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Microscopy imaging of mouse growth plates is extensively used in biology to understand the effect of specific molecules on various stages of normal bone development and on bone disease. Until now, such image analysis has been conducted by manual detection. In fact, when existing automated detection techniques were applied, morphological variations across the growth plate and heterogeneity of image background color, including the faint presence of cells (chondrocytes) located deeper in tissue away from the image’s plane of focus, and lack of cell-specific features, interfered with identification of cells. We propose the first method of automated detection and morphometry applicable to images of cells in the growth plate of long bone. Through ad hoc sequential application of the Retinex method, anisotropic diffusion and thresholding, our new cell detection algorithm (CDA) addresses these challenges on bright-field microscopy images of mouse growth plates. Five parameters, chosen by the user in respect of image characteristics, regulate our CDA. Our results demonstrate effectiveness of the proposed numerical method relative to manual methods. Our CDA confirms previously established results regarding chondrocytes’ number, area, orientation, height and shape of normal growth plates. Our CDA also confirms differences previously found between the genetic mutated mouse
*Smad1/5 ^{CKO}* and its control mouse on fluorescence images. The CDA aims to aid biomedical research by increasing efficiency and consistency of data collection regarding arrangement and characteristics of chondrocytes. Our results suggest that automated extraction of data from microscopy imaging of growth plates can assist in unlocking information on normal and pathological development, key to the underlying biological mechanisms of bone growth.

Microscopy imaging of mouse growth plates is extensively used to assess development and potential pathology. Such imaging confounds current automated methods for cell detection. Indeed, application to microscopy images of growth plates of each of the classic methods of image segmentation and processing (e.g. Canny segmentation [

Analysis of chondrocytes’ number, size, orientation, height and shape offer insights on the developmental effects of repressing various genes, the lack of which can lead to bone growth disorders (e.g. [

In this paper, we propose a method of automated multi-step image processing. These steps prepare an image for automated measurement of the characteristics of the chondrocytes located on the plane of focus of the original growth plate specimen. Rather than manually determining cell profiles, automated cell detection, and subsequent automated morphometry would aid biological research by increasing efficiency of measurements.

The ideal image should be of a specimen whose preparation involves a stain that maximizes the contrast between cell cytoplasm and background, and in focus throughout. We visualized chondrocytes in the growth plate of mice ex vivo with a BX60 microscope (Olympus America) set to bright-field. The animal protocol was approved by UCLA’s Animal Research Committee. Male mice on mixed background, predominantly 129/SvJ crossed with C57Bl6/J (WT for wild type), were sacrificed at perinatal 0, two weeks, four months. After isolation of hind limbs, we stained coronal sections of femora and tibiae to visualize cartilage (with Alcian blue or Safranin O), cell cytoplasm (Fast Green) and nucleus (Hematoxylin) [

We imported the TIFF images of growth plates into software XaraX version 1.1 (XaraX Group). The Shape Editor, Combine Shapes and Intersect Shapes allow tracing of the region of interest (ROI), cropping along the traced line to eliminate the image of the tissue surrounding the growth plate and possibly select a region of the growth plate, and replacement with a white background (

Our CDA is written with and runs on MATLAB version R2012a (Math Works), and is available for download as file CDAlg.m together with three of the full resolution images on which the CDA was run to prepare some of the figures in this paper, from https://sites.google.com/site/bthsca/automated-cell-detection-1. The input for the CDA consists of a TIFF image (

Step 1: We assign similar intensity values to cells in focus while we render the background, defined as cells out of focus and extracellular matrix, more homogeneous. We use the color blending method of Retinex, originally developed to simulate the visual cortex blending of colors [

where

is the discrete Laplacian at pixel, (i, j) and

where

is the hard thresholding function and τ the thresholding parameter. Thus, if a small gradient (of magnitude smaller than τ) is present at a pixel, it is replaced with zero. The Poisson equation then constructs the image whose gradient most closely matches the vector field that models the difference between each pixel and its neighboring pixels. We use the following semi-implicit scheme that converges rapidly to the steady state for local variations:

We apply this process iteratively, NI (number of iterations) times, each time to the previous resultant image.

Step 2: We enhance cell profiles (edges) with anisotropic diffusion. The anisotropic diffusion function needs to discriminate intensity changes by location: near vs. far from edges, and along vs. across edges [

for single time-steps

with g

Step 3: We extract cell profile information with gradient thresholding. We use the gradient matrix generated in Step 2, containing values scaled between 0 and 255. The hard threshold parameter, called S for “separation”, is chosen to identify single cells within packed clusters, while minimizing pixel loss from cell profile. We use the same Equation (4) with S replacing τ. We obtain a binary representation of the image with white cell profiles on black background suitable for subsequent morphometric analysis.

