 Research article
 Open Access
 Published:
Modified Gompertz equation for electrotherapy murine tumor growth kinetics: predictions and new hypotheses
BMC Cancer volume 10, Article number: 589 (2010)
Abstract
Background
Electrotherapy effectiveness at different doses has been demonstrated in preclinical and clinical studies; however, several aspects that occur in the tumor growth kinetics before and after treatment have not yet been revealed. Mathematical modeling is a useful instrument that can reveal some of these aspects. The aim of this paper is to describe the complete growth kinetics of unperturbed and perturbed tumors through use of the modified Gompertz equation in order to generate useful insight into the mechanisms that underpin this devastating disease.
Methods
The complete tumor growth kinetics for control and treated groups are obtained by interpolation and extrapolation methods with different time steps, using experimental data of fibrosarcoma Sa37. In the modified Gompertz equation, a delay time is introduced to describe the tumor's natural history before treatment. Different graphical strategies are used in order to reveal new information in the complete kinetics of this tumor type.
Results
The first stage of complete tumor growth kinetics is highly non linear. The model, at this stage, shows different aspects that agree with those reported theoretically and experimentally. Tumor reversibility and the proportionality between regions before and after electrotherapy are demonstrated. In tumors that reach partial remission, two antagonistic posttreatment processes are induced, whereas in complete remission, two unknown antitumor mechanisms are induced.
Conclusion
The modified Gompertz equation is likely to lead to insights within cancer research. Such insights hold promise for increasing our understanding of tumors as selforganizing systems and, the possible existence of phase transitions in tumor growth kinetics, which, in turn, may have significant impacts both on cancer research and on clinical practice.
Background
Tumors are complex biological systems, and, in spite of great therapeutic advances, many of these still do not respond to treatment and lead to death. Part of the complexity of the problem is the sheer consequence of the tumor size and its histogenic characteristics. The cancer phenomenon continues to challenge oncologists. The pace of progress has often been slow, in part because of the time required to evaluate new therapies. To reduce the time to approval, new paradigms for assessing therapeutic efficacy are needed [1]. This requires the intellectual energy of scientists working in the field of mathematics and physics, collaborating closely with biologists and clinicians. This essentially means that the heuristic experimental approach, which is the traditional investigative method in the biological sciences, should be complemented by a mathematical modeling approach [2].
Significant research has been done in the modeling of tumors using theoretical models and computer simulations in order to describe and predict various aspects of tumor growth kinetics (TGK). Predicting tumor growth is important in the planning and evaluation of screening programs, clinical trials, and epidemiological studies, as well as in the adequate selection of doseresponse relationships regarding the proliferative potential of tumors [2–5].
The biological behavior of a malignant tumor is highly influenced by its growth rate, which is determined by many intratumoral and microenvironmental factors. The spacetime permanent growth is probably the most characteristic feature of a malignant tumor.
Further advancement in mathematical modeling of TGK critically depends on a thorough testing of proposed models against new data as they become available with the development of experimental techniques [3–6]. Thus, it is apparent that theoretical mathematical models are needed to study cancer.
In electrotherapy (ET) with lowlevel direct electrical current (DEC), mathematical modeling has not been used. ET was revolutionary when first introduced and is a promising surgical technique for destroying tumors. It has been shown to be a very useful, alternative tool against cancer. Preclinical and clinical studies have shown that ET is simple, safe, effective, and, minimally traumatic, with few side effects. It provides a method for treating solid cancers that are conventionally inoperable, those that cannot be resected after thoracotomy, and those that are not responsive to chemotherapy or radiotherapy [7–10]. Similar results have been reported by our research group [11–15].
Although preclinical and clinical studies have shown that ET has a marked antitumor effect, it is not widely used in clinical practice. The reason is that ET is not a wellestablished therapy due to the lack of a standardized method and unclear knowledge concerning the mechanisms involved. As a result an optimal electrode distribution has not been determined for ET, nor has the doseresponse relationship been established. For this reason, we pay special attention to these two factors [16, 17].
Camué et al. [14] experimentally report that an increase in DEC intensity increases its antitumor effectiveness, and that Ehrlich and fibrosarcoma Sa37 tumors have a DEC threshold for which their complete remission is reached. These results have been theoretically corroborated through the use of a modified Gompertz equation (MGE), which has a good prediction capability to describe both unperturbed and DECperturbed TGK [17].
Many intrinsic processes that occur in both unperturbed and DECperturbed TGK are unknown. We believe that the MGE can be used as a tool to reveal some of these processes in order to improve DEC effectiveness. The aim of this paper is to describe the complete growth kinetics of unperturbed and DECperturbed fibrosarcoma Sa37 tumors through the MGE in order to generate useful insights into the mechanisms that underpin this devastating disease. In this study, we analyze this model taking into account the experimental data reported in [14] for fibrosarcoma Sa37 tumor. Also, we discuss the current limitations and potential implications of this model for further TGK research. It is important to note that the results reported in [14] and [17] support this paper.
Methods
This study is approved by the Committees of Ethics of the National Center of Electromagnetism Applied (CNEA) and the Conrado Benitez Oncologic hospital, Santiago de Cuba, Cuba.
Complete growth kinetics for unperturbed and DECperturbed tumors
