Chemoprevention in BRCA1 Mutation Carriers—A Proof-of-Concept Study

Background: Women with germline BRCA1 mutations have a high lifetime risk of breast cancer, with the only available risk-reduction strategies being risk-reducing surgery or chemoprevention. These women predominantly develop triple-negative breast cancers; hence, it is unlikely that selective estrogen receptor modulators (serms) will reduce the risk of developing cancer, as these have not been shown to reduce the incidence of estrogen receptor–negative breast cancers. Preclinical data from our laboratory suggest that exposure to estrogen and its metabolites is capable of causing dna double-strand breaks (dsbs) and thus driving genomic instability, an early hallmark of BRCA1-related breast cancer. Therefore, an approach that lowers circulating estrogen levels and reduces estrogen metabolite exposure may prove a successful chemopreventive strategy.

Aims: To provide proof of concept of the hypothesis that the combination of luteinizing-hormone releasing-hormone agonists (lhrha) and aromatase inhibitors (ais) can suppress circulating levels of estrogen and its metabolites in BRCA1 mutation carriers, thus reducing estrogen metabolite levels in breast cells, reducing dna dsbs, and potentially reducing the incidence of breast cancer.

Methods: 12 Premenopausal BRCA1 mutation carriers will undergo baseline ultrasound-guided breast core biopsy and plasma and urine sampling. Half the women will be treated for 3 months with combination goserelin (lhrha) plus anastrazole (ai), and the remainder with tamoxifen (serm) before repeat tissue, plasma, and urine sampling. After a 1-month washout period, groups will cross over for a further 3 months treatment before final biologic sample collection. Tissue, plasma, and urine samples will be examined using a combination of immunohistochemistry, comet assays, and ultrahigh performance liquid chromatography tandem mass spectrometry to assess the impact of lhrha plus ai compared with serm on levels of dna damage, estrogens, and genotoxic estrogen metabolites. Quality of life will also be assessed during the study.

Results: This trial is currently ongoing.

The proof of Kontsevich's periodicity conjecture on noncommutative birational transformations

For an arbitrary associative unital ring RR, let J1J1 and J2J2 be the following noncommutative, birational, partly defined involutions on the set M3(R)M3(R) of 3×33×3 matrices over RR: J1(M)=M−1J1(M)=M−1 (the usual matrix inverse) and J2(M)jk=(Mkj)−1J2(M)jk=(Mkj)−1 (the transpose of the Hadamard inverse).

We prove the surprising conjecture by Kontsevich that (J2∘J1)3(J2∘J1)3 is the identity map modulo the DiagL×DiagRDiagL×DiagR action (D1,D2)(M)=D−11MD2(D1,D2)(M)=D1−1MD2 of pairs of invertible diagonal matrices. That is, we show that, for each MM in the domain where (J2∘J1)3(J2∘J1)3 is defined, there are invertible diagonal 3×33×3 matrices D1=D1(M)D1=D1(M) and D2=D2(M)D2=D2(M) such that (J2∘J1)3(M)=D−11MD2(J2∘J1)3(M)=D1−1MD2.