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

Summary

Background

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.

Methods

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.

Results

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.

Conclusions

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.

Keywords:

apoptosis – Graphs – histology – Interpolation – Malignant tumors – Angiogenesis – Fractals – Fibrosarcoma

Introduction

One of the most interesting problems of current oncology is the understanding of the growth kinetics of a malignant tumor, named TGK (TGK), which follows a sigmoidal law. The TGK analysis is equally made by means of graphs of the number of cancer cells (n) versus time t, named n(t); tumor volume (V) versus t, named V(t); and/or the tumor mass (m) versus t, named m(t). This is due to the close relationship between these three physical quantities. Additionally, the sigmoidal form of TGK has been described by different equations, such as Gompertz, Logistics, Bertalanffy-Richards, Kolmogorov-Johnson-Mehl-Avrami modified, being the Gompertz equation (GE) the most used [1–3].

Izquierdo-Kulich et al. [4] report the fractal origin of GE (see appendix A). This fractal origin has also been reported in [5–8] but in terms only of the fractal dimension D_f. Here, we have considered the one in [4] because it also takes into account the fractal structure of the boundary of the tumor.

In the different formulations of the GE [1–3] and in the experiment [9, 10] the starting point of TGK is considered when the initial number of tumor cells (n₀) and the initial tumor volume (V₀) satisfy the conditions n (t = 0) = n₀ and V (t = 0) = V₀, respectively. In preclinical studies, the researcher chooses n₀/V₀ depending on the purpose of the investigation. The time that elapses from the inoculation of the tumor cells in the host until the tumor reaches n₀/V₀ is named t₀ [1, 3, 9]. Nevertheless, in clinics, n₀/V₀ corresponds to the tumor detected for the first time by the doctor by means of clinical and/or imaging methods. For this case, t₀ is the time that elapses from the tumor formation in the organism (via chemical, biological and/or physical carcinogens) [10], until its detection for the first time. This supposes n₀ ≥ n_med, where n_med is the minimum number of quantifiable cancer cells contained in the smallest measurable tumor volume, named V_med (V₀ ≥ V_med). The post-inoculation time that elapses until the tumor reaches n_med/V_med is named t_med (t₀ ≥ t_med) [3].

In [4], it is considered the Gompertz equation given in Eq (1) (named GE₁)

According the considerations in the previous paragraph, GE₁ has two limitations: 1) n₀ = 1, which means that the tumor has only one cell when it reaches V₀, in contradiction with the experiment [9, 10]. 2) The maximum capacity of the tumor (n_∞) depends only on α and β and not on n₀ (n(t) = n_∞ = e^α/β when t → ∞). From the mathematical point of view, n_∞ is the upper asymptote of TGK. Nevertheless, in the preclinical, the condition t → ∞ is the post-inoculation time that elapses until the tumor reaches a certain volume, for which animals are sacrificed for ethical reasons [1]. In clinics, this condition means the time that elapses from the tumor formation in the organism until the patient dies.

Each undisturbed solid tumor histological variety, that grows in a type of syngeneic host to it, has its own natural history (only sigmoidal law), which does not depend on the selection of n₀/V₀, as observed in [3, 10–12]. In the experiment, once the researcher fixes n₀/V₀, t₀ can be estimated a priori when the tumor latency time is known, named t_obs (t_obs < t₀), which is the post-inoculation time that elapses until that the tumor is observed for the first time. In this case, the tumor is observable and palpable but not measurable. However, its size, named V_obs (V(t = t_obs) = V_obs), is estimated following the methodology reported in [1, 3]. When the tumor reaches V_obs, it contains a number of cells, named n_obs (n(t = t_obs) = n_obs).

The interest of including n_obs/V_obs (n_obs/V_obs < n_med/V_med ≤ n₀/V₀) in GE is because an important part of vital cycle of a solid tumor occur before it is clinically detected (V_med), as reported in [1, 3, 10]. Furthermore, a high cellular viability (≥ 95%) and a correct inoculation of the initial concentration of tumor cells (c_o) are guaranteed, t_obs can be known a priori for a tumor histological variety that grows in a certain type of syngeneic host to it [3, 9–11].

