Trial of Ado-Trastuzumab Emtansine for Patients With HER2 Amplified or Mutant Cancers

Inclusion Criteria: * Adults who are ≥18 years old. * Pathologically confirmed advanced solid tumor cancers * For Cohort 1, documented activating HER2 mutation in lung cancer by CLIA laboratory, specifically exon 20 insYVMA (Y772\_A775dup), insGSP (G778\_P780dup), insTGT (G776delinsVC), single base pair substitutions L755A, L755S, V777L, V659E, S310F, or another HER2 mutation approved by the Principal Investigator * For Cohorts 2, 3, 4, 5, 6 documented HER2 amplification identified through next generation sequencing by MSK-IMPACT or at another Clinical Laboratory Improvement Amendments (CLIA) laboratory, or documented HER2 amplification by in-situ hybridization (ISH) with HER2/CEP17 ratio ≥2.0 at a CLIA laboratory. Patients with HER2 amplification identified by another method or criteria must be approved by the Principal Investigator and may enroll in the "Other" Cohort 4. * Measurable or evaluable indicator lesion(s) as defined by RECIST v1.1. Patients without RECIST measurable disease will be eligible for enrollment to "Other" cohort provided their disease can be evaluated using another accepted response criteria (e.g. Gynecologic Cancer InterGroup (GCIG) CA125 Response Criteria, modified PET Response Criteria in Solid Tumors (PERCIST)). Patients with salivary gland cancers (Cohort 5) may be eligible on the basis of evaluable disease on modified PET. * Karnofsky Performance Status 70% or above. * Left ventricular ejection fraction (LVEF) ≥50% measured by echocardiogram (ECHO) or multiple gated acquisition scan (MUGA). * Negative β-human chorionic gonadotropin (hCG) pregnancy test within 2 weeks before enrollment for premenopausal women of reproductive capacity and for women less than 12 months after menopause. Pregnancy screening will be conducted for women up to the age of 50 years per institutional standard. * Women of childbearing potential must agree to use of a highly effective method of contraception. Effective contraception is required during treatment and for 7 months following the last dose for female participants of reproductive potential and during treatment and for 4 months following the last dose for male participants with female sexual partners of reproductive potential. Male participants should also refrain from donating sperm during treatment and for 4 months following the last dose. * Absolute neutrophil count ≥ 1,000/µL within 30 days prior to C1D1 * Platelet count ≥ 100,000/µL within 30 days prior to C1D1 * Total bilirubin ≤ 1.5 x institutional upper limit of normal (ULN), in case of Gilbert's syndrome, ≤ 2x ULN within 30 days prior to C1D1 * Aspartate aminotransferase and/or alanine aminotransferase ≤ 3 x ULN (≤ 5 x ULN if liver metastases are present) within 30 days prior to C1D1 * Provide written, informed consent to participate in the study and follow the study procedures Exclusion Criteria: * Prior therapy resulting in cumulative epirubicin dose ≥ 900mg/m2 or cumulative doxorubicin dose ≥ 500mg/m2 or equivalent dose of another anthracycline. * Prior therapy with ado-trastuzumab emtansine (patients who had prior trastuzumab or other HER2 targeted agents are eligible). * Symptomatic congestive heart failure (New York Heart Association Classification II-IV). * Myocardial infarction or unstable angina within 6 months of enrollment. * Unstable ventricular arrhythmia requiring treatment. * Grade 3 or worse peripheral neuropathy as defined by CTCAE v4.1. * Women who are pregnant or breast-feeding. * Known hypersensitivity to any component of ado-trastuzumab emtansine. * History of interstitial lung disease or pneumonitis.