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

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.


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. PLOS

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][2][3].
Izquierdo-Kulich et al. [4] report the fractal origin of GE (see appendix A). This fractal origin has also been reported in [5][6][7][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][2][3] and in the experiment [9,10] the starting point of TGK is considered when the initial number of tumor cells (n 0 ) and the initial tumor volume (V 0 ) satisfy the conditions n (t = 0) = n 0 and V (t = 0) = V 0 , respectively. In preclinical studies, the researcher chooses n 0 /V 0 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 n 0 /V 0 is named t 0 [1,3,9]. Nevertheless, in clinics, n 0 /V 0 corresponds to the tumor detected for the first time by the doctor by means of clinical and/or imaging methods. For this case, t 0 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 n 0 � n med , where n med is the minimum number of quantifiable cancer cells contained in the smallest measurable tumor volume, named V med (V 0 � V med ). The post-inoculation time that elapses until the tumor reaches n med /V med is named t med (t 0 � t med ) [3].
In [4], it is considered the Gompertz equation given in Eq (1) (named GE 1 ) According the considerations in the previous paragraph, GE 1 has two limitations: 1) n 0 = 1, which means that the tumor has only one cell when it reaches V 0 , in contradiction with the experiment [9,10]. 2) The maximum capacity of the tumor (n 1 ) depends only on α and β and not on n 0 (n(t) = n 1 = e α/β when t ! 1). From the mathematical point of view, n 1 is the upper asymptote of TGK. Nevertheless, in the preclinical, the condition t ! 1 is the postinoculation 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 n 0 /V 0 , as observed in [3,[10][11][12]. In the experiment, once the researcher fixes n 0 /V 0 , t 0 can be estimated a priori when the tumor latency time is known, named t obs (t obs < t 0 ), 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 � n 0 /V 0 ) 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][10][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 , n 0 /V 0 , 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 u 2 (the constant of the velocity of apoptosis) and n obs are shown.

Conventional Gompertz equation
Eq (2), named GE 2 , is the conventional GE and the most used when TGK starts at n 0 /V 0 , given by According to GE 2 , n 1 depends on n 0 , α and β (n(t) = n 1 = n 0 e α/β when t ! 1) and results from solving the ordinary differential Eq (3) with its initial condition, given by GE 2 suggests that n 0 (constant in time) has to be included in Eq (A2). Tjørve and Tjørve [2] report that n 0 acts as a parameter of shape (n 1 changes with n 0 ) or location (n 1 remains constant).

Inclusion of n 0 in Eq (A2)
In this topic was followed the methodology exposed in [4] and the initial number of tumor cells at t = 0, named n 00 , 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 GE 5 , agrees with GE 2 when n 0 ¼ ðn 00 Þ e À bt . In addition, the parameters n 00 and n 0 coincided exactly at t = 0. The constant parameter n 00 (n 00 � n med ) constituted the starting point of TGK for GE 5 and reached for t = t 0 . Therefore, it was convenient to New formulation of the Gompertz equation differentiate n 0 and n 00 to compare GE 2 and GE 5 in order to avoid confusion in the interpretation of these two parameters. GE 5 revealed that n 1 depends only on α and β and not on n 00 (n (t) = n 1 = e α/β for t ! 1).

Inclusion of n obs in GE
Eq (3) was rewritten as where n 000 was the number of tumor cells that the researcher selected at t = t 0 . The analytical solution of Eq (7) was given by Eq (8), named GE 8 , agreed with GE 5 at t = 0 (for all n obs ) and when n obs = 1 (for all t). The GE 8 coincided with the GE 2 at t = 0 (for all n obs ) and when n 0 ¼ n obs ðn 000 =n obs Þ e À bt . The parameter n obs (n obs < n med � n 000 ) was the starting point of TGK. In general, n 000 did not coincide with n 0 (GE 2 ) or n 00 (GE 5 ). Therefore, it was convenient to differentiate the parameters n 0 , n 00 and n 000 . In addition, the GE 8 evidenced that n 1 depends on n obs , α and β, but not on n 000 (n (t) = n 1 = n obs e α/β for t ! 1). The parameters α and β in terms of u 2 , U 1 , 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 = (u 2 / U 1 ) 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 u 2 (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 u 2 (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.

