Consolidated mathematical growth model of the primary tumor and secondary distant metastases of breast cancer (CoMPaS)

The goal of this research is to improve the accuracy of predicting the breast cancer (BC) process using the original mathematical model referred to as CoMPaS. The CoMPaS is the original mathematical model and the corresponding software built by modelling the natural history of the primary tumor (PT) and secondary distant metastases (MTS), it reflects the relations between the PT and MTS. The CoMPaS is based on an exponential growth model and consists of a system of determinate nonlinear and linear equations and corresponds to the TNM classification. It allows us to calculate the different growth periods of PT and MTS: 1) a non-visible period for PT, 2) a non-visible period for MTS, and 3) a visible period for MTS. The CoMPaS has been validated using 10-year and 15-year survival clinical data considering tumor stage and PT diameter. The following are calculated by CoMPaS: 1) the number of doublings for the non-visible and visible growth periods of MTS and 2) the tumor volume doubling time (days) for the non-visible and visible growth periods of MTS. The diameters of the PT and secondary distant MTS increased simultaneously. In other words, the non-visible growth period of the secondary distant MTS shrinks, leading to a decrease of the survival of patients with breast cancer. The CoMPaS correctly describes the growth of the PT for patients at the T1aN0M0, T1bN0M0, T1cN0M0, T2N0M0 and T3N0M0 stages, who does not have MTS in the lymph nodes (N0). Additionally, the CoMPaS helps to consider the appearance and evolution period of secondary distant MTS (M1). The CoMPaS correctly describes the growth period of PT corresponding to BC classification (parameter T), the growth period of secondary distant MTS and the 10-15-year survival of BC patients considering the BC stage (parameter M).


Introduction
Breast cancer (BC) is the main cause of cancer mortality in women. BC accounts for approximately 20-25% of all the cancer types in women [1].
Finding algorithms to predict the growth of tumors has piqued the interest of researchers ever since the early days of cancer research. Several studies were carried out as an attempt to a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 All BC patients received comprehensive PT treatment. Therefore, the whole natural history of BC should include the period of secondary distant MTS growth (Fig 1): 1. the non-visible period of PT growth; 2. the visible period of PT growth, diagnostics and removal of PT; 3. the non-visible period of secondary distant MTS growth; 4. the visible period of secondary distant MTS growth, diagnostics, treatment and patient's death.
The non-visible period of PT growth is from the appearance of the first tumor cell (diameter = 10 μm) until it reached a visible size (diameter = 1-5 mm).
The visible period of PT growth is from the point that it reached a visible size (diameter = 1-5 mm) up to the pre-surgery size.
The non-visible period of MTS growth can be calculated as the period from diagnosis (date of PT resection) to visible size (diameter = 1-5 mm) of at least one MTS.
The visible period of MTS growth can be calculated as the period from diagnosis of the visible size (diameter = 1-5 mm) to achievement of the lethal size (death).
Thus, descriptions of the whole natural history of BC require building a consolidated mathematical BC growth model of PT and secondary distant MTS.
The CoMPaS may describe both the PT growth and secondary distant MTS growth considering the histopathological classification. Additionally, the CoMPaS might facilitate the survival (lifetime) and, therefore, make predictions about the future metastatic manifestation after the resection of PT.
The diameter of the first tumor cell equals 10 μm. After the 10 th doubling, the diameter of the tumor equals 100 μm (the number of cancer cells equals 10 3 ). After the 20 th doubling, the diameter of tumor equals 1 mm (the number of cancer cells equals 10 6 ), 10 mm (the number of cancer cells equals 10 9 ) after the 30 th doubling, and 103 mm (the number of cancer cells equals 10 12 ) after the 40 th doubling (Figs 1 and 2).

The influence of the first period were MTS appear on the life forecast for patient
A possible life forecast for the patients who have PT was designated as being: a. favourable-the appearance period of the first MTS is more than five years; b. moderately favourable-the appearance period of the first MTS is from three up to five years; c. unfavourable-the appearance period of the first MTS is less than three years.

