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A semi-automatic algorithm for measurement of basement membrane thickness in kidneys in electron microscopy images
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 9 7 ( 2 0 1 0 ) 223–231
journal homepage: www.intl.elsevierhealth.com/journals/cmpb
A semi-automatic algorithm for measurement of basement membrane thickness in kidneys in electron microscopy images Hai-Shan Wu ∗ , Steven Dikman, Joan Gil Department of Pathology, Box 1194, Mount Sinai School of Medicine, One Gustave L. Levy Place, New York, NY 10029, USA
a r t i c l e
i n f o
a b s t r a c t
Article history:
In this paper, we present a semi-automatic algorithm for measurement of the glomerular
Received 10 November 2008
basement membrane thickness in electron microscopy kidney images. A string of sparsely
Received in revised form 2 July 2009
spaced points are manually inputted along the central line of the basement membrane
Accepted 6 July 2009
(lamina densa) to be measured. The gaps between successive input points are lineally interpolated. A nonlinear mapping is applied to straighten the curved central line. Two dis-
Keywords:
tance functions of edges to the central line are constructed. The smooth envelope lines are
Image segmentation
obtained by repetitive applications of a linear low-pass filtering followed by a comparing and
Electron microscopy
selecting process. The boundaries of the glomerular basement membrane are obtained from
Kidney
the inverse mapping of the envelope functions. The average basement membrane thick-
Tissue
ness is estimated as the ratio of the basement membrane area to the length of the central
Glomerular basement membrane
line. © 2009 Elsevier Ireland Ltd. All rights reserved.
thickness
1.
Introduction
The basement membrane (BM) thickness is a very important feature in diagnosis of diseases [1–13]. The thickness of capillary basement membrane (CBM) may increase in diabetic patients and in animal models of diabetes [2]. The glomerular basement membrane (GBM) thickness increases significantly after 5 years of diabetes [4]. Hematuria of glomerular origin implies the development of alterations in the GBM such as thin basement membrane disease (TBMD) [5,6]. Patients with proteinuria and thick glomerular basement membrane must be differentiated from minimal-change nephropathy for therapeutic implications, and specifically managed for its strong association with prediabetes or early diabetes [7]. The onsets of albuminuria in early diabetes mellitus are
∗
accompanied by GBM alterations with a significant increase in true harmonic mean GBM thickness [8]. Although the pathogenesis of diabetic basement membrane disease is not completely understood, GBM thickening is widely regarded as a morphological consequence of hyperglycemia [9]. A detailed morphometric analysis of GBM thickness confirms the presence of thin GBMs in some glomerular diseases associated with hematuria and GBM thinning has been reported in half of patients with a diagnosis of minimal-change disease [10]. Focal segmental glomerulosclerosis and minimal-change nephropathy are often associated with incidental thin GBM nephropathy [11]. In kidney transplantation, the multilayering of the peritubular CBM by electron microscopy has been recognized as a novel diagnostic marker of chronic rejection [12].
Corresponding author. Tel.: +1 212 241 0468. E-mail address: [email protected] (H.-S. Wu). 0169-2607/$ – see front matter © 2009 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.cmpb.2009.07.002
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Automatic measurement of basement membrane thickness is difficult. Algorithms for the measurements are still performed manually with participations of users. Sue et al. [1] estimated the thickness of the GBM by using the arithmetic mean of a series of orthogonal intercept measurements. In an effort to randomize points of measurement, Carlson et al. [2] used a clear plastic sampling grid superimposed on micrographs and measured the GBMs wherever they intersected the sampling grid lines. Osawa et al. [3] measured the thickness of basement membranes at 100 positions in each tissue using the high powered electron micrographs. Thomsona et al. [4] calculated the GBM thickness by drawing a perpendicular line from the endothelial to the epithelial (podocyte) edge of the GBM. For each baboon, an average of 215 intercepts were measured [4]. Manual measurements of the thickness of basement membranes were also used in studies in glomerular diseases [5–8]. The orthogonal intercept procedures of GBM thickness measurements were also carried out for studies in GBM thickening [9,12], in thin glomerular basement membranes [10,11]. Marquez et al. [14] estimated the GBM thickness using a maximum number of measurements taken along the GBM at every 1/3 m actual size. Basta-Jovanovic utilized an interactive image analysis system with a touch sensitive screen and a grating replica of the same magnification as the electron micrographs [6]. Basically, current methods used for estimation of the GBM thickness rely mostly on manual measurements [1–14]. The
workload of such manual methods is very expensive and the results are sensitive to users’ inputs and may vary significantly from one use to another. There is a need to develop an algorithm that requires minimum manual inputs and is insensitive to the slight difference in manual inputs. In this paper, we intend to introduce such an algorithm producing an average thickness of GBM segment defined by a limited number of central points specified by a user. The algorithm is insensitive to the locations of the manually specified central points since it finds the GBM boundaries based on the image contents.
2.