As far as we reviewed, few experimental works report the analysis of TGK from V_obs [1, 3] and none of equations used to describe TGK includes n_obs/V_obs. In addition, in the literature a relationship of α and β in terms of D_f, d_f and n_obs/V_obs has not been reported in the literature. Therefore, the aim of this paper is to propose a new formulation of the GE that includes n_obs/V_obs, n₀/V₀, α, β, and to study the relation of these parameters with the fractal dimensions D_f and d_f. The validity of this new mathematical formulation and the estimation of its parameters are determined from volumes of the Ehrlich and fibrosarcoma Sa-37 tumors that grow in BALB/c/Cenp mice, previously reported in [9]. Furthermore, the graphs of α versus d_f and β versus d_f/D_f for different values of u₂ (the constant of the velocity of apoptosis) and n_obs are shown.

Methods

Conventional Gompertz equation

Eq (2), named GE₂, is the conventional GE and the most used when TGK starts at n₀/V₀, given by

According to GE₂, n_∞ depends on n₀, α and β (n(t) = n_∞ = n₀ e^α/β when t → ∞) and results from solving the ordinary differential Eq (3) with its initial condition, given by

GE₂ suggests that n₀ (constant in time) has to be included in Eq (A2). Tjørve and Tjørve [2] report that n₀ acts as a parameter of shape (n_∞ changes with n₀) or location (n_∞ remains constant).

Inclusion of n₀ in Eq (A2)

In this topic was followed the methodology exposed in [4] and the initial number of tumor cells at t = 0, named n₀₀, was included in Eq (A2), resulting the following problem

The exact solution of Eq (3) was given by

Two inconsistencies were found in [4]: 1) the coefficient 1.5 in the parameter α of Eq (A3) was not correct but 2/3, as in Eq (6). 2) Different types of experimental tumors with the same values of d_f and D_f had different values of α/β (we refer to the reader see Table 1 of [4]), in contrast to Eq (A3).

Eq (5), named GE₅, agrees with GE₂ when n0=(n00)e−βt. In addition, the parameters n₀₀ and n₀ coincided exactly at t = 0. The constant parameter n₀₀ (n₀₀ ≥ n_med) constituted the starting point of TGK for GE₅ and reached for t = t₀. Therefore, it was convenient to differentiate n₀ and n₀₀ to compare GE₂ and GE₅ in order to avoid confusion in the interpretation of these two parameters. GE₅ revealed that n_∞ depends only on α and β and not on n₀₀ (n(t) = n_∞ = e^α/β for t → ∞).

Inclusion of n_obs in GE

Eq (3) was rewritten as

Eq (8), named GE₈, agreed with GE₅ at t = 0 (for all n_obs) and when n_obs = 1 (for all t). The GE₈ coincided with the GE₂ at t = 0 (for all n_obs) and when n0=nobs(n000/nobs)e−βt. The parameter n_obs (n_obs < n_med ≤ n₀₀₀) was the starting point of TGK. In general, n₀₀₀ did not coincide with n₀ (GE₂) or n₀₀ (GE₅). Therefore, it was convenient to differentiate the parameters n₀, n₀₀ and n₀₀₀. In addition, the GE₈ evidenced that n_∞ depends on n_obs, α and β, but not on n₀₀₀ (n(t) = n_∞ = n_obs e^α/β for t → ∞). The parameters α and β in terms of u₂, U₁, θ, d_f, D_f and n_obs were given by

Eq (9) resulted from assuming that the value of n in the steady state was n_ss = n_obs e^α/β = (u₂/U₁)^1/(θ−1) and Eqs (7) and (8) were taken into account.

Simulations

Simulation of Eq (9)

Eq (9) coincided with Eq (6) for n_obs = 1. The simulation of α (in days^-1) versus d_f was shown for D_f = 5 and four values for u₂ (1, 10, 50 and 100 days^-1) and n_obs (1, 5, 10 and 20 cells). For this, values of d_f were varied from 0 to 5 with a step of 0.5, taking into account that d_f < D_f. The simulation of β (in days^-1) against d_f/D_f was shown for four values of u₂ (1, 10, 50 and 100 days^-1) and the values of d_f/D_f were ranged from 0 to 5 with a step of 0.5.