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][10][11] and clinical [10] studies; and 3) the graphs of V(t) and n(t) shown sigmoidal shapes. Consequently, n(t) in GE 1 , GE 2 , GE 5 and GE 8 was replaced by V(t); n 0 in GE 2 by V 0 ; n 00 in GE 5 by V 00 ; n 000 and n obs in GE 8 by V 000 and V obs , respectively. In addition, n med was replaced by V med and n 1 by V 1 . The parameter V 1 was the tumor volume when t ! 1.
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 u 2 from experimental data
Values of α and β (GE 1 , GE 2 , GE 5 and GE 8 ) and V obs (GE 8 ) were obtained from the individual fitting of each tumor volume (Ehrlich and fibrosarcoma Sa-37). The value of V obs estimated directly with GE 8 was named V obs(α,β) . The value V 0 = V 00 = V 000 = 0.5 cm 3 was the tumor volume chosen to describe TGK. This volume value was reached 15 days after 2x10 6 cells for the Ehrlich tumor and 5x10 5 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 u 2 resulted when Eq (6) was substituted in GE 1 , GE 2 and GE 5 . 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 GE 8 and resulted an equation in terms of d f , D f , u 2 and V obs , from which their values were estimated from fitting experimental data. Once known the values of d f , D f , u 2 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 GE 1 , GE 2 , GE 5 and GE 8 .
The estimation errors for α, β, d f , D f , u 2 , 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.

Criteria for model assessment
Five quality-of-fit criteria were used for fitting of experimental data with GE 1 , GE 2 , GE 5 and GE 8 : 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, r 2 a (Eq (12)), that depended on goodness-of-fit coefficient r 2 (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 SE ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi PRESS ¼ MPRESSðmÞ ¼ where V � j was the j-th measured tumor volume at discrete time t j , j = 1, 2, . . ., n 1 ;V � j was the j-th estimated tumor volume by GE 1 , GE 2 , GE 5 and GE 8 ; n 1 the number of experimental points (n 1 = 10) and k the number of parameters (k = 2 for GE 1 , GE 2 and GE 5 , and k = 3 for GE 8 ). The fitting was considered to be satisfactory when r 2 a > 0:98. Higher r 2 a meant a better fit. ðV � j Þ � a was the estimated value of V � j when GE 1 /GE 2 /GE 5 /GE 8 was obtained without the j-th observation. MPRESS removed the last n 1 −m measurements. Each equation (GE 1 , GE 2 , GE 5 and GE 8 ) was fitted to the first m measured experimental points (m = 3, 4 or 5) and then from calculated model parameters the error between tumor volume estimated and measured values in the remaining n 1 −m points was calculated. Least Sum of Squares of Errors was obtained when SSE was minimized in the Marquardt-Levenberg optimization algorithm.
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 RMSE ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where M was the number of interpolated data of tumor kinetics (graph of V(t)). F i was the i-th tumor volume of the experimental data, which was chosen as reference. G i was the i-th tumor volume calculated with GE 1 , GE 2 , GE 5 and GE 8 . Each fit with the GE 1 /GE 2 /GE 5 /GE 8 was performed for each animal growth curve. A computer program was implemented in the Matlab 1 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) , u 2 , d f , D f , V obs(u2,df,Df) , α c , β c ), fit criterion (SE, PRESS, MPRESS, r 2 a , 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.

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 u 2 (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.  Fig 3B) and GE 8 (Fig 3C and 3D) were used. Fig 3A revealed that the highest value of n 1 and the fastest TGK occurred for the highest values of n 0 and α. Fig 3B showed that TGK was faster with the increase of n 00 and all TGK tended to the same value of n 1 for all value of n 00 , keeping constant values of α and β. In this case, TGK was faster when the value of n 00 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 n 0 = n 00 ( Fig 3B) and n 0 = n 000 (Fig 3C and 3D).
The results of Fig 3D showed that TGK grows slower (when n < n 000 ) and then faster (when n > n 000 ) for the greater value of n obs ; all TGK were cut at t = 0 (same value of n 000 ), for all value of n obs ; and the value of n 1 depended on n obs and not n 000 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 GE 1 , GE 2 , GE 5

and GE 8 and estimation of values of α, β, d f , D f and u 2
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 (GE 1 , GE 2 , GE 5 and GE 8 ) 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 < u 2 < 1; the highest values of α, u 2 and the lowest values of d f and D f for GE 8 ; the lowest SE values for GE 5 and GE 8 ; the lowest values of PRESS, MPRESS, RMSE and D max ; the highest values of r 2 and r 2 a for GE 2 , GE 5 and GE 8 ; and values of the parameter α differed when GE 1 , GE 2 , GE 5 and GE 8 were used. Nevertheless, the parameter β was the same when GE 2 , GE 5 and GE 8 were used, but not for GE 1 .

