## 2 Materials and Methods
### Risk Assessment  
Incorporating bioaccessibility requires the modification of the approach usually undertaken to carry out HHRA at contaminated sites. In this discussion, we make reference to the approach followed in Italy [44], which essentially relies on the RBCA-ASTM approach [7, 8]. Namely, in forward mode, Risk or Hazard Index is estimated from the total soil concentration (C), which is calculated by normalizing the concentration measured on the soil particle size fraction below 2 mm (\(C_{TOT\;2 mm}\)) to the soil fraction with particle diameter between 2 mm and 2 cm (\(F_{2 mm}\)):  
\[C=C_{TOT2 mm}\cdot F_{2 mm} \tag{1}\]  
This means to assume that all contaminants are present in the particle size fraction below 2 mm, whereas the fraction between 2 mm and 2 cm is assumed to be clean.  
For the soil ingestion pathway, risk (\(R\)) and hazard index (\(HI\)) are given by  
\[R=EM\cdot SF\cdot C \tag{2}\]  
\[HI=\frac{EM\cdot C}{RfD} \tag{3}\]  
where \(SF\) (mg/kg/d)\({}^{-1}\) and \(RfD\) (mg/kg/d) are the carcinogenic slope factor and reference dose of a given chemical, respectively.  
\(EM\), which includes the exposure parameters, is given by  
\[EM=\frac{ED\cdot EF\cdot IR}{BW\cdot AT} \tag{4}\]  
where \(ED\) is the exposure duration (y), \(EF\) the exposure frequency (d/y), \(IR\) the soil ingestion rate (mg/d), \(BW\) the body weight (kg), and \(AT\) the averaging time (y).  
The backward HHRA allows for the estimation of the cleanup goal for a given chemical, here named \(CSR\) by the Italian legislation [45], by simply reversing the risk equations shown above, thus leading to  
\[CSR_{canc}=\frac{TR}{EM\cdot SF} \tag{5}\]  
\[CSR_{non\;canc}=\frac{THI\cdot RfD}{EM} \tag{6}\]  
where \(TR\) and \(THI\) are the target carcinogenic risk (10\({}^{-6}\) for the Italian legislation) and the target hazard index (1 for the Italian legislation), respectively. When incorporating the bioaccessibility (\(BA\)) in the estimate of the cleanup goals for the soil ingestion pathway, we obtained the following equations:  
\[CSR_{canc}=\frac{TR}{EM\cdot SF\cdot BA} \tag{7}\]  
\[CSR_{non\;canc}=\frac{THI\cdot RfD}{EM\cdot BA} \tag{8}\]The question now becomes how to estimate the bioaccessible fraction of hazardous compounds to be considered in the above-mentioned equations.  
One option proposed in the literature (e.g., [28; 29]) relies on estimating the bioavailability with reference to the total concentration measured in the same particle size fraction used for measuring the bioaccessible concentration, as shown in the following equation.  
\[BA_{1}=\frac{C_{BA}}{C_{TOT\,FINE}}\cdot 100 \tag{9}\]  
where \(C_{BA}\) (mg kg\({}^{-1}\)) is the bioaccessible concentration of each inorganic measured adopting for instance the UBM or SBET methods, and \(C_{TOT\,FINE}\) is the total inorganic content (mg kg\({}^{-1}\)) measured on the same soil fraction used for the bioaccessibility test (i.e., d \(<\) 250 \(\upmu\)m for the UBM or d \(<\) 150 \(\upmu\)m for the SBET).  
Another option could be to calculate the bioaccessibility (\(BA\)) with reference to the total soil concentration (C), i.e., the concentration in the soil fraction < 2 mm normalized to the to the soil fraction with particle diameter between 2 mm and 2 cm, which is usually considered in the HHRA procedure. However, for consistency, in this case, the particle size distribution also had to be taken into account, normalizing the bioaccessible concentration to the soil fraction used in the in vitro methods, thus leading to the following equation:  
\[BA_{2}=\frac{C_{BA}\cdot F_{BA}}{C}\cdot 100 \tag{10}\]  
where \(C_{BA}\) (mg kg\({}^{-1}\)) is again the bioaccessible concentration of each element measured adopting either the UBM or the SBET method, \(F_{BA}\) is the soil fraction used in the in vitro methods (i.e., d \(<\) 250 \(\upmu\)m for the UBM or d \(<\) 150 \(\upmu\)m for the SBET), and \(C\) is the total soil concentration (Equation (1)).