Formula - Days of Life Gained

Days of Life Gained is a formula that estimates the potential health improvement of having access to clean indoor air based on PM2.5 values.

1. What is Days of Life Gained

The purpose of this algorithm is to extend the methods of calculating “Years of Life lost”, or YLL, to quantify the positive effect of having access to clean air on human health with “Days of Life Gained” (DLG).

4 parameters are taken into consideration:

Outdoor PM2.5
Indoor PM2.5
Daily Number of Hours Spent Indoor
Number of Days Spent Indoor Per Year

Guo et al. [1] and Schwela [2] each described in their corresponding studies the correlations of PM2.5 with YLL, and with mortality rate. Both of which appeared to be linear in the region of our interest (PM2.5 < 140 μg/m³). We use the models developed in these studies to determine the values of YLL and mortality rate given a certain PM2.5 value. We use the values provided in ref [1] to be the starting point for our algorithm.

2. Days of Life Gained Formula

We compile the calculation of days of life gained into the following formulas:

where:

YLpD: year lost per death
YLL * 5 * 365: total death count over the course of 5 years given a specific value of PM2.5
D_ref: reference value of death count (from ref [1])
YLL: year of life lost, is a function of PM2.5 (ref [1])
MR: mortality rate, is a function of PM2.5 (ref [2])

and converting YLpD to days of life gained per year, we arrive at:

where:

ΔYLpD: change in year lost per death from values calculated with PM2.5 outdoor average to those calculated with indoor average

3. Formula Logic and Sample Scenario

We illustrate the step-by-step process of arriving at the DLG value over the course of one year with the below sample scenario.

Step 1 - Inputs and Assumptions

The average outdoor PM2.5 level reads 60 μg/m³ in Shanghai (ref [3]). It is assumed that a person spends around 6 hours a day, 250 days a year working in the office where filtration system is on and the average indoor PM2.5 level reads 10 μg/m³. To summarize, we have the following parameters:

Outdoor PM2.5 = 60 μg/m³
Indoor PM2.5 = 10 μg/m³
Daily Number of Hours Spent Indoor = 6 hours
Number of Workdays Per Year = 250 days

Step 2 - Calculating the Average Daily PM2.5 Exposure

We first calculate the average of daily exposure to PM2.5 weighted by daily hour spent indoor and number of days indoor per year:

Weighted Workday Indoor = 6/24 * 250/365 * PM2.5_indoor
Weighted Workday Outdoor = (24-6)/24 * 250/365 * PM2.5_outdoor
Weighted Offday Outdoor = (365-250)/365 * PM2.5_outdoor

Average Daily Exposure = Weighted Workday Indoor
+ Weighted Workday Indoor
+ Weighted Offday Outdoor = 51.4 μg/m³

The result is then used to calculate the change in mortality based on the relationship shown in the following equation taken from ref [2]:

where y is the percent increase in daily mortality and x is the value of PM2.5.

Step 3 - Updating Hypothetical Death Count and YLL

A hypothetical death count value (reference death count = 80515 per 5 years) is introduced here to gauge the effect of the change in mortality shown above. It assumes the mortality rate of the reference case to be 1. The average PM2.5 from the reference is 105.1 μg/m³.

Change in mortality from daily average: 0.151 * 51.4 - 0.151 * 105.1 = -8.1%
Change in mortality from outdoor average: 0.151 * 60 - 0.151 * 105.1 = -6.8%
Hypothetical death count from daily average: (100 - 8.1)% * 80515 = 73991
Hypothetical death count from outdoor average: (100 - 6.8)% * 80515 = 75032

YLLs corresponding to daily average and from outdoor average of PM2.5 are calculated from a linear regression model based on data from ref [1]:

YLL from daily average: 2.32 * 51.4 + 487.2 = 606.44
YLL from outdoor average: 2.32 * 60 + 487.2 = 626.29

Step 4 - Combining the results for DLG

From the hypothetical death count and the YLLs we just calculated, the value of YLL per death (YLpD) can be obtained with the following equation:

Note that the afore-calculated hypothetical death count is a value over the course of 5 years due to the reference data being the case. The change in YLLs per death (ΔYLpD) calculated from daily average and outdoor average then quantifies the improvement of human health due to good indoor air.

YLL per death from daily average: 606.44 / (73991 / (5 * 365)) = 14.96
YLL per death from outdoor average: 626.29 / (75031 / (5 * 365)) = 15.23
ΔYLpD: 14.96 - 15.23 = -0.27

The value of ΔYLpD means that per death, the year of life lost is decreased by 0.27 years. Given that we calculated every value based on a reference study that surveyed data for 5 years, it should be implied that a scenario created for this algorithm surveys for the same length. The final result in DLG per year, nevertheless, should be communitive. Hence, the final conversion of change in ΔYLpD to DLG per year follows:

-0.27 years lost per 5 years = 20.4 days gained per year
Days of Life Gained (DLG) = 20.4 days

We hereby consolidate our formulations with information from a recent study by Qi and Ruan et al. [4] in 2020. They have estimated that an improvement of indoor air PM2.5 levels to the WHO guideline level of 25 μg/m³ led to 0.14 years of gains in life expectancy for each death. With a period of investigation just shy of 4 years, their yearly average approximates to 0.035 years life gained per year. A quick re-calculation with our formulas with an improvement of indoor air quality to the same PM2.5 level yields 0.038 years life gained per year. So far, we have based our calculations on conservative assumptions and simple models. Yet the result we arrived at aligns beautifully with the result from ref [4] that implements more recent data and a more complex mathematic model. Therefore, we conclude that our formulation of days of life gain is a sound estimate.

4. Glossary

Days of Life Gained (DLG): describes how many days of life is gained over the course of 1 year of having access to an improved indoor air compared to outdoor
Years of Life Lost (YLL): describes the years of life lost from premature mortality
Mortality Rate (MR): measures the percentage of death in a given population
Year Lost per Death (YLpD): describes the average years of life lost per death in a given population

5. References

Guo Y, Li S, Tian Z, Pan X, Zhang J, Williams G et al. The burden of air pollution on years of life lost in Beijing, China, 2004-08: retrospective regression analysis of daily deaths; BMJ 2013; 347 :f7139
Schwela D. Air pollution and health in urban areas. Reviews on Environmental Health 2000 Jan-Jun;15(1-2):13-42
Yu Shang, Zhiwei Sun, Junji Cao, et al. Systematic review of Chinese studies of short-term exposure to air pollution and daily mortality. Environment International, 54, pp.100–111
Qi J, Ruan Z, Qian Z(, Yin P, Yang Y, Acharya BK, et al. (2020) Potential gains in life expectancy by attaining daily ambient fine particulate matter pollution standards in mainland China: A modeling study based on nationwide data. PLoS Med 17(1): e1003027