A review on mathematical models for estimating indoor radon concentrations
© Park et al. 2016
Received: 18 January 2016
Accepted: 18 February 2016
Published: 25 February 2016
Radiation from natural sources is one of causes of the environmental diseases. Radon is the leading environmental cause of lung cancer next to smoking. To investigate the relationship between indoor radon concentrations and lung cancer, researchers must be able to estimate an individual’s cumulative level of indoor radon exposure and to do so, one must first be able to assess indoor radon concentrations. In this article, we outline factors affecting indoor radon concentrations and review related mathematical models based on the mass balance equation and the differential equations. Furthermore, we suggest the necessities of applying time-dependent functions for indoor radon concentrations and developing stochastic models.
KeywordsMathematical model Indoor concentration Radon entry Exposure
Exposure to radiation from natural sources leads to various environmental diseases. Radon, which is the primary constituent of natural radiation , is the leading environmental cause of lung cancer next to smoking . Studies in Europe, the United States, Canada, and China have effectively shown a relationship between indoor radon concentrations and lung cancer .
There had been several studies to estimate the indoor radon concentrations with mathematical models. Ramola et al.  estimated the indoor radon concentration using a mathematical model based on the radon flux in soil and groundwater. Arvela et al.  calculated the indoor radon concentrations with a mathematical model which considers diffusion from soil and BM sources and air exchange simultaneously. These calculations have been compared with the measured variations of Finnish dwellings. Font  and Font and Baixeras  developed a generic and dynamic model of Radon Generation, Entry, and Accumulation indoors (RAGENA) which describes all the known sources including soil, BM, and water, and this model has been adapted to a Mediterranean climate house and Swedish house [9, 10]. Mathematical models for estimating indoor radon concentrations in high-rise buildings have been conducted by Man and Yeung  and Shaikh et al. . These models included the contributions from BM and outdoor air. Moreover, model of indoor radon concentration in thermal spas was developed by Vogiannis and Nikolopoulos .
Radon concentration in building material (BM)
Indoor radon concentration
Radon concentration of outdoor air
Radon concentration in soil
Radon concentration in water supply
Bq/(m2 · s)
Total radon flux from soil into building
m/(s · Pa)
Advection transfer coefficient of soil
Diffusion transfer coefficient of BM
Diffusion transfer coefficient of soil
Air current from compartment i to compartment j
Indoor surface area of radon containing BM
Building area towards ground
Radon transfer efficiency of water supply
Use rate of water
Volume of the indoor
ΔP s − i
Soil-indoor pressure difference
Radon decay constant
According to the above, changes in indoor radon concentrations are affected by the sources of radon entry, building ventilation (radon exchange), and decay reaction. First, we introduce a model for assessing indoor radon concentrations attributed to the three main sources of radon entry (soil, BM, and water) and decay reaction.
Radon entry and decay reaction
Although radon from groundwater contributes very little to indoor radon, it can constitute an important source in specific instances; the total radon flux from the ground, considering both radon in soil and groundwater, can be found in Ramola et al. . In the meantime, Font  and Font and Baixeras  suggested evaluating total radon flux from the soil into the building F s differently: the soil can be divided into two compartments, disturbed soil (DS, the volume of soil underneath a building from which radon can reach the basement of the building by diffusion and pressure driven flow) and undisturbed soil (US, the soil attached to the DS that is not influenced by the presence of the building). The radon concentrations in the US and in the DS are described in [5, 6]. However, for simplicity, throughout this article we develop a model using a total radon flux into the building that consider only DS, namely F s in Eq. (2).
Water for living
where dC i = C i (t + dt) − C i (t) and dt denotes an infinitesimal time.
Ventilation between indoor and outdoor air
Ventilation among the compartments
Summary of references reviewed in this article
Kusuda et al. 
Capra et al. 
An environmental chamber
Ramola et al. 
Budhakedar area of Garhwal Himalaya in summer and winter
Man and Yeung 
Newly constructed, uninhabited high-rise buildings in Hong Kong
Shaikh et al. 
A multi-storey building in Mumbai, India over four seasons
Vogiannis and Nikolopoulos 
Thermal spas in Greece
Arvela et al. 
Finnish dwellings in summer and winter
Mediterranean climate house and Swedish houseb
Font and Baixeras 
Mathematical models for estimating indoor radon concentrations provide not only a method of assessment but also an understanding between parameters that govern indoor radon levels. In order to estimate an individual’s cumulative indoor radon exposure, assessing indoor radon concentrations is necessary. To do so, one should seek to understand the mechanisms of radon entry and exchange in relation to side-specific surroundings and characteristics of a building. Accordingly, resultant models in consideration of said characteristics may be simpler than Eq. (8) or may necessitate additional research on factors of indoor radon concentrations (e.g., age of the BM and relative humidity).
Most of the models reviewed in this article utilized solutions at a steady state and considered parameters to be constant in order to estimate indoor radon concentrations. Such approaches are often adequate, as a steady state is rarely attained in actual buildings because of time-dependent factors (e.g., air exchange rates, which change with atmospheric conditions). For this reason, time-dependent solutions may be more accurate than steady state solutions in assessment of actual building conditions. Additionally, all of the models included in this review considered only deterministic models. However, a stochastic model designed to account for unrecognized factors and noise-corrupted measurements may offer better approximation of indoor radon levels.
This subject is supported by Korea Ministry of Environment (MOE) as “the Environmental Health Action Program” (Grant Number 2015001350002).
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