Background
Asymptotic growth indicates that a system shifts from positive feedback (which generates exponential growth) to negative feedback (which produces stabilizing growth). This shift is known as sigmoidal (“S-curve” or S-shaped growth). Systems that exhibit S-shaped growth-time behavior are characterized by constraints or limits to growth, as sickle cell disease [
1], tumors [
2], bacteria and microorganisms [
3], among others. Other systems produce S-shaped transformation-time behavior, as crystals [
4,
5].
Tumor growth kinetics (TGK) is not well understood so far. TGK has three well-defined stages: the first (Lag stage) is associated with the establishment of the tumor in the host. The second (Log or exponential stage) is related to rapid tumor growth. The third (Stationary stage) shows slow tumor growth asymptotically converging to a final volume [
2]. It is expected a fourth stage (Death stage) of TGK, in which tumor dies because the nutrients are depleted by anorexia of animal or human host, showing a decline. This fourth stage is not considered in TGK due to ethical considerations [
6,
7]. In mice, tumor burden should not usually exceed 10% of the host animal’s normal body weight [
6].
During the last decades, tremendous efforts have been made by both experimentalists and theoreticians to search a suitable growth law for tumors, one of the most striking and interesting issues in cancer research [
2,
8–
11]. The Logistic equation has been used to describe TGK and the interactions among different competing populations with and without an external perturbation [
12,
13]. The Logistic and von Bertalanffy equations have been reported to provide excellent fits for patients and mice bearing tumors, respectively [
8]. In contrast, Marušic et al. [
9] and Miklavčič et al. [
11] show that the standard Gompertz model outperforms both Logistic and von Bertalanffy models. Marušic et al. [
9] explain this disparity because the fit is dependent on the applied least squares fitting method. The Gompertz model is the most used to describe TGK [
2,
8,
10,
11,
14].
The standard Logistic equation (Eq.
1) and the standard Gompertz equation (Eq.
2) are given by [
8–
11]:
$$ V(t)=\frac{K^{*}{V}_o{e}^{r^{*} t}}{K^{*}+{V}_o\left({e}^{r^{*} t}-1\right)} $$
(1)
$$ V(t)={V}_o{e}^{\left(\frac{\alpha}{\beta}\right)\left(1-{e}^{-\beta t}\right)} $$
(2)
where V(t) represents the untreated tumor volume at time t and V
_{o} its initial volume at the beginning of observation (t = 0). Experimentally, V
_{o} (reached in a time t
_{o}) is any tumor volume that satisfies the condition V
_{o} ≥ V
_{meas}. V
_{meas} is the minimum measurable tumor volume and reaches in a time, t
_{meas}. The constant r
^{*} defines the growth rate and K
^{*} is the carrying capacity [
8,
12]. The parameter α is the intrinsic growth rate of the unperturbed tumor related to the initial mitosis rate. The parameter β is the growth deceleration factor related to the endogenous antiangiogenesis processes [
11,
15] by an overexpression of different antiangiogenic molecules (i.e., Angiostatin, Thrombospondin-1 molecules) [
15,
16]. As tumors are not perturbed with an external agent, this parameter β is not related to therapy-induced antiangiogenis [
12]. Despite the interpretation of the parameter β, authors of this study believe that this parameter may be related to other endogenous antitumor processes, as cellular death processes (apoptosis, necrosis, metastasis and exfoliation) and interactions between tumor cells and immune cells [
17]. Further experiments are required for a correct interpretation of this parameter.
An important part of tumor vital cycle has already happened long before V
_{meas} is reached [
17] and therefore it cannot be described with the Eqs. (
1) and (
2). However, this part of TGK may be fitted if an effective delay time (τ) is introduced in the Eqs. (
1) and (
2) [
2,
18–
20]. Besides, τ has been included in these two equations to describe Lag stage of bacteria- and microorganism growths [
3]. τ has a crucial role in the modeling of biological processes [
21]. The interesting question is if in the case with delay the Logistic model, named modified Logistic model (Eq.
3), or the Gompertz model, named modified Gompertz model (Eq.
4), is the best one for describing early tumor growth as it is believed in the case without delay (Eqs.
