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This phase II trial studies how well vorinostat and combination chemotherapy before donor stem cell transplantation work in treating patients with aggressive B-cell or T-cell non-Hodgkin lymphoma that has come back (relapsed). Vorinostat may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth. Drugs used in chemotherapy, such as busulfan, gemcitabine, and clofarabine, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving vorinostat together with combination chemotherapy before donor stem cell transplantation may help to control lymphoma.

PRIMARY OBJECTIVE:

I. Estimate the progression-free survival (PFS) time.

SECONDARY OBJECTIVES:

I. Estimate the day 100 non-relapse mortality (NRM) of allogeneic stem cell transplantation (allo SCT) using vorinostat/gemcitabine/clofarabine/busulfan (SAHA/Gem/Clo/Bu) with post-transplant cyclophosphamide (PT-CY).

II. Estimate the graft versus host disease (GVHD)-free/relapse free survival (GRFS) in patients with refractory lymphoma receiving an allo SCT with SAHA/Gem/Clo/Bu with PT-CY.

III. Estimate the overall survival (OS) in patients with refractory lymphoma receiving an allo SCT with SAHA/Gem/Clo/Bu with PT-CY.

IV. Assess the 1-year NRM in patients with refractory lymphoma receiving an allo SCT with SAHA/Gem/Clo/Bu with PT-CY.

V. Assess the relapse rate in patients with refractory lymphoma receiving an allo SCT with SAHA/Gem/Clo/Bu with PT-CY.

VI. Assess the graft failure rate in patients with refractory lymphoma receiving an allo SCT with SAHA/Gem/Clo/Bu with PT-CY.

VII. Assess the time to neutrophil and platelet engraftment in patients with refractory lymphoma receiving an allo SCT with SAHA/Gem/Clo/Bu with PT-CY.

VIII. Assess the incidence of grade 2-4 and grade 3-4 acute graft versus host disease (GVHD) in patients with refractory lymphoma receiving an allo SCT with SAHA/Gem/Clo/Bu with PT-CY.

IX. Assess the overall and severe chronic GVHD in patients with refractory lymphoma receiving an allo SCT with SAHA/Gem/Clo/Bu with PT-CY.

X. Determine the incidence of grade 3 and 4 nonhematological adverse events in patients with refractory lymphoma receiving an allo SCT with SAHA/Gem/Clo/Bu with PT-CY.

TERTIARY OBJECTIVE:

I. Describe changes of deoxyribonucleic acid (DNA) damage response and repair, poly (ADP-ribose) polymerase (PARP) inhibition and downstream cellular effects in peripheral blood mononuclear cells (PBMNC), and, when available, malignant lymphocytes obtained by fine needle aspiration (FNA) of peripheral lymph nodes of patients with refractory lymphoma receiving an allo SCT with SAHA/Gem/Clo/Bu with PT-CY.

OUTLINE:

Patients receive a low-level "test" dose of busulfan intravenously (IV) over up to 1 hour on days -15 to -9, vorinostat orally (PO) once daily (QD) on days -8 to -4, gemcitabine IV over about 90 minutes on days -7 and -5, clofarabine IV over about 1 hour and high-dose busulfan IV over 3 hours on days -7 to -4. Patients with CD20 positive (+) lymphoma also receive rituximab IV over 3 to 6 hours on days -15, -8, 1, and 8. Patients undergo allogeneic hematopoietic stem cell transplantation (HSCT) on day 0. Patients then receive cyclophosphamide IV over 2 hours on days 3 and 4. Beginning day 5, patients receive standard of care tacrolimus IV over 24 hours and mycophenolate mofetil IV over 2 hours three times daily (TID) until they can be tolerated PO. Once tolerated PO, patients receive tacrolimus PO twice daily (BID) for 6 months and mycophenolate mofetil PO TID for up to 30 days in the absence of disease progression or unacceptable toxicity. After 30 days, patients who develop GVHD continue treatment with mycophenolate mofetil at physician's discretion.

After completion of study treatment, patients are followed up at 3, 6, and 12 months after stem cell transplant, then every 6 months for 4 years.

Inclusion Criteria:

• Relapsed aggressive B-cell non-Hodgkin lymphoma (B-NHL) (diffuse large B-cell lymphoma [DLBCL], transformed B-NHL, high-grade B-cell lymphoma [HGBL] or Burkitt) or T-cell non-Hodgkin lymphoma (T-NHL) who meet both of the following criteria: a. High or very high disease risk index (DRI), and b. No response to at least 1 line of salvage chemotherapy, or relapse after a prior autologous SCT
• An 8/8 human leukocyte antigen (HLA) matched (high resolution typing at A, B, C, DRB1) sibling or unrelated donor, or a haploidentical donor
• Left ventricular ejection fraction (EF) >= 45%
• Forced expiratory volume in 1 second (FEV1), forced vital capacity (FVC) and corrected diffusion capacity of the lung for carbon monoxide (DLCO) >= 50%
• Estimated serum creatinine clearance >= 50 ml/min (using the Cockcroft-Gault formula)
• Serum bilirubin =< 2 x upper limit of normal
• Serum glutamate pyruvate transaminase (SGPT) =< 2 x upper limit of normal
• Able to sign informed consent
• Men and women of reproductive potential must agree to follow accepted birth control methods for the duration of the study. Female subject is either post-menopausal or surgically sterilized or willing to use an acceptable method of birth control (i.e., a hormonal contraceptive, intra-uterine device, diaphragm with spermicide, condom with spermicide, or abstinence) for the duration of the study. Male subject agrees to use an acceptable method for contraception for the duration of the study

Exclusion Criteria:

• Patient with active central nervous system (CNS) disease
• Pregnancy (positive beta human chorionic gonadotropin [HCG] test in a woman with child bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization) or currently breast-feeding. Pregnancy testing is not required for post-menopausal or surgically sterilized women
• Active hepatitis B, either active carrier (hepatitis B surface antigen positive [HBsAg +]) or viremic (hepatitis B virus [HBV] DNA >= 10,000 copies/mL, or >= 2,000 IU/mL)
• Evidence of either cirrhosis or stage 3-4 liver fibrosis in patients with chronic hepatitis C or positive hepatitis C serology
• Human immunodeficiency virus (HIV) infection
• Active uncontrolled bacterial, viral or fungal infections
• Exposure to other investigational drugs within 4 weeks before enrollment
• Grade >= 3 non-hematologic toxicity from previous therapy that has not resolved to =< grade 1
• Radiation therapy to head and neck (excluding eyes), and internal organs of chest, abdomen or pelvis in the month prior to enrollment
• Prior whole brain irradiation
• Prior autologous SCT in the prior 3 months