The blood tests don't actually show cancer cells, they pick up proteins that our body sometimes produces when you have cancer. They are notoriously unreliable as other conditions can also trigger things the tests will pick up. I've had elevated markers when I had the flue. This shit is not helpful.
Ten years ago they were very popular but less so now--particulary as a diagnostic tool. The tests can be really usful when monitoring people who have active disease as they give an indication about how treatments are working.
My onc, bless him, says 'No, no. No blood test. They make everybody crazy.'