1 and
2). Equations (
3) and (
4) result of the substitution of t by (t-τ) in the Eqs. (
1) and (
2):
$$ V\left( t-\tau \right)=\frac{K^{*}{V}_{\tau}{e}^{r^{*}\left( t-\tau \right)}}{K^{*}+{V}_{\tau}\left({e}^{r^{*}\left( t-\tau \right)}-1\right)} $$
(3)
$$ V\left( t-\tau \right)={V}_{\tau}{e}^{\left(\frac{\alpha}{\beta}\right)\left(1-{e}^{-\beta \left( t-\tau \right)}\right)} $$
(4)
where V(t-τ) represents tumor volume at time (t-τ), meaning that the growth at present time t depends on the previous time (t-τ). Parameters τ and V
_{τ} are the time and tumor volume corresponding to inflection point of TGK, respectively. Parameters r
^{*}, K
^{*}, α and β have been defined above in Eqs.
1 and
2.
Different findings have been documented in cancer, as: heterogeneity, anisotropy, fractal property, stiffness, surface roughening, curved surface, high macroscopic shear elastic modulus, among others [
17,
21–
25]. These findings have been also reported in crystals, despite noticeable differences between tumors and crystals, and in their growth mechanisms [
26–
30].
The classical Kolmogorov-Johnson-Mehl-Avrami model, named KJMA model (Eq.
5), and modified Kolmogorov-Johnson-Mehl-Avrami model, named mKJMA model (Eq.
6), have been used to fit entire sigmoidal curve of a crystal [
26], given by
$$ p(t)=1-{e}^{-{(Kt)}^n} $$
(5)
$$ p(t)=1-{\left[1+\left(\lambda -1\right){(Kt)}^n\right]}^{-1/\left(\lambda -1\right)} $$
(6)
With
$$ K(T)={K}_o{e}^{-{E}_a/ RT}\;\left(\mathrm{Arrhenius}\ \mathrm{equation}\right) $$
(7)
where p(t) is the transformed fraction at t (fraction of grains that is transformed to crystal phase). n (n ≥ 0), K(T), λ (λ ≥ 1), K
_{o}, E
_{a}, R and T are the Avrami exponent, specific rate of transformation process that depends on temperature, impingement factor, the pre-exponential factor, effective (overall) activation energy of the transformation (or activation energy barrier to crystal formation), Boltzmann constant and temperature, respectively. RT represents the thermic kinetics energy. Arrhenius Eq. (
7) is substituted in the Eqs. (
5) and (
6) to know K
_{o} and E
_{a}.
In crystals, K is constant, proportional to the transforming volume/surface area and results of unbalanced diffusion processes (linked to heterogeneity). λ represents impingement mechanisms, as: capillarity effect, interfacial and superficial phenomena, among others. n is closely related to nucleation mechanisms, the existence of a lag stage, anisotropy, structural changes, vacancy annihilation, stiffness, surface roughening, curved surface, change of shape and high macroscopic shear elastic modulus of the forming and growing crystal. Additionally, n is inversely proportional to fractal dimension of the crystal.
n ≥ 3 has been related to spherical shape of crystals, formation of micro-clusters of crystal seeds, high anisotropy and higher vacancies number [
26–
30].
On the other hand, nucleation and impingement mechanisms emerge to eliminate high energetic instabilities (by thermal fluctuation) during forming and growing crystal structure. Nucleation sites (or vacancy numbers) disorder the interior of forming and growing system and need be filled to guarantee their stability and growth. Deviation from integer value for n has been explained as simultaneous development of two (or more) types of crystals, or similar crystals from different types of nuclei (sporadic or instantaneous). Nucleation is either instantaneous, with nuclei appearing all at once early on in the process, or sporadic, with the number of nuclei increasing linearly with time [
26–
30].
KJMA and mKJMA models are phenomenological and not valid when T varies with time [
31]. Furthermore, they are developed for the kinetics of phase changes to describe the rate of transformation of the matter from an old phase to a new one, taking into account that the new phase is nucleated by germ nuclei that already exists in the old phase. The Eq. (
6) can be reduced to the Eq. (
5) when λ tends to 1. Wang et al. [
26] report that KJMA model cannot be applied to crystal growth when λ > 1 because there are phenomena (i.e., capillarity effects, vacancy annihilation, blocking due to anisotropic growth) that may cause violations to KJMA. Consequently, a misinterpretation of the kinetics may be given if these phenomena are ignored.
