Correspondence between formulations of Avrami and Gompertz equations for untreated tumor growth kinetics

Abstract

The classical and modified equations of Kolmogorov-Johnson-Mehl-Avrami are compared with the equations of conventional Gompertz andMontijano-Bergues-Bory-Gompertz, in the frame of growth kinetics of tumors. For this, different analytical and numerical criteria are usedto demonstrate the similarity between them, in particular the distance of Hausdorff. The results show that these equations are similar fromthe mathematical point of view and the parameters of the Gompertz equation are explicitly related to those of the Avrami equation. It isconcluded that Modified Kolmogorov-Johnson-Mehl-Avrami and Montijano-Bergues-Bory-Gompertz equations can be used to describe thegrowth kinetics of unperturbed tumors.

Simulations of surface charge density changes during the untreated solid tumour growth

“…Furthermore, the integral characterization of cancer patients by means of an integrated analysis of clinical-biological(tumour and patient)-functional-bioelectrical parameters [61] is possible from these larger networks. The cancer fractality at submicron [60] and tissue [1,4,5] levels confirms the close relation of the multiscale hierarchies in malignant tumours.…”

“…The time variation of σ 12 at Σ corresponds to the change from the quick tumour growth phase to asymptotic phase of TGK and follows a sigmoidal behaviour in time, as TGK [1][2][3][4][5]. This nonlinear time behaviour of σ 12 may be explained because η 1 and η 2 exhibit nonlinear behaviour due to biological tissues being nonlinear systems [1,4], and η k , ε k (k = 1,2), τ 1 , τ 2 and the relaxation time of the interfacial polarization (τ p ) change in time [49].…”

“…The cancer bioelectric handling has been suggested as a useful tool to understand bioelectrical fields that change dynamically during cancer growth and possible anti-cancer therapeutic targets, aspects that remain unclear as yet [8,43]. (5) What relationship exists between σ 12 and the tumour contour fractal dimension reported in [1,4,5]? (6) What implication does non-homogeneous distribution of σ 12 at Σ have during tumour growth?…”

“…Several equations are used to describe TGK, such as: the conventional Gompertz (CGE, the most accepted) [1,3], Montijano-Bergues-Bory-Gompertz [4], modified Kolmogorov-Johnson-Mehl-Avrami, Logistic & Bertalanffy [1]. The parameters of the first three equations are interconnected [5]. These equations involve different biological parameters (i.e.…”

“…Several findings have been revealed, such as: (1) differences between electrical conductivities (η k , k = 1,2) and electrical permittivities (ε k , k = 1,2) of the untreated malignant tumour (k = 1) and the surrounding healthy tissue (k = 2) [11]; (2) these two physical properties as a potential diagnostic method [16]; (3) differences between ionic current (due to the movement of charged ions) and faradic current (produced by electrons exchange from reduction and/or oxidation of biochemical molecules) in cancer and surrounding healthy tissue [8]; (4) the existence of chemical and electrical (charged negatively) environments in cancer cells and untreated tumours [1,4,14] and their key roles in the genesis, growth, progression, metastasis and treatment of cancer [17]; (5) the impact of tumour microenvironment on its electrical properties [16]; (6) the breakdown of intercellular communication (gap junction) in the tumour due to low regulation in expression of the connexin [12,18]; (7) negative electrical biopotentials in the tumour and positive electrical biopotentials in the surrounding healthy tissue [12][13][14]; (8) cancer cells and some cells of the immune system negatively charged [14]; (9) weaker electrical coupling among cancer cells and the association of deregulation of intercellular communication with tumourigenicity and metastasis of the cancer [14,18]; (10) bioelectronic cancer regulator as an initiator of the mitosis and deoxyribonucleic acid synthesis; and (11) correction of alterations in the electrical communication system of cancer by manipulating its bioelectrical properties, known as bioelectronic medicine [8].…”