Figure 2. Effect of different half-lives on percent of patients with symptoms on various days, showing weighted average durations (arrows).
The effect of a hypothetical zinc and placebo treatment having different half-lives on percentage of patients during first week of treatment and projected values beyond the week of treatment are shown in Figure 2. Consider the situation where 50 percent of patients are well by day 2.2 with zinc treatment and by day 7 with placebo; 75 percent are well by day 4.4 with zinc treatment, and by day 14 with placebo; 86.5 percent are well by day 6.6 with zinc treatment, and by day 21 with placebo. Using half-life theory, the expected number of patients recovered can be projected beyond duration of studies. Estimates after the week of treatment are predicated upon half-life of colds continuing to decay at same rate as during the week of treatment.
Related to the half-life of common colds is the average duration. Although average duration is not the same as half-life, they are frequently confused with each other. The average duration is equal to the number of days colds persist, where the sum of days is taken over the collection of patients, divided by total number of patients. The average duration is mathematically related to half-life (H) of common colds. For common colds decaying at a set exponential rate, average duration of common colds is provided by the mathematical expression where N is initial number of patients, and H is half-life of colds observed in study group.
The expression simplifies to H/ln 2, and ln 2 equals 0.6931. Therefore, once the half life (H) is determined and the decay rate is found to be exponential, the average duration can be directly determined. For example, zinc-treated colds having a half life of 2.2 days have an average duration of 3.2 days, and placebo-treated colds with a half-life of 7 days have an average duration of 10 days. Arrows in Figure 2 show weighted average duration. Differences in weighted average duration between treated and untreated colds directly follow. This method is not reliable for colds not decaying at an exponential rate.
Other methods of determining average duration must be used for non-exponentially decaying colds, which would involve observing the duration of each cold, an arduous task for placebo-treated colds. In the case of nonexponentially decaying colds, half-life analysis and comparison of decay rates (plotted as the number of colds remaining on each day of the study) are probably sufficient.
The above method of determining half-life and estimating average durations for exponentially decaying common colds should be adopted as the favored means to measure effects of zinc lozenges on shortening the duration of common colds. Half-life and average duration analyses may be used to supplement all other methods of measuring effects of treatments including mean clinical scores, total nasal mucus weights, and number of facial tissues used. Each of the reports in Chapter 4 have been re-analyzed using published facts and factual details from lozenge manufacturers, and half-life and average reductions (or increases) in duration have been calculated using all the available information.
Chapter 3
References
Chapter 4