Our experiences in preclinical and clinical studies have indicated that DECtreated TGK is complex, with two welldefined regions (REGI and REGII). REGI (defined before DEC treatment is performed) includes the initial time of tumor cell inoculation (t = 0 days) up to the moment that tumor is perturbed by a DEC stimulus, which occurs when it reaches a volume V_{o} (initial volume selected by the therapist). REGII (defined after DEC treatment is performed) includes the time at which the tumor is perturbed by DEC stimulus up to the end of the experiment.
In preclinical studies, the end of the experiment is fixed by the researcher, whereas in clinical studies, it can occur at multiple events: 1) the patient dies, 2) the patient leaves the clinical trial, or 3) the patient is completely cured [9, 10, 15]. It is important to point out that this REGII is only reported in the field of ET in cancer [9–15, 18–21].
Electrochemical treatment
Once fibrosarcoma Sa37 tumors have reached approximately V_{o} = 0.5 cm^{3} in BALB/c mice, four platinum electrodes are inserted into their bases and a singleshot electrotherapy is supplied. V_{o} is reached 15 days after viable tumor cells are inoculated in the dorsolateral region of the animals. Four groups (one control group and three treated groups), each consisting of ten mice, were randomly formed: the control group (CG), a treated group with 36 C/cm^{3} (18 C in 0.5 cm^{3}) and 6.7 mA for 45 min (TG1), a treated group with 63 C/cm^{3} (31.5 C in 0.5 cm^{3}) and 11.7 mA for 45 min (TG2), and a treated group with 80 C/cm^{3} (40.0 C in 0.5 cm^{3}) and 14.8 mA for 45 min (TG3). The experimental details are discussed by Camué et al. [14].
Modified Gompertz equation
A feature of the MGE is that it is developed after the experiments. It is implemented in order to fit the experimental data corresponding to REGII for the Ehrlich and fibrosarcoma Sa37 TGK [17], given by
where
with
and
All parameters involved are real and positive. V*(t') represents the tumor volume (TV) at time t' after DEC treatment. The parameter α (α > 0) is the intrinsic growth rate of the tumor related to the initial mitosis rate. The parameter β (β > 0) is the growth deceleration factor related to the antiangiogenic process. The parameter α* is the modified tumor growth rate due to DEC action. i (i > 0) is the DEC intensity that flows through the tumor by the application of an external electric field. i _{ o }(i _{ o }> 0) is the polarization current (or electric current distributed into the tumor by DEC action). The parameter γ is the firstorder exponential decay rate of the net effect induced in the solid tumor after the DEC is removed and its inverse is the decay constant (or decay time) that characterizes the duration of such an effect. a _{ 1 }and a _{ 2 }are dimensionless parameters that depend only on the (i/i _{ o }) ratio.
The results obtained from fitting the experimental data for REGII of fibrosarcoma Sa37 TGK for CG, TG1, TG2, and TG3 are shown in Table 1. The numerical information is obtained upon fitting the individual tumor growth data and then fitting the data for each experimental group. The mean values and standard errors of these optimized parameters after fitting were computed [17].
Interpolation of data corresponding to REGII of fibrosarcoma Sa37 TGK
Interpolation of experimental data corresponding to REGII of fibrosarcoma Sa37 TGK was developed. From an experimental point of view, to perform such a study, the information in ET is reported in terms of a non equidistant time dependence of TV (V*(t') vs. t' plot, named TV plot) spaced by one day or more. As a result, the TGK details are not revealed. At the experimental level, it is difficult to show the TGK for a small time step like one day because such a study would be cumbersome, expensive in resources, timeconsuming, and requiring excessive handling of animals, which is not permitted by the ethics code care and use of Laboratory Animals Committee. For this reason, we interpolated the experimental data corresponding to REGII for this tumor type using different time steps, Δt (1, 1/3, 1/8, 1/24, and 1/48 days). In this case, we take into account the mean values of each parameter in the MGE for each experimental group (Table 1).
Reconstruction of REGI for fibrosarcoma Sa37 TGK
In the ET framework, neither experimental nor theoretical reports have taken into account REGI of TGK which, for the former, can be very important for understanding the fibrosarcoma Sa37 natural history before DEC treatment and its future influence on therapeutic effectiveness after DEC treatment. For this reason, we reconstructed this first region using an extrapolation method (to find unknown values for TV in points that are outside the typical studied range) for each Δt. In order to obtain the complete TGK for CG, we substitute t' with (tτ) in Equation 1, keeping in mind the α, β, and V_{o} parameters (Table 1) and the interpolated experimental data for REGII. In this case, τ is a time delay that represents the time interval from the point at which the tumor cells are inoculated in the host until the solid tumor reaches V_{o}.
The considerations included in the MGE are:
1. For unperturbed tumors (α* = α for i = 0), α in Equation 1, is constant during TGK. In this case, the MGE coincides with the conventional Gompertz equation [1, 17, 22, 23].
2. REGI for fibrosarcoma Sa37 TGK is the same for CG, TG1, TG2, and TG3. As a result, α in Equation 1, is the same for all of the experimental groups. This assumption has been experimentally corroborated, since the tumors in CG, TG1, TG2, and TG3 reach V_{o} at approximately the same time τ (τ = 15 days) [14].
The MGE can be rewritten in the form
where t is the time that elapses from the initial moment at which tumor cells are inoculated in the host (t = 0 days) up to the end of the experiment. t' is the time that elapses from the moment of DEC application up to the end of the experiment.
Graphical strategies for the analysis of TGK of the experimental groups
Different graphical strategies are used in order to obtain further timedependent information for both untreated and DECtreated TGK that is not revealed in a simple TV plot. For this reason, we use the following plots: first derivative of tumor volume (FDTV) versus t, named the FDTV plot (or dV*(t)/dt vs. t plot); TV dependence of FDTV, named FDTVTV plot (or dV*(t)/dt vs. V*(t) plot), the time consecutive dependence on TV plot, named CTV plot (or V*(t) vs. V*(tΔt) plot); and the modules and loglog plots for TV and FDTVTV in order to analyze whether REGI and REGII for TG3 are the same.