We are not aware that KJMA model and mKJMA model have been used to describe TGK. Nevertheless, in principle, these two models can be used to fit S-shaped growth of tumors, taking into account that “S-curve” is universal, the Eqs. (
1,
2,
3,
4,
5 and
6) are phenomenological and the above-mentioned findings are common for both tumors and crystals. The application of the Eqs. (
5) and (
6) may reveal whether other findings not yet described are involved in TGK. Elucidating underlying mechanisms in entire TGK is of great importance for both understanding and planning antitumor therapies. The aim of this paper is to use, for the first time, KJMA and mKJMA models to describe the untreated fibrosarcoma Sa-37 TGK. Also, KJMA and mKJMA models are compared with modified Gompertz and Logistic models.
Discussion
The results of this study are valid for the unperturbed fibrosarcoma Sa-37 tumor, experimentally transplanted to BALB/c mice. As shown, parameter n
_{loc} is a better descriptor than n for the entire TGK. The plausibility of V(t) versus t plot and/or p(t) versus t plot for TGK analysis is also suggested, in agreement with [
34]. Equations (
1,
2,
3,
4,
5 and
6) can be used to fit normalized experimental data from Sa-37 tumor, as assessed by the high
r
_{ a } ^{2} values, low values of SSE, SE, PRESS, MPRESS as well as overall estimation accuracy. Each equation has high prediction capability and good missing data handling. This further supports sigmoid laws universality [
3,
35].
Despite mentioned in the previous paragraph, a weighted least square minimization in formula (
6) may be proposed for selection of the best model, taking into account the uncertainty of the individual measurements of the tumor volume and the fact that the larger the volume, the larger the standard deviation. This and other statistical criteria [
33] in tumor volumes with smaller and larger standard deviations will be included in a further study.
As obtained, V
_{o} can be indistinctly chosen as V
_{obs}, V
_{oo} or V
_{meas} since Eq. (
2) behaves similarly when any of them is used in experimental data fitting. Unlike Eqs. (
2) and (
4), the parameters of Eq. (
6) depend on the first point of TGK, indicating that it senses the microstructural changes from beginning of TGK (t = 0).
The good fits yielded by Eqs. (
1) and (
3) are in contrast with [
8,
9,
11,
33]. This can be due to the omission of larger tumors, since mice were slaughtered earlier, following [
6]. That is why, p(t) and n
_{loc} do not reach the values of 1 and 0, respectively. In crystals, p(t) = 1 and n
_{loc} = 0 [
26].
Equation (
5) should not be used for TGK interpretation, since λ > 1; its parameters differ respect to those of Eq. (
6) (Tables
2 and
3, and Fig.
3) and graphical strategies are noticeably different for these two equations. This agrees with Wang et al. [
26]. Accordingly, results obtained with Eq. (
5) have not been exposed here.
The close relationship between fibrosarcoma Sa-37 tumor spherical shape and n
_{loc} ≥ 3 is corroborated in this study. Similar finding is reported in crystals [
28–
30]. This tumor spherical shape may be vital for tumor growth due to a lower surface curvature, in agreement with [
2,
36–
38]. Jump of n
_{loc} and the change from spherical to non-spherical shape may be related to a shift from avascular (before 10 days) to vascular growth phase (after 11 days). Transition between these two phases has been previously reported [
36,
37]. The observed n
_{loc} jump corresponds to a transition of high (before n
_{loc} jump) to low (after n
_{loc} jump) value of n
_{loc}, suggesting the occurrence on TGK of two types of growth mechanisms that happen at different time scales: nucleation (below 10 days) and pure growth (above 11 days). Nucleation is expected at vascular growth phase, mainly at its very early stages, by high values of n
_{loc} and it is the stochastic stage of a forming and growing system. This later may be due to the Brownian motion (a fractal stochastic process) of thermally fluctuating and energetically unstable tumor cells in suspension at t = 0.
