Appendix
Randomized controlled trials provide the most rigorous bias free method for quantifying lifespan gain
from ICD implantation. However, the duration of randomization in a trial is limited, typically 3-5
years in ICD trials. Even when trials are nominally continued in their randomised allocations, as time
passes there will be a tendency for patients in the control arm to undergo ICD implantation which will
lead to attenuation of later differences between the two arms. For these reasons, it is difficult to obtain
primary data from trials assessing benefit of ICDs over a prolonged period of 10, 20 or 40 years.
Wanting to know lifespan gain over such prolonged periods is increasingly relevant with dramatic
improvements in medical therapy of heart failure, especially if devices are implanted into young
patients.
To explore longer-term lifespan gain therefore requires some element of projection into the post-trial
period. We used a conservative approach to minimise overestimation of benefit. The basis of this
estimation is to recognize that deaths may be from cardiac or non cardiac causes. Even the cardiac
deaths should be considered to be partitioned into deaths which an ICD has the potential to prevent
(sudden cardiac death) and those which it cannot prevent (non sudden cardiac death). The latter
includes deaths from progressive pump failure. Deaths from non cardiac causes, such as cancer or
generalized sepsis, are also very unlikely to be prevented by an ICD and are termed “non cardiac
deaths”.
The projection therefore uses three parallel streams: the first is sudden cardiac death, which is
potentially defibrillator preventable. It is assumed that defibrillators are dramatically effective in
preventing these deaths with a hazard ratio of, for example, 0.41 in the fixed hazard ratio model. The
second is non sudden cardiac death, such as death from progressive pump failure. For these, the hazard
ratio from defibrillators is taken to be 1.0, i.e. the devices have no effect. In order to be conservative,
these two components of mortality are considered not to increase with age. We set the relative balance
between the sudden and non sudden cardiac death components based on clinical trial data from the
Seattle Heart Failure Model (SHFM) data-set15 where the proportion of sudden cardiac death decreased
with increasing baseline mortality (90% at 5% baseline mortality to 75% at 10% baseline mortality)
with a corresponding increase in non-sudden cardiac death with increased baseline mortality.
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The third stream is non cardiac mortality. This we considered to start at a baseline value derived from
the SHFM data set (15%) and then increase progressively with every year of aging. This increased
hazard from non cardiac causes during longitudinal follow up is hard to see within actual clinical trial
data, because the numbers of patients, even in large trials, are insufficient to distinguish this
biologically inevitable trend. It requires country wide data to become evident, for example National
Statistics for the United States of America, which tracks tens of millions of people, observes that the
annual hazard of death increases with each year of ageing by a scale factor of approximately 1.1 fold
per year. We implemented this using a Gompertz-Makeham function13.
Equations used for modelling the post-trial period
The equations for survival gain from this model are described below in a step wise fashion.
Baseline annual hazard is derived from baseline annual mortality as follows:
𝐵𝑎𝑠𝑒𝑙𝑖𝑛𝑒 𝑎𝑛𝑛𝑢𝑎𝑙 ℎ𝑎𝑧𝑎𝑟𝑑 = −log(1 − 𝑏𝑎𝑠𝑒𝑙𝑖𝑛𝑒 𝑎𝑛𝑛𝑢𝑎𝑙 𝑚𝑜𝑟𝑡𝑎𝑙𝑖𝑡𝑦)
This baseline hazard is the sum of the hazard for each of the three streams:
𝐵𝑎𝑠𝑒𝑙𝑖𝑛𝑒 𝑎𝑛𝑛𝑢𝑎𝑙 ℎ𝑎𝑧𝑎𝑟𝑑 = 𝑠𝑢𝑑𝑑𝑒𝑛 𝑐𝑎𝑟𝑑𝑖𝑎𝑐 𝑑𝑒𝑎𝑡ℎ ℎ𝑎𝑧𝑎𝑟𝑑 + 𝑛𝑜𝑛 𝑠𝑢𝑑𝑑𝑒𝑛 𝑐𝑎𝑟𝑑𝑖𝑎𝑐 𝑑𝑒𝑎𝑡ℎ ℎ𝑎𝑧𝑎𝑟𝑑 + 𝑛𝑜𝑛 𝑐𝑎𝑟𝑑𝑖𝑎𝑐 𝑑𝑒𝑎𝑡ℎ ℎ𝑎𝑧𝑎𝑟𝑑
At baseline, 15% of the annual hazard is in all groups considered to be non cardiac. The remaining
85% is allocated between sudden cardiac and non sudden cardiac, according to the findings of the
SHFM data.