It is important to point out that the results shown in this paper are in long format (scaled fixed point with 15 digits after the decimal point).
Results
Analysis of complete unperturbed fibrosarcoma Sa37 TGK
The complete growth kinetics of unperturbed fibrosarcoma Sa37 tumors are generated by interpolation of the experimental data for REGII and the extrapolation process for REGI using Equation 5 with values for α, β, τ, and V_{o} from the CG (Table 1). TGK exhibits a characteristic S shape with three stages (SI, SII, and SIII), which are well defined for all Δt values, as shown in Figure 1 for Δt = 1/3 days.
The results show that SI is nonlinear and that there are two intersection points that separate each of the stages. The first point (V_{s} in Figure 1) is obtained from the interception between SI and SII, and it represents the beginning of SII (TGK is triggered). The second point (V_{ic} in Figure 1) is obtained by the interception of SII and SIII, representing the beginning of SIII (TV tends to a limit value, V_{f}). V_{ic} represents the irreversible TV from which it growth up to V_{f}. The V_{s}, V_{ic}, and V_{f} points are characterized by an ordered pair (t, TV) and are estimated as (12.34 days, 0.069 cm^{3}), (25.99 days, 3.169 cm^{3}) and (60 days, 3.536 cm^{3}) for all values of Δt, respectively.
The interpolation and extrapolation processes reveal that unperturbed fibrosarcoma Sa37 TGK has a point of inflection, V_{i} at (17.56 days, 1.301 cm^{3}). This value may also be analytically corroborated by making i = 0 and setting the second derivative of Equation 1 to zero. V_{i} is a point in the TGK at which the curvature changes from concave upwards (positive curvature) to concave downwards (negative curvature). Additionally, these processes predict three other TV values, which are observed in the experiment: V_{o} = 0.5 cm^{3} at 15 days; 0.02 cm^{3} at 11.29 days; and 0.03 cm^{3} at 11.60 days [14]. In preclinical studies, our experience shows that 0.02 cm^{3} is the smallest measurable TV, designated as V_{m} [11–14]. V_{m} for fibrosarcoma Sa37 tumors is experimentally observed at 12 days [14]. The difference in time is 0.71 days, which is not significant from an experimental point of view.
We macroscopically observe the first nonzero volume, V_{ob}, for fibrosarcoma Sa37 at 8 days (Figure 1). This tumor size is observable and palpable, but not measurable. Equation 5 estimates V_{ob} to be 0.000016 cm^{3} (0.031 cm in diameter) for this time value.
The FDTVTV plot (Figure 2) shows that TV increases between V_{obs} and V_{f}; however, FDTV increases from V_{s} to V_{i} and then decreases to zero (from V_{i} to V_{f}). Figure 2 illustrates that reaches its maximum value, FDTV_{max} when TV reaches V_{i}. The ordered pair (V_{i}, FDTV_{max}) is (1.301 cm^{3}, 0.341 cm^{3}/day), which is observed at 17.56 days.
Analysis of REGII DECperturbed TGK for TG2
We directly present the results for REGII TGK for tumors treated with DEC for 45 minutes because the REGI is similar to that of the CG. The complete fibrosarcoma Sa37 TGK treated with 11.7 mA (TG2) is shown in Figure 3, for Δt = 1/3 days, in agreement with other values of Δt = 1, 1/8, 1/24, and 1/48 days (results not shown). This figure reveals that REGII TGK (from V_{o} up to end of the experiment) is characterized by two subregions (REGIIa and REGIIb). In REGIIa, TV decreases from V_{o} to its minimum volume, V_{min}, whereas in REGIIb, it increases from V_{min} until the end of the experiment. Additionally, V_{min} is estimated by the interception of REGIIa and REGIIb, resulting in a value of 0.0698 cm^{3} that is reached at 20.58 days, for all Δt values, as shown in Figure 3, for Δt = 1/3 days. V_{min} is analytically corroborated through the following transcendent equation, constructed by minimizing V*(t') in Equation 1, given by
For this, we substitute the values of α, β, γ, i, and i _{ o }for TG2 (Table 1) in Equation 6. V_{min} is experimentally observed to be 0.07 cm^{3}, reached at 21 days after the inoculation process. The differences between the estimated and analytical values are 0.0002 cm^{3} for TV and 0.42 days for time, neither of which are significant at the experimental level.
Figure 4 shows that when TV decreases from V_{o} to V_{min}, FDTV first decreases from 0.5 to 0.376 cm^{3} (positive slope) and then it increases from 0.376 to 0.069 cm^{3} (negative slope). The FDTV values for 0.376 and 0.069 cm^{3} are  0.3069 cm^{3}/days (at 15.33 days) and 0.000068 cm^{3}/days (at 20.58 days), respectively. The values of  0.3069 cm^{3}/days corresponds to the minimum negative value of FDTV, FDTV_{min}, which is observed at 15.33 days and between V_{o} and V_{min} (Figure 4); however, this is not revealed in the TV plot (Figure 3). The TV correspondent to FDTV_{min} in Figure 4, reveals that when the TV reaches V_{min}, it always increases from V_{min} up to V_{f}; however, FDTV increases from V_{min} up to V_{i} (predicted as (1.461 cm^{3}, 0.121 cm^{3}/days) and reached at 42.22 days). Then, FDTV decreases to zero from V_{i} up to V_{f}.
Analysis of REGII DECperturbed TGK for TG3
A completely different picture is observed in REGII of the DECperturbed fibrosarcoma Sa37 TGK for TG3, as shown in Figure 5. This TGK is characterized by a decrease of TV from V_{o} to 0 cm^{3}, with two welldefined subregions (REGIIc and REGIId). In REGIIc, TV rapidly changes from V_{o} to TV from which begins its complete destruction, named V_{d} (estimated as 0.014 cm^{3} at 21.58 days); however, in REGIId, it slowly varies from V_{d} to zero, reached at 30 days after the inoculation process (15 days after DEC treatment), as shown in Figure 5, for Δt = 1/3 days. These results are in agreement with those obtained for the other values of Δt. This figure reveals that in REGIIc, there is a point of inflection V_{id}, which is predicted at (16.15 days, 0.355 cm^{3}).
Figure 6 reveals, that TV and FDTV decrease from V_{o} to V_{id} (estimated as 0.355 cm^{3}, 0.163 cm^{3}/day) at 16.15 days (positive slope), and then both magnitudes decrease up to a state of the tumor characterized by the ordered pair (0.014 cm^{3},  0.0072 cm^{3}/day) at 21.58 days (negative slope). TV and FDTV abruptly decrease to zero for TV smaller than V_{d} (negative slope).
