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A Mathematical Model to Predict Diagnostic Periods for Secondary Distant Metastases in Patients with ER/PR/HER2/Ki-67 Subtypes of Breast Cancer

Previously, a consolidated mathematical model of primary tumor (PT) growth and secondary distant metastasis (sdMTS) growth in breast cancer (BC) (CoMPaS) was presented. The aim was to detect the diagnostic periods for visible sdMTS via CoMPaS in patients with different subtypes ER/PR/HER2/Ki-67 (Estrogen Receptor/Progesterone Receptor/Human Epidermal growth factor Receptor 2/Ki-67 marker) of breast cancer. CoMPaS is based on an exponential growth model and complementing formulas, and the model corresponds to the tumor-node-metastasis (TNM) staging system and BC subtypes (ER/PR/HER2/Ki-67). The CoMPaS model reflects (1) the subtypes of BC, such as ER/PR/HER2/Ki-67, and (2) the growth processes of the PT and sdMTSs in BC patients without or with lymph node metastases (MTSs) in accordance with the eighth edition American Joint Committee on Cancer prognostic staging system for breast cancer. CoMPaS correctly describes the growth of the PT in the ER/PR/HER2/Ki-67 subtypes of BC patients and helps to calculate the different diagnostic periods, depending on the tumor volume doubling time of sdMTS, when sdMTSs might appear. CoMPaS and the corresponding software tool can help (1) to start the early treatment of small sdMTSs in BC patients with different tumor subtypes (ER/PR/HER2/Ki-67), and (2) to consider the patient almost healthy if sdMTSs do not appear during the different diagnostic periods.