Simulations of GE₂, GE₅ and GE₈

GE₅ was used as reference because GE₅ and GE₈ were reported for the first time in the literature. The simulations of GE₂, GE₅ and GE₈ were shown in a graph of n(t). Simulation of GE₂ was made for different values of n₀ (1x10³, 1x10⁴, 1x10⁵ and 1x10⁶ cells). Additionally, GE₈ was simulated for three different situations: 1) n_obs = 1 cell (GE₈ and GE₅ coincided) and different values of n₀₀ (5, 10, 15, 20 and 25 cells); 2) n_obs = 1x10⁴ cells and different values of n₀₀₀ (1x10⁴, 5x10⁴, 1x10⁵ and 2x10⁵ cells); and 3) n₀₀₀ = 1x10⁵ cells and different values of n_obs (5x10³, 1x10⁴, 5x10⁴ and 1x10⁵ cells). In all these simulations, α = 1.0 days^-1 and β = 0.3 days^-1.

Experimental groups

In this study, V(t) was used by three reasons: 1) V(t) is related to n(t) and can be used interchangeably; 2) V(t) is less cumbersome to estimate than n(t) and it is frequently used in preclinical [9–11] and clinical [10] studies; and 3) the graphs of V(t) and n(t) shown sigmoidal shapes. Consequently, n(t) in GE₁, GE₂, GE₅ and GE₈ was replaced by V(t); n₀ in GE₂ by V₀; n₀₀ in GE₅ by V₀₀; n₀₀₀ and n_obs in GE₈ by V₀₀₀ and V_obs, respectively. In addition, n_med was replaced by V_med and n_∞ by V_∞. The parameter V_∞ was the tumor volume when t → ∞.

Two experimental groups were formed, each consisting of 10 male BALB/c/Cenp mice. The first group corresponded to the Ehrlich tumor, denominated G1, while the second group to the fibrosarcoma Sa-37 tumor, denominated G2. Experimental data of V(t) for Ehrlich and fibrosarcoma Sa-37 tumors were reported in [9], corresponding to their control groups.

Interpolation of experimental data

The Hermite interpolation method [13] was used to interpolate volume data of each individual tumor, in G1 and G2.

Estimation of values of α, β, d_f, D_f and u₂ from experimental data

Values of α and β (GE₁, GE₂, GE₅ and GE₈) and V_obs (GE₈) were obtained from the individual fitting of each tumor volume (Ehrlich and fibrosarcoma Sa-37). The value of V_obs estimated directly with GE₈ was named V_obs(α,β). The value V₀ = V₀₀ = V₀₀₀ = 0.5 cm³ was the tumor volume chosen to describe TGK. This volume value was reached 15 days after 2x10⁶ cells for the Ehrlich tumor and 5x10⁵ cells for the fibrosarcoma tumor Sa-37 were inoculated in the BALB/c/Cenp mouse (see details in [9]).

Three equations in terms of d_f, D_f and u₂ resulted when Eq (6) was substituted in GE₁, GE₂ and GE₅. The values of these three parameters were determined when each of these equations was used to fit experimental data. Besides, Eq (12) was substituted in GE₈ and resulted an equation in terms of d_f, D_f, u₂ and V_obs, from which their values were estimated from fitting experimental data. Once known the values of d_f, D_f, u₂ and V_obs, they were substituted in their respective Eqs (6) and (9) to calculate their corresponding values of α and β. Values of α, β and V_obs obtained by this way were denominated α_c, β_c and V_{obs(u2,df,Df)}, respectively, to distinguish these values from those that were directly obtained from fitting of the experimental data with GE₁, GE₂, GE₅ and GE₈.

The estimation errors for α, β, d_f, D_f, u₂, V_obs and V_{obs(u2,df,Df)} were denominated e_α, e_β, e_df, e_Df, e_u2, e_Vobs and e_{Vobs(u2,df,Df)}, respectively. The estimation error for each parameter was reported for each individual tumor of Ehrlich and fibrosarcoma Sa-37.

The difference between α and α_c, named Δα (Δα = α—α_c), was calculated for each equation (GE₁, GE₂, GE₅ and GE₈) and experimental group (G1 and G2). In addition, it were computed differences between β and β_c, denominated Δβ (Δβ = β—β_c), and V_obs(u₂,d_f,D_f) and V_obs(α,β), denominated ΔV_obs (ΔV_obs = V_obs(α,β)—V_obs(u₂,d_f,D_f)).