We propose a CTV plot in order to demonstrate whether the tumor can be completely reversible, as shown in Figure 7. In this figure, a closed loop appears for Δt = 1/3 and 1/24 days, being narrower for Δt = 1/24 days. This figure reveals that REGI and REGII for fibrosarcoma Sa37 TGK are not symmetric. In addition, an analysis of the FDTV module versus TV plot (Figure 8) and the loglog plot (log TV versus log t, log FDTV versus log t, and log V(t) versus V(tΔt)) are conducted to demonstrate whether REGI and REGII are proportional. The figures indicate that these two regions are not symmetrical or equal.
Eliminating the nonlinear part in both REGI and REGII shown in Figure 8, we may fit each one of these regions to a straight line. For REGI, the slope ± its error and the intercept ± its error are 0.724 ± 0.011 and  0.729 ± 0.024, respectively. These respective parameters are 0.992 ± 0.031 and  0.524 ± 0.068, for REGII. The ratio between the slopes is 1.37.
In the CG, the CTV plot shows that V(t) increases with increasing V(tΔt), as expected. Additionally, the CTV plot for TG2 reveals that both V(t) and V(tΔt) decrease to 0.376 cm^{3} beyond V_{o}. Then, V(t) increases with decreasing V(tΔt) until reaching V_{min}. Beyond this value, both V(t) and V(tΔt) increases.
The patterns shown in Figure 3 and Figure 4 for TG2 are similar to those for TG1. Furthermore, the results obtained for fibrosarcoma Sa37 TGK are similar to those obtained for Ehrlich tumors in the three experimental groups. For this reason, in the present study, such results are not included.
Discussion
We show that the macroscopic behavior of both untreated and DECtreated fibrosarcoma Sa37 TGK can be realistically modeled using Equation 5. For this, we use previous experimental data for CG, TG1, TG2, and TG3 [14]; the parameters are obtained from fitting these data (Table 1) [17], and both interpolation and extrapolation methods for different time steps Δt (1; 1/3; 1/8; 1/24; and 1/48 days) are used.
Unperturbed fibrosarcoma Sa37 TGK
The TV plot corroborates that the complete untreated fibrosarcoma Sa37 TGK (i = 0) exhibits an S shape with three welldefined stages (SI, SII, and SIII) (Figure 1). SI is common to each experimental group, and it is associated with the establishment of the tumor in the host. SII is related to rapid tumor growth. SIII of this kinetic shows slow tumor growth and its behavior towards V_{f} (asymptotic value).
In SI, V_{ob} for fibrosarcoma Sa37 tumor at 8 days is experimentally observable and palpable but not measurable [14]; however, Equation 5 predicts this value. Nerterets et al. [24], reported tumor diameters below 0.025 cm via imaging with Xray phasecontrast microCT inline holography. The extrapolation of SI estimates a tumor size of 0.0000082 cm^{3} to be reached at 7.79 days, with the first approximation assumption that 0.025 cm^{3} is the smallest volume measured for all tumor types. The differences between these values and those estimated for this tumor type are 0.0000078 cm^{3} for TV and 0.21 days for time, which are not significant at the experimental level.
Experimentally, TV is measured with a vernier caliper with a precision of 0.005 cm, and the thickness of the mouse skin (between 0.1 and 0.2 cm) is taken into account. Our experience indicates that above 0.02 cm^{3}, the mouse skin thickness is negligible as compared with the tumor size [11–14]. Below 0.02 cm^{3}, this thickness is comparable and larger than the tumor size, being more evident when the TV approaches V_{ob}.
Equation 5 is continuous and smooth for all t (from t = 0 up to the end of the experiment), in contrast with the experiment. The tumor sizes are smaller than 10^{6} cm^{3}, below V_{ob}, which cannot be observed or measured with any of the current experimental techniques for measuring TV, and therefore, to a first approximation, the sizes are considered as zero, in agreement with our experimental observations [11–14]. This suggests that REGI consists of two parts: from t = 0 up to t_{ob} (t_{ob}: observable time, in days, for which V_{ob} is observed) and from t_{ob} up to τ. As a result, Equation 5 can be rewritten as
Equation 7 suggests that the MGE is continuous for t ≥ t_{ob}. Our experience indicates that V_{ob} and t_{ob} depend on the tumor histogenic characteristics, the host type, and the initial concentration of tumor cells inoculated in the host [11–14].
We experimentally observe that fibrosarcoma Sa37 solid tumors are spheroids between 8 and 10 days (SI of TGK), which are also palpable and observable but not measurable. Our model predicts that TV at 10 days is 0.0025 cm^{3} (0.17 cm in diameter). It is surprising that this volume range (0.031 to 0.17 cm in diameter) for which the tumor is spherical coincides with that reported by other authors for the avascular phase (0.025 up to 0.2 cm in diameter) [5, 25–32]. Our model estimates that a tumor 0.2 cm in diameter (0.0042 cm^{3}) is reached at 10.28 days. The differences for volume and time are 0.0017 cm^{3} and 0.28 days, which are not significant at the experimental level.
The fact that the tumors are spheroids (between 0.000016 and 0.0025 cm^{3}) may be explained by a central force field of the Coulomb type due to the fact that the cancer cells are negative charged [33]. It is important keep in mind that a force field is central if and only if it is spherically symmetric. An increase in the tumor cell number occurs when the tumor grows, and as a result, these cells are closer. Since they have the same electrical charge, they are repelled and the tumor is deformed, a fact that explains why the tumor has an ellipsoidal shape after 10 days.
The results show that TGK for SII changes quickly at first (from V_{s} up to V_{i}: concave upwards) and then slowly (from V_{i} up to V_{ic}: concave downwards). This pattern occurs because FDTV first increases and then decreases with increasing TV. In the first case (when both TV and FDTV increase), several factors are involved, such as local growth that is facilitated by enzymes (e.g., proteases) that destroy adjacent tissues and, tumor angiogenesis factors that are produced to promote formation of the vascular supply required for further tumor growth, among others [1]. In the second case (when FDTV decreases with increasing TV), the tumor itself generates different mechanisms that oppose its own growth (i.e., antiangiogenic substances). If the tumor does not generate such mechanisms, its growth would be exponential and, as a consequence, the tumorhost relationship would be broken, which is not observed in oncological practice [1]. This may indicate that unperturbed tumors intelligently regulate their own growth. This means that the tumor selforganizes, and as a result, new emergent variables appear in order for the tumor to grow, evade the immune system, and achieve maximum survival.