The current guidelines on BC follow-up recommend regular visits. The European Society of Medical Oncology (ESMO) recommends follow-up visits every 3-4 months after resection of the PT for the first two years [27]. The American Cancer Society/American Society of Clinical Oncology (ASCO) recommends follow-up visits every 3-6 months after resection of the PT for the first three years, every 6-12 months for the next two years, and every 12 months after the first five years [28].
The National Comprehensive Cancer Network (NCCN) recommends follow-up visits every 4-6 months after resection of the PT for the first five years and every 12 months after the first five years [29]. The Associazione Italiana di Oncologia Medica (AIOM) recommends follow-up visits every 3-6 months after resection of the PT for the first five years and every 12 months after the first five years [29].
The current guidelines for multimodal examinations are generalized to all patients [43]. Consequently, there is a lack of personalized recommendations for multimodal examinations to detect sdMTSs in patients with BC with regard to the stage and/or the growth rate of the PT. All patients with BC worry about the appearance of sdMTSs after resection of the PT. Thus, the most important question is whether sdMTSs will appear.
If sdMTSs do appear, when will the sdMTSs first materialize? The available clinical studies provide no information about the period of manifestation of sdMTSs after PT resection in each patient as a function of the size of the PT and the stage of the BC. The problem becomes very complex because the patients must obtain a personalized approach to build a schedule of multimodal examinations to detect sdMTSs at the early stage and to start early treatment, which can increase the patient's lifetime. Currently, the possibility of calculating the earliest diagnostic period of sdMTSs in patients with BC, taking into account the stage and/or the growth rate of the PT, was not proposed in any mathematical model in the available studies.
Moreover, if sdMTSs do not appear, when will the patient be considered healthy? Currently, there is no answer to this question.
The purpose of this study was to answer these questions using the model CoMPaS, considering the TNM stage and ER/PR/HER2/Ki-67 subtypes. Therefore, the aim of the research was to detect the personalized diagnostic periods for visible sdMTS via CoMPaS in patients with different subtypes (ER/PR/HER2/Ki-67) of breast cancer. The personalized diagnostic periods for multimodal examinations during the MTS-free period were calculated in BC patients depending on the TVDT of PT and TVDT of sdMTSs. The CoMPaS model helps to determine the causes of BC appearance, which can lead to the development of prevention methods and a deeper understanding of the BC process ( Figure 1; Table 1) [21]. sdMTSs are formed from the metastatic cells of the metastasizing PT. The diameter of the metastasizing PT may vary in size from 1 mm up to the diameter of the PT at the time of resection (non-visible MTS-I period) (Figure 1).      If the diameter of the PT at its resection, d PT , was 15.1 mm ( Figure 2; Table 1), and the TVDT PT was 10-135 days (a rapid growth rate of PT in patients with HR(−)/HER2(−) (triple-negative) tumors), then the diameter of the sdMTS at PT resection, d MTS , could be 0.01-1.00 mm.
For patients in the rapid growth rate group, TVDT MTS was equal to 10-135 days. The non-visible growth period of sdMTS(1-X)-I was 0.55-4.34 years ( Figure 2; Table 1). The non-visible growth period of sdMTS(1-X)-II was 0.26-6.56 years. The visible growth period of sdMTS(1-X) was 0.29-3.90 years. The survival of patients with BC was 0.55-10.47 years.
In summary, in this period of rapid growth rate of sdMTS, patients with T1cN0-3M0 BC must undergo multimodal examination every three months 0.26 years after resection of the PT for 6.30 years.
The total period of MTS(1-X) diagnosis may be 10.64 years ( Figure 2; Table 1). BC patients with d PT = 15.1 mm and TVDT PT = 10-135 days (T1aN0-3M0) may be considered healthy only after 10.64 years if sdMTSs were not diagnosed in this period. In summary, in this period of rapid growth rate of sdMTS, patients with T1cN0-3M0 BC must undergo multimodal examination every three months 0.26 years after resection of the PT for 6.30 years.
The total period of MTS(1-X) diagnosis may be 10.64 years ( Figure 2; Table 1). BC patients with dPT = 15.1 mm and TVDTPT = 10-135 days (T1aN0-3M0) may be considered healthy only after 10.64 years if sdMTSs were not diagnosed in this period. The total period of MTS(1-X) diagnosis may be 13.07 years ( Figure 3; Table 1). BC patients with dPT = 15.1 mm and TVDTPT = 136-165 days (T1aN0-3M0) may be considered healthy only after 13.07 years if sdMTSs were not diagnosed in this period. The total period of MTS(1-X) diagnosis may be 13.07 years ( Figure 3; Table 1). BC patients with d PT = 15.1 mm and TVDT PT = 136-165 days (T1aN0-3M0) may be considered healthy only after 13.07 years if sdMTSs were not diagnosed in this period.
136-165 days (intermediate growth rate of PT in patients with HR(−)/HER2(+) (HER2-positive) tumors), then the diameter of the sdMTS at PT resection, dMTS, could be 0.01-1.00 mm. TVDTMTS was equal to 10-135 days for rapid growth rates and/or 136-165 days for intermediate growth rates.
The total period of MTS(1-X) diagnosis may be 13.07 years ( Figure 3; Table 1). BC patients with dPT = 15.1 mm and TVDTPT = 136-165 days (T1aN0-3M0) may be considered healthy only after 13.07 years if sdMTSs were not diagnosed in this period.    The total period of MTS(1-X) diagnosis may be 18.32 years ( Figure 5; Table 1). BC patients with d PT = 15.1 mm and TVDT PT = 196-230 days (T1aN0-3M0) may be considered healthy only after 18.32 years if sdMTSs were not diagnosed in this period.  Table 1). These calculated data may help define the minimal number of patient examinations in the non-visible MTS-II period (MTS-free period) depending on the growth rate of the PT and sdMTS (rapidly growing MTS, intermediately growing MTS, and slowly and very slowly growing MTS) (Figures 2-6; Table 1).  Table 1). These calculated data may help define the minimal number of patient examinations in the non-visible MTS-II period (MTS-free period) depending on the growth rate of the PT and sdMTS (rapidly growing MTS, intermediately growing MTS, and slowly and very slowly growing MTS) (Figures 2-6 Table 1). These calculated data may help define the minimal number of patient examinations in the non-visible MTS-II period (MTS-free period) depending on the growth rate of the PT and sdMTS (rapidly growing MTS, intermediately growing MTS, and slowly and very slowly growing MTS) (Figures 2-6; Table 1).