Measurement of GBM thickness
Let an electron microscopy image be represented by a 2D discrete sequence x = {x(n1 , n2 )} whose elements are valued between 0 and 255 and defined in a rectangular domain of S = {(n1 , n2 )|0 ≤ n1 < N1 ; 0 ≤ n2 < N2 }. For convenience of explanation, we use the original image in Fig. 1(a) as an example. If we manually specify a sequence of center points along the center of the GBM as shown in Fig. 1(b) the cross marks, we have a vector sequence c = {c(k)|0 ≤ k < K}, where c(k) = (c1 (k), c2 (k)), and K is the total number of center points manually inputted. The vectors are sparsely located along a segment of the GBM. To have a continuous central line of the GBM, the gap between two neighboring vectors is filled with a linear line. The procedure is as follows:
Fig. 1 – (a) Original image acquired at magnification of 2k, (b) the image superimposed with the manually specified vector sequence along a segment of the GBM, (c) the sequence, , the smooth middle line, and the inverse-mapped f+ and f− lines on its two sides, (d) the segmented GBM segment indicated by the shape.
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The procedure produces a sequence of positions, = {(m)|m = 0, 1, 2, . . ., M − 1}, where the component (m) is a vector, i.e., (m) = { 1 (m), 2 (m)). If placing a white point at the location of each vector in the sequence, we will have a smooth and unbroken white line along the central area of the GBM as shown in Fig. 1(c). The line indicates the central line in the segment of the GBM to be measured. If we straighten the smoothly curved line and map the image accordingly, we will have a segment of straight GBM as shown in Fig. 2(a). Let a nonlinear mapping, ϕ/4 , transform a portion of the image in the neighborhood into a rectangular sub-image, y, such as y = ϕ/4 (x). The subscript, /4, is used for this particular segment of membrane for the reason that the angles between the line and the horizontal axis are around /4. The mapped image y y y = {y(n1 , n2 )|0 ≤ n1 < N1 , 0 ≤ n2 < N2 }, where the image width y y N1 = M, the image height, N2 , is an odd integer selected significantly higher than half of the maximum width of the basement membrane, and the image components are determined by
y(n1 , n2 ) = x
y
n2 + 1 (n1 ) −
N2 − 1 2
y
, n2 + 2 (n1 ) −
N2 − 1 2
.
(1)
Fig. 2(b) shows the mapped image of y from the original image x in Fig. 1(a) based on the line in Fig. 1(c). The central line in the GBM in Fig. 2(b) appears straight. At the edges of the GBM, there are noticeable intensity discontinuities. These intensity discontinuities, in contrast to the homogeneities among the intensities inside membrane regions, can be used as stoppers when determining the membrane boundaries. Since the central line is located in the middle area of the GBM, the line and its vicinity pixels are expected to be inside the GBM. If an integer, , is small enough, y those pixels in the image y for which |n2 − ((N2 − 1)/2)| ≤ can be considered belonging to the GBM. Let this narrow rectangular region be represented by Sstrip , i.e., Sstrip = y {(n1 , n2 )|(0 ≤ n1 < M)&(|n2 − ((N2 − 1)/2)| ≤ )|}. Since the GBM
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Fig. 2 – (a) Local image neighboring the GBM central line after the line squeezed straight, (b) the converted image in (a) without the superimposed straight white central line, (c) the absolute error image between (b) and the average in the neighborhood of central line segment, (d) smoothed image of (c), (e) an automatic threshold based on the highest value in the central neighborhood in image (d), (f) the discrete functions f+ (n1 ) and f− (n1 ), as shown in white dots at y y positions (n1 , f + (n1 ) + (N2 − 1)/2) and (n1 , (N2 − 1)/ − 2 − f (n1 )), corresponding to the floor and ceiling of the non-GBM pixels above and below the central line, respectively, where n1 denotes the horizontal axis.
has a relatively homogeneous intensities, the average intensity inside Sstrip , a narrow and long strip of dimension of M × (2 + 1) at the center of the image y, can be an appropriate approximation to the actual intensity average of the GBM. Thus, the average intensity of the GBM is estimated by M−1 y = (1/(M(2 + 1))) m =0 m =− y(m1 , m2 + ((N2 − 1)/2)). We 1 2 compose an image, e, from the absolute error between the image y and the average value , such as e(n1 , n2 ) = |y(n1 , y y n2 ) − | for 0 ≤ n1 < N1 and 0 ≤ n2 < N2 . It is expected that a pixel inside the GBM will have a low intensity in the image e. Fig. 2(c) shows the image e, the absolute difference of the image in Fig. 2(b) and , the estimate of the GBM average intensity.
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Fig. 3 – (a) Dashed line for f+ (n1 ) above the central line y n2 = (N2 − 1)/2 = 50, and solid line for its lower envelope + e (n1 ), (b) dashed line for f− (n1 ) below the central line and solid line for its upper envelope e− (n1 ).