The study of the correlations between the non-visible growth period of the secondary distant MTS and 10-15-years survival
The mathematical models of Michaelson et al. [29][30][31] were used to validate the compatibility of the CoMPaS and the histopathological classification of BC with the 15-year survival correlation. According to the Michaelson et al. [29][30][31] equation, the data on the 15-year survival was calculated considering the PT volume (from 1 mm to 75 mm). According to the CoMPaS, the data on the non-visible MTS growth period was obtained considering the PT volume (from 1 mm to 75 mm). The correlations between the data on the 15-year survival and the data on the non-visible MTS growth period were analysed. The mathematical models of Holzel et al. [24] and Engel et al. [32][33][34][35] was used to validate the compatibility of the CoMPaS and the histopathological classification of BC and 10-year survival correlation. According to the CoMPaS, the data on the non-visible MTS growth period was calculated considering the TNM-classification (from 7.5 mm to 47.5 mm). The correlations between the 10-year survival data and the data on the non-visible MTS growth period were analysed.

Implementation in software
The CoMPaS was implemented as an original software application for iOS devices. It is necessary to collect predictions in one database to compare forecasts with real data and estimate the quality of the proposed model. Consequently, the CoMPaS can be connected to a database that allows us to test the model implementation. As it turns out, the new predictive tool: 1) is a solid foundation to develop future studies of BC models; 2) does not require any expensive diagnostic tests; 3) is the first predictor that makes forecasts using only current patient data, whilst the others are based on the additional statistical data.

Statistical analysis of data
The obtained results were analysed on PC by using Python 3.5. Statistical relations between the parameters of the dataset were researched using a correlation analysis.
Statistical relations between the dataset parameters researched using a correlation analysis. The direction of the correlation was estimated by the Pearson correlation coefficient sign (r): (+)-positive and (-)-negative. The level of correlation was determined by the correlation coefficient value (r): 0 < r 0.2 -very low correlation; 0.2 < r 0.5 -low correlation; 0.5 < r 0.7 -medium correlation; 0.7 < r 0.9 -high correlation; 0.9 < r 1.0 -very high correlation. The indicator is considered significant at p < 0.05. Benzekry et al. (2014) [11] completed an experimental studying about the growth of the primary BC and lung cancer. As research indicates, both the Gompertz and exponential growth models describe primary BC more voraciously. Moreover, Gompertz and the exponential models describe the PT growth of BC with excellent prediction scores ! 80% [11]. However, the Gompertz model has some disadvantages. In particular, the model gives no opportunity a) to determine which stage the PT and secondary distant MTS are or b) to calculate the doubling time of the tumor [3,8,11,23].