Criteria for model assessment

Five quality-of-fit criteria were used for fitting of experimental data with GE₁, GE₂, GE₅ and GE₈: the sum of squares of errors, SSE (Eq (10)); standard error of the estimate, SE (Eq (11)); adjusted goodness-of-fit coefficient of multiple determination, ra2 (Eq (12)), that depended on goodness-of-fit coefficient r² (Eq (14)); predicted residual error sum of squares, PRESS (Eq (14)); and multiple predicted residual sum error of squares, MPRESS (Eq (15)) [1, 3, 14], given by

The Root Means Square Error, RMSE (Eq (16)) and the maximum distance, D_max (Eq (17)) were also calculated following the methodology suggested in [1, 3, 14], given by

Each fit with the GE₁/GE₂/GE₅/GE₈ was performed for each animal growth curve. A computer program was implemented in the Matlab^® software (version R2012b 64-bit, Institute for Research in Mathematics and Applications, University of Zaragoza, Spain) to calculate the tumor volume. In addition, the mean ± standard error of each parameter of the equation (α, β, V_obs(α,β), u₂, d_f, D_f, V_{obs(u2,df,Df)}, α_c, β_c), fit criterion (SE, PRESS, MPRESS, ra2, RMSE and D_max) and estimation error (e_α, e_β, e_df, e_Df, e_u2, e_Vobs and e_{Vobs(u2,df,Df)}) were calculated from their individual values, in each experimental group, following the methodology reported in [1, 3]. These calculations were performed on a PC with an Intel(R) core processor (TM) i7-3770 at 3.40 GHz with a Windows 10 operating system. All calculations took approximately 10 min, for each equation.

Results

Simulation of Eq (6)

Fig 1 showed the simulations of β versus d_f/D_f (Fig 1A) and α versus d_f (Fig 1B) for different values of u₂. The positive values of α (in the interval 0 ≤ d_f < 1) and β (in the interval d_f/D_f < 1) increased non-linearly with the increase of d_f and decreased linearly with the increase of d_f/D_f, respectively. The negative values of α increased non-linearly with the increase in d_f (d_f > 1.5). The negative values of β decreased linearly with the increase of d_f/D_f (d_f/D_f > 1). These behaviors were noticeable for the greater value of u₂. Additionally, the parameter α had a discontinuity in the interval 1 < d_f < 1.5 and β = 0 when d_f/D_f = 1 for all values of u₂.

Simulation of Eq (9)

Results of the simulation of β versus d_f/D_f in Eq (11) coincided with that shown in Fig 1A (see Eqs (6) and (9)). The simulation of α versus d_f for n_obs = 1 (Fig 2A) reproduced the same result as in Fig 1B. However, values of α were more negative, in the interval 0 ≤ d_f < 1, when n_obs increased, being noticeable for the higher value of u₂ (Fig 2B, 2C and 2D). In Fig 2A, 2B, 2C and 2D, as in Fig 1B, it was observed a discontinuity of α in the interval 1 < d_f < 1.5.

Simulations of GE₂, GE₅ and GE₈

Fig 3 showed the behavior of n(t) when GE₂ (Fig 3A, GE₅ (Fig 3B) and GE₈ (Fig 3C and 3D) were used. Fig 3A revealed that the highest value of n_∞ and the fastest TGK occurred for the highest values of n₀ and α. Fig 3B showed that TGK was faster with the increase of n₀₀ and all TGK tended to the same value of n_∞ for all value of n₀₀, keeping constant values of α and β. In this case, TGK was faster when the value of n₀₀ increased with respect to n_obs (Fig 3B), being noticeable when n_obs increased with respect to 1 (Fig 3C). It is important to note that n₀ = n₀₀ (Fig 3B) and n₀ = n₀₀₀ (Fig 3C and 3D).

The results of Fig 3D showed that TGK grows slower (when n < n₀₀₀) and then faster (when n > n₀₀₀) for the greater value of n_obs; all TGK were cut at t = 0 (same value of n₀₀₀), for all value of n_obs; and the value of n_∞ depended on n_obs and not n₀₀₀ for each TGK. The results shown in Fig 3 were noticeable when the value of α increased with respect to that of β (results not shown).