Although the intensities inside GBM vary in a very limited range, occasional spikes may cause errors if a thresholding operation is applied on the intensities. To reduce the effects of local spikes, a low-pass filter is applied. Fig. 2(d) is the image g, the moving average of the input image in Fig. 2(c) filtered by a moving window of size 3 × 3, such 1 1 as g(n1 , n2 ) = (1/9) m =−1 m =−1 y(n1 − m1 , n2 − m2 ) for 0 ≤ 1
y
2
y
n1 < N1 and 0 ≤ n2 < N2 . The range of the GBM intensities in g can be written as [G− , G+ ], estimated by G− = 0 and G+ = G + max{g(n1 , n2 )|(n1 , n2 ) ∈ Sstrip } where G is a small value purposely added to increase the intensity range in Sstrip by a small amount. Pixels of g beyond this intensity range will be considered outside of the GBM. Thus, we have a binary image, g , converted by
g (n1 , n2 ) =
0, if g(n1 , n2 ) ≤ G+ , 1, otherwise, y
y
for 0 ≤ n1 < N1 and 0 ≤ n2 < N2 . The pixel at (n1 , n2 ) is a point outside of GBM if g (n1 , n2 ) is true or 1. Converted from the gray-level image in Fig. 2(d), Fig. 2(e) shows the resulting binary image in which black is for zero and white is for one. It is observed from g that white pixels are located y in some distances from the centerline of n2 = (N2 − 1)/2. We + − compose two functions, f (n1 ) and f (n1 ), for the two distances to the central line with respect to the horizontal axis of n1 , such as f + (n1 ) = min−1 n −(Ny −1)/2 {(g (n1 , n2 ) = 1)&(n2 >
y (N2 y (N2
− 1)/2)} and
f − (n1 )
= min
−1
2
2
y {(g (n1 , n2 ) (N2 −1)/2−n2
= 1)&(n2 <
− 1)/2)}. The function is the smallest positive integer y of n2 − (N2 − 1)/2 for which the image pixel g (n1 , n2 ) is white, y and f− (n1 ) is the smallest positive integer of (N2 − 1)/2 − n2 for which the image pixel g (n1 , n2 ) is white. Fig. 2(f) shows y the two functions where pixels at (n1 , f + (n1 ) + (N2 − 1)/2) and y − (n1 , (N2 − 1)/2 − f (n1 )) are marked with white dots. Fig. 3(a) shows, in dashed line, the function f+ (n1 ) above the central line y of n2 = (N2 − 1)/2 = 50, while Fig. 3(b) shows, also in dashed line, the function f− (n1 ) below the central line. With the inverse −1 , the functions of f+ (n1 ) and f− (n1 ) will be mapping of ϕ/4 f+ (n1 )
mapped to the two lines on both sides of the line in Fig. 1(c), respectively. It is assumed that any pixel at (n1 , n2 ) for which g (n1 , n2 ) is white is not a pixel in the GBM. Since the GBM has smooth boundaries on both sides, we extract a smooth envelope line for each side of boundaries such that each envelope line is in tangent touch to the peaks of the waveform it covers. If e+ (n) is a lower envelope for f+ (n), it has to satisfy the following three conditions: (1) e+ (n) ≤ f+ for 0 ≤ n < M; (2) e+ (n) is a smooth function of n, meaning that |e+ (n + 1) − e+ (n)| for 0 ≤ n < M − 1 are relatively small; and (3) e+ (n) line is in tangent with f+ (n) line at some lowest points, i.e., e+ (n) = f+ (n) for some points among the lowest trough of f+ (n) line. We present an iterative procedure to obtain, from the function f+ (n), its envelope e+ (n). The other envelope function e− (n) can be obtained from the function f− (n) in a similar manner. Let h(n) be the finite impulse response (FIR) of a low-pass filter. If the input is fk (n), the output of the filter will be k (n) = h(n) ∗ fk (n), where the sign, ‘*’, represents the discrete convolution operation. Compare k (n) with fk (n) and select the smaller one as the system output fk+1 (n), i.e., fk+1 (n) = min{k (n), fk (n)}.
(2)
This output, fk+1 (n), is fed back to the same system as the new input if the termination condition is not satisfied. The initial iteration number is set as k = 0 and the upper distance function, f+ (n), is set to be the initial image, such as of iterations, the diff0 (n) = f+ (n). With the increasing number y ference εk+1 = max{ |fk (n) − fk+1 (n)| 0 ≤ n < N1 } will decrease and finally converge to zero if iteration number goes to infinite high. When the difference is small enough, say εk+1 < 0.1, the iterations can be terminated and the system produces the envelope function e+ (n) = fk+1 (n). Fig. 3(a) shows the distance function of f+ (n) in the dashed line with its envelope, e+ (n), in the solid line. In a similar procedure except replacing the min{·, ·} operation with the max{·, ·} operation in Eq. (2), we obtain the other envelope function, e− (n), in the solid line in Fig. 3(b), from the distance function, f− (n), in the dashed line. −1 , the solid lines of e+ (n) Taking the inverse mapping of ϕ/4 − and e (n) are mapped to the smooth lines along the GBM boundaries on both sides of the GBM. Filling the two ends of the curves of e+ and e− with two straight lines, we have an enclosed belt region as shown in Fig. 1(d). The length of the belt can be estimated by
L=
K−2
2
2 1/2
((c1 (k + 1) − c1 (k)) + (c2 (k + 1) − c2 (k)) )
.