The FDTV behavior may suggest that the tumor doubling time and α are not constant during unperturbed fibrosarcoma Sa37 TGK, in agreement with Steel [22]. This result is in contrast with the fact that these two kinetic parameters are constant during all TGK, as we assume in this paper and as reported previously by our group [17] and other authors [1, 23]. Additionally, the TV dependence of FDTV indicates that, V_{i} may have important implications in DEC planning, if we take into account the fact that the tumor is more sensitive to DEC than healthy tissue [7–15, 18–21], the Steel equation [22], and the results of Smith et al. [34].
In SIII, the tumor behavior is explained by the fairly slow rate of growth due to the amount of nutrients and O_{2} needed for quick expansion of the tumor [1, 22].
Both interpolation and extrapolation methods estimate V_{m}, V_{s}, V_{o}, and V_{f} with good accuracy as well as their respective times, which are experimentally observed [14]. This is reasonable because the differences between the experimental and theoretically predicted values for these volumes and times at the experimental level are not significant. Furthermore, these methods predict V_{i} and V_{ic} and their respective times, which are not available from a TV plot. These points may have important implications in TGK and tumor treatment. The existence of V_{ic} establishes the irreversibility of TGK.
Our experience in preclinical studies indicates that a good DEC effectiveness is obtained for TV smaller than 1.5 cm^{3} [11, 12, 14]; however, it markedly decreases for TV bigger than 1.5 cm^{3} although DEC treatment is repeated several times [13]. In clinical studies, DEC effectiveness decreases when TV ≥ 8 cm^{3} [9, 10, 15]. It is interesting that 1.5 cm^{3} is near to V_{i}, fact that may suggest that DEC treatment is effective for TV below V_{i}, indicating that is important to know this TV in TGK. V_{i} may be a criterion of application for this therapy. We suggest to apply electrotherapy for TV below V_{i}.
DECtreated fibrosarcoma Sa37 TGK for TG2
In TG2, REGIIa (from V_{o} up to V_{min}) is related to the rapid tumor inhibition resulting from DEC cytotoxic action, and REGIIb (from V_{min} up to V_{f}) represents the tumor prevalence (tumor regrowth). However, FDTVTV plot reveals that FDTV first decreases up to FDTV_{min} and then increases with decreasing TV in REGIIa. This may suggest that in this region the tumor selforganizes whereas its volume decreases, indicating that DEC dose is not effective, an aspect not addressed in the literature. As a result, FDTV tends to 0.000068 cm^{3}/days corresponding to V_{min}, from which TGK triggers.
Tumor destruction (when both TV and FDTV decrease) is caused by DEC cytotoxic action, which induces toxic products in the tumor, generated by electrochemical reactions [19], and it potentiates humoral and cellular components of the immune system [20]. At this time interval, necrosis, apoptosis, chronic inflammation, polymorphous nuclear, monocytes, vascular congestion, and the activation of macrophages and T lymphocytes have been observed [7–15, 18–21].
Tumor selforganization is not observed in the TV plot and occurs when FDTV changes of slope independently of the decrease of TV. This timing may occur because the DEC dose used does not induce significant damage to the tumor. As a result, the tumor potentiates its existing mechanisms and/or generates other new mechanisms for its own protection, growth, and metastasis processes in order to reach its maximum survival. This second process can also be explained from the point of view of the complexity theory because the tumor is selforganized and new emergent variables appear [35–38]. This selforganization process of the tumor dominates the process of tumor destruction caused by DEC action, with TV reaching V_{min} and consequential tumor regrowth (REGIIb).
V_{min} observed in the TV plot for TG2 is very important from a therapeutic point of view because when TV reaches this value, DEC should be repeated [9, 10, 13]; however, the results shown in this study indicate that the tumor is selforganized when it reaches V_{min}. For us, the existence of FDTV_{min} (corresponding to TV = 0.376 cm^{3}) on the FDTVTV plot is surprising because this tumor selforganization process is not observed in the TV plot and therefore its explanation is not possible from this plot. This is relevant, at the therapeutic level because DEC stimulus alone or combined should be repeated when the TV reaches this value.
This procedure may be implemented in practice through two possible ways: 1) by weekly measuring (once or twice) the TV during the first three months after DEC treatment by means of a vernier caliper (for superficial tumors) or ultrasound (for visceral tumors) and 2) by knowing the tumor relaxation time (T_{rt}) of a small sample treated with DEC by means of Nuclear Magnetic Resonance method.
In the first way, we observe a significant decrease of TV in DEC treated patients during the first three months, after this time, a tumor regrowth is observed if the dose is not effective [15]. We suggest two measurements/week of TV to obtain various experimental points in the first three months of observation so that the values of the parameters: α, β, γ, and i _{ o }can be calculated knowing the values of V_{o} and TV on the first four measurements. Then, a numerical method is used to solve a nonhomogeneous system of four nonlinear equations with these four unknown parameters. This is possible because MGE has a good prediction capability to describe both unperturbed and perturbed tumor growths [17]. We can predict the temporal behavior of TV (TV plot) and of its derived (FDTV plot) once the values of these four parameters are wellknown and then estimate FDTV_{min} in a FDTVTV plot. If FDTV changes the sign of its slope (positive to negative) although TV continues increasing, we suggest to repeat this therapy and/or to combine it with another therapeutic procedure, as shown in this study. Therefore, we do not recommend the use Tomography Axial Computerized and Imaging Nuclear Magnetic Resonance, because of their high costs and the regulatory norms established for the use of each one of these imaging techniques.
