## 3 Results  
As seen in Figure 1A, a circle of a 6 unit radius and an area of 113.1 unit\({}^{2}\) were calculated via the area formula for a circle. Use of ImageJ to measure the area of the circle yielded a value of 112.3 unit\({}^{2}\). Populating this circle with unit squares (Figure 1B) and counting the number of squares lying within the circle yielded a measured area of 88 unit\({}^{2}\). On the other hand, populating this circle with a mixed lattice comprising hexagons and triangles and then counting the total number that were within the circle yielded a measured area of 93.5 unit\({}^{2}\). By contrast to the circle shown in Figure 1, there was no preset formula to calculate the area of the arbitrary irregular curvilinear region shown in Figure 2A. Use of ImageJ to measure this area yielded 87.3 unit2. Populating this region with unit squares (Figure 2B) and counting their number yielded a calculated area of 72 unit2. Use of the mixed-unit lattice approach (Figure 2C) yielded a calculated area of 74.4 unit2. Next, areas of several glomeruli and renal cysts were calculated using the unit-squares and the mixed-unit lattice methods, and results were compared with the areas obtained using ImageJ. An image of a glomerulus, as seen in Figure 3A, was populated with either unit squares (Figure 3B) or the mixed-unit lattice (Figure 3C). Areas of a number of glomeruli were measured using ImageJ and then calculated. As seen in Table 1, in each case, the mixed-unit lattice returned a value closer to that measured by ImageJ, and it outperformed the unit squares method (\(p<0.01\)). Figure 1: Area of a circle. (**A**) Circle of radius 6. (**B**) Circle populated with unit squares. Blue but not pink, squares were completely enclosed within circle and counted. (**C**) Circle populated with lattice comprising unit hexagons and unit triangles. Hexagons were filled in first, and then triangles. Figure 3: Area of a glomerulus. (**A**) Region of interest (outlined by pink line). (**B**) Region populated with unit squares. Only blue/turquoise squares were completely enclosed within the region and counted. (**C**) Region populated with lattice comprising unit hexagons and unit triangles. Hexagons were filled in first, then triangles (green). All hexagons and green triangles lay within region of interest and were therefore counted. Figure 2: Area of irregular curvilinear region. (**A**) Region of interest (outlined in black). (**B**) Region populated with unit squares. Blue but not pink squares were completely enclosed within region and counted. (**C**) Region populated with lattice comprising unit hexagons and unit triangles. Hexagons were filled in first, then triangles (green). Pink triangles lay outside the region and were not counted. An image of a renal cyst, as seen in Figure 4A, was populated with either unit squares (Figure 4B) or the mixed-unit lattice (Figure 4C). Areas of a number of renal cysts were measured and calculated. As seen in Table 2, in each case the mixed-unit lattice returned a value closer to that measured by ImageJ and outperformed the unit-squares method (\(p<0.01\)). In each of the above instances, the area of the region measured using ImageJ was substantially greater than that when using either the unit-squares method or the mixed-unit lattice method. A better approximation to the ImageJ values could be made by shrinking the physical size of the unit square or the mixed-unit lattice, akin to shrinking a segment of 1 in length to 1 cm or 1 mm. Use of ImageJ to \begin{table}
\begin{tabular}{c c c c c} \hline
**Gloperulus** & **Image] (unit2)** & **Squares (unit2)** & **Lattice (unit2)** & **Improvement (\%) over Unit Squares** \\ \hline
1 & 42.5 & 29 & 31 & 7 \\
2 & 53 & 40 & 42.4 & 6 \\
3 & 21.3 & 15 & 15.6 & 4 \\
4 & 29.7 & 20 & 22.5 & 13 \\ \hline \end{tabular}
\end{table} 
Table 1: Glomerval-area determination using ImageJ, unit squares, and mixed-unit lattice.