**Study Overview:** This study is testing a new injection called **177Lu-rhPSMA-10.1** for men with a type of prostate cancer that has spread and no longer responds to hormone therapy. The study will check how safe it is, how well it works, and the correct dose. **Key Points:** - **Study Length and Design:** The study has two parts. Phase 1 finds the dose and checks safety, Phase 2 tests how well the treatment works. It includes people who have or have not had certain chemotherapies. - **Eligibility:** Participants must be men aged 18 or older with prostate cancer that shows a specific marker (PSMA). They should have tried other treatments before and meet specific health criteria. - **Risks and Requirements:** Participants should not have severe allergies to the treatment, certain types of cancer, or serious health problems. They also shouldn't have participated in other studies recently. **Participation Considerations:** The study requires multiple visits and tests to evaluate safety and effectiveness. Participants must ensure they meet all health criteria and be prepared for the potential side effects and demands of the study.