In the second way, the knowledge of T_{rt} is important because we know the time for which the tumor recovers after DEC treatment, and the times that DEC treatment should be repeated in order to the tumor is not selforganize (for example, at a time smaller than T_{rt}). The knowledge concerning to these two facts will allow us to determine the exact time at which the DEC should be repeated, and as a result, it will allowed one to avoid unnecessary DEC stimulus to the patient. The tumor selforganization process is slower if the duration of the DEC cytotoxic effect induced into the tumor is greater than T_{rt}. In a previous study, we corroborate theoretically that DEC effectiveness increases with the increase of the duration of DEC cytotoxic effect induced into the tumor [17]. The introduction of any of these two possible ways in our experiments will lead to a high antitumor effectiveness, which suggests that our future researchers should take this fact into account.
DECtreated fibrosarcoma Sa37 TGK for TG3
In TG3, REGIIc (when TV and FDTV both rapidly decrease) may be explained from a biological point of view by DEC cytotoxic action, as we propose above. It should be noted that in the FDTVTV plot, just before the tumor reaches V_{d}, there is a change of slope for FDTV with V_{id}, implicating that other antitumor mechanisms have been activated (e.g., the activation of cellular and humoral components of the immune system mentioned above and others unreported until now). In contrast to TG2, in TG3, this change of slope for FDTV does not change its negative sign between V_{o} and V_{d}.
The net rate of the antitumor processes involved between V_{o} and V_{id} is higher than that resulting from other antitumor processes induced between V_{id} and V_{d}. From a biophysical point of view, this indicates the existence of at least two other unknown main antitumor mechanisms, which can occur simultaneously. Each one of these mechanisms has its own time constant, in agreement with previous reports [11]. As a result of these antitumor mechanisms, the tumor is completely destroyed (or reversible). This is corroborated, as TV and FDTV tend to zero when TV is smaller than V_{d}; in agreement with our results [17].
The fact that the complete TGK for TG3 is a closed loop suggests the reversibility of the tumor. We believe that this is true if TV is comprehended between V_{m} and V_{i}. This fact corroborates the above discussion regarding the goal of V_{id} in DEC treatment. Some additional experiments are required to prove this statement.
This loop shows that REGI of TGK (before DEC treatment) and REGII (after DEC treatment) are asymmetric for all Δt values. The linear fits of these two regions suggest that the slope of the curve for REGII is 1.37 times higher than that for REGI, a fact that corroborates that the TV regression rate is proportional to the rate of growth, in agreement with the NortonSimon hypothesis [39]. Prior to this study, we think that these rates are equal.
This new paradigm forces us to reconsider our knowledge and to modify our traditional approach to research and treatment. This statement is relevant for ET because it completely changes the conception of cancer treatment. The actual idea behind in vitro and in vivo studies is to treat the tumor and then to observe its evolution, which is not known to priori [7–15, 18–21]. However, the existence of V_{d} establishes that fibrosarcoma Sa37 tumors have a DEC threshold for which the tumor is completely destroyed, as demonstrated experimentally and theoretically for Ehrlich and fibrosarcoma Sa37 tumors [14, 17], in agreement with other studies [9, 10, 18, 21, 33]. This is possible if we establish an explicit dependence of V_{d} as a function of the parameters of Equation 7, the host type, and the ET parameters (dosage and exposure time of DEC, electrode array, and times that DEC is repeated). This is very complex at the experimental and theoretical levels; however, mathematical modeling may be a useful tool for finding an approximate solution (analytical or numerical) to this problem. Such modeling will lead to further improvement in the treatment of solid tumors, and it can also help guide treatment decisions for therapists treating patients (or animals) with this disease. In addition, this statement will contribute to standardizing this therapy.
New predictions and hypothesis for TGK
In the physical sciences, mathematical theory and experimental investigation have always worked together. Mathematical theory can help to direct experimental research, while the results of experiments help to refine the modeling [2]. This is precisely one of the intentions of this manuscript.
Although Equation 5 (or 7) does not reveal other information, we can propose hypothesistesting (or hypothesisgenerating) methods from it and our experimental observations. The fact that in REGI of TGK, specifically in SI, for all experimental groups, Ehrlich and fibrosarcoma Sa37 solid tumors are not observed below V_{ob} (tumor cells in suspension) and are observed above V_{ob} (solid tumor or tumor mass) may suggest the existence of a phase transition. It is more evident for these tumor types in REGII of TGK for TG3 when the solid tumor passes from its active phase (below V_{d}) to the phase in which the tumor is completely destroyed due to DEC action (above V_{d}) [14]. In both cases, these transitions are named PT_{1} and PT_{2}, respectively, as schematically represented in Figure 9 and Figure 10. This is also supported if we remember that a phase transition has the characteristic of taking a medium with given properties and transforming some (or all) of it into a new medium with new properties (i.e., the transformation of a thermodynamic system from one phase to another) [40, 41].
We know from thermodynamics that at the phase transition point, physical properties may undergo abrupt changes: for instance, the volumes of the two phases may be vastly different, as observed in SI (below and after V_{ob}) and REGII in TG3 (below after V_{d}), a fact that could suggest the existence of a critical TV in SI, V_{c1}, and another in REGII in TG3, V_{c2}, as schematically represented in Figure 9 and Figure 10. We believe that when V_{c1} (V_{c2}) is reached; the tumor begins to grow (completely destroyed).
It is possible that such a phase transition involves a large amount of energy (a dissipative system) accompanied by fluctuations, chaos, and/or selforganization processes with the presence of emergent variables, in agreement with other authors [5, 35–38, 40–46].