Step 4: We use the convex hull operation [

Step 5: We eliminate profiles that do not reflect actual cell size or shape with hard tresholding. Overly large or overly small or misshaped profiles are often created during convexification because of proximity of incomplete profiles or proximity of an incomplete profile with the boundary of the growth plate region. Alo (area lower) and Aup (area upper) bound in pix^{2} define a range out of which the profiles are excluded. Because these boundaries depend on image size and magnification, they need to be set by the user. Further, we use set up shape thresholding in terms of isoperimetric ratio (IR)

where A denotes area and P perimeter of the shape. The threshold was set at 0.5 because an NI greater than 0.5 demonstrate geometriesS different from ellipses.

The proposed CDA, implemented with MATLAB R2012a with 64-bit Windows-7, an Intel^{@}Core^{TM} i3-370M processor (2.40 GHz) and 4 GB of RAM, takes between

MCD was conducted on 390 cells on a total of three images with XaraX software, for comparison with the CDA. Because chondrocytic two-dimensional profiles throughout the growth plate are best modeled as ellipses [

We conducted morphometry on images from each of CDA and MCD with MetaMorph software (Molecular Devices). The Color Threshold was set from 127 to 255 for each of Red, Green and Blue. We calibrated MetaMorph in conjunction with the magnification used to generate the image to measure lengths in real microns. The output parameters were cell number, area, orientation (measured with respect to the long bone axis, counterclockwise from −90˚ to 90˚), height, and shape in terms of isoperimetric ratio IR (see Equation (8)).

The robustness of the morphometric MCD was measured in terms of inter- and intra-observer errors. The magnitude of these errors was assessed for cell number, area, orientation, height and IR. To assess the magnitude of such errors, the cell profile of six randomly chosen cells in focus was manually drawn with XaraX and measured with MetaMorph forty-two times by each of two observers, twice, two months apart. Further, to detect significant differences between CDA and MCD, and between adjacent zones of a given image, we used the t-test for each of the measured parameters, and the Dice index to measure similarity of cell shapes [

We have developed an automated method for analysis of growth plate images.

The CDA depends on the five parameters τ, NI, S, Alo, and Aup whose values depend on the image characteristics: τ (Step 1, Equation (4)) is the threshold value at which small visual differences between a given pixel and its neighborhood are eliminated. At small values, we obtain numerous white dots corresponding to either miscellaneous background details or cell fragments. At high values, all cells are homogenized into the background and therefore not detected. NI, the number of iterations of Step 1, has an upper bound, beyond which the cell profiles become too blurry for correct detection. When NI is appropriately smaller than such upper bound, it sufficiently homogenizes the background noise without blurring the cell profile. S is used to identify the cell profile (Step 3). S has an upper bound below which it identifies cell profiles with small pixel gradients. The use of small pixel gradients has the benefit of identifying the complete cell, at the expense of multiple cell aggregation. Higher values of S tend to identify only the high contrast between the nucleus and the stained background leading to reduced cell aggregation, but also to partial cell detection. For stains that do not provide good contrast between cell on plane of focus and background, a low value for S is necessary to detect the small pixel gradients defining the cell profiles. For stains that provide high contrast between the majority of the cells in focus and background, a higher value of S is suitable. Alo and Aup (Step 5) control the area of the detected shape: Alo eliminates leftover minute fragments and Aup eliminates oversize shapes that cannot represent cells.

With appropriate values of the parameters, the CDA analysis confirmed known facts regarding the mouse growth plate in tibia and femur [

The CDA captures more cell area (per cell) in the proliferative zone because it picks up the parts of the cell which are stained: the CDA is far more sensitive to small gradients in color change, hence it is better able to pick up the color transition between the background and cytoplasm even if the stain bleeds into the cell. We

Zones | Cell data | |||
---|---|---|---|---|

Cell count | Area (µm^{2}) | Orientation (degrees) | Shape (IR, 0 to 1) | |

Resting | 352 | 516.75 ± 472.71^ | 5.13 ± 45.18^ | 0.79 ± 0.18 |

Proliferative | 1002 | 760.24 ± 738.05^ | −4.78 ± 25.86^ | 0.71 ± 0.17 |

Hypertrophic | 181 | 1492.39 ± 1840.17^ | −6.05 ± 31.90^ | 0.66 ± 0.17^ |

found that the mean cell area is significantly smaller in the proliferative zone than in the hypertrophic zone; and that the mean orientation differs significantly between resting and proliferative, but not between proliferative and hypertrophic zones. Further, the mean isoperimetric ratio IR is significantly different between resting and proliferative zones, and between proliferative and hypertrophic zones, indicating that the cells become less circular as they progress from the resting to proliferative, and from proliferative to hypertrophic zone. Furthermore, our CDA confirms the rounder shape, and the orientation off the horizontal (medial-lateral) direction, of chondrocytes in the Smad1/5^{CKO} mutant in comparison with the control WT mouse (