High energetic instabilities at avascular growth phase are mitigated by nucleation mechanisms, suggesting a high micro-anisotropy, confirmed by n
_{loc} ≥ 5. Micro-anisotropy leads to random formation of non-uniform and energetically unstable cellular micro-clusters, which establish a space-time competence for nutrients, oxygen and energy, resulting in high micro-heterogeneities, as reported in multicellular spheroid models [
36–
38]. This may explain the existence of the entropy production [
39] and the diffusion limited aggregation at avascular tumor growth (mainly at its very early stages of TGK) because the tumor cells move randomly in Brownian motion, forming fractal clusters when diffusion is the main transport mechanism. Brownian motion and diffusion limited aggregation are stochastic rather than deterministic processes with random fractal dynamics. This diffusion limited aggregation may have an impact in TGK [
40] and result in tumor cells packed in a multicellular spheroid not yet connected to the host’s blood supply, in agreement with [
36–
39,
41].
The formation of these cellular micro-clusters discards the occurrence of a burst nucleation, which means that all nucleation sites are immediately saturated at t = 0. Burst nucleation is reached for K → ∞, λ = 1, n → ∞ and/or DT → 0, in contrast with the results of this paper and with duration of Lag stage of TGK observed in preclinical (several days) and in clinical (several months and years) studies. Additionally, the existence of cellular micro-clusters may suggest that a tumor solid seed (or smallest size of a solid tumor), long before of V
_{obs}, may be essentially formed via heterogeneous nucleation mechanisms, as previously hypothesized Cabrales et al. [
2]. This via is confirmed in this study by non-integer values of n and n
_{loc}, as in crystals [
28–
30].
Nucleation mechanisms may help to form these cellular micro-clusters by filling the high nucleation sites (or vacancies), which may correspond with unoccupied sites of the cancer cells. The existence of these sites may be justified because n
_{loc} ≥ 5; this can lead to a higher number of heterogeneous sites, making unstable both the forming cellular system and the cellular micro-clusters. This process may be stabilized and ordered by both inter-cellular interactions and the overlapping of diffusion fields of tumor cells, a matter that agrees with [
19,
36,
41], suggesting the existence of soft impingement mechanisms during the avascular growth phase. These mechanisms are also confirmed because λ > 2, as in crystals [
26–
28]. Nucleation and soft impingement mechanisms may explain, in part, why a slightly better binding of cancer cells with less detachment, in agreement with [
42].
The filling of vacancies may explain why n
_{loc} drops up to the jump of n
_{loc}. After n
_{loc} jump, n
_{loc} increases probably because pure growth mechanisms emerge and prevail over nucleation mechanisms. If pure growth mechanisms do not emerge, nucleation sites are completely saturated (n
_{loc} tends to 0) in less than 30 days, in contrast with the results shown in Fig.
3. It should be expected that n
_{loc} tends to 0 for larger tumors (≥3 cm
^{3}, which is reached long past 30 days) because TGK decelerates at stationary stage of TGK (cell-production-to-cell-loss rate is very slow or unalterable). This ratifies that TGK cannot be linear nor exponential (the host cannot fully sustain solid tumors due to their sizes would be bigger than host size). Accordingly, solid tumors are cooperative boundless systems, in agreement with the S-shape of tumor growth, and the fact that Eq. (
6) has to level off at both extremes to represent almost no binding at the beginning of TGK and saturated binding at the final of TGK.
Heterogeneity and anisotropy of the fibrosarcoma Sa-37 tumor at vascular growth phase are confirmed by palpation; time changes of n
_{loc}, L
_{1}, L
_{2}, L
_{3}, FF and R
_{c}; irregular border, deformation and surface roughening [
2,
17,
25] and are associated with compactness, stiffness and surface tension of the tumor [
23,
24,
43]. Anisotropy produces preferred directions of growth, minimizing surface tension.
Brownian motion and cellular micro-clusters at very early stages of TGK; Figs.