Fitting of experimental data with GE₁, GE₂, GE₅ and GE₈ and estimation of values of α, β, d_f, D_f and u₂

The mean ± standard deviation of each parameter of the equation, fit criterion and estimation error were shown in Tables 1 and 2 of each equation (GE₁, GE₂, GE₅ and GE₈) used to fit experimental data of the Ehrlich and fibrosarcoma Sa-37 tumors, respectively. Tables 1 and 2 shown for these two tumor histological varieties: 0 < d_f < 1; 1 < D_f < 2; 0 < u₂ < 1; the highest values of α, u₂ and the lowest values of d_f and D_f for GE₈; the lowest SE values for GE₅ and GE₈; the lowest values of PRESS, MPRESS, RMSE and D_max; the highest values of r² and ra2 for GE₂, GE₅ and GE₈; and values of the parameter α differed when GE₁, GE₂, GE₅ and GE₈ were used. Nevertheless, the parameter β was the same when GE₂, GE₅ and GE₈ were used, but not for GE₁.

For the Ehrlich tumor, Δα = 0.003, 0.005, 0.001 and 0.005 days^-1 for GE₁, GE₂, GE₅ and GE₈, respectively. The variable Δβ = 0.012, 0.026, 0.014 and 0.000 days^-1 for these respective equations and ΔV_obs = 0.007 cm³. For the tumor fibrosarcoma Sa-37, Δα = 0.009, 0.003, 0.006 and 0.019 days^-1 for GE₁, GE₂, GE₅ and GE₈, respectively. The variable Δβ = 0.025, 0.038, 0.028 and 0.000 days^-1 for these respective equations and ΔV_obs = 0.006 cm³.

Discussion

This study shows that GE₂, GE₅ and GE₈ can be used interchangeably to describe experimental data of Ehrlich and fibrosarcoma Sa-37 tumors, taking into account their higher values of r² and ra2, and lower values of each parameter of the equation, fit criterion, estimation error, Δα, Δβ and ΔV_obs (ΔV_obs is only calculated for GE₈).

The theoretical and experimental results of this work confirm different findings reported previously in the literature, such as: 1) the fractal origin of GE₁, GE₂, GE₅ and GE₈, as reported in [4, 15]; 2) the fractal property of tumors once reached n_med/V_med, a matter that agrees with [16, 17]; 3) the role of the fractal dimension for the understanding of TGK, as suggested by Sokolov [18] and Breki et al. [19]; and 4) 1 < D_f < 2, in agreement with [4, 20, 21] and the preferential growth along the largest diameter of the tumor, despite its ellipsoidal geometry [1, 3, 9, 11]. This fourth finding is in contradiction with 2 < D_f < 3 reported by Breki et al. [19] in patients with metastatic melanoma; 5) The condition 0 < u₂ < 1 for both types of tumors is consistent with the Steel equation [12]. If u₂ = 0, then the tumor growth fraction must be high so that its mean doubling time (TD) is short, in contrast to [10, 12]. If u₂ = 1 day^-1 (all cancer cells are in apoptosis), TD → ∞ and α = 0 (tumor self-destruction), in contrast to the failure of the apoptosis mechanism in malignant tumors (because of the gene p-53 is repressed) and the existence of other cell loss mechanisms (metastasis, necrosis and exfoliation) [10, 11, 22]. The increase in u₂ brings about a decrease in TD and therefore a higher value of α (Figs 1B, 2A, 2B, 2C and 2D).

Other novel findings have been revealed in this investigation that may be of interest for understanding of TGK, such as: 1) TGK sigmoidal form and n_∞/V_∞ do not depend on n₀ and if on α, β and n_obs/V_obs, when a given tumor histological variety grows in a certain type of syngeneic host to it. In this way, the action form of parameter n₀/V₀ (form or location) is eliminated in GE₂, as reported in [2]. 2) The GE₈ states that n₀ in the GE₂ is not a constant parameter but depends non-linearly with n_obs/V_obs, n₀₀₀/n_obs (V₀₀₀/V_obs), β and t. 3) The growth of a malignant tumor occurs for 0 < d_f < 1 and not when d_f = 0 (α = 0: the tumor does not form), 1 < d_f < 1.5 (discontinuity of α due to forbidden conformations or very unlikely tumor) and d_f > 1.5 (α < 0: the tumor self-destructs), in contrast to the values of d_f (1 < d_f < 2) reported in [4, 14, 23]. The forbidden conformations of the tumor can be explained by its stereochemistry due to the steric collides between all its elements and the tumor-host interaction. 4) The increase of α with the increase of d_f, at 0 < d_f < 1, confirms that the growth efficiency of a malignant tumor increases with its d_f, in agreement with [17, 24]. 5) Eq (11) states that this increase of α with d_f occurs if n_obs satisfies strictly the condition nobs<[(2/3df−1)/(df−1)]u2/β; otherwise, α < 0 for all β positive (Fig 2B, 2C and 2D). The case α < 0 means that the tumor self-destructs, in contrast to the experiment.