Examinations during the MTS-Free Period
Therefore, every patient obtains personalized data on an adequate minimal number of examinations during the MTS-free period (non-visible MTS-II period). Examples are as follows: In the period of a rapid growth rate of sdMTSs (group V with subtype V of BC-triple-negative tumors = HR(−)/HER2(−)), patients must undergo multimodal examination every three months; In the period of an intermediate growth rate of sdMTSs (group IV with subtype V of BC-HER2-positive tumors = HR(−)/HER2(+)), patients must undergo multimodal examination every five months; In the period of an intermediate growth rate of sdMTSs (group III with subtype III of BC-luminal B = HR(+)/HER2(−)), patients must undergo multimodal examination every six months; In the period of a slow growth rate of sdMTSs (group II with subtype II of BC-luminal B = HR(+)/HER2(+)), patients must undergo multimodal examination every eight months; In the period of a very slow growth rate of sdMTSs (group I with subtype I of BC-luminal A = HR(+)/HER2(−)), patients must undergo multimodal examination every nine months.
Hence, every patient obtains personalized data on an adequate maximal quantity of examinations during this period for the early diagnosis of visible sdMTSs (diameter = 5-9 mm). This may help oncologists start early treatment of small sdMTSs in BC patients (T1-3N0-3M0) and increase the survival of BC patients with sdMTSs (T1-3N0-3M0).
The CoMPaS model calculates the different diagnostic periods of sdMTS in patients with BC (T1-3N0-3M0) and facilitates the understanding of the periods of appearance and materialization of sdMTSs.

Application of Consolidated Mathematical Growth Model of Primary Tumor and Secondary Distant Metastases (CoMPaS) in Patients with ER/PR/HER2/Ki-67 Subtypes and Stage I/II/III Breast Cancer in Clinical Practices
This mathematical model used to predict the different diagnostic periods of sdMTS in patients with BC may help explain difficult clinical cases of BC patient survival (Figures 2-6; Table 1).
The 5-15-year survival of patients with BC depends on the diameter of the PT at resection and TVDT MTS from 10 to 270 days (Figures 2-6; Table 1).
Moreover, if patients with BC have a TVDT PT from 10 to 135 days (rapid growth rate of the PT-group V with subtype V of BC-triple-negative tumors = HR(−)/HER2(−)), they have a high risk of death within five years (rapid (TVDT MTS = 10-135 days) growth rate of sdMTSs) (Figures 7 and 8).

Application of Consolidated Mathematical Growth Model of Primary Tumor and Secondary Distant Metastases (CoMPaS) in Patients with ER/PR/HER2/Ki-67 Subtypes and Stage I/II/III Breast Cancer in Clinical Practices
This mathematical model used to predict the different diagnostic periods of sdMTS in patients with BC may help explain difficult clinical cases of BC patient survival (Figures 2-6; Table 1).
The 5-15-year survival of patients with BC depends on the diameter of the PT at resection and TVDTMTS from 10 to 270 days (Figures 2-6; Table 1).
Moreover, if patients with BC have a TVDTPT from 10 to 135 days (rapid growth rate of the PTgroup V with subtype V of BC-triple-negative tumors = HR(−)/HER2(−)), they have a high risk of death within five years (rapid (TVDTMTS = 10-135 days) growth rate of sdMTSs) (Figures 7 and 8).

Software Tool for Personalized Scheduling of Multimodal Examinations
CoMPaS was updated with a feature that includes a recommendation to have a multimodal examination every 3-4 months k 1 years after resection of the PT for k 2 years. The recommendation can help oncologists assign early treatment and increase survival.
In 2010, Holzel et al. [14] and Engel et al. [10,11,15] studied the 10-and 18-year survival from the diagnosis date to PT resection using a large patient group (n = 33475) with regard to the BC stage (parameter T-size of the PT). It was demonstrated that the 10-year mortality linearly increases with the increasing diameter of the PT [15]. In patients with stage pT3, the 18-year mortality was higher than that in patients with stage pT2 and so forth, i.e., the 18-year mortality was much higher in patients with pT3 > pT2 > pT1c > pT1b > pT1a [14].
The duration of the non-visible MTS-I period depends on the number of doublings and the TVDT [21]. According to early research results from a sequence of sdMTS appearances in BC patients after multimodal therapy, (1) a total of 75% of the recurrences were found in the first five years, (2) a total of 20% of the recurrences were found between 5-15 years, and (3) the remaining 5% of the recurrences were found in 15-25 years [20]. Therefore, the growth rate of the metastatic tumor can increase 2.2 times that of the PT; meanwhile, the TVDT can decrease 2.2 times [20].

