Axitinib and Bosutinib in Treating Patients With Chronic, Accelerated, or Blastic Phase Chronic Myeloid Leukemia

This phase I/II trial studies the side effects and best dose of axitinib and bosutinib and how well they work in treating patients with chronic, accelerated, or blastic phase chronic myeloid leukemia. Axitinib and bosutinib may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth.

PRIMARY OBJECTIVES:

I. To assess the rate of major cytogenetic response (MCyR) of an alternating schedule of axitinib and bosutinib in patients with chronic myeloid leukemia, chronic phase (CML-CP) after failure of/intolerance to \>= 3 tyrosine kinase inhibitors (TKIs) using standard response criteria. (Chronic Phase Cohort) II. To determine the recommended phase II doses (RPTDs) of axitinib and bosutinib in combination in patients with chronic myeloid leukemia (CML) in accelerated phase (CML-AP) or blast phase (CML-BP). (Advanced phase \[AP\] patients must have received \>= 1 prior TKI). (Advanced Phase Cohort - Phase I Portion) III. To evaluate the rate of major hematologic response (MaHR) of combined treatment with axitinib and bosutinib in patients with CML-AP or CML-BP using standard response criteria. (AP patients must have received \>= 1 prior TKI). (Advanced Phase Cohort - Phase II Portion)

SECONDARY OBJECTIVES:

I. To determine the rate of complete cytogenetic response (CCyR), breast cancer gene (BCR-ABL/ABL) =\< 10% and =\< 1%, major molecular response (MMR), molecular response 4-log (MR4), molecular response 4.5-log (MR4.5), and complete molecular response (CMR), overall and at different time points. (Chronic Phase Cohort) II. To determine the duration of response (DOR), event-free survival (EFS), transformation-free survival (TFS), failure-free survival (FFS) and overall survival (OS) for patients with CML-CP treated with alternating axitinib and bosutinib after failure of/intolerance to \>= 3 TKIs. (Chronic Phase Cohort) III. To determine the safety and tolerability of alternating therapy with axitinib and bosutinib after failure of/intolerance to \>= 3 TKIs. (Chronic Phase Cohort) IV. To establish the response rate of concurrent administration of axitinib and bosutinib to patients with CML-AP or CML-BP. (AP patients must have received \>= 1 prior TKI). (Advanced Phase Cohort - Phase I Portion) V. To determine the rate of complete hematologic response (CHR), complete cytogenetic response (CCyR), BCR-ABL/ABL =\< 10% and =\< 1%, major molecular response (MMR), molecular response 4-log (MR4), molecular response 4.5-log (MR4.5), and complete molecular response (CMR), overall and at different time points of combined treatment with axitinib and bosutinib in patients with CML-AP or CML-BP. (AP patients must have received \>= 1 prior TKI). (Advanced Phase Cohort - Phase I Portion) VI. To determine the DOR, EFS, TFS, FFS and OS for patients with CML-AP or -BP treated with combined axitinib and bosutinib. (AP patients must have received \>= 1 prior TKI). (Advanced Phase Cohort - Phase II Portion) VII. To evaluate the probability of developing Abl kinase domain and other somatic mutations in patients with CML treated with alternating (CP) or concurrent (AP/BP) axitinib and bosutinib. (Advanced Phase Cohort - Phase II Portion) VIII. To analyze differences in response rates, duration and survival according to pre-treatment mutations and patient characteristics in both the CP and AP/BP cohorts. (Advanced Phase Cohort - Phase II Portion) IX. To characterize mechanisms of resistance in patients who develop resistance to alternating (CP) or concomitant (AP/BP) therapy with axitinib and bosutinib. (Advanced Phase Cohort - Phase II Portion) X. To evaluate symptom burden in patients with CML receiving axitinib and bosutinib, whether alternating (CP) or in combination (AP/BP). (Advanced Phase Cohort - Phase II Portion)

OUTLINE: This is a dose-escalation followed by a phase II study. Patients are assigned to 1 of 2 cohorts.

COHORT I: Patients with chronic phase CML receive either bosutinib orally (PO) once a day (QD) or axitinib PO twice a day (BID) alone for 3 months. Patients then switch to the other drug for 3 months and alternate between the two every 3 months in the absence of disease progression or unacceptable toxicity.