(3)
k=0
Let A represent the area in the shape drawn by the two boundary lines on both sides of and the two straight end lines as shown in Fig. 1(d). The area, A, can be estimated by simply counting the number of pixels inside the enclosure. The average thickness, , can then be estimated by = A/L. The unit of is the size of a single pixel. The actual GBM thickness is obtained by applying on the factor of the metric width of a single pixel. We summarize the algorithm as the following. At the beginning, the user specifies a sequence of scattered points (cross
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marks in Fig. 1(b)) along the central line of the GBM segment. The gap between the two neighboring user-specified points is filled by a linear interpolation. Thus, a consecutive sequence of locations is obtained as the central line (Fig. 1(c)). Mapping the central line into a straight horizontal line, we have a smaller local image vertically centered at the central line (Fig. 2(b)). Evaluating the average intensity in the narrow region along the central line, we then calculate the absolute difference (Fig. 2(c)) between the image (Fig. 2(b)) and the average intensity of the
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GBM segment. To reduce the effects of local spikes, a low-pass filter is applied to obtain a smoother image (Fig. 2(d)). The intensity range in the narrow region along the central line is evaluated. All pixels whose intensities are out of this intensity range are considered outside of GBM and indicated by white color in Fig. 2(e). We find the two sequences of points in the white regions vertically closest to the central line on both the upper and lower sides of the central line as shown the white dots in Fig. 2(f). Two 1-D functions for the upper and lower
Fig. 4 – (a) Original image acquired at magnification of 2k, (b) user specified sequence shown by the cross signs along the segment of BM to be measured, (c) two segmented sub-BM segments indicated by the shapes.
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dots are plotted in Fig. 3 as the dashed lines. Their respective smooth envelopes are plotted as the solid lines in Fig. 3. The inverse mappings convert the envelope lines back to the original image space as the GBM edges in Fig. 1(d). The GBM segment region is obtained after adding the two straight end lines. The GBM thickness is estimated as the ratio of the area of the extracted GBM region to the length of the central line.
3.
GBM with a wide-angled sequence
If a straight line determined by two separate points (c1 (k), c2 (k)) and (c1 (k + 1), c2 (k + 1)), for 0 ≤ k < K − 2, the slope of the line is (k) = (c2 (k + 1) − c2 (k))/(c1 (k + 1) − c1 (k)). The angle between the line and the horizontal axis is k = tan−1 ( (k)) if c1 (k + 1) − c1 (k) > 0, and k = tan−1 ( (k)) + if c1 (k + 1) − c1 (k) < 0. The angle is between −/2 and −3/2. To have the angle range of [0, 2), we increase k by 2 when c1 (k + 1) − c1 (k) > 0 and c2 (k + 1) − c2 (k) < 0. When c1 (k + 1) − c1 (k) = 0, k is either /2 or 3/2 depending on the sign of c2 (k + 1) − c2 (k). The angle range, [min{ k |0 ≤ k < K − 2}, max{ k |0 ≤ k < K − 2}], for the GBM segment in Fig. 1(c) is relatively narrow. A single mapping works well since the inverse mapping is unique. However, if the angle range is very wide, one mapping may not be enough. Fig. 4(a) shows an original image with a long segment of GBM. In Fig. 4(b), the GBM segment is manually marked with a sequence of locations, c, whose angle ranges from just above /4 to just below 5/4 as shown in Fig. 5. Since the angle ranges close to , we have to partition the GBM segment into two or more sub-segments so that each GBM sub-segment has a narrower angle range to have a unique inverse mapping. Let = min{ k |0 ≤ k < K − 2}, the minimum angle, and = min{ k |0 ≤ k < K − 2} the maximum angle. Both and are inside [0, 2). The total range of [0, 2) is partitioned into eight periods, each of which covers /4. Thus, we have a discrete sequence { k |0 ≤ k < K − 2} whose component k = [4 k /], val-
Fig. 5 – k , the angle between the central line and the horizontal axis vs. k, the index of the input points, for 0 ≤ k < K − 1, where K is the total number of central points inputted manually.
ued as an integer between 0 and 7, where the floor function [·] yields the highest integer below the float variable. Since each integer in k represents /4 in angle, the sequence in a single GBM sub-segment should have a range narrower than 4 or an angle range less than or equal to 3/4. If the total angle range is wider than 3/4, we have to partition the whole sequence of the input central points into multi sub-sequences so that each sub-sequence has an angle range narrow enough. We start with k = 0 and then search through the { k } sequence until a given angle range is about to be exceeded the limit. The procedure for the first GBM sub-segment is as the following: (1) Initialization step: Let k = 0. (2) Iteration step: If max{ m |0 ≤ m ≤ k} − min{ m |0 ≤ m ≤ k} ≥ 2, terminate and output k, otherwise let k = k + 1 and repeat this iteration step. The above procedure separates the first GBM sub-segment from the whole segment. The procedure can be repetitively applied to separate the subsequent sub-segments, one by one, until the whole GBM segment is exhausted. The angle sequence for the GBM segment in Fig. 4(b) is depicted in Fig. 5 with a range of [/4 + 0.0238, 5/4 − 0.0052]. With the above procedure, the GBM segment is separated into two sub-segments with cut-off angle at 3/4. The middle solid horizontal lines for the two separate angle ranges are /2 and , respectively. Thus, the two mappings are ϕ/2 and ϕ . Each GBM sub-segment is processed in a similar way as that in the previous section where an individual GBM segment is segmented. The average GBM thickness is then calculated as the ratio of the total area enclosed in the multiple shapes and the estimated total length of the GBM segment.
4.