Several authors have reported various phenomena that occur in SI of TGK, such as: a transition from the tumor avascular phase to the vascular phase (angiogenesis), which is accompanied by fluctuations [5, 25–28, 31, 32]; the existence of a stochastic transition at the change between these two tumor phases [29]; the disruption of normal blood vessels of the organs in which the tumor is developing caused by chaotic growth [25]; the existence of a threshold under which sprouts cannot reach the tumor during the growth of the vascular network [46]; among others. It is interesting that our model reveals that SI is highly non linear, a fact that could be associated with the presence of chaos [5, 42, 43], in agreement with other authors [5, 42, 43]. This corresponds with established nonequilibrium thermodynamics, in which systems driven out of equilibrium (as solid tumors and biological systems generally are) often exhibit fluctuations or phase transitions [35, 44]. In addition, these systems can develop from disorder (systems known as dissipative) because they are formed and maintained by dissipative processes that take place due to an exchange of energy and matter between the system and its environment, and they disappear if that exchange ceases. From Equation 1 (5 or 6), it may be corroborated that a tumor is a dissipative system because i _{ o }is much lower than i [17]. The biological processes that are constantly receiving, transforming, and dissipating chemical energy can, and do, exhibit properties of selforganization far from thermodynamic equilibrium [35–38, 44, 45].
MGE offers information of the global dynamics of unperturbed and DEC treated tumors and therefore only gives a limited understanding about the selforganization processes in TGK. However, we believe that these processes are involved in unperturbed and DEC treated TGK for discussed above and the following facts, which are implicitly in MGE: 1) Selforganization makes sense only in relation to the whole: it is the whole that selforganizes into a multitude of interacting levels. At the same time, the whole cannot sustain its integrity, if the process of selforganization does not work. This suggests that selforganization has an important role in the formation, maintenance, and function of cells, tissues, organs and the complete human body. 2) A key requirement for a selforganizing system is nonlinearity and therefore the selforganizing systems are governed by nonlinear dynamics [47], in agreement with our results. 3) Gompertzian dynamics emerges as a result of the fractalstochastic dualism, which is a universal natural law of biological complexity [48], in agreement with Brú et al. [49]. 3) System changes from nonorder to order, from lowgrade order to advanced order, basis on the principle of autoorganization adaption [50]. 4) Cancer is a reflection of a failing system; preventive steps should involve rebalancing the entire system through lowering of disorderly complexity, entropy, and optimizing selforganization with orderly complexity [51]. 5) The malignant tumor is a complex system and therefore this complexity expresses its functionality and reflects a high degree of resilience and robustness to environmental challenges through their selfadaptation and internal selforganization [51]. 6) The process of tumor cell growth, invasion and metastasis involves a selforganized cascade of multiple tumorhost and tumorimmune interactions [52]. Selforganization might be a general principle in cellular organization and an elegant, efficient way to optimally organize cellular structures [53]. 7) Selforganization occurs when a real system evolves toward a higher differentiation from its initial state (or presystem phase) [51]. These two phases are revealed with MGE: pretumor phase (below V_{ob}) and solid tumor phase (above V_{ob}). Also, this differentiation is observed in our pathological studies [11–15], and it is the cause of the aggressiveness and difference in the cellular/molecular patterns of the different types of malignant tumors [1]. In spite of these facts and others, more studies at cellular/molecular/atomic/quantum levels and new physicmathematical approaches are needed to have more meaningful results about the selforganization process in TGK.
During such a phase transition, a tumor either absorbs or releases a fixed (and typically large) amount of energy, which is characteristic of a firstorder phase transition. Because energy cannot be instantaneously transferred between the tumor and it's surrounding healthy tissue, firstorder transitions are associated with "mixedphase regimes" in which some parts of the system have completed the transition and others have not. Based on statistical physics, mixedphase systems are difficult to study, because their dynamics are violent and challenging to control [40].
The hypotheses proposed in this study can doubtlessly be seriously attacked by many; however, this study sets the basis to derive some practical understanding from our diverse (and often, at this time, empirical) experimental and clinical observations in cancer electrotherapy. The availability of powerful computers has already helped to bridge the gap between observations and predictions in many complex problems, and a few attempts have already been made to attack the problem of tumor growth with mathematical models.
We are recognizing biophysics principles that may be broadly applied in developing more useful programs of DEC treatment of solid tumors. To begin to understand the complexity of the proposed system, novel simulations must be developed, incorporating concepts from many scientific areas such as cancer research, statistical mechanics, applied mathematics, and nonlinear dynamical systems.
Our results suggest that the MGE should be modified, or a new mathematical approach should be proposed in order to describe TGK and explain the presence of at least one of these phenomena. These results are, in agreement with Bellomo et al. [2], who proposed that "future research will definitely refine and improve the existing models, while the analysis of the inherent mathematical problems will hopefully lead to new mathematics, allowing us to tackle problems presently beyond our technical abilities".
Conclusion
In conclusion, the modified Gompertz equation is likely to lead to insights within cancer research. Such insights hold promise for increasing our understanding of tumors as selforganizing systems and, the possible existence of phase transitions in tumor growth kinetics, which, in turn, may have a significant impact both on cancer research and on clinical practice.
Abbreviations
 ET:

