## 4 Discussion  
In this study, we outlined a developed mixed-unit lattice comprising hexagons and triangles that outperformed the conventional unit squares or pixels method in determining the area of curvilinear regions. The mixed-unit lattice method proved superior to the unit-squares method in determining the areas of biological ultrastructures, including glomeruli and renal cysts. Figure 5: Refining area calculations. (**A**) Renal cyst enclosed by wall (region of interest lay within wall). (**B**) Region populated with relatively large unit squares. (**C**) Region populated with smaller unit squares. (**D**) Region populated with mixed but relatively large unit lattice. (**E**) Region populated with mixed but smaller unit lattice. Changes in the dimensions of cells and subcellular structures could be associated with a number of diseases [8,9]. Cardiomyocyte hypertrophy occurs during ventricular remodeling and could signal cardiomyopathyopathy [10], and glomeruli hypertrophy can occur in kidney disease. Both adriamycin- and puromycin-induced models of proteinuria kidney disease are accompanied by glomerular hypertrophy, with the amount of urine protein proportional to the extent of glomerular remodeling [1,11]. In fact, the extent of glomerular expansion informs a number of clinical outcomes, including risk of relapse in MCD, and the risk of transition from MCD to FSGS [2, 3, 4, 12]. Both glomerular size and the number of hypertrophied glomeruli are biomarkers for progression to end-stage renal disease [1]. Availability of serial biopsies in the setting of proteinuria kidney disease makes the determination of glomerular changes possible. Given that, in many mammalian models, glomerular diameters are of the order of 50 to 70 \(\upmu\)m with even a 10% increase in glomerular dimensions assuming significance, accurate determination of the glomerular area and changes therein is critical [1]. In PKD, expansion of fluid-filled cysts drives an increase in total kidney volume (TKV) [5]. A number of studies [13, 14, 15] have shown that increasing TKV corresponds to a more rapid decline in renal dysfunction. Measurement of the cystic index (i.e., the ratio of cystic area or cystic volume to renal parenchymal area or TKV, respectively) is critical to inform of disease progression [5]. Furthermore, at least in preclinical PKD models, the efficacy of a number of therapeutic strategies is determined by the ability of these regimens to retard the progressive increase in cystic index [13, 14, 15]. Measurement of the cystic area and time-dependent changes therein is therefore critical to inform of both disease status and drug efficacy. Cellular and subcellular bodies rarely assume well-defined geometric shapes that lend themselves to the easy calculation of their cross-sectional areas. More often than not, their cross-sectional areas are curvilinear regions. Planimetric techniques currently use the method of counting the number of unit squares lying within a region of interest to estimate this area [6,7]. Nonetheless, given the inherent shape of a square vs. that of a curvilinear plane, there is often underestimation of the region's true area. Reported herein is an improvised solution for the same via use of a mixed lattice method comprising unit hexagons and unit triangles. Across a wide variety of curvilinear shapes, including a circle, a randomly drawn figure, glomeruli, and renal cysts, the mixed-unit lattice method outperformed the conventional unit-squares method. In fact, after shrinking the size of the unit segment, performance of the mixed-unit lattice approached that of the ImageJ method, which utilizes multifold smaller unit squares or pixels to determine areas. It can be inferred that shrinking the size of the unit hexagon and the unit triangle down to that of a pixel further improves area-measurement accuracy. The greater the number of its vertices, the better the calculation of the approximate area of a given geometric shape inscribed within a circle [16].