**Core Concept**
The patient presents with urinary retention and back pain, suggesting a potential diagnosis of benign prostatic hyperplasia (BPH) or a more serious condition like prostate cancer, which can cause obstructive urinary symptoms and referred pain in the back.

**Why the Correct Answer is Right**
The correct answer is a transrectal ultrasound (TRUS) of the prostate. This investigation is essential in evaluating the prostate gland for any abnormalities, such as an enlarged prostate or a palpable nodule, which could be suggestive of prostate cancer. TRUS also allows for the measurement of the prostate-specific antigen (PSA) level, which is a critical marker for prostate cancer. Additionally, TRUS-guided biopsy can be performed to obtain tissue samples for histopathological examination.

**Why Each Wrong Option is Incorrect**
**Option A:** A plain radiograph of the spine may show vertebral fractures or spinal stenosis, but it is not the primary investigation for urinary retention and back pain in this context.
**Option B:** A CT scan of the abdomen and pelvis may be useful in evaluating the retroperitoneal space or assessing for any potential complications, but it is not the initial investigation of choice for prostate-related symptoms.
**Option C:** A urine flow study may be helpful in assessing the urinary flow patterns, but it is not a primary diagnostic tool for prostate-related symptoms.
**Option D:** A cystoscopy may be useful in evaluating the urethra and bladder, but it is not the initial investigation of choice for prostate-related symptoms.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that prostate cancer can present with symptoms similar to BPH, and a high index of suspicion is necessary, especially in older men with persistent urinary retention and back pain.

**Correct Answer:** C.