Results
The studies detailed below regarding normal glomerular basement membranes utilized a core needle biopsy from a 63year-old man who had a liver transplant 4 years prior to the biopsy and recently developed proteinuria. The biopsy showed focal segmental glomerulosclerosis and the basement membranes of the non-sclerotic glomeruli were normal by light and electron microscopy. Fig. 1(a) shows an original image of a human renal biopsy specimen fixed in glutaraldehyde acquired by a transmission electron microscope (Hitachi H700) at magnification of 2k. The digital dimension of the image is 469 × 432. The component of GBM evaluated is the lamina densa which appears as a central broader zone of gray intensities with limited fluctuations and sandwiched by subepithelial and subendothelial thin lighter regions (lamina rara interna and lamina rara externa) situated beneath the endothelial cell and epithelial cell (podocyte). Along the GBM central line, we manually specified a sequence of sparsely spaced points as shown in Fig. 1(b) with the plus signs. There are totally 17 manually inputted points in this case. Sparser points can be used at the smooth portion of the GBM. However, in the places of the more curved GBM, denser points are needed so that the central line, , drawn along the specified points, is laid approximately in the middle of the GBM as shown in Fig. 1(c). Fig. 1(c) also shows the two edge
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lines located at both sides of from the inverse mapping of the f+ (n) and f− (n) functions, respectively. The GBM segmentation based on the input central points in Fig. 1(b) is illustrated in Fig. 1(d) as the shape enclosed by the two curved side lines from the inverse mappings of the envelope functions and the two straight end lines. The area, which is the total number of pixels inside the enclosed shape, can be counted automatically, while the length of the GBM segment can be estimated by L in Eq. (3) based on the input central points. The ratio of the area to the length L gives the estimate of the average GBM thickness. The total number of pixels in the enclosure in Fig. 1(d) is counted to be 8140 pixels, while the length of the central line is estimated as 457.24. The average GBM thickness is then calculated as 8140/457.24 = 17.803 pixel units. Applying the factor of 0.020838 m/pixel, we have the average metric GBM thickness of 17.803 × 0.020838 = 0.37098 m or 371 nm. Fig. 4(a) shows another digital GBM image of size 350 × 579. A series of 23 central points are manually inputted as shown
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in Fig. 4(b) marked by the 23 crosses. The angle of the line segments angels from /4 to 5/4. The GBM is partitioned into two sub-segments. Starting from the first point of the lowest cross in Fig. 4(b), moving up along the central line and stopping just before the angle reaches 3/4. The first sub-segment includes the lower 13 points, and the second one includes the upper 11 points. The 13th point is the last in the first subsegment and also the first in the second sub-segment. The central line of the first sub-segment by linearly connecting the lower 13 points is 436.67 pixels long, while the central line of the second sub-segment by linearly connecting the lower 11 upper points is 218.89 pixels long. The complete length of the GBM segment is then 436.67 + 218.89 = 655.56 pixels long. Fig. 4(c) shows the two shapes for the two subsegments. The numbers of the pixels in the two shapes are counted as 7600 and 4499, respectively. Thus, the average GBM thickness measured by the ratio of the total area to the length of central line is 7600 + 4499/655.56 = 18.456 pixel
Fig. 6 – (a) An original GBM image acquired at magnification of 5k, (b) the original image in (a) superimposed with the crosses for the input central points and the shape for the resulting GBM segmentation, (c) another original GBM image acquired at magnification of 3k, and (d) the image in (c) superimposed with the crosses for the input central points and the shape for the resulting GBM segmentation.
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units. The average metric GBM thickness is calculated as 18.456 × 0.020838 = 0.384586 m or 384.6 nm. Fig. 6 shows two more GBM images in (a) and (c) and their respective resulting segmentations in (b) and (d). In Fig. 6, (a) displays an original digital image, size of 563 × 765, acquired at magnification of 5k, while (c) shows another original digital GBM image, size of 698 × 742, acquired at magnification of 3k. In Fig. 6(b), the central line is calculated as 434.74, and the shape contains 21,646 pixels. The GBM thickness is then 21,646/434.74 = 49.79 in pixel units. At the magnification of 5k the scale is 0.0083352 m/pixel. Thus, the average metric GBM thickness is calculated as 49.79 × 0.0083352 = 0.41501 m or 415 nm. In Fig. 6(d), the central line is 832.20 pixels long, and the closed shape contains 22,351 pixels. The average GBM thickness is then calculated as 22,351/832.20 = 26.86 pixels thick. At the magnification of 3k the scale is 0.013892 m/pixel. Thus, the average metric GBM thickness is calculated as 26.86 × 0.013892 = 0.373139 m or 373.1 nm. Fig. 7 shows the application to the case of thin basement disease. The patient studied was from a 39-year-old woman
with a family history indicating that both her mother and sister had hematuria. The patient had microscopic hematuria and mild proteinuria. Electron microscopy demonstrated thin lamina densa of the glomerular basement membranes. An evaluation of the patient’s biopsy tissue for abnormalities of type IV collagen to evaluate for Alport’s syndrome utilized monoclonal antibodies against alpha 1, 3 and 5 chains of type IV collagen (Wieslab reagents: normal human kidney and skin utilized as controls) indicated no abnormality. The image was taken with the magnification of 7k×. The digital dimension is 712 × 1170 and the metric dimension is 5.8361 m × 9.5902 m. The scattered cross marks along the central line in the GBM segment are the locations manually specified by a user. The two curved lines along the GBM boundaries are the derived edges. The estimated GBM thickness, the ratio of the area enclosed by the surrounding boundaries to the length of the interpolated central line, is 193.6 nm.
5.
