BCMA-Specific CAR T-Cells Combined With a Gamma Secretase Inhibitor (JSMD194) to Treat Relapsed or Persistent Multiple Myeloma

Inclusion Criteria:

* Eastern Cooperative Oncology Group (ECOG) performance status score =\< 2
* Have measurable disease by International Myeloma Working Group (IMWG) criteria based on one or more of the following findings:

  * Serum monoclonal immunoglobulin (M-protein) \>= 1 g/dL
  * Urine M-protein \>= 200 mg/24 hour
  * Involved serum free light chain (sFLC) level \>= 10 mg/dL with abnormal kappa/lambda ratio
  * Measurable biopsy-proven plasmacytomas (\>= 1 lesion that has a single diameter \>= 2 cm)
  * Bone marrow plasma cells \>= 30%
* Have a diagnosis of multiple myeloma (MM); the MM diagnosis must be confirmed by internal pathology review of a fresh biopsy specimen at the Fred Hutchinson Cancer Research Center/Seattle Cancer Care Alliance (FHCRC/SCCA)
* Have relapsed or treatment refractory disease with \>= 10% CD138+ malignant plasma cells immunohistochemistry (IHC) on BM core biopsy, either:

  * Following autologous stem-cell transplantation (ASCT)
  * Or, if a patient has not yet undergone ASCT, the individual must:

    * Be transplant ineligible, due to age, comorbidity, patient choice, insurance reasons, concerns of rapidly progressive disease, and/or discretion of attending physician and principal investigator and,
    * Demonstrate disease that persists after \> 4 cycles of induction therapy and that is double refractory (persistence/progression) after therapy with both a proteasome inhibitor (PI) and immunomodulatory drug (IMiD) administered either in tandem, or in sequence, or demonstrate intolerance to both classes of agents (IMiD and PI); \> 4 cycles of therapy are not required for patients with a diagnosis of plasma cell leukemia
* Male and female patients of reproductive potential must be willing to use an effect contraceptive method before, during, and for at least 4 months after the BCMA CAR T cell infusion

Exclusion Criteria:

* History of another primary malignancy that requires intervention beyond surveillance or that has not been in remission for at least 1 year (the following are exempt from the 1-year limit: non-melanoma skin cancer, curatively treated localized prostate cancer, curatively treated superficial bladder cancer and cervical carcinoma in situ on biopsy or a squamous intraepithelial lesion on PAP smear)
* Active hepatitis B, hepatitis C at the time of screening
* Patients who are human immunodeficiency virus (HIV) seropositive
* Subjects with uncontrolled active infection
* \> 1 hospital admission for infection in prior 6 months
* Presence of acute or chronic graft-versus-host disease (GVHD) unless limited to skin involvement and managed with topical steroid therapy alone
* History of any one of the following cardiovascular conditions within the past 6 months: class III or IV heart failure as defined by the New York Heart Association (NYHA), cardiac angioplasty or stenting, myocardial infarction, unstable angina, or other clinically significant cardiac disease as determined by the principal investigator (PI) or designee
* History of clinically relevant central nervous system (CNS) pathology such as epilepsy, seizure, paresis, aphasia, stroke, severe brain injuries, dementia, Parkinson's disease, cerebellar disease, organic brain syndrome, or psychosis, active central nervous system MM involvement and/or carcinomatous meningitis; subjects with previously treated central nervous systems involvement may participate, provided they are free of disease in the CNS (documented by flow cytometry performed on the cerebrospinal fluid \[CSF\] within 14 days of enrollment) and have no evidence of new sites of CNS activity
* Pregnant or breastfeeding females
* Allogeneic hematopoietic stem cell transplantation (HSCT) or donor lymphocyte infusion within 90 days of leukapheresis
* Use of any of the following:

  * Therapeutic doses of corticosteroids (defined as \> 20 mg/day prednisone or equivalent) within 7 days prior to leukapheresis; physiologic replacement, topical, and inhaled steroids are permitted
  * Cytotoxic chemotherapeutic agents within 1 week of leukapheresis; oral chemotherapeutic agents are allowed if at least 3 half-lives have elapsed prior to leukapheresis
  * Lymphotoxic chemotherapeutic agents within 2 weeks of leukapheresis
  * Experimental agents within 4 weeks of leukapheresis unless progression is documented on therapy and at least 3 half-lives have elapsed prior to leukapheresis
  * Daratumumab (or other anti-CD38 therapy) within 30 days of leukapheresis
* Absolute neutrophil count (ANC) \< 1000/mm\^3, per PI discretion if cytopenia thought to be related to underlying myeloma
* Hemoglobin (Hgb) \< 8 mg/dl, per PI discretion if cytopenia thought to be related to underlying myeloma
* Platelet count \< 50,000/mm\^3, per PI discretion if cytopenia thought to be related to underlying myeloma
* Active autoimmune disease requiring immunosuppressive therapy
* Creatinine clearance \< 20 ml/min
* Significant hepatic dysfunction (serum glutamic-oxaloacetic transaminase \[SGOT\] \> 5 x upper limit of normal; bilirubin \> 3.0 mg/dL)
* Forced expiratory volume in one second (FEV1) of \< 50% predicted or carbon monoxide diffusing capacity (DLCO) (corrected) \< 40% (patients with clinically significant pulmonary dysfunction, as determined by medical history and physical exam should undergo pulmonary function testing)
* Anticipated survival of \< 3 months
* Contraindication to cyclophosphamide or fludarabine chemotherapy
* Patients with known amyloidosis (AL) subtype amyloidosis
* Uncontrolled medical, psychological, familial, sociological, or geographical conditions that do not permit compliance with the protocol, as judged by the investigator; or unwillingness or inability to follow the procedures required in the protocol
* Documented malabsorptive syndromes including enteropathies, gastroenteritis (acute or chronic) or diarrhea (acute or chronic)