1,
2 and
3; and the irregular border, surface roughening and stiffness of the tumor at vascular growth phase may suggest that forming, growing and transforming cellular system along TGK happens in a fractal space-time; as a consequence the fractional Hausdorff dimension (D
_{H}) is higher than the topologic dimension (D
_{T}), as it corresponds to a fractal space [
44]. This means that although D
_{T} = 0 for tumor cells in suspension (considered as a set of points) at t = 0, 0 < D
_{H} < 1. It is expected that the forming and growing cellular system on TGK pass through different spatial patterns, starting from worm-like linear structures (D
_{T} = 1 and 1 < D
_{H} < 2); then, fish-like plane structures (D
_{T} = 2 and 2 < D
_{H} < 3); spatial solid-like structures (D
_{T} = 3 and 3 < D
_{H} < 4); and lastly, space-time structure (D
_{T} = 4 and 4 < D
_{H} < 5 or higher dimensions). It is possible that these two later structure types are reached once the tumor solid seed and vascular growth phase are formed, respectively. This is in contrast with [
45] and agrees with [
46]. Waliszewski and Konarski [
45] obtain that the value of the mean temporal fractal dimension decreases along the curve approaching integer value because the fractal structure is lost with tumor progression. Shim et al. [
46] correlate the S-shaped time increase of tumor fractal dimension, with textural parameters (i.e., hardness) and the growth in the time of space-time branching structures (or patterns). These structures are linked to the abnormal network of blood vessels, in agreement with the findings of the present study. This and the inverse relation between p(t) and n
_{loc} (Figs.
1,
2 and
3) may suggest that n
_{loc} and the tumor fractal dimension are inversely related, as it takes place in crystals [
30]. Time changes in D
_{T} and D
_{H} may explain, in part, why tumor cells in vitro form colonies and grow in layers, unlike the normal cells, which do not form colonies [
47].
Fractal properties of tumors have been correlated with its microstructure, microscopic coherent local deformation processes (or local dynamical rearrangements), mitosis rate, heterogeneity, anisotropy, complexity degree, spatial-temporal coherence, self-organization, self-stabilization, self-symmetry, self-ordering, self-similarity, mechanical properties (stiffness, compactness and surface roughening), temporal changes of nontrivial shape and dynamical structural intrinsic transformations [
21,
35,
48–
51].
Tumor fractal dimension may suggest that the tumor is a type of fractal, named space-filling fractal that continuously attempts to fill in the area leaving no empty holes. The space-filling pattern is formed by placing some non-overlapping units of smaller sizes. This may confirm the existence of annihilation of vacancies, in agreement with Molski and Konarski [
48], confirming that solid phase of TGK is spatially coherent and therefore, all tumor cells co-operate collectively producing spatial-temporal organization and complex patterns.
The above discussed suggests that TGK is a fractal from its beginning (t = 0), unprecedented in the literature. This statement agrees with [
52], in which is demonstrated the fractal origin of the Gompertz equation. Izquierdo-Kulich et al. [
52] explain their results because α and β are connected with morphology of the tumors, specifically with the fractal character of them.
Small values of E
_{a} and minimal tumor surface tension [
23] may explain time evolution of V(t), p(t), n
_{loc}, D
_{T}/D
_{H}, shapes of the fibrosarcoma Sa-37 tumor. In addition, small values of E
_{a} corroborate that vacancies require small amount of energy for their creations. They have received insufficient attention and may have an important role in the carcinogenesis, production/lost rate of tumor cells, and in mechanics [
23,
24] and dielectric [
53] properties of a forming and growing cellular system, during entire TGK. This may be explained because vacancies disorder these cellular systems, leading to structural, morphological and electrical instabilities [
21,
48,
49,
51]. As a result, impingement mechanisms emerge during TGK in order to guarantee the most efficient space-filling and to stabilize the forming and growing cellular system, in order to maximize its exchanges of nutrients in the minimum amount of space and therefore to maximize the tumor growth and its stabilization. After n
_{loc} jump, soft impingement mechanisms guarantee the growth, stabilization and survival of the tumor by branching structure (abnormal vascular network) [
14–
17], the different abnormal signaling pathways, the interactions that happen in the tumor and/or other uncontrolled environmental factors [
17,
23,
36,
54,
55]. As a result, the tumor cells do not multiply in an unregulated manner, as reported in [
17], but they are regulated by the number of vacancies available to be filled. Furthermore, intrinsic local dislocations lead to dynamical rearrangements of tumor cells, suggesting that dynamical structural intrinsic transformations take place along the entire TGK. This indicates that the forming and growing cellular system passes through different dynamical conformational states or meta-stable configurations. These configurations are rearrangements of the cancer cells that take place over a wide energy range due to the large number of stabilized and ordered cellular configurations. This agrees with Guha [
56], who reports that a change of state takes place if there is an unbalanced force anywhere within the system, or between the system and its surrounding, leading to variations in pressure or elastic stress which give rise to the tumor expansion. This brings about that TGK may be limited and controlled by vacancies, which are governed by nucleation/growth and impingement mechanisms, and dynamical structural intrinsic transformations. As a result, cancer self-renews constantly and TGK is a highly coordinated dynamic multi-step process, in agreement with [
57].
