluded that this new formulation has two parameters that are directly estimated from the experiment, describes well the growth kinetics of unperturbed Ehrlich and fibrosarcoma Sa-37 tumors, and confirms the fractal origin of the Gompertz formulation and the fractal property of tumors.

Klíčová slova:

Angiogenesis – Apoptosis – Fractals – Graphs – Histology – Interpolation – Malignant tumors – Fibrosarcoma


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2019 Číslo 11