Using vorinostat and a radioactive treatment for low PSMA metastatic prostate cancer

Inclusion Criteria:

* Documented histologically confirmed adenocarcinoma of the prostate.
* Patient must have evidence of castration resistant prostate cancer as evidenced by PSA progression (per Prostate Cancer Working Group 3 \[PCWG3\] criteria) and a castrate serum testosterone level (i.e., ≤ 50 mg/dL).
* PSMA SUVmean \< 10 as determined by 68Ga-PSMA-11 PET.
* Patients must have received a next-generation androgen receptor-signaling inhibitor (e.g. abiraterone, enzalutamide, apalutamide, darolutamide). There must be at least a 2-week washout period after stopping these agents. Patients should be weaned off steroids at least 1 week prior to starting treatment.
* Eastern Cooperative Oncology Group (ECOG) performance status 0-2.
* At least one lesion (measurable and/or non-measurable) that can be accurately assessed at baseline by CT and/or bone scan and is suitable for repeated assessment.
* Hemoglobin ≥ 10 g/dL (measured within 28 days prior to administration of study treatment)
* Absolute neutrophil count (ANC) ≥ 1.5 x 10\^9/L (measured within 28 days prior to administration of study treatment)
* Platelet count ≥ 100 x 10\^9/L (measured within 28 days prior to administration of study treatment)
* Total bilirubin ≤ 1.5 x institutional upper limit of normal (ULN) (measured within 28 days prior to administration of study treatment)
* Aspartate aminotransferase (AST) and alanine transaminase (ALT) ≤ 2.5 x ULN, unless liver metastases are present in which case they must be ≤ 5 x ULN (measured within 28 days prior to administration of study treatment) . For patients with known Gilbert's Syndrome they must be ≤ 3 x ULN (measured within 28 days prior to administration of study treatment)
* Calculated creatinine clearance ≥ 50 mL/min (using Cockcroft-Gault formula) (measured within 28 days prior to administration of study treatment)
* Patients and their partners, who are sexually active and of childbearing potential must agree to the use of two highly effective forms of contraception in combination throughout the period of taking study treatment and for 3 months after last dose of study drug to prevent pregnancy in a partner.
* Patient is willing and able to comply with the protocol for the duration of the study including undergoing treatment and scheduled visits and examinations.

Exclusion Criteria:

* Evidence of serious and/or unstable pre-existing medical, psychiatric or other condition that could interfere with patient safety or provision of informed consent to participate in this study.
* Evidence of metastatic neuroendocrine/small cell prostate cancer (NEPC). Note: baseline biopsy is not required but is strongly encouraged if a patient is found to have an FDG-positive/PSMA-negative lesion on baseline imaging.
* Patients receiving any systemic therapy (aside from an luteinizing hormone-releasing hormone \[LHRH\] analogue) or radiotherapy within 2 weeks prior to study treatment.
* Any previous treatment with an HDAC inhibitor (including valproic acid) or 177Lu-PSMA-617.
* Persistent toxicities (CTCAE grade \>2) from prior cancer therapy, excluding alopecia and stable neuropathy.
* Patients considered a poor medical risk due to a serious, uncontrolled medical disorder or active, uncontrolled infection. Examples include, but are not limited to uncontrolled seizure disorder, unstable spinal cord compression, superior vena cava syndrome, or any psychiatric disorder that prohibits obtaining informed consent.
* Patients who are known to be serologically positive for human immunodeficiency virus (HIV) and a CD4 count \< 200.
* Patients with known active hepatitis (i.e. Hepatitis B or C). Prior Hep C infection is allowed as long as polymerase chain reaction (PCR) is negative.
* Major surgery within 2 weeks of starting study treatment and patients must have recovered from any effects of any major surgery.
* Patients unable to swallow orally administered medication and patients with gastrointestinal disorders likely to interfere with absorption of the study medication.
* Deep vein thrombosis or pulmonary embolism diagnosed within the past six months.
* Active use of coumarin-derived anticoagulant medication (i.e. warfarin).
* Serious cardiac disorder, including but not limited to uncontrolled ventricular arrhythmia, recent (within 12 months) myocardial infarction, resting electrocardiogram (ECG) indicating Fridericia's corrected QT interval prolongation \> 500ms, or congenital long QT syndrome.