HSA KIT: AI tools for the detection of tumor markers and other biomarkers in cancer

Total Area

The sum of all polygon areas.
$$\text{Total Area} = \sum_{i=1}^{n} \text{Area}_i$$

Percentage Area of Base ROI

The percentage of the total area of polygons relative to the base ROI.
$$\text{Percentage Area} = \left( \frac{\text{Total Area}}{\text{Base ROI Area}} \right) \times 100 \%$$

Minimum Area

The smallest polygon area.
$$\text{Minimum Area} = \min(\text{Area}_1, \text{Area}_2, \ldots, \text{Area}_n)$$

Maximum Area

The largest polygon area.
$$\text{Maximum Area} = \max(\text{Area}_1, \text{Area}_2, \ldots, \text{Area}_n)$$

Average Area

The mean area of all polygons.
$$\text{Average Area} = \frac{\sum_{i=1}^{n} \text{Area}_i}{n}$$​​

Median Area

The median area of all polygons.
$$\text{Median} =
\begin{cases}
\text{Area}_{\left(\frac{n+1}{2}\right)} & \text{if } n \text{ is odd} \\
\frac{\text{Area}_{\left(\frac{n}{2}\right)} + \text{Area}_{\left(\frac{n}{2} + 1\right)}}{2} & \text{if } n \text{ is even}
\end{cases}$$​​

Mode Area

The most frequently occurring area value.
$$\text{Mode Area} = \text{The value that appears most frequently in the dataset}$$

Standard Deviation (StdDev) Area

The measure of dispersion of polygon areas.
$$\text{StdDev} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (\text{Area}_i – \text{Average Area})^2}$$​

Variance of Area

The square of the standard deviation.
$$\text{Variance} = \frac{1}{n} \sum_{i=1}^{n} (\text{Area}_i – \text{Average Area})^2$$

Mean Absolute Deviation of Area

The average of the absolute deviations from the mean.
$$\text{MAD} = \frac{1}{n} \sum_{i=1}^{n} |\text{Area}_i – \text{Average Area}|$$

Skewness

The measure of the asymmetry of the distribution of area values.
$$\text{Skewness} = \frac{\sum_{i=1}^{n} (\text{Area}_i – \text{Average Area})^3}{n \cdot \text{StdDev}^3}$$

Kurtosis

The measure of the “tailedness” of the distribution of area values.
$$\text{Kurtosis} = \frac{\sum_{i=1}^{n} (\text{Area}_i – \text{Average Area})^4}{n \cdot \text{StdDev}^4}$$

Object Size Quartiles (1st, 2nd, 3rd)

The quartile values of polygon areas.
$$\text{Q1} = \text{Area}_{\left(\frac{n+1}{4}\right)}$$
$$\text{Q3} = \text{Area}_{\left(\frac{2(n+1)}{4}\right)}$$
$$\text{Q3} = \text{Area}_{\left(\frac{3(n+1)}{4}\right)}$$​

Interquartile Range (1st – 3rd)

The range between the first and third quartile.
$$\text{IQR} = \text{Q3} – \text{Q1}$$

Decile Sizes (1st to 9th)

Values that divide the dataset into ten equal parts.
$$\text{D1} = \text{Area}_{\left(\frac{n+1}{10}\right)}$$
$$\text{D2} = \text{Area}_{\left(\frac{2(n+1)}{10}\right)}$$
$$\text{D3} = \text{Area}_{\left(\frac{3(n+1)}{10}\right)}$$
$$\text{D4} = \text{Area}_{\left(\frac{4(n+1)}{10}\right)}$$
$$\text{D5} = \text{Area}_{\left(\frac{5(n+1)}{10}\right)}$$
$$\text{D6} = \text{Area}_{\left(\frac{6(n+1)}{10}\right)}$$
$$\text{D7} = \text{Area}_{\left(\frac{7(n+1)}{10}\right)}$$
$$\text{D8} = \text{Area}_{\left(\frac{8(n+1)}{10}\right)}$$
$$\text{D9} = \text{Area}_{\left(\frac{9(n+1)}{10}\right)}$$

Confidence Intervals of Area

The range within which the true mean of the population lies with a certain confidence level.
$$\text{CI} = \text{Average Area} \pm (z \times \frac{\text{StdDev}}{\sqrt{n}})$$

Shape Factor

A measure of the compactness of the shape.
$$\text{Shape Factor} = \frac{4\pi \times \text{Area}}{\text{Perimeter}^2}$$​

Area to Perimeter Ratio

The ratio of area to perimeter.
$$\text{Ratio} = \frac{\text{Area}}{\text{Perimeter}}$$​

Eccentricity

The ratio of the distance between the foci of the ellipse and its major axis length.
$$\text{Eccentricity} = \sqrt{1 – \frac{b^2}{a^2}}$$

Object count

The number of polygons

Object density

The number of polygons per unit area.
$$\text{Object Density} = \frac{\text{Number of Polygons}}{\text{Total Area}}$$

Dispersion index

A measure of how polygons are dispersed over the area.
$$\text{Dispersion Index} = \frac{\text{Variance of Polygons per Unit Area}}{\text{Mean Polygons per Unit Area}}$$

Clustering analysis

A measure of how polygons are clustered together.
$$\text{Clustering Index} = \frac{\text{Variance of Polygons in Sub-regions}}{\text{Mean Polygons in Sub-regions}}$$

Maximum perimeter

The largest perimeter of the polygons.
$$\text{Maximum Perimeter} = \max(\text{Perimeter}_1, \text{Perimeter}_2, \ldots, \text{Perimeter}_n)$$

Average perimeter

The mean perimeter of all polygons.
$$\text{Average Perimeter} = \frac{\sum_{i=1}^{n} \text{Perimeter}_i}{n}$$

Median perimeter

The median perimeter of all polygons.
$$\text{Median Perimeter} =
\begin{cases}
\text{Perimeter}_{\left(\frac{n+1}{2}\right)} & \text{if } n \text{ is odd} \\
\frac{\text{Perimeter}_{\left(\frac{n}{2}\right)} + \text{Perimeter}_{\left(\frac{n}{2} + 1\right)}}{2} & \text{if } n \text{ is even}
\end{cases}$$stdev perimeter

The measure of dispersion of the perimeters of the polygons.
$$\text{StdDev Perimeter} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (\text{Perimeter}_i – \text{Average Perimeter})^2}$$

Variance of perimeter

The square of the standard deviation of the perimeters.
$$\text{Variance of Perimeter} = \frac{1}{n} \sum_{i=1}^{n} (\text{Perimeter}_i – \text{Average Perimeter})^2$$