Final answer:
The structure that thickens to regulate urine flow from the bladder is the internal urethral sphincter, which works with the external urethral sphincter to control urination.
Step-by-step explanation:
The structure formed by the thickening of the smooth muscle cell wall surrounding the bladder that regulates the flow of urine from the bladder is the internal urethral sphincter. The internal urethral sphincter is made of smooth muscle at the juncture of the bladder and urethra, functioning to retain urine in the bladder by staying contracted and to allow passage into the urethra by relaxing as the bladder fills. This sphincter, along with the external urethral sphincter, is crucial for urinary continence and the control of urination.
The external urethral sphincter is under voluntary control, allowing the individual to consciously hold in urine until they are ready to urinate. Conversely, the detrusor muscle in the bladder wall contracts in response to the autonomic nervous system to expel urine when voiding. However, the key player in regulating urine flow from the bladder is the internal urethral sphincter, making it the final answer for this question.