This randomized phase II trial studies how well afimoxifene works in reducing the risk of breast cancer in women with mammographically dense breast. Estrogen can cause the growth of breast cancer cells. Hormone therapy using afimoxifene may fight breast cancer by blocking the use of estrogen by the tumor cells.

PRIMARY OBJECTIVES:

I. To estimate and compare the percent change in mammographic breast density (using Cumulus software) from baseline to month 12 in women applying 4mg afimoxifene (4-hydroxytamoxifen [4-OHT]) gel per breast versus placebo.

SECONDARY OBJECTIVES:

I. To compare the Cumulus versus (vs.) Volpara breast density measurement methods to estimate percent change in mammographic breast density from baseline to month 12 in women applying 4mg of 4-OHT gel per breast vs. placebo.

II. To compare the percentage of women who underwent a change in Breast Imaging Reporting and Data System (BIRADS) category, comparing pre-and post- treatment measurements, for recipients of active agent versus placebo.

III. To estimate percentage of women with >= 10% absolute decrease in quantitative mammographic density percentage between baseline and 12 months, comparing between treated group 4mg per breast 4-OHT gel to placebo.

IV. To describe symptoms assessed by breast cancer prevention trial (BCPT) eight symptom scale (BESS) questionnaire and laboratory toxicity assessment (factor VIII [F VIII], Von Willebrand [vWB] factor, sex hormone-binding globulin [SHBG], lipid profile).

V. To evaluate serum measurements of 4-OHT and related metabolite levels and factors related to tamoxifen exposures, such as insulin-like growth factor (IGF) pathway members, C-reactive protein (CRP), estradiol.

VI. To evaluate tissue biomarkers (among women undergoing optional pre- and post-treatment biopsies): terminal duct lobular unit (TDLU) involution; collagen structural changes; SETER/PR index: estrogen related transcription, Ki-67, COX-2, p16, CD68.

VII. To examine whether any reductions in mammographic density seen after 1 year of 4-OHT vs. placebo gel application persist at 24 months, one year after gel application has stopped.

OUTLINE: Patients are randomized to 1 of 2 arms.

ARM A: Patients apply placebo gel topically to each breast once daily (QD) for up to 52 weeks.

ARM B: Patients apply afimoxifene gel topically to each breast QD for up to 52 weeks.

After completion of study treatment, patients are followed up at 24 months.

Inclusion Criteria:

• Women age 40-69 years, or less than 40 years if 5-year breast cancer Gail risk is ≥ 1.66%.
• Mammographically dense breast (heterogeneously dense [C] or extremely dense [D], based on American College of Radiology [ACR] BIRADS fifth edition classification or heterogeneously dense [3] or extremely dense [4], based on ACR BIRADS fourth edition classification) in either breast
• Eastern Cooperative Oncology Group (ECOG) performance status =< 1 (Karnofsky >= 70%)
• White blood cells >= 3,000/microliter
• Absolute neutrophil count >= 1,500/microliter
• Platelets >= 100,000/microliter
• Total bilirubin within normal institutional limits
• Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT]) =< 1.5 × institutional upper limit of normal (ULN)
• Creatinine within normal institutional limits
• Participant must have a gynecology examination within the last 3 years, with no atypical hyperplasia and no cancer
• Premenopausal women taking non hormonal intra-uterine device (IUD) birth control method will be eligible, if they have been on the same IUD for at least 3 months prior to enrollment and plan to continue using the same method throughout the study
• Women of child-bearing potential must agree to use a reliable nonhormonal contraceptive method during the study and for 2 months after completing study medications; reliable nonhormonal methods of contraception include barrier contraception and an intra-uterine device (IUD); Note: women who had tubal ligation or had a partner who had undergone a vasectomy (and are monogamous) are eligible for the study and are not required to use barrier contraception
• If the participant is of childbearing potential, she must have a documented negative urine pregnancy test within 7 days prior to randomization
• Willingness to avoid exposing breast skin to natural or artificial sunlight (i.e. tanning beds) for the duration of the study
• Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

• History of allergic reactions attributed to compounds of similar chemical or biologic composition to 4-OHT gel
• Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, thromboembolic disease, or psychiatric illness/social situations that would limit compliance with study requirements
• Pregnant, or had given birth, or nursed at any time during the last 12 months
• Women with a previous history of invasive breast cancer or bilateral ductal carcinoma in situ (DCIS) or current untreated DCIS; women with a history of cancer within the last 3 years, except for non-melanoma skin cancer; women with unilateral DCIS (with or without radiation therapy) are eligible as long as they have an unaffected breast
• Prior bilateral breast surgery (mastectomy, segmental mastectomy, or breast augmentation surgery including breast implants or breast reductions)
• Women with "mosaic mammographic screening views", i.e., whose larger breast size precludes being imaged within a single mammographic screening view
• Women with active liver disease, abnormal uterine bleeding, or prior diagnosis of endometrial hyperplasia
• Prior use of selective estrogen receptor modulators (SERMS) and aromatase inhibitors (AIs) for prevention or therapy
• Skin lesions on the breast that disrupt the stratum corneum (e.g., eczema, ulceration)
• Treatment with any investigational drug or investigational biologic within 30 days of initiating study treatment or during the study