Consolidated mathematical growth model of PT and secondary distant MTS of patients without lymph nodes MTS, I-II stage (CoMPaS)
The CoMPaS describes the growth of PT well and fits the BC classification (TNM). For comparing the CoMPaS and BC classification, it was first determined that the CoMPaS correctly describes all the sizes of PT in conformity with the survival of BC patients [1]. In general, the non-visible period of the secondary distant MTS growth follows the resection of PT. According to the CoMPaS, the increasing 5-year survival of BC patients depends on PT volume.
Calculations of the whole natural history on CoMPaS. Given all the above information, it is relevant to dwell upon building the whole natural history of the BC stages T1aN0M0, T1bN0M0, T1cN0M0, T2N0M0 and T3N0M0, provided that the formulas allow for calculating: • the number of doublings for the secondary distant MTS considering only two measurements of the PT sizes; • the correct coefficient of the secondary distant MTS spreading rate in patients without lymph node MTS related with the PT growth rate; • the doubling time of the secondary distant MTS.
Data on the mean diameter of PT for each stage (T1, T2, T3, T4) were obtained from Table 1 of Engel et al. [33].  Fig 8). In general, the non-visible growth period of the secondary distant MTS continues after resection PT (Fig 8).
For the first time, the sizes of secondary distant MTS were detected in the range of 0.08 and 0.64 mm at the date of resection PT (Table 1; Fig 8). In other words, the size of the secondary distant MTS was lower than 1 mm (the minimum size for diagnostic of tumor). Consequently, secondary distant MTS could not be detected (Table 1; Fig 8). The diameter of the PT increases from 4.5 mm to 64.5 mm, and the diameter of the secondary distant MTS increases from 0.08 mm to 0.64 mm (Table 1; Fig 8).
For the first time, it was determined that ( Table 2; Fig 9): • the TVDT of the MTS was in range of 60 and 70 days for pT1, pT2, and pT3 considering the sizes of the PT;  • the TVDT of the MTS was in range of 46 and 48 days for pT4. In other words, the tumor grew aggressively (pT4) considering sizes of PT; • the whole natural history of the PT and MTS growth (the period from the appearance of the first tumor cell up to death) was in range of 9.9 and 11.5 years for pT1, pT2, and pT3 (Table 2; Fig 9); • the whole natural history of the PT and MTS growth (the period from the appearance of the first tumor cell up to death) was in range of 7.5 and 7.9 years for pT4.
The survival of BC patients included both the non-visible and visible periods of MTS growth, whereas the non-visible was the period from the date of PT resection to the date of  For the first time, it was determined that the survival was in range of 3.6 to 6.4 years for pT1, pT2, and pT3 (Table 2; Fig 9).
For the first time, it is possible to calculate an approximate period of appearance of the first MTS cell considering the size of the PT (parameter T) for pT1, pT2, pT3.    For the first time, it is possible to calculate an approximate period of manifestation of the secondary distant MTS considering the sizes of PT (parameter T) for pT1, pT2, pT3.
For the first time, it was determined that: • the non-visible period of MTS growth was in range of 1.9 to 4.4 years for pT1, pT2, and pT3 (Table 2; Fig 9); • the visible period of MTS growth was in range of 1.9 to 2.0 years for pT1, pT2, and pT3 (Table 2; Fig 9); • the visible period of MTS growth was in range of 1.8 to 1.9 years for pT4. In other words, the visible period of MTS growth for pT4 almost equalled the visible period of MTS growth for pT1, pT2, and pT3.
In 2002, Michaelson et al. [29][30][31] determined the high negative correlation between the 15-year survival of BC patients and the sizes of the BC PT (r = -0.98; p<0.001). In other words, the larger the size of the PT was, the lower the patient's chance to live 15 years after PT resection was. Additionally, a high positive correlation between the probability of appearance of the secondary distant MTS and the sizes of PT were determined as (r = +0.94; p <0.001, respectively), by [29][30][31]. In other words, the larger the PT size of the patient was, the higher the probability of the appearance of secondary distant MTS after PT resection was.
We compared data on the non-visible growth period of the secondary distant MTS and the date of the 15-year survival of the BC patients, which were obtained from the Michaelson et al. [29][30][31] function. We obtained a high positive correlation between the non-visible growth period and the 15-year survival of the BC patients (r = +0.94; p<0.001). In other words, the longer the non-visible growth period of the secondary distant MTS that the patient has, the higher their chance to live 15 years after PT resection (Figs 9 and 10).
In 2010, Holzel et al. [24] and Engel et al. [32][33][34][35] researched the 10-and 15-year survival from diagnosis date to PT resection on a large patient group (n = 33475) considering the BC stage (parameter T-sizes of PT). As it was demonstrated, the 15-year survival decreases with increasing stage (parameter T). However, all the differences were disappearing when patients on different stages were compared by the date of appearance of the secondary distant MTS.
We compared the data from Holzel et al. [24] and Engel et al. [32][33][34][35] on the non-visible growth period of the secondary distant MTS and the 10-year survival of BC patients. We obtained high positive correlation between the non-visible growth period of the secondary distant MTS and the 10-year survival of the BC patients (r = +0.97; p< 0.001). In other words, the longer the patient's non-visible growth period for the secondary distant MTS, the higher their chance to live 10 years after PT resection (Figs 9 and 10).