Conclusions
In this paper, we have described an approach for estimation of average GBM thickness. Contrary to the usual manual measurements whose reliability is largely dependent on the skills of users, the proposed algorithm only requires minimum user inputs whose positions are much less critical and determines the GBM thickness based on the local contents in the images. In the case of a narrow angle range, an appropriate mapping of the local image into a smaller rectangular image converts the usually curved GBM segment into a straight one. Two distance functions of the non-GBM pixels to the straight central line on the two sides are evaluated and their smooth envelope functions are derived. The inverse mappings of the two envelopes produce the boundaries of the GBM segment. The average GBM thickness is estimated as the ratio of the area inside the recognized GBM segment to the length of the central line. Long GBM segments with wide-angled vector sequence of manual inputs are partitioned into multiple narrow-angled sub-segments prior to the segmentation procedures. The results on images of human renal biopsy specimen fixed in glutaraldehyde show the effectiveness of the proposed algorithm.
Conflict of interest statement There is no conflict of interest. We do not have any actual or potential conflict of interest including any financial, personal or other relationships with other people or organizations within that could inappropriately influence (bias) this work.
references
Fig. 7 – Image of thin basement membrane disease, magnification 7k×. Digital dimension 712 × 1170, metric dimension 5.8361 m × 9.5902 m. Estimated thickness is 193.6 nm.
[1] Y.-M. Sue, J.-J. Huang, R.-Y. Hsieh, F.-F. Chen, Clinical features of thin basement membrane disease and associated glomerulopathies, Nephrology 9 (2004) 14–18. [2] E.C. Carlson, J.L. Audette, N.J. Veitenheimer, J.A. Risan, D.I. Laturnus, P.N. Epstein, Ultrastructural morphometry of capillary basement membrane thickness in normal and transgenic diabetic mice, The Anatomical Record, Part A 271 (2003) 332–341.
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[3] T. Osawa, M. Onodera, X.-Y. Feng, Y. Nozaka, Comparison of the thickness of basement membranes in various tissues of the rat, Journal of Electron Microscopy 52 (2003) 435–440. [4] S.E. Thomsona, S.V. McLennana, P.D. Kirwand, S.J. Heffernanb, A. Hennessya, D.K. Yuea, S.M. Twigga, Renal connective tissue growth factor correlates with glomerular basement membrane thickness and prospective albuminuria in a non-human primate model of diabetes: possible predictive marker for incipient diabetic nephropathy, Journal of Diabetes Complications 22 (2008) 284–294. [5] S.M. Jalalah, I.H. Alzahrani1, P.N. Furness, Glomerular changes in microscopic haematuria, studied by quantitative immunoelectron microscopy and in situ zymography, Nephrology Dialysis Transplantation 17 (2002) 1586–1593. [6] G. Basta-Jovanovic, V.S. Venkataseshan, J. Gil, D.U. Kim, S.H. Dikman, J. Churg, Morphometric analysis of glomerular basement membranes (GBM) in thin basement membrane disease (TBMD), Clinical Nephrology 33 (1990) 110–114. [7] F.M.-M. Lai, C.-C. Szeto, P.C.L. Choi, K.K.L. Ho, N.L.S. Tang, K.-M. Chow, P.K.T. Li, K.-F. To, Isolate diffuse thickening of glomerular capillary basement membrane: a renal lesion in prediabetes? Modern Pathology 17 (2004) 1506–1512. [8] C. El-Aouni, N. Herbach, S.M. Blattner, A. Henger, M.P. Rastaldi, G. Jarad, J.H. Miner, M.J. Moeller, R. St-Arnaud, S. Dedhar, L.B. Holzman, R. Wanke, M. Kretzler, Podocyte-specific deletion of integrin-linked kinase results in severe glomerular basement membrane alterations and
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progressive glomerulosclerosis, Journal of the American Society Nephrology 17 (2006) 1334–1344. E.C. Carlson, R.C. Vari, J.L. Audette, M.A. Finke, M.J. Ressler, Significant glomerular basement membrane thickening in hyperglycemic and normoglycemic diabetic-prone bb wistar rats, Anatomical Record Part A 281 (2004) 1308–1318. B. Marquez, F. Stavrou, I. Zouvani, E. Anastasiades, C. Patsias, A. Pierides, K. Kyriacou, Thin glomerular basement membranes in patients with hematuria and minimal change disease, Ultrastructural Pathology 23 (1999) 149–156. M. Haas, Thin glomerular basement membrane nephropathy incidence in 3471 consecutive renal biopsies examined by electron microscopy, Archives of Pathology & Laboratory Medicine 130 (2006) 699–706. K. Aitaa, Y. Yamaguchib, S. Horitac, M. Ohnoc, K. Tanabed, S. Fuchinouee, S. Teraokae, H. Tomad, M. Nagataa, Thickening of the peritubular capillary basement membrane is a useful diagnostic marker of chronic rejection in renal allografts, American Journal of Transplantation 7 (2007) 923–929. A. Stacchiotti, A. Lavazza, R. Rezzani, R. Bianchi, Cyclosporine A-induced kidney alterations are limited by melatonin in rats: an electron microscope study, Ultrastructural Pathology 26 (2002) 81–87. B. Marquez, I. Zouvani, A. Karagrigoriou, E. Anastasiades, A. Pierides, K. Kyriacou, A simplified method for measuring the thickness of glomerular basement membranes, Ultrastructural Pathology 27 (2003) 409–416.
