Meters from 2D Images: comparisons with real 3D estimatesTo test the accuracy of estimating microtubule Ornipressin web parameters from 2D images, we applied our new 2D method (see Methods) using the LED 209 site central slice (at half height of the cell) of 3D HeLa cell images and compared the estimated parameters with those from the 3D method. The half height was chosen as the preferred slice because the 2D images used later were also acquired at half the height of the cell. We computed the mean absolute percentage error (MAPE) in each of the parameters estimated from the 2D images assuming that the estimated parameters from the 3D method were correct. Results are shown in Table 1 for 42 cells. From the table, we can see that the estimates of the number of microtubules and collinearity from a single 2D slice are reasonably close to those from the entire 3D image. However, the MAPE for the mean length appears to be somewhat larger. We will aim to reduce this discrepancy in future work. However, we note that most cells were estimated to have mean length of 10 or 15 microns (see the section of library generation in Methods) using the 3D method on the original 3D images. Therefore a small deviation in the estimates of 5 microns (the increment of the range of allowed values of mean length) would cause a MAPE of 50 or 33. The table also shows aFigure 1. Growth model for generating microtubules dependent on cell and nuclear shapes. Each microtubule starts from the centrosome, and randomly grows to the second point on the lateral surface of a cone whose aperture is 2a. Then the microtubule grows the same way until it hits the cell or nuclear shape boundary and is not able to step further within the cytosolic area. At this time, we relax the collinearity requirement but still confine the next direction under the local constraint alocal. Moreover, we also keep on checking a consecutive multiple (30) steps, and require that there are less than or equal to 3 pairwise vector angles that are larger than the global constraint aglobal. Beginning with an empty (black) cytosolic area (shaped by cell and nuclear boundary), we add one to the intensity of the pixel which a microtubule crosses. In this paper, we used every step of growth to be 0.2 microns (1 pixel). For the two constraints on the collinearity which controls the curvature of each microtubule and the local and global rebounding issues, we used alocal to be 63.9 degrees and aglobal to be 120 degrees. The figure only illustrates the procedure of growth in 2D for better visualization but can be easily imagined to extend to 3D.Comparison of Microtubule DistributionsFigure 2. An overview of the framework introduced in this paper. The framework contains two sub-systems, one for generating 3D synthetic images of distributions of microtubules (A), and one for estimating and comparing the model parameters of distribution of microtubules from real 2D images of eleven cell lines (B). doi:10.1371/journal.pone.0050292.gcomparison of the true cell heights and the estimated ones, with the results showing that they are reasonably close.Recovering 3D Microtubule Generative Model Parameters from 2D Images: simulated experimentsWe estimated how well our recovery method can perform using simulated images for which the correct parameters were known. For one cell geometry (cell shape and nucleus shape), a library of 3D synthetic images was generated with predefined parameters as a validation bed; then 5 other testing libraries were generated.Meters from 2D Images: comparisons with real 3D estimatesTo test the accuracy of estimating microtubule parameters from 2D images, we applied our new 2D method (see Methods) using the central slice (at half height of the cell) of 3D HeLa cell images and compared the estimated parameters with those from the 3D method. The half height was chosen as the preferred slice because the 2D images used later were also acquired at half the height of the cell. We computed the mean absolute percentage error (MAPE) in each of the parameters estimated from the 2D images assuming that the estimated parameters from the 3D method were correct. Results are shown in Table 1 for 42 cells. From the table, we can see that the estimates of the number of microtubules and collinearity from a single 2D slice are reasonably close to those from the entire 3D image. However, the MAPE for the mean length appears to be somewhat larger. We will aim to reduce this discrepancy in future work. However, we note that most cells were estimated to have mean length of 10 or 15 microns (see the section of library generation in Methods) using the 3D method on the original 3D images. Therefore a small deviation in the estimates of 5 microns (the increment of the range of allowed values of mean length) would cause a MAPE of 50 or 33. The table also shows aFigure 1. Growth model for generating microtubules dependent on cell and nuclear shapes. Each microtubule starts from the centrosome, and randomly grows to the second point on the lateral surface of a cone whose aperture is 2a. Then the microtubule grows the same way until it hits the cell or nuclear shape boundary and is not able to step further within the cytosolic area. At this time, we relax the collinearity requirement but still confine the next direction under the local constraint alocal. Moreover, we also keep on checking a consecutive multiple (30) steps, and require that there are less than or equal to 3 pairwise vector angles that are larger than the global constraint aglobal. Beginning with an empty (black) cytosolic area (shaped by cell and nuclear boundary), we add one to the intensity of the pixel which a microtubule crosses. In this paper, we used every step of growth to be 0.2 microns (1 pixel). For the two constraints on the collinearity which controls the curvature of each microtubule and the local and global rebounding issues, we used alocal to be 63.9 degrees and aglobal to be 120 degrees. The figure only illustrates the procedure of growth in 2D for better visualization but can be easily imagined to extend to 3D.Comparison of Microtubule DistributionsFigure 2. An overview of the framework introduced in this paper. The framework contains two sub-systems, one for generating 3D synthetic images of distributions of microtubules (A), and one for estimating and comparing the model parameters of distribution of microtubules from real 2D images of eleven cell lines (B). doi:10.1371/journal.pone.0050292.gcomparison of the true cell heights and the estimated ones, with the results showing that they are reasonably close.Recovering 3D Microtubule Generative Model Parameters from 2D Images: simulated experimentsWe estimated how well our recovery method can perform using simulated images for which the correct parameters were known. For one cell geometry (cell shape and nucleus shape), a library of 3D synthetic images was generated with predefined parameters as a validation bed; then 5 other testing libraries were generated.