The 390 cells were chosen on the plane of focus of a total of three images by human eyesight judgment. The chosen cells showed presence of the 92 to 100 range of Red, 62 to 100 of Green and 55 to 100 for Blue. The out-of focus cells appearedoverall darker with 86 to 100 range of Red, 49 to 97 of Green, and 45 to 100 of Blue. The inter- and intra-observer errors for a unique measurement were found to be smaller than 1.5%. We show the data of the calculation of inter- and intra-observer errors concerning the cell area from

The best detection of the CDA in comparison to MCD occurs with a 5% difference on 390 cells detected on three images, a Dice similarity index of 0.88 in cell number, and no significant differences for cell position, area, orientation, height, shape factor in terms of isoperimetric ratio. In particular, this means that the convexification step in the CDA did not produce a shape with significant distortion. The aggregation of adjacent cells was present in all zones of the growth plate when single values were assigned to each of τ and S for the whole growth plate. To decrease such aggregation, the parameters τ and S need to vary across the growth plate, specified for each of resting, proliferative and hypertrophic zones, due to variation of relative distance among cells.

Parameters | Cell area (µm^{2}) | |||
---|---|---|---|---|

Obs1, I | Obs1, II | Obs 2, I | Obs 2, II | |

Min | 486.00 | 485.60 | 482.80 | 477.60 |

Max | 488.00 | 487.60 | 484.00 | 480.80 |

Mean | 486.90 | 486.30 | 483.20 | 479.40 |

Stdev | 0.64 | 0.72 | 0.44 | 1.04 |

Comparison between | I and II by Obs 1 | I and II by Obs 2 | Obs 1 and 2 for I | Obs 1 and 2 for II |

% error p | 0.49 | 1.09 | 0.99 | 1.34 |

% error u | 0.08 | 1.42 | 0.83 | 0.67 |

The hypertrophic zone presents the highest challenge to the CDA because the cells are tightly packed and their profiles frequently touch (^{*}indicates significant difference in height with

We have developed a specialized algorithm, implemented in MATLAB, which performs the automated detection of cells in growth plate images. We have used the algorithm to verify known biological characteristics of the mouse growth plate and provide a tool for further research regarding, e.g., implications of height of hypertrophic chondrocytes on limb length.

During the process of developing CDA, we considered current methods for image detection. Canny segmentation, cartoon-texture decomposition (Figures 6(a)-(c); [

Recently, Buggenthin et al. [

Parameters | Hypertrophic chondrocytes | ||||
---|---|---|---|---|---|

S | Alo | Cell count | Height | ||

7 | 130 | 80 | 181 | 34.33 ± 22.57 | |

7 | 120 | 400 | 136 | 40.16 ± 21.56^{*} | |

5 | 120 | 400 | 161 | 37.05 ± 30.64 | |

5 | 145 | 400 | 148 | 32.33 ± 16.61^{*} |

a gray-scale image obtained under bright field microscopy. This algorithm uses a machine-learning based background correction method, which plays a role analogous to that of Retinex and anistropic diffusion in our CDA. After correction, both CDA and Buggenthin algorithm use a hard-thresholding method to identify cells. The Buggenthin algorithm includes a marker based water-shedding method to separate the clumped cells, while our CDA lacks such step. Therefore, when our CDA is applied to the image of hematopoietic stem cells, aggregation of cells occurs, and 5% of cells, small ones, were not detected. However, when we applied the Buggenthin algorithm to our colored growth plate images, it does not perform as well as the CDA. In fact, it produces a single-colored image of pure background because our colored growth plate images have a considerably high degree of background heterogeneity, and the color of many cells is not significantly different from their neighborhood background.

Complexity and local variations of the images are such that the automation of the CDA does not extend to the choice of the values of the parameters τ, NI, S, Alo and Aup that need to be chosen by the user. In fact, the optimal values of these parameters can only be determined by user’s experiment and experience on the specific image. We note that the cells on focus need to be resolved enough on the image to be clearly visible by human eye against the background, for the CDA to be able to detect them. Presence or absence of stain of the nucleus does not interfere with appropriate cell detection with the CDA. In fact, while the black stained nucleus causes detection of only the cell around the nucleus, the convex hull operation fills up the space occupied by the nucleus, therefore providing results similar to the results obtained for images with unstained nucleus. Also, if the

CDA leaves a few cells lumped together, a quick analysis of the morphometry data outliers can assess whether such lumping affects the mean height of the distribution. Such outliers can be excluded if needed.

