Inclusion Criteria:

          -  Adults (male and female) age >18

          -  Eastern Cooperative Group (ECOG) Performance Scale 0-1 (see Appendix I)

          -  Histologically or cytologically confirmed invasive breast cancer of the following
             subtype:

          -  TRIPLE NEGATIVE (ER-negative, PR-negative, and HER2-negative disease). Triple-negative
             patients will be defined per ASCO-CAP Guidelines (American Society of Clinical
             Oncology-College of American Pathologists).

          -  HER2-POSITIVE: HER2-positive patients will be defined per ASCO-CAP Guidelines.

          -  HORMONE REFRACTORY: Patients with ER/PR-positive disease according to ASCO-CAP
             guidelines above may be considered if they have disease progression after two lines of
             hormonal therapy (administered in the adjuvant or metastatic setting), or are deemed
             clinically hormone-resistant taking into consideration the rate of progression of
             disease or a short interval of time on first line hormonal therapy before progression.
             Clinically hormone-resistant patients MUST also be discussed with the investigator,
             co-investigator or designee in advance for approval.

        NOTE: ASCO-CAP guidelines state that ER and PR assays be considered positive if there are
        at least 1% positive tumor nuclei in the sample on testing in the presence of expected
        reactivity of internal (normal epithelial elements) and external controls. HER2-positive is
        defined as HER2 IHC 3+, ISH ≥ 2.0, or average HER2 copy number ≥ 6.0 signals.

        NOTE: A patient who has a change in receptor status (e.g. PR negative to positive) may be
        stratified as triple negative or hormone positive, contrary to the most recent receptor
        testing, for the purposes of the study, based upon the clinical course at the discretion of
        the Study Chair, Study co-Chair, or designee in advance for approval.

          -  Cytologic or unequivocal radiographic confirmation of leptomeningeal metastasis by
             dural puncture and/or neuroimaging with or without known brain metastasis

          -  Adequate organ function as follows:

          -  Estimated creatinine clearance >70 cc/min (calculated by Cockcroft-Gault formula)

          -  White blood cell counts >3000 cells/mcL

          -  Absolute neutrophil count >1500 cells/mcL

          -  Platelet count >100,000 cells/mcL

          -  Hematocrit >30%

          -  Serum bilirubin <1.5 x the ULN

          -  Alanine aminotransferase or aspartate aminotransferase <2.5x the ULN

          -  Alkaline phosphatase <2.5x the ULN or <5x the ULN if secondary to liver metastasis

          -  Able to provide confirmed consent

        Exclusion Criteria:

          -  Prior allergy or adverse reaction to methotrexate

          -  New York Heart Association Heart Failure Class >3 (see Appendix II)

          -  Active diabetes insipidus

          -  Active mucositis

          -  Chemotherapy or stereotactic radiotherapy within the last 2 weeks

          -  Partial brain radiotherapy (i.e., less than or equal to 40% of total brain volume)
             within the last two weeks

          -  Whole brain radiotherapy within the last 6 months or partial brain radiotherapy
             exceeding greater than 40% of total brain volume in the last 6 months

          -  Prior treatment with any methotrexate containing systemic regimen within 1 year
             (excluding intrathecal methotrexate)

          -  Concurrent or planned systemic chemotherapy, radiotherapy, new hormonal or anti-HER2
             directed therapy directed at management of breast cancer (existing anti-HER2 therapy
             can be continued as recently recommended in the National Consensus Guidelines

          -  Uncontrolled or progressive systemic disease or other concurrent condition which in
             the Investigator's opinion makes HD-MTX an undesirable treatment option for the
             patient or would jeopardize compliance

          -  Contraindication to MRI

          -  Use of salicylates, non-steroidal anti-inflammatory drugs, or sulfonamide medications
             within one week of start of methotrexate

          -  Pregnant women or women who are breastfeeding.

          -  Patients with significant visceral fluid collections including ascites, pericardial
             effusions, pleural effusions or others may experience delayed clearance of
             methotrexate because of third space accumulation which could result in methotrexate
             toxicity and inability to tolerate the proposed study treatment. While these are not
             absolute exclusions the investigators or co-investigators should be contacted to
             discuss possible enrollment. Patients with significant ascites defined as European
             Association for the Study of the Liver > grade 2, or with asymptomatic pleural
             effusions with an estimated size >200 mL, or with symptomatic pleural effusion of any
             size will be excluded.

        NOTE: Systemic staging of the chest/abdomen/pelvis is required for study entry. See
        Sections 8.1.9. Body fluid will be assessed based on this study.