The Relationship between the TVDT and the Subtypes ER/PR/HER2/Ki-67 of BC
The following observations were made: (1) BC patients with axillary lymph node MTS have a shorter TVDT than BC patients without axillary lymph node MTS (p < 0.05); (2) BC patients positive for the estrogen receptor ER(+) in the PT have a longer TVDT than BC patients negative for the ER(−) in the PT (p < 0.05); (3) BC patients positive for the progesterone receptor PR(+) in the PT have a longer TVDT than BC patients negative for the PR(−) in the PT (p < 0.05); (4) BC patients negative for the Ki-67(−) receptor in the PT have a longer TVDT than BC patients positive for the Ki-67(+) receptor in the PT (p < 0.05); (5) BC patients positive for HER2(+) in the PT have a much longer TVDT than BC patients with triple-negative HR(−)/HER2(−) expression in the PT (p < 0.05) [52]. In other words, this study proposes that BC patients with regional lymph node MTSs have more aggressive BC and a higher risk for the appearance of sdMTSs than patients without regional lymph node MTSs who have a lower risk for the appearance of sdMTSs.
In addition, BC patients with triple-negative (HR(−)/HER2(−)) BC have more aggressive BC and a higher risk of sdMTS appearance than BC patients positive for the progesterone receptor PR(+) who have a lower risk of sdMTS appearance. Ryu et al. [49] reported the following: (1) BC patients positive for ER in the PT (ER(+)) and BC patients positive for HER2 in the PT (HER2(+)) have a much longer TVDT than BC patients with triple-negative expression in the PT (p < 0.05).

The Relationship between the Different Diagnostic Periods and Subtypes ER/PR/HER2/Ki-67 of BC
Mathematical models could lead to more precise results and more meaningful prognostic risk scores, and they could integrate the planning of multimodal examinations into outcome-oriented clinical decision-making. While many useful stand-alone models are already in clinical use for forecasting the development of the BC process, the following question remains: will the patient have sdMTSs? If yes, when will the earliest period of the clinical manifestation of sdMTSs occur? If no, when will the patient be considered healthy? To answer these questions, CoMPaS was chosen as the main research tool.
Consequently, this study concentrated on calculating the different diagnostic periods (rapid growth rates in patients with triple-negative tumors (HR(−)/HER2(−)), intermediate growth rates in patients with HER2-positive tumors (HR(−)/HER2(+)), intermediate growth rates in patients with luminal B tumors (HR(+)/HER2(−)), slow growth rates in patients with luminal B tumors (HR(+)/HER2(+)), and very slow growth rates in patients with luminal A tumors (HR(+)/HER2(−))) before the manifestation of sdMTSs (MTSs) after resection of the PT. The broad implementation of the model in everyday oncology requires versatile software platforms that can be easily integrated into existing workflows and information technology (IT) architectures. Therefore, the CoMPaS model was integrated into an iOS application with input parameters such as patient data from examinations and was updated with the possibility of calculating the earliest diagnostic period.
The consolidated mathematical growth model (CoMPaS) and the corresponding software tool may be used for work with the eighth edition American Joint Committee on Cancer (AJCC) prognostic staging system for breast cancer [69][70][71][72].

The Relationship between the Mortality and the Convalescence and Subtypes ER/PR/HER2/Ki-67 of BC
In patients with triple-negative tumors (HR(−)/HER2(−)), the five-year mortality was higher than that in patients with HER2-positive tumors (HR(−)/HER2(+)) and so forth, i.e., the five-year mortality was much higher in patients with triple-negative tumors (HR(−)/HER2(−)) > patients and very slow growth rate in patients with luminal A subtype tumors (HR(+)/HER2(−))), the patient could be considered to be almost healthy, and she could be classified into the survival group. The consolidated mathematical growth model of the PT and sdMTSs of BC (CoMPaS) can calculate the total period of MTS(1-X) diagnosis and determine the time when patients may be considered healthy (Figures 1-8; Table 1).