The established condition for n_obs suggests that: 1) n_obs/V_obs depends on d_f and the ratio u₂/β; 2) the fractal property of a malignant tumor also happens before or long before its detection (n_med/V_med), as reported in [1, 25]; 3) the ratio u₂/β may be an indirect indicator of the apoptosis-angiogenesis relationship reported in [26, 27]; 4) endogenous anti-angiogenic factors or inhibitors of angiogenesis (endostatin, angiostatin, among others) are present in the tumor before or long before reaching n_med/V_med; 5) the term e^−βt (see GE₈ and the established condition for n_obs) and the decrease of the parameter β with the increase of d_f/D_f corroborate the essential role of angiogenesis process and the displacement of the balance between endogenous anti-angiogenic factors and endogenous pro-angiogenic factors towards these latter, when the tumor volume grows at time t, consistent with [10, 17, 22, 28, 29].

From the mathematical point of view, the condition 0 < d_f < 1 may suggest that the contours of Ehrlich and fibrosarcoma Sa-37 malignant tumors have zero area and/or they are totally disconnected. The first assumption confirms that these two types of tumors can be delimited from their surrounding healthy tissue, as in [9, 11]. The second hypothesis is based on proposition 2.5 [30]: “A set F⊂ℜn with dim_H F<1 is totally disconnected”. In this proposition, F is any set and dim_H is the fractal dimension Hausdorff. It is important to note that, although the tumor boundary is wide, d_f < 1 if its only fractality is given by a totally disconnected line contained in that wide band.

From the biophysical point of view, the tumor contour totally disconnected can indicate the existence in it of pores/channels formed randomly of different sizes and shapes, changing in the time. This porous contour of a tumor may be related to the angiogenesis process (neo-formation of blood vessels), the formation of spicules by fragmentation of the contour into simple forms of molds (for example, triangles), roundness, irregular edge, anisotropy, roughness and compactness, findings reported in [1, 3, 10, 22, 31–34]. We believe that the tumor angiogenesis process can be regulated by the amount of pores/channels existing in its contour to interconnect with the surrounding healthy tissue. This hypothesis can corroborate that the angiogenesis of a malignant tumor is an emergency and regulated by the structural and conformational dynamic transformations that occur during TGK, as reported in [1]. On the contrary, if these pores/channels do not exist, the tumor would behave as an isolated system and would self-destruct, in contrast to the experiment.

Fig 3 deserves a careful interpretation, taking into account experimental results reported in the preclinical [1, 3, 9, 11, 14] and clinical [10] studies. The result of Fig 3A corresponds with the selection of different values of n₀/V₀ in the same TGK for different instants t₀. For this case, in the experiment is guaranteed fixed c_o, cell viability, the tumor histological variety and the type of syngeneic host to it. The higher value of n₀/V₀ in the same TGK means a larger tumor size, which is reached at a higher t₀.

Results of Fig 3B and 3C are associated to the same tumor histological variety that grows in several types of syngeneic hosts to it. For this case, c_o and cell viability fixed are guaranteed, taking into account the role of the immune system in the delay of TGK, depending on its immunocompetence degree [10, 11, 22, 35]. As a result, tumors reach different values of n₀₀/V₀₀ o n₀₀₀/V₀₀₀ at the same time t₀. The higher value of n₀₀/V₀₀ (n₀/V₀ in Fig 3B) or n₀₀₀/V₀₀₀ (n₀/V₀ in Fig 3C) corresponds to the lower immunocompetence degree of the host (e.g., an immunosuppressed host).