For example, a group with a 5% annual baseline mortality approximately 85% x 5% = 4.25% mortality
is considered cardiac and this is partitioned 90:10 between sudden and non sudden since this is the
partition found in the SHFM data when the annual risk is 5%. As the baseline mortality increases, the
proportion changes so that at a 10% baseline mortality, approximately 85%x10%=8.5% mortality is
considered cardiac and this is partitioned 75:25 between sudden and non sudden.
For non cardiac mortality, the hazard increases with time from device implantation using the scale
factor derived from the National Statistics for the United States of America (1.1 fold per year).
Therefore, the hazard of non cardiac mortality at time t is:
𝑛𝑜𝑛 𝑐𝑎𝑟𝑑𝑖𝑎𝑐 ℎ𝑎𝑧𝑎𝑟𝑑 = 𝑏𝑎𝑠𝑒𝑙𝑖𝑛𝑒 𝑎𝑛𝑛𝑢𝑎𝑙 𝑛𝑜𝑛 𝑐𝑎𝑟𝑑𝑖𝑎𝑐 ℎ𝑎𝑧𝑎𝑟𝑑 × 𝑒 𝑎𝑡
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Where ea is the annual scale factor in non cardiac hazard, which is 1.1.
The ICD was considered to decrease the hazard of sudden cardiac death. This scale factor is
represented byR, the hazard ratio. For the fixed hazard ratio model this was 0.41, which was the
weighted mean we obtained from trial data. For the baseline adjusted hazard ratio model, R was
adjusted to match the findings of various strata of baseline mortality in SCD-HeFT16. Thus for the
patients in the ICD arm, the sudden cardiac hazard will decrease by a scale factor of R, the hazard ratio.
The annual mortality at time t can therefore be calculated from the hazard from each of the streams at
time t as follows:
𝐴𝑛𝑛𝑢𝑎𝑙 𝑚𝑜𝑟𝑡𝑎𝑙𝑖𝑡𝑦 𝑖𝑛 𝐼𝐶𝐷 𝑎𝑟𝑚 = 1 − 𝑒 −(𝑹 ×𝑠𝑢𝑑𝑑𝑒𝑛 𝑐𝑎𝑟𝑑𝑖𝑎𝑐 ℎ𝑎𝑧𝑎𝑟𝑑+ 𝑛𝑜𝑛 𝑠𝑢𝑑𝑑𝑒𝑛 𝑐𝑎𝑟𝑑𝑖𝑎𝑐 ℎ𝑎𝑧𝑎𝑟𝑑+𝑛𝑜𝑛 𝑐𝑎𝑟𝑑𝑖𝑎𝑐 ℎ𝑎𝑧𝑎𝑟𝑑)
And in the non-ICD arm as:
𝐴𝑛𝑛𝑢𝑎𝑙 𝑚𝑜𝑟𝑡𝑎𝑙𝑖𝑡𝑦 𝑖𝑛 𝑛𝑜 𝐼𝐶𝐷 𝑎𝑟𝑚 = 1 − 𝑒 −(𝑠𝑢𝑑𝑑𝑒𝑛 𝑐𝑎𝑟𝑑𝑖𝑎𝑐 ℎ𝑎𝑧𝑎𝑟𝑑+ 𝑛𝑜𝑛 𝑠𝑢𝑑𝑑𝑒𝑛 𝑐𝑎𝑟𝑑𝑖𝑎𝑐 ℎ𝑎𝑧𝑎𝑟𝑑+𝑛𝑜𝑛 𝑐𝑎𝑟𝑑𝑖𝑎𝑐 ℎ𝑎𝑧𝑎𝑟𝑑)
The Kaplan-Meier curve is calculated from the annual mortality as follows, where t is time in years:
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑎𝑙𝑖𝑣𝑒 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑝𝑜𝑖𝑛𝑡 𝑡 = 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑎𝑙𝑖𝑣𝑒 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑝𝑜𝑖𝑛𝑡 (𝑡 − 1) × (1 − 𝑎𝑛𝑛𝑢𝑎𝑙 𝑚𝑜𝑟𝑡𝑎𝑙𝑖𝑡𝑦)
Using this data, a Kaplan-Meier curve may be constructed for the ICD and the non-ICD arms as
follows:
Supplemental Figure 1: A Kaplan Meier curve for post-trial survival. The lifespan gain is the grey
shaded area and may be calculated as follows:
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𝑇
𝑇
∫ (𝐼𝐶𝐷 𝑎𝑟𝑚)𝑑𝑡 − ∫ (𝑁𝑜 𝐼𝐶𝐷 𝑎𝑟𝑚)𝑑𝑡
0
0
4