journal homepage: www.intl.elsevierhealth.com/journals/cmpb
A semi-automatic algorithm for measurement of basement membrane thickness in kidneys in electron microscopy images Hai-Shan Wu ∗ , Steven Dikman, Joan Gil Department of Pathology, Box 1194, Mount Sinai School of Medicine, One Gustave L. Levy Place, New York, NY 10029, USA
a r t i c l e
i n f o
a b s t r a c t
Article history:
In this paper, we present a semi-automatic algorithm for measurement of the glomerular
Received 10 November 2008
basement membrane thickness in electron microscopy kidney images. A string of sparsely
Received in revised form 2 July 2009
spaced points are manually inputted along the central line of the basement membrane
Accepted 6 July 2009
(lamina densa) to be measured. The gaps between successive input points are lineally interpolated. A nonlinear mapping is applied to straighten the curved central line. Two dis-
Keywords:
tance functions of edges to the central line are constructed. The smooth envelope lines are
Image segmentation
obtained by repetitive applications of a linear low-pass filtering followed by a comparing and
Electron microscopy
selecting process. The boundaries of the glomerular basement membrane are obtained from
Kidney
the inverse mapping of the envelope functions. The average basement membrane thick-
Tissue
ness is estimated as the ratio of the basement membrane area to the length of the central
Glomerular basement membrane
line. © 2009 Elsevier Ireland Ltd. All rights reserved.
thickness
1.
Introduction
The basement membrane (BM) thickness is a very important feature in diagnosis of diseases [1–13]. The thickness of capillary basement membrane (CBM) may increase in diabetic patients and in animal models of diabetes [2]. The glomerular basement membrane (GBM) thickness increases significantly after 5 years of diabetes [4]. Hematuria of glomerular origin implies the development of alterations in the GBM such as thin basement membrane disease (TBMD) [5,6]. Patients with proteinuria and thick glomerular basement membrane must be differentiated from minimal-change nephropathy for therapeutic implications, and specifically managed for its strong association with prediabetes or early diabetes [7]. The onsets of albuminuria in early diabetes mellitus are
∗
accompanied by GBM alterations with a significant increase in true harmonic mean GBM thickness [8]. Although the pathogenesis of diabetic basement membrane disease is not completely understood, GBM thickening is widely regarded as a morphological consequence of hyperglycemia [9]. A detailed morphometric analysis of GBM thickness confirms the presence of thin GBMs in some glomerular diseases associated with hematuria and GBM thinning has been reported in half of patients with a diagnosis of minimal-change disease [10]. Focal segmental glomerulosclerosis and minimal-change nephropathy are often associated with incidental thin GBM nephropathy [11]. In kidney transplantation, the multilayering of the peritubular CBM by electron microscopy has been recognized as a novel diagnostic marker of chronic rejection [12].
Corresponding author. Tel.: +1 212 241 0468. E-mail address: [email protected] (H.-S. Wu). 0169-2607/$ – see front matter © 2009 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.cmpb.2009.07.002
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Automatic measurement of basement membrane thickness is difficult. Algorithms for the measurements are still performed manually with participations of users. Sue et al. [1] estimated the thickness of the GBM by using the arithmetic mean of a series of orthogonal intercept measurements. In an effort to randomize points of measurement, Carlson et al. [2] used a clear plastic sampling grid superimposed on micrographs and measured the GBMs wherever they intersected the sampling grid lines. Osawa et al. [3] measured the thickness of basement membranes at 100 positions in each tissue using the high powered electron micrographs. Thomsona et al. [4] calculated the GBM thickness by drawing a perpendicular line from the endothelial to the epithelial (podocyte) edge of the GBM. For each baboon, an average of 215 intercepts were measured [4]. Manual measurements of the thickness of basement membranes were also used in studies in glomerular diseases [5–8]. The orthogonal intercept procedures of GBM thickness measurements were also carried out for studies in GBM thickening [9,12], in thin glomerular basement membranes [10,11]. Marquez et al. [14] estimated the GBM thickness using a maximum number of measurements taken along the GBM at every 1/3 m actual size. Basta-Jovanovic utilized an interactive image analysis system with a touch sensitive screen and a grating replica of the same magnification as the electron micrographs [6]. Basically, current methods used for estimation of the GBM thickness rely mostly on manual measurements [1–14]. The
workload of such manual methods is very expensive and the results are sensitive to users’ inputs and may vary significantly from one use to another. There is a need to develop an algorithm that requires minimum manual inputs and is insensitive to the slight difference in manual inputs. In this paper, we intend to introduce such an algorithm producing an average thickness of GBM segment defined by a limited number of central points specified by a user. The algorithm is insensitive to the locations of the manually specified central points since it finds the GBM boundaries based on the image contents.
2.
Measurement of GBM thickness
Let an electron microscopy image be represented by a 2D discrete sequence x = {x(n1 , n2 )} whose elements are valued between 0 and 255 and defined in a rectangular domain of S = {(n1 , n2 )|0 ≤ n1 < N1 ; 0 ≤ n2 < N2 }. For convenience of explanation, we use the original image in Fig. 1(a) as an example. If we manually specify a sequence of center points along the center of the GBM as shown in Fig. 1(b) the cross marks, we have a vector sequence c = {c(k)|0 ≤ k < K}, where c(k) = (c1 (k), c2 (k)), and K is the total number of center points manually inputted. The vectors are sparsely located along a segment of the GBM. To have a continuous central line of the GBM, the gap between two neighboring vectors is filled with a linear line. The procedure is as follows:
Fig. 1 – (a) Original image acquired at magnification of 2k, (b) the image superimposed with the manually specified vector sequence along a segment of the GBM, (c) the sequence, , the smooth middle line, and the inverse-mapped f+ and f− lines on its two sides, (d) the segmented GBM segment indicated by the shape.
