MLN0128 Compared to Sorafenib in Advanced or Metastatic Hepatocellular Carcinoma

Inclusion Criteria: * Male or female subjects 18 years or older at the time of informed consent. * Voluntary written consent must be signed before performance of any study related procedure not part of standard medical care, with the understanding that the subject may withdraw consent at any time without prejudice to future medical care. * Females of childbearing potential must agree to practice 2 effective methods of contraception, at the same time, from the time of signing the informed consent through 90 days after the last dose of study drug, or agree to completely abstain from heterosexual intercourse. * Females of childbearing potential must have a negative serum pregnancy test within 7 days prior to prior to registration for protocol therapy. NOTE: Female subjects are considered of childbearing potential unless they are surgically sterile (they have undergone a hysterectomy, bilateral tubal ligation, or bilateral oophorectomy) or they are naturally postmenopausal for at least 12 consecutive months. * Male subjects, even if surgically sterilized (i.e., status post-vasectomy), must agree to practice effective barrier contraception during the entire study treatment period and through 120 days after the last dose of study drug, or agree to completely abstain from heterosexual intercourse. * Subjects must have a diagnosis of measurable advanced or metastatic hepatocellular carcinoma (HCC). Advanced HCC is defined as disease not amenable to surgery, ablation, transplant, or embolic therapy. * Phase II subjects must be willing to provide a tissue biopsy prior to registration if archived HCC tumor tissue is not available for correlative studies. * For the phase I cohort, subjects with one prior systemic treatment are eligible. * Eastern Cooperative Oncology Group (ECOG) performance status 0, 1, or 2. * Adequate organ function, as specified below, within 28 days before study registration: * Bone marrow reserve consistent with: absolute neutrophil count (ANC) ≥ 1.5 x 10\^9/L; platelet count ≥ 50 x 10\^9/L; hemoglobin ≥ 9 g/dL; * Hepatic: total bilirubin ≤ 2 x upper limit of normal (ULN), transaminases (aspartate aminotransferase/serum glutamic oxaloacetic transaminase-AST/SGOT and alanine aminotransferase/serum glutamic pyruvic transaminase-ALT/SGPT) ≤ 5 x ULN * Renal: creatinine clearance ≥50 mL/min based either on Cockcroft-Gault estimate or based on urine collection (12 or 24 hour); * Metabolic: Glycosylated hemoglobin (HbA1c) ≤7.0%, fasting serum glucose (≤ 130 mg/dL) and fasting triglycerides ≤ 300 mg/dL. * Ability to swallow oral medications. * Measurable disease according to RECIST v1.1 and obtained by imaging within 28 days prior to registration for protocol therapy. * Subjects who have a history of brain metastasis are eligible for the study provided all the following criteria are met: * Must have completed their treatment for brain metastasis * Must be asymptomatic * Must not have evidence of disease progression for ≥3 months or hemorrhage after treatment; * Must be off-treatment from dexamethasone for 4 weeks prior to study registration and * Must not have an ongoing requirement for dexamethasone or anti-epileptic drugs. * Prior locoregional liver directed therapy is allowed as long as treatment was at least 6 weeks prior to study registration, and clear progression is demonstrated by RECIST v1.1 criteria. Subject must have recovered from the acute toxic effects (≤ grade 1 CTCAE v4) of previous anti-cancer treatment prior to study enrollment; the only exception is that grade 2 neuropathy is permitted. * Prior radiation therapy is allowed to \< 25% of the bone marrow, but is not permitted within 28 days prior to study registration. * Estimated life expectancy \> 3 months as determined by the treating physician. Exclusion Criteria: Subjects meeting any of the following exclusion criteria are not to be enrolled in the study: * Female subjects who are both lactating and breastfeeding * Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol. * Treatment with any investigational products within 28 days prior to study registration. * No prior systemic treatment is allowed, except for subjects in the phase I cohort who are permitted one prior systemic treatment. * Failed to recover from the reversible effects of prior anticancer therapies with the exception of alopecia and grade 2 neuropathy. * Have initiated treatment with bisphosphonates less than 30 days prior to study registration. Concurrent bisphosphonate use is only allowed if the bisphosphonate was initiated at least 30 days prior to study registration. * Manifestations of malabsorption due to prior gastrointestinal (GI) surgery, GI disease, or for an unknown reason that may alter the absorption of MLN0128. * No condition that could affect the absorption of study drug, including any of the following: * Malabsorption syndrome * Disease significantly affecting gastrointestinal function * Bowel obstruction or sub-obstruction * History of any of the following within the last 6 months prior to study registration: * Ischemic myocardial event, including angina requiring therapy and artery revascularization procedures * Ischemic cerebrovascular event, including transient ischemic attack (TIA) and artery revascularization procedures * Requirement for inotropic support (excluding digoxin) or serious (uncontrolled) cardiac arrhythmia (including atrial flutter/fibrillation, ventricular fibrillation or ventricular tachycardia) * Placement of a pacemaker for control of rhythm * New York Heart Association (NYHA) Class III or IV heart failure * Pulmonary embolism * Significant active cardiovascular or pulmonary disease at the time of study registration, including: * Uncontrolled high blood pressure (i.e., systolic blood pressure \>160 mm Hg, diastolic blood pressure \> 95 mm Hg) * Pulmonary hypertension * Uncontrolled asthma or O2 saturation \< 90% by ABG (Arterial Blood Gas) analysis or pulse oximetry on room air * Significant valvular disease; severe regurgitation or stenosis by imaging independent of symptom control with medical intervention, or history of valve replacement * Medically significant (symptomatic) bradycardia * History of arrhythmia requiring an implantable cardiac defibrillator * Baseline prolongation of the rate-corrected QT interval (QTc) (e.g., repeated demonstration of QTc interval \> 480 milliseconds, or history of congenital long QT syndrome, or torsades de pointes) * Initiation of treatment with hematopoietic growth factors, transfusions of blood and blood products, or systemic corticosteroids (either IV or oral steroids, excluding inhalers) within 1 week prior to study registration (subjects already receiving erythropoietin on a chronic basis for ≥ 28 days are eligible). * Other clinically significant co-morbidities, such as uncontrolled pulmonary disease, active central nervous system disease, active infection, or any other condition that could compromise participation of the subject in the study. * Cirrhosis with Child-Pugh score \> 7 * Variceal bleeding within 1 month prior to study registration. * Refractory encephalopathy or ascites * Known HIV positivity * Hepatitis B surface antigen (HBsAg) positivity without active treatment. A subject found to be HBsAg positive should be on antiviral therapy for at least two weeks prior to study registration. * Treatment with strong inhibitors and/or inducers of cytochrome P450 (CYP) 3A4, CYP2C19 or CYP2C19 within 7 days prior to study registration. * Subjects requiring daily or chronic use of a proton pump inhibitor (PPI) and/or having taken a PPI within 7 days prior to study registration.