Aperiodicolleague Kit Yates has recently had a new book out: The Maths of Life and Death. He’s kindly agreed to share a sample chapter with us, explaining the God Equation: it’s used by NICE to decide whether to fund new drugs.

In my new book, The Maths of Life and Death, I explore the true stories of life-changing events in which the application (or misapplication) of mathematics has played a critical role: patients crippled by faulty genes and entrepreneurs bankrupt by faulty algorithms; innocent victims of miscarriages of justice and the unwitting victims of software glitches. I follow stories of investors who have lost fortunes and parents who have lost children, all because of mathematical misunderstanding. I wrestle with ethical dilemmas from screening to statistical subterfuge and examine pertinent societal issues such as political referenda, disease prevention, criminal justice and artificial intelligence. I aim to demonstrate that mathematics has something profound or significant to say on all of these subjects, and more.

Rather than just pointing out the places in which maths might crop up, I also try to arm my readers with simple mathematical rules and tools which can help them in their everyday life: from getting the best seat on the train, to keeping one’s head when on the receiving end of an unexpected test result from the doctor. I suggest simple ways to avoid making numerical mistakes and get my hands dirty with newsprint when untangling the figures behind the headlines. I also get up close and personal with the maths behind consumer genetics and display maths in action as I highlight the steps we can all be taking to help halt the spread of deadly diseases.

The God Equation

In November 2016, Daniella and John Else’s 14-month-old son, Rudi, was rushed to the Sheffield Children’s Hospital. He was hooked up to a ventilator to keep him breathing, with doctors telling Daniella and John that Rudi might not last the night. The cause of the alarm was a common chest infection that most children would fight off. Most children, however, do not have Spinal Muscular Atrophy (SMA).

When Rudi was six months old, after doctors had failed to figure out what was wrong with him, Daniella and John helped to diagnose their son with SMA after finding out that John’s cousin had been afflicted by the same disorder. Rudi’s type of the progressive muscle wasting disease comes with a life expectancy of just two years. Miraculously, there is a drug, Spinraza, developed by a company called Biogen, which can halt and even reverse some of the debilitating effects of SMA. This drug has the potential to improve and extend the lives of SMA sufferers like Rudi, but in England in 2016, when Rudi was fighting for his life in hospital, it was not available free of charge.

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Mathematics is becoming an increasing important tool for modern healthcare provision. The extent of mathematical interventions range from simple fractions for calculating drug doses to complex algorithms for reconstructing images from CAT scans. In UK healthcare, there is perhaps one formula that stands out above all others in its contentiousness, importance and wide-ranging implications. The ‘God equation’ dictates which new drugs will be paid for by the NHS: it literally determines who lives and who dies. If you have a child who is terminally ill, you might argue that no price is too high to pay in order to buy you some more time with your little one. The ‘God equation’ says otherwise.

In theory, in the US, as soon as the FDA approves a drug for sale it is made available to patients. Spinraza was approved by the FDA in December 2016. In practice, most insurance companies have a ‘prior authorization’ list for expensive or potentially risky drugs. For each treatment, the list stipulates a range of conditions that must be met before it is released for a particular patient. Spinraza is on every insurance company’s prior authorization list. Of course, access to healthcare in the US also depends on being able to afford medical insurance. In 2017, 12.2% of Americans were uninsured, and the US remains the only industrialised nation without universal healthcare coverage.

By contrast, in England, healthcare is available to everyone, free at the point of use, and largely paid for by general taxation. The European Medicines Agency (EMA) and the Medicines and Healthcare Products Regulatory Agency are responsible for approving the safety and efficacy of drugs in England. In May 2017, the EMA approved Spinraza for use. However, since the NHS has only a limited budget, it is not able to sanction every new treatment that comes on the market. Decisions taken one way or another might, for example, lead to cuts in social care provision, lack of diagnostic or treatment equipment for cancer patients, or understaffing of neonatal care units. The National Institute for Clinical Healthcare Excellence (NICE) is the body responsible for making these tough choices. When it comes to drugs, there is a well-established formula by which NICE ensures that its decisions are objective.

The God equation attempts to balance how much extra ‘health benefit’ a drug gives to a patient against how much extra the NHS is being asked to pay for it. Assessing the former is a difficult task. How can one compare the advantages of a drug that reduces the incidence of heart disease, for example, against the benefits of a drug that prolongs the life of a cancer patient?

NICE uses a common benchmark known as a quality-adjusted life year or QALY. When comparing a new treatment to the existing therapy, the QALY accounts not just for how much a drug might extend life, but also the quality of life that it affords. A single QALY might result from a cancer drug that extends life by two years, but leaves patients in only 50% health, or it might result from knee replacement surgery, which does nothing to extend a patient’s remaining 10-year life expectancy but improves their quality of life by 10%. Successful treatment of testicular cancer might garner a large number of QALYs, as the typically young patients have a dramatically extended life expectancy without a reduction in their quality of life.

Once a reliable figure for QALYs has been established, the difference in QALYs and the change in costs between the new and old treatments can be compared. If the QALYs decrease, then the new treatment will be rejected out of hand. If the QALYs increase and the cost decreases then clearly funding a more effective, cheaper new treatment is a no-brainer. However, if, as is most often the case, both the QALYs and the cost increase, NICE is left with a decision to make. In these cases the incremental cost-effectiveness ratio (ICER) is calculated by dividing the increase in QALYs by the increase in cost. The ICER tells us the increased cost per QALY gained. Typically, NICE set their threshold for the maximum ICER they will fund to be between £20,000 and £30,000 per QALY.

US policy-makers are not bound by such cost-effectiveness analyses. Indeed, the 2010 Patient Protection and Affordable Care act, more commonly known as Obamacare, prohibits the federal government from using a dollars-per-QALY threshold . Nevertheless, QALYs are instrumental in US healthcare judgements, with the US Panel on Cost-Effectiveness in Health and Medicine endorsing their use . QALYs are also used informally by decision-makers in the US in order to determine whether investments represent value for money. $50,000 per QALY is often cited as the threshold below which ‘favourable interventions’ are approved .

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In August 2018, SMA sufferers and their families, including Daniella, John and Rudi, waited anxiously to find out whether NICE would sanction Spinraza for use on the NHS. NICE recognised that Spinraza ‘provides important health benefits’ for patients with SMA. The results of quality of life improvement were also extremely positive. Spinraza was expected to generate an additional 5.29 QALYs. The additional cost, however, ran to an enormous £2,160,048, giving an ICER of over £400,000 per QALY gained: way, way above NICE’s threshold. Despite compelling testimony from SMA sufferers and the carers of SMA patients, the God equation meant that the only option was to prohibit the use of Spinraza on the NHS.

Fortunately for the Else family, Rudi is enrolled in an expanded access programme run by manufacturer Biogen, allowing infants with type 1 SMA to receive the drug. In February 2019, he received his 10th injection and is now a thriving thee-year-old, far exceeding the Spinraza-less life expectancy of type 1 SMA sufferers. In November 2018, however, the programme was closed to new participants. There is still no long term agreement for the NHS to fund Spinraza, a life-saving and life-extending drug, to treat SMA sufferers in England. Sometimes, even relatively simple mathematics can be a matter of life and death.