Discussion
This study shows that GE 2 , GE 5 and GE 8 can be used interchangeably to describe experimental data of Ehrlich and fibrosarcoma Sa-37 tumors, taking into account their higher values of r 2 and r 2 a , and lower values of each parameter of the equation, fit criterion, estimation error, Δα, Δβ and ΔV obs (ΔV obs is only calculated for GE 8 ).
The theoretical and experimental results of this work confirm different findings reported previously in the literature, such as: 1) the fractal origin of GE 1 , GE 2 , GE 5 and GE 8 , 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 < u 2 < 1 for both types of tumors is consistent with the Steel equation [12]. If u 2 = 0, then the tumor growth fraction must be high so that its mean doubling time (TD) is short, in contrast to [10,12]. If u 2 = 1 day -1 (all cancer cells are in apoptosis), TD ! 1 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]. New formulation of the Gompertz equation The increase in u 2 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 1 /V 1 do not depend on n 0 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 n 0 /V 0 (form or location) is eliminated in GE 2 , as reported in [2]. 2) The GE 8 states that n 0 in the GE 2 is not a constant parameter but depends non-linearly with n obs /V obs , n 000 /n obs (V 000 /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 n obs < ½ð2=3d f À 1Þ=ðd f À 1Þ� u 2 =b ; 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 u 2 /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 u 2 /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 GE 8 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 � < n 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 (neoformation 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][32][33][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.  [1,3,9,11,14] and clinical [10] studies. The result of Fig 3A corresponds with the selection of different values of n 0 /V 0 in the same TGK for different instants t 0 . 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 n 0 /V 0 in the same TGK means a larger tumor size, which is reached at a higher t 0 .
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 n 00 / V 00 o n 000 /V 000 at the same time t 0 . The higher value of n 00 /V 00 (n 0 /V 0 in Fig 3B) or n 000 /V 000 (n 0 /V 0 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 n 000 /V 000 at the same time t 0 . 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) < n 000 (V(t) < V 000 ) and then faster for n(t) > n 000 (V(t) > V 000 ). By contrast, the tumor that has the lowest n obs /V obs is the fastest growing for n(t) < n 000 (V(t) < V 000 ) and then its TGK is slowest for n(t) > n 000 (V(t) > V 000 ).
The advantages of GE 8 over the various formulations of GE [2,3], the Hahnfeldt model [36][37][38] and mKJMA equation [1], used to describe undisturbed TGK, are: 1) inclusion of two parameters (n obs /V obs y n 000 /V 000 ) that are measured and estimated from experimental data. 2) TGK and n 1 /V 1 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][6][7][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 GE 8 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 , u 2 , 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]. GE 8 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 where m represents the number of tumor cells at the boundary of the tumor, u 1 is the constant of the velocity of the mitosis and u 2 is the constant of the velocity of apoptosis. Assuming that the boundary has a fractal structure with dimension d f , then m ¼ k 1 r d f , 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 ¼ k 2 r D f . The morphological constants k 1 and k 2 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.

dn dt
¼ U 1 n y À u 2 n ¼ n u 2 n ss n � � 1À y À 1 � � ; where n ss = (u 2 /U 1 ) 1/(1−θ) is the value of n at the steady state, the dimensionless morphological parameter θ is defined by θ = d f /D f and U 1 is given by U 1 ¼ u 1 k 1 =k y 2 . Taking into account that ln x ¼ lim s!1 s x This approximation is valid when θ!1 or n!n ss . In [36] it is justified that the quotient U 1 /u 2 can be expressed as a function of d f and in [4] it is shown that the solution of the differential system (2) ð Þð1À e À bt Þ with the intrinsic growth rate of the undisturbed tumor, named α (α > 0), and the deceleration factor, named β (β > 0), related to the tumor fractal dimensions by 8 > > > > > < > > > > > : Supporting information S1 Data. Supporting information. (TXT)