Consider this equation.

Early Diagnosis = Early Diagnosis + Overdiagnosis (1.1)

This sort of unequal algebra will fail GCSE mathematics. A new NHS initiative is arithmetic defying as well. Patients who think they have symptoms of cancer will be allowed to book medical imaging directly, without seeing their GP. This is to catch cancer early. The logic is impenetrable: early diagnosis of cancer saves lives.

Here is the problem. Cancer does not unequivocally announce its arrival. Early cancer presents with non-specific symptoms, such as an uncomfortable niggle in the back.

Tom

Let’s take Tom, who has advanced pancreatic cancer. He recalls that three years earlier he noticed a dull pain in his back during a misguided drinking binge. He would be correct in thinking that had he attended the emergency department and had a CAT scan of his abdomen, the cancer would have been smaller and would not have spread to other organs. He is right in contending that had the cancer been removed then, he would have a longer survival than presently.

The rationale has implications if extrapolated to everyone. To understand the consequences of extrapolation let’s visit a logical fallacy.

1) All Mr Smiths are over six feet tall.

2) He is above six feet tall so he must be Mr Smith.

This is affirming the consequent. Not all men above six feet in height are Mr Smiths. In fact, most are not.

3) Early pancreatic cancer presents with back pain.

4) All patients with back pain have early pancreatic cancer.

Similarly, (3) doesn’t imply (4).

Cancer often presents with non-specific symptoms, such as a vague discomfort, early on. But the majority of people with a vague discomfort do not have cancer. That is, the chance that someone with pancreatic cancer has dull back pain should not be confused with the chances that someone with dull back pain has pancreatic cancer. The chances of the latter are much lower than the former.

To pick up Tom with back pain who has pancreatic cancer, we will image many Toms with back pain who don’t have pancreatic cancer. Because we don’t know which Tom with back pain has cancer and which Tom does not have cancer.

Dick

In the search for a Tom with back pain with pancreatic cancer, we will pick up a Dick with back pain who does not have pancreatic cancer. Dick has a small kidney tumour. The tumour was minding its business and would not have caused Dick any issues in his life. But we will now remove Dick’s kidney to deal with an incidental finding, which has nothing to do with his symptoms. Thus, in order to save Tom from pancreatic cancer, Dick loses a kidney. This is collateral damage. Tom’s early diagnosis leads to Dick’s overtreatment.

Harry

We will also pick up a Harry with back pain and a small cyst in his pancreas, which may or may not be cancer. We will then scan Harry repeatedly, perhaps every six months, to make sure that it really is a harmless cyst. It probably is, but you can never be too careful.

To save Tom from pancreatic cancer, we end up wasting Harry’s time by CAT scanning him every six months, driving his anxiety off the scale. Because when the doctor said “probably not cancer,” Harry heard “cancer.” Harry is a false positive. He won’t know of the “false” bit for a very long time.

The equation can be modified to:

Early diagnosis = Early Diagnosis + Overdiagnosis + False Positives (1.2)

Tom Jones

A flamboyant Tom Jones also has back pain. His CAT scan fails to reveal the cancer that presents later. CAT scans are good. Very good. But they are not perfect. Tom Jones is very peeved because he believed the “early diagnosis saves life” mantra. As far as he is concerned, he did his bit. He complied.

But we lied. Tom Jones is a false negative.

The equation becomes:

Early Diagnosis = Early Diagnosis + Overdiagnosis + False Positive + False Negative (1.3)

The right side of the equation has costs. Opportunity costs. There will be less money for other services, such as emergency departments.

Tom, Dick, Harry, and Tom Jones

Tom, Dick, Harry, and Tom Jones will Google “back pain + cancer.” The search will yield “pancreatic cancer.” After a while this will become a self fulfilling prophecy, thanks to Google’s ranking algorithms, which do not distinguish between the incidence of and anxiety from cancer.

Del Trotter

When Tom et al can’t get a CAT scan within 48 hours, they will throng the emergency department because of anxiety. This will affect the care of Del Trotter from underserved Peckham. Del is having a heart attack. Del will die if his left anterior descending artery is not promptly opened. Del’s doctors are being distracted by the demanding Toms.

Britain’s NHS has problems—structural problems, through systematic under investment. Emergency departments and intensive care units are in crisis. Is the pursuit of early diagnosis of cancer really its highest priority? Why are politicians promising cake when there is not enough bread?

Actually, this pursuit is rewarding. Why? Because the public will be grateful. Tom is happy that his pancreatic cancer was detected early. The lucky Toms, the true negatives, are even happier that they don’t have cancer. Dick is ecstatic that his kidney tumour was taken out. He does not know the whole exercise was an utter waste of his time and an innocent kidney was sent to the gallows (surgical pathology). He does not know that he was overdiagnosed and overtreated.

Harry, the false positive, is relieved. After several surveillance CAT scans, his doctors broke the news that the pancreatic lesion was a false alarm. He is not only relieved but prostrating with gratitude. He is suffering from a medical variant of the Stockholm syndrome. Del Trotter from Peckham, the opportunity cost, has other things on his mind. This leaves Tom Jones. He is annoyed. But a root cause analysis, with considerable assistance of hindsight, will have found an element missing from the diagnostic pathway. His anger has been redirected to the imperfect medical profession, which refuses to adequately self-regulate.

We can show:

Early Diagnosis = Early Diagnosis + Overdiagnosis + False Positives + False Negative + Opportunity Costs + Votes ++++ (1.4)

Quod Erat Demonstrandum (“that which was to be proved”).

If you were a politician, would you honestly see a downside to this?

Competing interests: I declare that I have read and understood BMJ policy on declaration of interests and I would like to declare the following: I have received a speaker’s fee from Toshiba.

Saurabh Jha is an assistant professor of radiology at the University of Pennsylvania School of Medicine. Follow him on Twitter @RogueRad