A Study of BXQ-350 in Children With Newly Diagnosed Diffuse Intrinsic Pontine Glioma (DIPG) or Diffuse Midline Glioma (DMG)

Inclusion Criteria: Each subject must meet the following criteria: 1. Provide signed, written informed consent prior to the initiation of any study-specific procedures (consent from guardians for minor children and patient assent according to institution and Institutional Review Board (IRB) standards) 2. Age ≥ 1 year of age and ≤ 30 years of age at the time of study entry 3. Have radiographically suspected or histologically confirmed newly diagnosed DIPG or DMG with the following defining disease characteristics/features: * Part 1 and Part 2: * DIPG: radiographic imaging showing a diffuse expansion of the brainstem/pons by a tumor that is poorly defined but abnormally bright in signal on T2-weighted images and abnormally dark on T1-weighted images; contrast enhancement, when present, is typically mild; there may be some indication of necrosis, and the basilar artery is commonly engulfed by tumor. * DMG: the same imaging characteristics as noted above for DIPG are present, but the lesion can extend superiorly up into the midbrain and thalami, and/or inferiorly to the medulla. * In cases of disease that is not classic for DIPG or DMG, biopsy may be necessary in order to obtain histological confirmation of eligibility. * Part 2: newly-diagnosed DIPG or DMG planning to undergo biopsy at the recommendation of the treating physician; to be considered evaluable for PK tissue concentration analysis, tumor samples must have at least 50% viable tumor cells with \<30% necrosis or hemorrhage as determined by the study neuropathologist 4. If receiving glucocorticoid therapy: dexamethasone allowed with maximum allowable dose 16mg/day 5. Have measurable or non-measurable disease per RANO criteria 6. Have Lansky (age 1 - 15) / Karnofsky (age ≥ 16) Performance Score of ≥ 50% or Eastern Cooperative Oncology Group (ECOG) Performance Status (age ≥ 18) of 0 - 2 * Subjects unable to walk because of paralysis, but who can sit without assistance/restraint and control a wheelchair, will be considered ambulatory for the purpose of assessing the performance score 7. Have acceptable liver function defined as: * Total serum bilirubin ≤ 1.5 x upper limit of normal for the study site (ULN) (in subjects with known Gilbert Syndrome, total bilirubin ≤ 3 x ULN, with direct bilirubin ≤ 1.5 x ULN) * Aspartate Transaminase (AST), Serum Glutamic Oxaloacetic Transaminase (SGOT), Alanine Transaminase (ALT), Serum Glutamic-Pyruvic Transamine (SGPT) ≤ 3 x ULN (if liver metastases are present, then ≤ 5 x ULN is allowed) * Serum albumin ≥ 3 grams/deciliter (g/dL) 8. Have acceptable renal function defined as: * Creatinine clearance or radioisotope glomerular filtration rate (GFR) ≥70 milliliters (mL)/min/1.73 meters squared (m2) or a serum creatinine based on age/gender as follows: 1 to \< 2 years: 0.6 (male); 0.6 (female) 2 to \< 6 years: 0.8 (male); 0.8 (female) 6 to \< 10 years: 1 (male); 1 (female) 10 to \< 13 years: 1.2 (male); 1.2 (female) 13 to \< 16 years: 1.5 (male); 1.4 (female) * 16 years: 1.7 (male);1.4 (female) * Threshold creatinine values derived from the Schwartz formula for estimating GFR utilizing child length and stature data published by the Center for Disease Control (Schwartz 2009) 9. Have acceptable bone marrow function defined as: * Absolute neutrophil count (ANC) ≥ 1,500 cells/ cubic millimeter (mm3) * Platelet count ≥ 100,000 cells/mm3 (unsupported, no transfusion within 7 days of enrollment) * Hemoglobin \> 9.0 g/dL (unsupported, no transfusion within 7 days of enrollment) 10. Have acceptable coagulation parameters (anti-coagulation allowed) defined as: * International normalized ratio (INR) ≤ 2 x ULN unless on anticoagulation or prothrombin time (PT) within normal limits * Activated partial thromboplastin time (aPTT) within normal limits 11. Have a negative serum pregnancy test result at screening (for females of child bearing potential (FCBP); not applicable to subjects who are unable to become pregnant, including those with tubal ligation, bilateral oophorectomy and/or hysterectomy, post-menopausal is defined as \> 12 months since last menstrual cycle) 12. FCBP and male subjects whose sexual partner(s) are FCBP must agree to abstain from heterosexual activity or use a double barrier method of contraception (e.g., condom and occlusive cap with spermicide) or highly effective contraception (intrauterine device or system, established hormonal contraceptive methods on a stable dose from the time of the last menstrual cycle, or vasectomized partner with confirmed azoospermia) from the time of study entry to 1 month after the last day of treatment Exclusion Criteria: Subjects must not meet any of the following criteria: 1. Have a concurrent or secondary malignancy 2. Have a low-grade glioma (Grade II or less) 3. Have received prior treatment with radiation, chemotherapy, anti-neoplastic, or investigational agents 4. Have had major surgery other than a minor outpatient procedure within 28 days prior to dose assignment or have not recovered from major side effects of the surgery if more than 4 weeks have elapsed since surgery 5. Have poorly controlled hypertension despite the use of antihypertensive agents defined as blood pressure ≥95th percentile for age and weight or \>160/90 on at least 2 repeated determinations on separate days within 2 weeks (14 days) prior to initiation of screening 6. Have a history of cardiac dysfunction including: * myocardial infarction within 6 months prior to initiation of screening * history of documented congestive heart failure (New York Heart Association functional classification III-IV) within 6 months prior to initiation of screening * active cardiomyopathy * electrocardiogram (ECG) with QTc \>470 msec at screening * echocardiogram with ejection fraction \<50% or a decrease in the left ventricular shortening fraction to \<27% 7. Have a known history of Human Immunodeficiency Virus (HIV) seropositivity 8. Have active (acute or chronic) or uncontrolled severe infections 9. Have active poor wound healing (delayed healing, wound infection or fistula) 10. Have evidence of active clinically significant bleed (e.g., gastrointestinal bleed, hemoptysis, or gross hematuria) at screening 11. Are pregnant or nursing, where pregnancy is defined as the state of a female after conception and until the termination of gestation, confirmed by a positive serum human chorionic gonadotropin (hCG) laboratory test 12. Have a known sensitivity to any component of BXQ-350 13. Have other concurrent severe and/or uncontrolled medical condition that would, in the Principal Investigator's judgment contraindicate the subject's participation in the clinical study or obscure assessment of toxicity