Consolidated Mathematical Growth Model of Primary Tumor and Secondary Distant Metastases (CoMPaS)
To describe the growth processes of PTs and sdMTSs at stages I and II, the CoMPaS model was developed. A detailed description and the limitations of CoMPaS, as well as the influence of the appearance of the first sdMTS on the survival prognosis of a patient, were provided previously. Figure 1 demonstrates the model in terms of the whole natural history of PT and sdMTS growth [21].
The whole natural growth history of the PT and sdMTSs includes the non-visible growth period of the PT, the visible growth period of the PT, the non-visible growth period of the sdMTSs, and the visible growth period of the sdMTSs [21]. The non-visible period of PT growth is from the appearance of the first tumor cell (diameter = 10 µm) until it reaches a visible size (diameter = 1-5 mm) [21]. The visible period of PT growth is from the time that it reaches a visible size (diameter = 1-5 mm) up to the time that it reaches pre-surgery size. The non-visible period of MTS growth can be calculated as the period from diagnosis (date of PT resection) to the time that at least one MTS reaches a visible size (diameter = 1-5 mm) [21]. The visible period of MTS growth can be calculated as the period from diagnosis of the visible size (diameter = 1-5 mm) to when it reaches lethal size (death) [21]. Thus, descriptions of the whole natural history of BC require building a consolidated mathematical BC growth model of the PT and secondary distant MTSs [21].
The following updated formulas illustrate the mathematical side of the CoMPaS [21]: TVDT non = TVDT vis = log 2 N mtsII days + log 2 N mtsII−vis days log 2 N mtsII + log 2 N mtsII−vis , where log 2 DT is the fraction of proliferative cell times, θ drives the linear phase (θ = 1), Npt is the number of PT doublings, NmtsII is the number of doublings for the non-visible growth period of sdMTS, Nmts-vis is the number of doublings for the visible growth period of sdMTS, TVDT is the tumor volume doubling time, and 60 doublings represent the whole nature growth history of the PT and sdMTSs.
The appearance of the first metastatic cell of the first sdMTS coincides with the 20th doubling of the PT of BC, which allows defining the non-visible growth period of the sdMTS and the initial period of sdMTS manifestation [7,14,15,21,55,56]. The appearance of the last metastatic cell of sdMTS coincides with the date of BC PT resection [21].
The model describes the following: (a) PT growth from 1 mm up to 60 mm (exponential growth of the PT) without/with MTSs in the lymph nodes; (b) a TVDT PT from 10 days to 310 days; (c) sdMTSs growth from 1 mm up to 60 mm (exponential growth of the sdMTSs); (d) a TVDT MTS from 10 days to 310 days. TVDT MTS must be bigger than 10 days, but TVDT MTS must be less than or equal to TVDT PT .
However, the growth rate of the secondary distant MTS may be higher than the natural growth rate of the BC PT [2][3][4]6,7]. For the first time, the growth rates of secondary distant MTSs were divided into five groups depending on the ER/PR/HER2/Ki-67 subtypes and growth rates [49,50,[52][53][54]

Limitations
The model does not describe or explain an appearance of the secondary distant MTSs (M1) in patients with stage T4N0-3M0.

Implementation Software
The application was built using Swift and references CoMPaS, where the input data consist of the following fields that a user (doctor) must fill: the first diagnostic data (date of diagnostics, diameter of the PT in mm) and the second diagnostic data (date of diagnostics, diameter of the PT in mm, subtype). As a result, the output data provide the following: prognosis (the category of prognosis: favorable, mid-favorable, unfavorable; the number of months before the manifestation of the sdMTSs) [21].

Calculation Method
The obtained results were calculated on a personal computer (PC) using Python 3.8.

Conclusions
The implementation of CoMPaS and the corresponding software tool offers fascinating prospects for personalized diagnostics and early treatment by detecting the earliest diagnostic period of sdMTSs in BC patients (T1-3N0-3M0 and ER/PR/HER2/Ki-67 subtypes) with regard to the eighth edition AJCC prognostic staging system for breast cancer and the growth rate of the PT and sdMTSs in BC [22][23][24][25][26][69][70][71][72][101][102][103][104][105][106]. Such mathematics-based approaches could better identify high-risk patients in the future and help prevent unnecessary treatments. Therefore, this approach could integrate diagnostic oncology more closely with outcome-oriented clinical decisions by increasing the survival of BC patients with sdMTSs (T1-3N0-3M0 and ER/PR/HER2/Ki-67 subtypes). These kinds of gains in efficiency will become increasingly important, given the growing demand for less toxic treatments and the discovery of almost healthy patients.

Patents
The predictor of the whole natural history of breast cancer (COMBREC): Certificate of the state registration of the computer program No. 2018612104. Date of the state registration in the register of computer programs: 12.02.2018. Authors: Neznanov AA, Tyuryumina EYa.