COHORT II: Patients with accelerated or blastic phase CML receive bosutinib PO QD and axitinib PO BID for 3 months. Courses repeat every 3 months in the absence of disease progression or unacceptable toxicity.

After completion of study treatment, patients are followed up at 1 month.

Eligibility Criteria:
Inclusion Criteria:

* Diagnosis of Philadelphia chromosome positive (Ph+) (by cytogenetics or FISH) or BCR-ABL+ (by polymerase chain reaction \[PCR\]) CML in CP (cohort 1), AP (cohort 2) or BP (cohort 2)
* Patients should have failed (demonstrated resistance, intolerance or treatment discontinuation for any other reason of) at least 3 Food and Drug Administration (FDA)-approved TKIs if in CP (cohort 1), or at least 1 FDA-approved TKI if in AP (cohort 2); resistance will be defined as meeting the criteria for failure or warning by the European Leukemia Net (ELN); no prior therapy is necessary for patients in BP (cohort 2); patients in CP who have failed \< 3 TKIs, but are ineligible to receive other FDA-approved TKIs, may also be enrolled in cohort 1; at least 10 CP patients with the T315I mutation affecting the kinase domain of Bcr-Abl will be enrolled in cohort 1, as well as in the phase II portion of cohort 2
* Eastern Cooperative Oncology Group (ECOG) performance status of 0-2
* Total bilirubin =\< 1.5 x upper limit of normal (ULN) (unless due to Gilbert syndrome, in which case it should be =\< 3.0 x ULN)
* Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 x ULN
* Serum creatinine =\< 1.5 x ULN
* Patients must sign the Institutional Review Board (IRB)-approved informed consent document for this trial
* Reliable telephone access so as to be able to receive calls from an interactive voice response (IVR) system (only applicable to patients participating in the optional symptom burden assessment portion)
* Women of childbearing potential (WOCBP) must practice 2 effective methods of birth control during the course of the study; male patients who are partners of WOCBP should also practice an effective method of contraception: postmenopausal women must be amenorrheic for \>= 12 months to be considered of non-childbearing potential; women and men must continue birth control for the duration of the trial and \>= 3 months after the last dose of study drug; all WOCBP MUST have a negative pregnancy test prior to first receiving study medication(s)
* Patients should have discontinued therapy with imatinib, dasatinib, nilotinib, ponatinib, omacetaxine or other anti-leukemia therapy (except hydroxyurea) \>= 48 hours prior to start of study therapy and recovered from any toxicity due to these therapies to grade =\< 1; hydroxyurea may be received up to the time of enrollment and for the first 6 weeks of study treatment if necessary

Exclusion Criteria:

* Prior therapy with axitinib; prior therapy with bosutinib is allowed, except in the following circumstances: the subject is currently on bosutinib; bosutinib is the subject's most recent TKI for CML; the subject has a history of intolerance to bosutinib
* Active gastrointestinal conditions that are expected to impair absorption of orally administered medications
* Patients who currently have or have a history of the following within 6 months preceding study entry are not eligible: unstable angina (UA), myocardial infarction (MI), transient ischemic attack (TIA), stroke, deep vein thrombosis (DVT), acute peripheral or pulmonary arterial thromboembolism (PE); clinically significant ventricular arrhythmias (e.g., ventricular tachycardia, ventricular fibrillation, or torsades de pointes); New York Heart Association class III or IV heart failure
* Patients with active, uncontrolled psychiatric disorders including: psychosis, major depressive, and bipolar disorders
* Patients with uncontrolled hypertension (defined as sustained systolic blood pressure \> 160 mmHg or diastolic blood pressure \> 100 mmHg)
* Pregnant or breast-feeding women are excluded
* Inability to understand a written informed consent document
* Patients receiving anticoagulants that are unable to be discontinued
* Patients with active, uncontrolled infection
* Patients with a history of hypersensitivity to bosutinib or axitinib
* Patients on proton pump inhibitors, potent CYP3A or P-glycoprotein substrates, inhibitors or inducers a minimum 7 day period washout required unless discontinuation or substitution is not in the best interests of the patient as determined by the investigator; in instances where use of these agents is felt to be required for optimal management, inclusion of such patients should be discussed with the principal investigator (PI) and the rationale documented; these patients, if enrolled on study, may require dose modifications for both axitinib and bosutinib