In this protocol, the investigators hypothesize that the combination of intravenous busulfan and melphalan with carfilzomib will be an effective preparative regimen with acceptable toxicity for participants with multiple myeloma who are candidates for autologous stem cell transplantation. To test this hypothesis, the investigators designed a phase I/II trial combining IV busulfan 130 mg/m2 plus melphalan 140 mg combined with escalating doses of carfilzomib ranging from 20 mg/m2 to 45 mg/m2. These results will be compared with the center's historical controls of participants treated with melphalan, busulfan and bortezomib.

Participants enrolled in this study protocol will receive daily intravenous (IV) infusions of carfilzomib for a total of 4 days (Day-9, -8 and Days -2, -1). The first two daily infusions will be given at a fixed dose of 20 mg/m2 and the final two doses will be escalated from the standard dose of 27 mg/m2 to 56 mg/m2 in a Phase I design, based on toxicity. The busulfan will be administered for 2 days over 3 hours from D-7, -6, at 130 mg/m2 . This dose was found to be safe and equivalent to the standard daily dose of 3.2 mg/kg. The 3rd and 4th daily doses of IV Busulfan will be adjusted in order to yield a systemic plasma drug exposure represented by a daily area under the plasma concentration versus time curve (AUC) of approximately 5,000 millimoles-minute per dose (mM-min). These targeted plasma concentration of IV busulfan will be based on pharmacokinetics studies performed during the first day of IV busulfan. Melphalan will be given at a dose of 140 mg/m2 on Day -3. Each cohort will start with a goal of accruing three patients to determine the dose limiting toxicity.

Inclusion Criteria:

• Participants must be greater than or equal to 18 years of age.
• Participants must have been diagnosed with multiple myeloma in a first or subsequent remission and have undergone a successful pre-transplant work up and are otherwise eligible for an autotransplant with a busulfan/melphalan preparative regimen at Loyola University Medical Center.
• Participants may receive this preparative regimen if in first or subsequent remission. Participants may enter if they have received a prior autologous stem cell transplant and this therapy produced a remission that lasted greater than 18 months before progression of disease. Participants who have undergone prior allogeneic transplantation are excluded.
• All participants must have responsive disease as defined by a Partial Response or greater to most recent conventional regimen.
• Participants receiving prior carfilzomib will be eligible for inclusion provided they demonstrated responsive disease to this agent and either had a remission that lasted greater than 6 months after its discontinuance, or if in remission after a carfilzomib containing regimen administered to qualify for transplant.
• Participants must have an Eastern Cooperative Oncology Group (ECOG) Performance Status (PS) less than or equal to 2.
• Acceptable heart function test.

Exclusion Criteria:

• Participants must not have below normal kidney function.
• Participants must not have below normal liver function.
• Participants must not have active bacterial, fungal, or viral infection.
• Participants must not have severe lung function.
• Participants must not have Grade 2 or greater peripheral neuropathy.
• Participants must not have uncontrolled hypertension.