In conclusion, we have presented here an innovative algorithm that performs accurate cell detection of growth plate images. In particular, our CDA can provide rapid measurement of height of hypertrophic chondrocytes, for comparison to height of hypertrophic zone, of growth plate size and limb length, for a first-screening before investigations on phases of volume increase. This method is applicable to specific genetic mice models to investigate the biological mechanism of limb lengthening.

The authors thank F. D’ Almeida for use of Nonlinear Diffusion Toolbox (http://www.mathworks.com/matlabcentral/fileexchange/3710-nonlinear-diffusion-toolbox); J. Kimmel for algorithm verification; and L. Vese for discussions. This study was partially supported by National Institutes of Health Grant AR044528 to K. M. Lyons and National Science Foundation Grant DMS-1045536 to A. Bertozzi.

We used a smaller and a larger image to compute running time and memory usage of the CDA for Steps 1 - 5, separately, with τ = 7, S = 130, Alo = 80, Aup = 20,000. Note that there is no difference in memory usage between different values for NI. Further, the increasing number of iterations increases the running time of Step 1 and reduces the run time of the last Steps.

The computational cost of running the CDA on an M × N pixel image equals

where P(MN) denotes a positive integer smaller or equal to MN. The contribution of each step to the computational cost is explained here. In Step 1, because we are only looking at local variations, we only consider 4 paths for each pixel, which are from the pixel of interests to its left, right, top and bottom pixels (2). By counting the paths in terms of pixels, we obtain, for NI = 1, the computational cost of 4(MN – 2M – 2(N – 2)) – 3 (2(M – 2)) – 3(2(N – 2)) – 2(4) = 4MN – 2M – 2N. For NI > 1, we have NI (4MN – 2M – 2N).

Step 2 contributes with at most 10(12MN − 4M − 4N + 2(P(MN))). The function “aosiso” employs an additive operator splitting scheme to generate the solution of (10). Its input consists of the image matrix x (of the size M × N), diffusivity matrix D of size M × N (11) and time step t (constant). The following procedure is repeated ten times, each time with updated image matrix x and diffusivity matrix D, obtained from (11) with new λ.

1) A zero matrix of size M × N is defined as y and a zero matrix of size M × N is defined as p, at no computational cost (variable initialization).

2) A matrix q, of size (M − 1) × N, is constructed by adding two sub-matrices of D together. The first sub-matrix is of size (M − 1) × N, which contains M − 1 rows (the first row to the second to last row of D). The second sub-matrix is also of size (M − 1) × N, which also contains M − 1 rows (the second row to the last row of D). The computational cost is (M − 1)N.

3) The first row of p is set equal to the first row of q and the last row of p is set equal to the last row of q, at no computational cost.

4) The M − 2 rows of p, which are not first and last rows, are constructed by adding two sub-matrices of q obtained from 2). The two sub-matrices are both of size (M − 2) × N. The first one is q without the last row; the second one is q without the first row. The computational cost is (M − 2)N.

5) Matrix a is constructed by multiplying p obtained from 4) by the time step t and then adding it to a matrix of size M × N whose entries equal 1. Matrix a will be used in the Thomas algorithm. The computational cost is 2MN.

6) Matrix b is constructed by multiplying q by the scalar −t. Matrix b will be used in the Thomas algorithm. The computational cost is (M − 1)N.

7) The Thomas algorithm is used to solve a tridiagonal linear system for each column of the input image matrix. The solution is a new matrix, y, of size M × N. For each column, the computational cost is at most M, P(M). Because there are N columns, the computational cost is P(MN).

8) Repeat 1) to 7) using the transpose of D and the transpose of x. We obtain a matrix y of size N × M. The computational cost is the same as 2) to 7) with M and N switched.

9) Add the y obtained from G) to the transpose of y obtained from 8). We define the result as the new matrix y of size M × N. The computational cost is MN.

10) Divide the y from 9) by 2. The computational cost is MN.

Step 3 has a computational cost of MN because the gradient threshold compares each pixel’s color code gradient to the gradient threshold. Step 4 has a computational cost of P(MN) because the number of the convex hull operations is at most MN. The black pixels that are inside the boundaries of each polygon generated through convex hull operation are then transformed into white pixels through MATLAB built-in function “poly 2 mask” and the computational cost is P(MN). Step 5 has also a computational cost of P(MN), since the number of polygons examined for size and shape thresholds equals the number of convex hull operations.