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## 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) pro- cess 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 con- sidering 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 con- sider 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).

This study is an attempt to obtain reliable data on the natural history of breast cancer growth. The opportunities for using classical mathematical models (exponential and logistic tumor growth models, Gompertz and von Bertalanffy tumor growth models) were analysed in order to describe growth of the primary tumor and the secondary distant metastases of human breast cancer. Our results suggest a new «Consolidated mathematical growth Model of the Primary tumor and the Secondary distant metastases» (CoMPaS). The CoMPaS is based on exponential tumor growth model and consists of a system of determinate nonlinear and linear equations. The CoMPaS describes correctly the primary tumor growth (parameter T) and the secondary distant metastases growth (parameter M). Also, CoMPaS associates with data of 10–15-year survival in patients with the different tumor stage. Analysis of the metastases «nonvisible period» growth indicate the case of discrepancy between 15-year survival depending on tumor stage. In conclusion, the CoMPaS and supporting computer program were build to improve the accuracy of the forecast on survival of breast cancer and facilitate the optimisation of diagnosing secondary distant metastases. This led to completely original results that show how the growth rate of the metastases can change in relation to the growth rate of the primary tumour, taking into consideration its size and diameter of the tumour.

The search for novel parameters to predict the risk of relapse in breast cancer was conducted. Significant correlation between the risk of relapse and α-2A adrenergic receptor (*ADRA2A*) expression was revealed using public microarray datasets. This relationship was confirmed by validation on independent microarray dataset. It was found that when assessing the risk of BC relapse, the accuracy of prediction based solely on the expression of *ADRA2A* gene is close to that made using OncotypeDX and MammaPrint test systems. In this case, addition of only one or two supplemental prognostic markers (for instance, expression of *SQLE* gene or *SQLE* and*DSCC1genes*) to *ADRA2A* ensures the accuracy of prediction not inferior to reliability of these test systems.

Genes with significant differential expression are traditionally used to reveal the genetic background underlying phenotypic differences between cancer cells. We hypothesized that informative marker sets can be obtained by combining genes with a relatively low degree of individual differential expression. We developed a method for construction of highly informative gene combinations aimed at the maximization of the cumulative informative power and identified sets of 2–5 genes efficiently predicting recurrence for ER-positive breast cancer patients. The gene combinations constructed on the basis of microarray data were successfully applied to data acquired by RNA-seq. The developed method provides the basis for the generation of highly efficient prognostic and predictive gene signatures for cancer and other diseases. The identified gene sets can potentially reveal novel essential segments of gene interaction networks and pathways implied in cancer progression.

We propose a new mathematical growth model of primary tumor and primary metastases which may help to improve predicting accuracy of breast cancer process using an original mathematical model referred to CoM-IV and corresponding software. The CoM-IV model and predictive software: a) detect different growth periods of primary tumor and primary metastases; b) make forecast of patient survival; c) have higher average prediction accuracy than the other tools; d) can improve forecasts on survival of BC and facilitate optimisation of diagnostic tests. The CoM-IV enables us, for the first time, to predict the whole natural history of primary tumor and primary metastases growth on each stage (pT1, pT2, pT3, pT4) considering only on primary tumor sizes. Summarising: CoM-IV a) describes correctly primary tumor and primary distant metastases growth of IV (T1-4N0-3M1) stage with (N1-3) or without regional metastases in lymph nodes (N0); b) facilitates the understanding of the appearance period and manifestation of primary metastases.

Specificity of RNAi to selected target is challenged by off-target effects, both canonical and non-canonical. Notably, more than half of all human microRNAs are co-expressed with hosting them proteincoding genes. Here we dissect regulatory subnetwork centered on *IGFBP6* gene, which is associated with low proliferative state and high migratory activity of basal-like breast cancer. We inhibited expression of *IGFBP6* gene in a model cell line for basal-like breast carcinoma MDA-MB-231, then traced secondary and tertiary effects of this knockdown to *LAMA4*, a laminin encoding gene that contributes to the phenotype of triple-negative breast cancer. *LAMA4*-regulating miRNA miR-4274 and its host gene *SORCS2* were highlighted as intermediate regulators of the expression levels of *LAMA4*, which correlated in a basal-like breast carcinoma sample subset of TCGA to the levels of *SORCS2* negatively. Overall, our study points that the secondary and tertiary layers of regulatory interactions are certainly underappreciated. As these types of molecular event may significantly contribute to the formation of the cell phenotypes after RNA interference based knockdowns, further studies of multilayered molecular networks affected by RNAi are warranted.

PRIMARY THERAPY OF EARLY BREAST CANCER

Evidence, Controversies, Consensus

This paper is devoted to mathematical modelling of the progression and stages of breast cancer. The Consolidated mathematical growth Model of primary tumor (PT) and secondary distant metastases (MTS) in patients with lymph nodes MTS (Stage III) (CoM-III) is proposed as a new research tool. The CoM-III rests on an exponential tumor growth model and consists of a system of determinate nonlinear and linear equations. The CoM-III describes correctly primary tumor growth (parameter T) and distant metastases growth (parameter M, parameter N). The CoM-III model and predictive software: a) detect di erent growth periods of primary tumor and distant metastases in patients with lymph nodes MTS; b) make forecast of the period of the distant metastases appearance in patients with lymph nodes MTS; c) have higher average prediction accuracy than the other tools; d) can improve forecasts on survival of breast cancer and facilitate optimisation of diagnostic tests. The CoM-III enables us, for the rst time, to predict the whole natural history of PT and secondary distant MTS growth of patients with/without lymph nodes MTS on each stage relying only on PT sizes.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.