Intrinsic Validity of Molecular Marker(s) Detection on Tissular Tumoral DNA to Predict the Efficacy of 177Lutetium-PSMA-617 (Lu-PSMA) Treatment for Castration-resistant Metastatic Prostate Cancer

Inclusion Criteria: * Male \>18 years of age * ECOG ≤ 2 * Patient with histologically confirmed of metastatic castration resistant prostatic adenocarcinoma and with tumor biological material available (prostatic biopsies or prostatectomy) * Patient who received at least one taxane line and a second generation hormone therapy line * Patient receiving androgen deprivation therapy with serum testosterone \< 50 ng/dL or \< 1.7 nmol/L or having undergone surgical castration * Progressive mCRPC based based on at least 1 of the following criteria : * Serum or plasma PSA progression defined as 2 consecutive increases in PSA measured at least 1 week prior. The minimal start value is 2.0 ng/mL ; 1,0 ng/mL is the minimal start value if confirmed increase in PSA is the only indication of progress * Soft-tissue progression by RECIST 1.1 criteria * Progression of bone disease : two new lesions ; only the positivity of bone scan defines metastatic bone disease, according to PCWG3 criteria. * Patients with at least one metastasis, bone and/or soft tissue and/or visceral, documented by the following methods in the 43 days prior to inclusion : * Bone metastasis (regardless of location) highlighted by bone scan AND/OR * Lymph nodes metastasis, regardless of size and location; if the metastasis are only lymph nodes, the short axis of at least one node should be at least 15 mm AND outside the pelvis ; AND/OR * Visceral metastasis, regardless of size and location; a history of visceral metastasis at any time prior to randomization should be encoded as the presence of visceral metastasis at baseline (i.e., a patient with visceral metastasis prior ADT introduction which are disappeared at baseline will be counted as having visceral metastasis and will be considered to have a high tumor volume during stratification) * Patient with Lu-PSMA treatment indication, confirmed by PET 68Ga-PSMA-11. Eligibility for 68Ga-PSMA-11 PET is defined as: * At least one lesion with a binding intensity greater than that of the liver parenchyma (definition of positivity), * All lymph node lesions larger than 25 mm in the short axis must be positive on PSMA PET * All bone metastases with a soft tissue component ≥ 10 mm in the largest diameter must be positive on PSMA-PET * All solid organ metastases (e.g., lung, liver, adrenal glands, etc.) ≥ 10 mm in the largest diameter must be positive on PSMA-PET. * Adequate organ function : * Bone marrow reserve : * Absolute neutrophil count ≥ 1.5 x 10\^9/L * Platelets ≥ 100 x 10\^9/L. * Hemoglobin ≥ 9 g/dL * Hepatic function : * Total bilirubin ≤ 2 x the upper limit of normal (ULN). For participants with known Gilbert's Syndrome ≤ 3 x ULN is permitted. * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) ≤ 3.0 x ULN OR ≤ 5.0 x ULN for patients with liver metastases. * Albumin \> 2.5 g/dL * Renal function : Glomerular Filtration Rate (GFR) ≥ 50 mL/min/1.73m2 according to MDRD equation. * Obtaining the patient's free and informed consent * Social security scheme or beneficiary. Exclusion Criteria : * Continuation of second-generation hormone therapy Patient * Other cancer in the last 3 years likely to change life expectancy or interfere with the assessment of the disease * Protected adult * History of somatic or psychiatric illness/condition that may interfere with study objectives and evaluations * Patient unable to understand and comply with study instructions and requirements * ECOG \> 2 * Dilation of pyelocalicial cavities not previously supported * Obstruction of bladder discharge or uncontrollable and simultaneous urinary incontinence * Symptomatic spinal cord compression or clinical or radiological findings indicating imminent spinal cord compression * Fractured risk of bone damage * Active and symptomatic brain injury * Concurrent participation in a therapeutic trial and administration of any investigational agent within 28 days of inclusion * Metastatic tumor tissue as the only material available for prostate cancer diagnosis * Previous treatment with any of the following in the 6 months prior to inclusion : Strontium-89, Samarium-153, Rhenium-186, Rhenium-188, Radium-223, hemi-cyclic irradiation * Previous treatment with radioligands targeting PSMA * Known hypersensitivity to one of the study treatments or its excipients or similar class drugs * Transfusion or use of bone marrow stimulating agents for the sole purpose of making a participant eligible for inclusion in the study