Discussion
In 2002, Michaelson et al. [29][30][31] proposed a high positive correlation between the risk of secondary distant MTS appearance in BC patients and the sizes of PT in BC (r = +0.94; p < 0.001). In other words, the larger the PT a patient has, the higher the risk the patient has for the appearance of the secondary distant MTS for 15-years after PT resection. For instance, if the size of PT equals 2 mm, then the risk of appearance of the secondary distant MTS equals 1.5% over the course of 15 years.
Michaelson et al. [29][30][31] determined that there was a high negative correlation between the 15-year survival of BC patients and the size of the PT (r = -0.98; p< 0.001). In other words, the larger the patient's PT is, the higher the patient's risk of dying 15 years after PT resection is. For instance, if the size of PT equals 2 mm, then the risk of dying for 15 years equals 1%.
In 2003, Coumans et al. [26] researched the influence of the BC stage (date of diagnosis and resection PT) on the risk of secondary distant MTS appearance using data on the BC patients (n = 38715). According to the research, the resection of a PT with a size equal to or less than 2.7 mm significantly decreases the risk of MTS appearance for 5 years (from 9.2% to 1.0%).
The CoMPaS explains the cause of the negative correlation between the 10-year survival and the diameter of the PT in BC at the time of PT resection. The CoMPaS explains the cause of the positive correlation between the 10-year survival and the date of secondary distant MTS diagnosis [24,[32][33][34][35].
The CoMPaS describes correctly the whole natural history of breast cancer growth. The primary tumor and secondary distant MTS growth periods contain: the non-visible growth period of PT; the visible growth period of PT; the non-visible growth period of secondary distant MTS; the visible growth period of secondary distant MTS, diagnostics, treatment and death of patients (Figs 9 and 10).
Relative survival (%) of the BC patients depends on the diameter of the PT (parameter T) (Fig 8) [24,[32][33][34][35]. Survival (lifetime) is the period between the date of diagnosis (TNM staging system of BC) and the date of a patient's death. Relative survival (%) of BC patients: 1) depends on the non-visible growth period of MTS (MTS free period); 2) does not depend on the visible growth period of MTS from the appearance of the first MTS to the date of a patient's death ( Table 2  The diameter of PT increases from 4.5 mm to 64.5 mm, and the diameter of secondary distant MTS increases from 0.08 mm to 0.64 mm (Table 1; Fig 8). This means that the non-visible period of secondary distant MTS growth decreases, which leads to a decline in the survival of the BC patients (Figs 9 and 10). Therefore, it could be concluded that the PT resection size can explain the survival time of the BC patients (Figs 9 and 10).
The analysis of the non-visible growth period of the secondary distant MTS helps to consider the cause of the differences between the 10-15-year survival of BC patients considering the BC stage (parameter M). The CoMPaS correctly estimates the survival in each group of patients who have a similar BC stage. It allows the prediction of future BC patients at the period of resection and the treatment of PT. Additionally, the CoMPaS allows us: 1) to determine the various critical growth periods of PT (the time of appearance of the first cancer stem cell, etc.) and secondary distant MTS (the time of the appearance of visible MTS, whose size is larger than 5-10 mm, etc.); and 2) to determine the groups of BC patients with high risk (during 1-5 years), medium risk (during 5-10 years), and low risk (during 10-15 years) of secondary distant MTS appearance.
As it should be highlighted, the CoMPaS can be used as a standard for the evaluation of the effectiveness of an individual treatment for each patient. If the patient lives longer with a personal treatment design, this means that the individual treatment has a positive effect. Otherwise, if a patient's lifetime becomes shorter, the individual treatment has a negative effect. Consequently, individual treatment allows for increasing the real effectiveness for each patient.

Conclusions
The CoMPaS allowed us to calculate: a) the number of doublings for the non-visible growth period of the secondary distant MTS; b) the number of doublings for the visible growth period of the secondary distant MTS; c) the tumor volume doubling time (days) for the non-visible growth period of the secondary distant MTS; d) the tumor volume doubling time (days) for the visible growth period of the secondary distant MTS.
A mathematical modelling of non-visible growth period of PT helps to detect the appearance period of the first tumor cell and to group more accurately the reasons for occurrence of PT. A mathematical modelling of visible growth period of PT helps to detect the appearance period of the first metastatic cell and to group more accurately the reasons for occurrence of secondary distant MTS. A mathematical modelling of non-visible growth period of secondary distant MTS helps to detect the period of diagnostic of secondary distant MTS and the period of manifestation of secondary distant MTS in different organs that may start treatment at the early stages and increase survival of patients with BC. A mathematical modelling of visible growth period of secondary distant MTS allows us to estimate a treatment effect if it increases the survival.
The CoMPaS correctly describes the growth of PT for the patients in the T1aN0M0, T1bN0M0, T1cN0M0, T2N0M0 and T3N0M0 stages, who do not have MTS in the lymph nodes (N0). Additionally, the CoMPaS helps to consider the period in which the secondary distant MTS might appear and evolve (M1).
The CoMPaS correctly describes the growth period of PT and corresponds to the BC classification (parameter T), the growth period of the secondary distant MTS and the 10-15-year survival of BC patients considering the BC stage (parameter M).
We determined via CoMPaS the variety of critical growth periods that influenced the survival forecast for BC patients. Consequently, the limits of the applicability of CoMPaS were imposed.
According to validation on the small clinical datasets, the mathematical model CoMPaS and the corresponding prediction software have a predictive capability that exceeds the predictive capability of the previous mathematical models. In further research, the pipeline includes 1) validation of the CoMPaS on a larger clinical dataset, 2) an extension of the applicability limits, and 3) software implementation in complex medical software for further oncological data analysis.