electrotherapy
 DEC:

direct electric current
 CG:

control group
 TG1:

treated group 1
 TG2:

treated group 2
 TG3:

treated group 3
 MGE:

modified Gompertz equation
 TV:

tumor volume
 TGK:

tumor growth kinetics
 REGI:

part of TGK before DEC treatment
 REGII:

part of TGK after DEC treatment
 REGIIa and REGIIb:

subregions of REGII for TG2
 REGIIc and REGIId:

subregions of REGII for TG3
 FDTV:

first derivative of tumor volume
 FDTV_{max} :

maximum FDTV observed in TGK for the CG
 FDTV_{min} :

minimum FDTV observed in TG2
 TV plot:

tumor volume versus t plot
 FDTV plot:

time dependence of first derivative of tumor volume plot
 FDTVTV plot:

first derivative of tumor volume versus tumor volume
 CTV plot:

time consecutive dependence of tumor volume plot
 V_{o} :

initial volume at which DEC is supplied
 V_{ob} :

first nonzero value of TV
 τ:

time delay
 T_{rt} :

tumor relaxation time
 V_{min} :

minimum TV
 V_{d} :

TV from which begins tumor complete destruction
 V_{id} :

inflection point in REGIIc
 V_{m} :

smallest measurable TV
 V_{f} :

final volume of TGK
 t_{ob} :

time at which V_{ob} is observed
 V_{i} :

point of inflection in TGK
 V_{s} :

TV that separates SI and SII
 V_{ic} :

TV that separates SII and SIII
 Δt:

time step
 SI:

SII, and SIII are the first, second, and third stages in TGK of the control group, respectively
 PT_{1} :

phase transition between the phases of tumor cells in suspension and a solid tumor
 V_{c1} :

critical volume for which PT_{1} occurs
 PT_{2} :

phase transition between an active solid tumor and a completely destroyed tumor
 V_{c2} :

critical volume for which PT_{2} occurs.
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Acknowledgements
The authors wish to thank Emilio Suárez and Dr. José Luis García Cuevas for their technical assistance. JJGN acknowledges the support of Dr. Isaac Hernández Calderón. This research was supported by Physics Department, Research Center and Advanced Studies of National Polytechnic Institute, México and the Ministry of Superior Education, Republic of Cuba. Also, we thank in a special way the reviewers for their invaluable recommendations and suggestions.
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Authors' contributions
LEBC planned the study, and participated in its design and coordination, and also assisted with the manuscript. JJGN participated in its design and discussion, and also contributed to the manuscript. ARA and JAGJ organized this study and participated in the elaboration of the software to obtain the results shown. HMCC, MMG, MFS, MVJ, TRG, MAOM, SCAB, FSP, LZO, MCCQ, SES, VCC, IBC, and GSG participated in the design of this study, and contributed to the analysis, interpretation, and discussion of the results. All authors read and approved the final manuscript.
It is important to point out that studies in cancer necessarily require the knowledge of a multidisciplinary group of researchers because cancer is not wellunderstand, and currently, no therapy that completely cures this illness has been reported.
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Cabrales, L.E.B., Nava, J.J.G., Aguilera, A.R. et al. Modified Gompertz equation for electrotherapy murine tumor growth kinetics: predictions and new hypotheses. BMC Cancer 10, 589 (2010). https://doi.org/10.1186/1471240710589
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DOI: https://doi.org/10.1186/1471240710589
Keywords
 Tumor Volume
 Fibrosarcoma
 Graphical Strategy
 Direct Electrical Current
 Antitumor Mechanism