# An Automatic Approach for Individual HU-Based Characterization of Lungs in COVID-19 Patients
## Materials and Methods
### Threshold Definition Methods  
Considering the typical HU-distribution, lungs may be divided into three regions: the aerated component, the consolidated component and an intermediate component. The HU density of the lungs of COVID-19 patients is generally characterized by the presence of two peaks with a shape similar to that of a Gaussian curve, as shown in Figure 1, one next to the air HU (\(-\)1000 HU) which defines the aerated, and therefore \"properly\" functioning, lung; and one next to the water HU (0 HU) [29] corresponding to the lung component with \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Equipment**} & \multirow{2}{*}{**Number of Patients**} & \multirow{2}{*}{**Voltage (kV)**} & \multicolumn{4}{c}{**Current (mA)**} \\ \cline{4-7}  & & & **Range** & **Median** & **1 st Quartile** & **3 rd Quartile** \\ \hline _Lightspeed_ & 146 & 120 & 149–549 & 352.5 & 219.5 & 451.5 \\ _Brilliance_ & 12 & 120 & 240–500 & 458.5 & - & - \\ _Incisive_ & 8 & 120 & 189–368 & 334 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Schematic of the CT chest present in the database based on the equipment used, the number of patients, and the voltage and current range. consolidated disease. Between these two peaks there is a quite evident and pronounced region with highly variable patterns from patient to patient. This region corresponds to the component of the lung affected by the disease, but not yet scarred, and mostly includes ground glass opacities (GGO) and crazy-paving regions. The corresponding region in the HU distribution plot often has a plateau-like trend, while in most cases it is characterized by a bell shape. Taking into account the shape of the HU-density distributions, different strategies were followed to individually define the best threshold values for distinguishing the different lung components. The first method consisted of parameterizing the distribution of the HU densities of the individual lungs with a curve given by the linear combination of three Gaussian curves, and considering the half-width half-maximum of the curves to assess the thresholds. This method was named the \"Gaussian method\". The three Gaussians used in the fit corresponded to the three lung components (see Figure 2). The resulting function used for the fit was as follows:  
\[fit(x)=a1\times\exp\left(-\left(\frac{x-b1}{c1}\right)^{2}\right)+a2\times\exp \left(-\left(\frac{x-b2}{c2}\right)^{2}\right)+a3\times\exp\left(-\left(\frac{x -b3}{c3}\right)^{2}\right)\]  
where:  
* \(x\) are the values of the densities HU;
* \(a1\), \(a2\), and \(a3\) are the multiplicative coefficients of the Gaussians;
* \(b1\), \(b2\), and \(b3\) represent the mean values of the Gaussians;
* \(c1\), \(c2\), and \(c3\) are related to the standard deviation from the relation \(c^{2}=2\sigma^{2;}\)
* Subscript 1 refers to the Gaussian related to the aerated component;
* Subscript 2 refers to the Gaussian related to the intermediate component;
* Subscript 3 to the Gaussian related to the consolidated component. Figure 1: Examples of Hounsfield unit (HU)-density histograms. The histograms shown correspond to the lungs of the same patient; in particular, (**a**) is the histogram relating to the right lung and (**b**) is the histogram relating to the left lung. Concerning the fit, it was imposed that the search for peaks corresponding to the aerated and consolidated components would start from the HU-density values typical of air (\(-1000\) HU) and water (0 HU), respectively. The standard deviation \(\sigma\) of the individual Gaussians was obtained from the fit and used to determine the thresholds.