Can a targeted drug boost standard chemo for a rare blood cancer?

Copied word for word from the study's registry entry, so the wording is the study team's rather than ours.

Inclusion Criteria: * Age 18-75 years * No prior therapy for mantle cell lymphoma (MCL) * MCL in need of systemic therapy, and potentially eligible for ASCT as assessed by the treating physician * Documented histological confirmation of MCL by local institutional review * Documented, fludeoxyglucose F-18 (FDG)-avid measurable disease (at least 1 lesion \>= 1.5 cm in diameter) as detected by positron emission tomography (PET)/computed tomography (CT) and as defined and includes measurable nodal and extranodal disease sites, or splenomegaly measuring more than 13 cm in vertical length * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1, 2 * Absolute neutrophil count (ANC) \>= 1000/mm\^3 or \>= 500/mm\^3 if due to lymphomatous marrow or spleen involvement (obtained =\< 30 days prior to registration) * Platelet count \>= 100,000/mm\^3 or \>= 75,000/mm\^3 if due to lymphomatous marrow or spleen involvement (obtained =\< 30 days prior to registration) * Total bilirubin =\< 1.5 x upper limit of normal (ULN) (unless documented Gilbert's syndrome, for which total bilirubin =\< 3 x upper limit of normal \[ULN\] is permitted) (obtained =\< 30 days prior to registration) * Aspartate transaminase (AST) =\< 3 x ULN (obtained =\< 30 days prior to registration) * Prothrombin time (PT)/international normalized ratio (INR) or partial thromboplastin time (PTT) =\< 2 x ULN, unless elevated due to a lupus anticoagulant (obtained =\< 30 days prior to registration) * Calculated creatinine clearance must be \>= 30 ml/min using the Cockcroft-Gault formula (obtained =\< 30 days prior to registration) * Negative pregnancy test done within =\< 14 days prior to registration for women of childbearing potential only * For women of childbearing potential (WOCBP, defined as premenopausal women capable of becoming pregnant): Must agree to use of highly effective method of birth control during study therapy and until 12 months after last dose of study therapy. NOTE: 'Acceptable' methods are not adequate. Highly effective methods are defined by Clinical Trials Facilitation and Coordination Group \[CTFG\] as having a failure rate of \< 1% per year * Men must agree to use barrier contraception starting with the first dose of study therapy and through 180 days after completion of study therapy * Provide informed written consent * Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study) * Hematologic labs must be obtained within =\< 14 days of registration * Willing and able to participate in all required evaluations and procedures in this study protocol * Ability to understand the purpose and risks of the study and provide signed and dated informed consent and authorization to use protected health information Exclusion Criteria: * Prior systemic treatment for mantle cell lymphoma. Short course of steroids (=\< 7 days) for symptom management or localized radiation is permissible, as long as measurable disease outside of the radiation field exists * Peripheral neuropathy or neuropathic pain of grade 2 or worse as assessed by the investigator * Prior exposure to bortezomib or a BTK inhibitor * Prior anthracycline exposure unless cumulative prior exposure is under 150 mg per square meter * Requiring anticoagulation with warfarin or equivalent vitamin k antagonist * Uncontrolled AIHA (autoimmune hemolytic anemia) or ITP (idiopathic thrombocytopenia purpura) * Active bleeding or history of bleeding diathesis (e.g. hemophilia or von Willebrand disease) * History of stroke or intracranial hemorrhage within 6 months prior to enrollment * Requires treatment with a strong cytochrome P450 3A4 (CYP3A4) inhibitor/inducer * Requiring treatment with a proton pump inhibitor. Examples include: dexlansoprazole, esomeprazole, lansoprazole, omeprazole, pantoprazole, rabeprazole, or therapeutic class equivalents * Note: H2-receptor agonists are not exclusionary * History of allergic reactions attributed to acalabrutinib, cytarabine, bortezomib, boron, or any of the other agents administered as part of the therapeutic regimen in this study * Active systemic fungal, bacterial, viral, or other infection that is worsening (defined as increasing signs/symptoms of infection during screening) or, requires intravenous antibiotic therapy * Active or chronic uncontrolled hepatitis B or hepatitis C infection. Patients with positive hepatitis B core antibody positive require negative polymerase chain reaction (PCR) prior to enrollment. Hepatitis B surface antigen positive or PCR positive patients will be excluded. Patients with hepatitis C must have negative hepatitis C virus (HCV) ribonucleic acid (RNA) for inclusion * Co-morbid systemic illnesses or other severe concurrent disease (including major surgery within 2 weeks) which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens * Known to be human immunodeficiency virus (HIV) positive since antiretroviral therapy has a potential for drug interactions with acalabrutinib * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure or low cardiac ejection fraction (New York Heart Association \[NYHA\] class 3-4 or ejection fraction \[EF\] \< 45%), unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm * Other active malignancy =\< 2 years prior to registration. EXCEPTIONS: Non-melanotic skin cancer localized prostate cancer, or carcinoma-in-situ of the breast or cervix. NOTE: If there is a history or prior malignancy, patients must not be receiving other specific treatment for their cancer * Pregnant and/or breastfeeding * Has difficulty with or is unable to swallow oral medication, or has significant gastrointestinal disease that would limit absorption of oral medication * Presence of a gastrointestinal ulcer diagnosed by endoscopy within 3 months before screening. unless directly due to MCL Involvement by endoscopic or histologic evaluation * Major surgical procedure within 28 days of first dose of study drug. NOTE: If a subject had major surgery, they must have recovered adequately from any toxicity and/or complications from the intervention before the first dose of study drug * Concurrent participation in another therapeutic clinical trial