On the other hand, these dynamical structural intrinsic transformations may explain, in part, immune resistance mechanisms and low effectiveness of some antitumor therapies (i.e., immunotherapy when solely applied), in agreement with [
58]. This may be explained because these transformations may be responsible for structural and stereochemical changes on membrane-bound receptor-ligand immune checkpoints that promote the tumor activity. As a result, ligand-receptor interactions are perturbed due to the expression of immune-checkpoint proteins which are disregulated [
58].
The results may evidence that entire TGK is not only due to imbalance between cell production and cell loss [
17] and other hallmarks of cancer [
17,
54,
55], but also to diffusion-controlled nucleation/growth and impingement mechanisms, and dynamical structural intrinsic transformations, which may be the key to understand how a solid tumor arises and grows. These findings are often ignored in literature and may indicate that TGK is about dynamical structural transformations, instead of pure growth kinetics. They may explain why K is an order smaller than α/r
^{*}; DT value estimated with the Eq. (
6) is smaller than that estimated experimentally and with the Eq. (
2); the differences between the values of K, n, λ and E
_{a} report in the Tables
2 and
3; and the difference of n
_{loc} versus ln(t) for KJMA and mKJMA models (Fig.
3). On the other hand, if these findings are not considered on entire TGK, then pure growth mechanisms prevail in it, meaning that K ≅ α ≅ r
^{*} and DT estimated experimentally and with Gompertz, Logistic and mKJMA models are equals, in contrast with results here shown.
Besides, the prevalence of these findings at avascular growth phase ratify that an important part of vital cycle of a solid tumor occur before it is clinically detected, in agreement with [
17]. On the other hand, the Eq. (
6) senses the microstructural changes that happen during the entire TGK, mainly at avascular growth phase, in contrast to Eqs. (
1,
2,
3 and
4).
Many questions may arise, as: how n, n
_{loc}, λ and E
_{a} depend on α, β and DT, which characterize the histological characteristics of a solid tumor? How an immune-deficient or immune-competent organism affect n, n
_{loc}, λ and E
_{a} values? Can the Eq. (
6) be modified to fit the perturbed tumor growth kinetics with an external agent? among others. This first study cannot give answers to all these questions. Relevant biological and clinical data may now be gathered in a systematic manner in order to test our theory or any other quantitative model derived using a methodology similar to ours, with the aim of helping to understand, and potentially handling, the process of tumor growth. Future studies will provide in-depth experimental findings that permit a best interpretation of the parameters of mKJMA model in cancer.
Acknowledgements
All authors are supported by the Ministry of Higher Education, Republic of Cuba. Our special thanks to Rosa Ivette Robles Matos and Yenia Infante Frómeta for their technical assistances. Further, Ana Elisa Bergues Pupo (at present in Max Planck Institute of Colloids and Interfaces, department Theory and Bio-Systems, Potsdam, Germany) thanks Dr. Jesús Manuel Bergues Cabrales (Universidad de San Jorge, Zaragoza, Spain) for his support. We would like to give our special thanks to the Editor in Chief and reviewers of this article for their expert help and invaluable feedback.
Authors’ contributions
Study concepts: MMG, JAGJ, LEBC. Study design: MMG, JAGJ, LEBC, AEBP. Data acquisition: MMG, JAGJ, LEBC, AEBP, HMCC, MVJ, MAOM. Quality control of data and algorithms: LEBC. Data analysis and interpretation: MMG, JAGJ, LEBC, AEBP, BS, SK, HMCC, JBR, MVJ, MAOM, TRG, SCAB, JLHC, GVSG. Statistical analysis: MMG, JAGJ, LEBC, AEBP. Manuscript preparation: MMG, JAGJ, LEBC, AEBP, JLHC. Manuscript editing: MMG, LEBC, AEBP. Manuscript review: MMG, JAGJ, LEBC, AEBP, BS, SK, HMCC, JBR, MVJ, MAOM, TRG, SCAB, JLHC, GVSG. All authors read and approved the final manuscript.