Joe Collier on: When does “1+1+1 = 1” become “1+1+1 = 3?”

Professor Joe Collier A key component of medicine is the diagnosis. The principle process we use for reaching a diagnosis is to identify the patient’s signs and symptoms and then look for a unifying explanation. Based on the explanation, we then label the disease. So, where an illness has three key features (viz oedema, albuminuria and hypoalbuminemia as in nephrotic syndrome), the unifying process can expressed mathematically as 1 (oedema) +1 (albuminuria) +1 hypoalbuminemia) = 1 (nephrotic syndrome), or more simply 1+1+1=1. 

Sometimes unification is a little more complex but the principle remains the same. In the schema for diagnosing rheumatoid arthritis devised by the American College of Rheumatology, the diagnosis is made when four of seven stipulated criteria are present (morning stiffness in and around joints, lasting more than 1 hour; arthritis of three or more joint areas involved simultaneously; arthritis of at least one area in a wrist, metacarpal or proximal interphalangeal joint; symetrical arthritis involving the same joint areas; rheumatoid nodules; positive serum rheumatoid factor;  radiological changes typical of RA on hand and wrist x-rays.). But again the sum comes to “1.”  Mathematically it gets slightly more complicated where some symptoms must be present (major, or key, symptoms) while others are not essential (minor symptoms), but the same equation holds 1+1+1 = 1. To all intents and purposes the number of 1’s is probably limitless.

Underlying such a unification theory is an assumption that all of the patient’s symptoms must be accounted for. However, while this is possible, indeed likely, in an otherwise healthy young person, things change as we get older. From a recent UK report on health and disease in the older old (Collerton et al BMJ 2009;339:b4904), we learn that men aged 85 live with an average of 4 diseases and women with 5.

So, as a minimum, when a new illness develops, men for instance start from a baseline of 1+1+1+1 = 4. And if one has to add in a further two features that have resulted from unwanted effects from medicines, then the baseline position for a new patient episode might be bolstered to 1+1+1+1+1+1 = 6.

Moreover, some of the signs or symptoms, such as those relating, for example, to raised blood pressure, atrial fibrillation or peripheral vascular disease, could well have been previously unknown to the patient.
The implications of the shift from unification (1+1+1 = 1) to summation (1+1+1 +1= 4), is not trivial as it could determine the mindset of clinicians and influence the extent to which they should look (or screen) for multiple as apposed to single pathologies.  And, of course, the shift will not only be determined by aging, in addition it is bound be influenced by gender, ethnicity and also probably class (wealth). Understanding the implications of such changes could well help to make diagnoses more reliable.  It is now over to the mathematicians amongst us to analyse (and quantify) this effect and provide workable equations to tell us how far we should look for multiple pathologies in any one clinical situation. 

Joe Collier is emeritus professor of medicines policy at St George’s, University of London

  • The mathematical ‘unification’ or ‘summation’ methodology may be helpful in the diagnosis of many “biomedical” disorders such as those explained in this article; but in the case of “bio-psychosocial” disorders such as behavior or mood or personality disorders, the diagnostic mathematical unification / summation quantum can be sizable.
    On the other hand certain disorders can be diagnosed on clinical history alone in spite of negative lab work up – e.g. migraine headache.
    So although the diagnosis of a disorder may be assisted by a ‘scoring’system, its a vary variable factor to depend on, especially in chronic, psycho-somatic, and elderly population.