Results of Fig 3B refer to two possible situations: 1) different tumor histological varieties that grow in the same type of syngeneic host to them. For this case, c_o is different so that each tumor histological variety reaches the same value of n₀₀₀/V₀₀₀ at the same time t₀. 2) A given tumor histological variety that grows in different types of syngeneic hosts to it. For this case, c_o is the same for each tumor histological variety. For these two cases, n_obs/V_obs for each tumor histological variety is reached in a different t_obs, in accordance with the experiment [9, 11]. These two situations become noticeable when β approaches α (results not shown). Furthermore, this figure reveals that for the highest value of n_obs/V_obs (reached in a greater t_obs) TGK is slower for n(t) < n₀₀₀ (V(t) < V₀₀₀) and then faster for n(t) > n₀₀₀ (V(t) > V₀₀₀). By contrast, the tumor that has the lowest n_obs/V_obs is the fastest growing for n(t) < n₀₀₀ (V(t) < V₀₀₀) and then its TGK is slowest for n(t) > n₀₀₀ (V(t) > V₀₀₀).

The advantages of GE₈ over the various formulations of GE [2, 3], the Hahnfeldt model [36–38] and mKJMA equation [1], used to describe undisturbed TGK, are: 1) inclusion of two parameters (n_obs/V_obs y n₀₀₀/V₀₀₀) that are measured and estimated from experimental data. 2) TGK and n_∞/V_∞ can be known a priori if n_obs/V_obs (starting point of TGK), reached at t_obs, is estimated for each type of tumor that grows in a syngeneic host to it, as reported in [1, 3, 11].

The relation of the tumor growth with d_f and D_f is previously obtained by using a mesoscopic formalism and fractal dimension [39]. Besides, Izquierdo-Kurlich [39] report the differences between d_f and D_f and propose a relation between d_f and the dynamic quotient on the interface, named k_c, (see Eq (48)). This relationship differs from that reported in [4] (see Eq (3)), which is used to obtain Eq (8). If the relation published in [39] is taken into account in this study, Eq (8) is also obtained, except a small change in α numerator (1/2 instead of 1). As a result, 0.75 and 1 are the discontinuities of α, instead of 1 and 1.5, respectively. Nevertheless, these change do not affect significantly the results of this manuscript and confirm that tumors exits for 0 < d_f < 1. It can be verified that d_f for Ehrlich and fibrosarcoma Sa-37 tumors are less than 0.75 and 1 when Eq (48) in [39] and Eq (3) in [4] are used.

In this study, the tumor growth in the time results of the complex interactions that happen in the tumor and between it and the surrounding healthy tissue, as in [3,14]. Nevertheless, in it does not explicitly discuss the interactions among the individuals neither the cooperative capacity of they in a population to explain its growth behavior, as in [25, 5–8]. These works confirm the fractal property of the tumors, as in this study. Therefore, an additional study may include these interactions for Eq (8).

Further studies can be carried out to validate GE₈ in TGK of different tumor histological varieties that grow in both immune-competent and immune-deficient organisms. This will allow us to know how D_f, d_f, u₂, V_obs(α,β) and V_{obs(u2,df,Df)} change when using different types of tumors and degrees of immune-competence of several organisms, as well as confirming the relationship of these five parameters with the aggressiveness [1], angiogenesis [17], coherence [15, 16], anisotropy, heterogeneity, hardness, changes in the mechanical-elastic-electrical properties of a tumor, among others findings [1].

Conclusions

GE₈ describes well the growth kinetics of the Ehrlich and fibrosarcoma Sa-37 tumors and includes two parameters that are directly estimated from the experiment that confirm the fractal property of the tumors and the fractal origin of different Gompertz formulations.

Appendix A

In [4] it is assumed that the growth ratio of the number n(t) of tumor cells obeys to the differential equation

Assuming that the boundary has a fractal structure with dimension d_f, then m=k1rdf, r being the average radius of the tumor. On the other side, n depends on the morphology of the tumor, described by the fractal dimension D_f, and n=k2rDf. The morphological constants k₁ and k₂ are related to the magnification of the image [4].

Substituting these values of m and n and eliminating r, Eq (1) can be written as a Bertalanffy-Richards equation.

Taking into account that

This approximation is valid when θ→1 or n→n_ss.

In [36] it is justified that the quotient U₁/u₂ can be expressed as a function of d_f and in [4] it is shown that the solution of the differential system (2)

Supporting information

S1 Data [txt]
Supporting information.