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The procedure produces a sequence of positions, = {(m)|m = 0, 1, 2, . . ., M − 1}, where the component (m) is a vector, i.e., (m) = { 1 (m), 2 (m)). If placing a white point at the location of each vector in the sequence, we will have a smooth and unbroken white line along the central area of the GBM as shown in Fig. 1(c). The line indicates the central line in the segment of the GBM to be measured. If we straighten the smoothly curved line and map the image accordingly, we will have a segment of straight GBM as shown in Fig. 2(a). Let a nonlinear mapping, ϕ/4 , transform a portion of the image in the neighborhood into a rectangular sub-image, y, such as y = ϕ/4 (x). The subscript, /4, is used for this particular segment of membrane for the reason that the angles between the line and the horizontal axis are around /4. The mapped image y y y = {y(n1 , n2 )|0 ≤ n1 < N1 , 0 ≤ n2 < N2 }, where the image width y y N1 = M, the image height, N2 , is an odd integer selected significantly higher than half of the maximum width of the basement membrane, and the image components are determined by
y(n1 , n2 ) = x
y
n2 + 1 (n1 ) −
N2 − 1 2
y
, n2 + 2 (n1 ) −
N2 − 1 2
.
(1)
Fig. 2(b) shows the mapped image of y from the original image x in Fig. 1(a) based on the line in Fig. 1(c). The central line in the GBM in Fig. 2(b) appears straight. At the edges of the GBM, there are noticeable intensity discontinuities. These intensity discontinuities, in contrast to the homogeneities among the intensities inside membrane regions, can be used as stoppers when determining the membrane boundaries. Since the central line is located in the middle area of the GBM, the line and its vicinity pixels are expected to be inside the GBM. If an integer, , is small enough, y those pixels in the image y for which |n2 − ((N2 − 1)/2)| ≤ can be considered belonging to the GBM. Let this narrow rectangular region be represented by Sstrip , i.e., Sstrip = y {(n1 , n2 )|(0 ≤ n1 < M)&(|n2 − ((N2 − 1)/2)| ≤ )|}. Since the GBM
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Fig. 2 – (a) Local image neighboring the GBM central line after the line squeezed straight, (b) the converted image in (a) without the superimposed straight white central line, (c) the absolute error image between (b) and the average in the neighborhood of central line segment, (d) smoothed image of (c), (e) an automatic threshold based on the highest value in the central neighborhood in image (d), (f) the discrete functions f+ (n1 ) and f− (n1 ), as shown in white dots at y y positions (n1 , f + (n1 ) + (N2 − 1)/2) and (n1 , (N2 − 1)/ − 2 − f (n1 )), corresponding to the floor and ceiling of the non-GBM pixels above and below the central line, respectively, where n1 denotes the horizontal axis.
has a relatively homogeneous intensities, the average intensity inside Sstrip , a narrow and long strip of dimension of M × (2 + 1) at the center of the image y, can be an appropriate approximation to the actual intensity average of the GBM. Thus, the average intensity of the GBM is estimated by M−1 y = (1/(M(2 + 1))) m =0 m =− y(m1 , m2 + ((N2 − 1)/2)). We 1 2 compose an image, e, from the absolute error between the image y and the average value , such as e(n1 , n2 ) = |y(n1 , y y n2 ) − | for 0 ≤ n1 < N1 and 0 ≤ n2 < N2 . It is expected that a pixel inside the GBM will have a low intensity in the image e. Fig. 2(c) shows the image e, the absolute difference of the image in Fig. 2(b) and , the estimate of the GBM average intensity.
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Fig. 3 – (a) Dashed line for f+ (n1 ) above the central line y n2 = (N2 − 1)/2 = 50, and solid line for its lower envelope + e (n1 ), (b) dashed line for f− (n1 ) below the central line and solid line for its upper envelope e− (n1 ).
Although the intensities inside GBM vary in a very limited range, occasional spikes may cause errors if a thresholding operation is applied on the intensities. To reduce the effects of local spikes, a low-pass filter is applied. Fig. 2(d) is the image g, the moving average of the input image in Fig. 2(c) filtered by a moving window of size 3 × 3, such 1 1 as g(n1 , n2 ) = (1/9) m =−1 m =−1 y(n1 − m1 , n2 − m2 ) for 0 ≤ 1
y
2
y
n1 < N1 and 0 ≤ n2 < N2 . The range of the GBM intensities in g can be written as [G− , G+ ], estimated by G− = 0 and G+ = G + max{g(n1 , n2 )|(n1 , n2 ) ∈ Sstrip } where G is a small value purposely added to increase the intensity range in Sstrip by a small amount. Pixels of g beyond this intensity range will be considered outside of the GBM. Thus, we have a binary image, g , converted by
g (n1 , n2 ) =
0, if g(n1 , n2 ) ≤ G+ , 1, otherwise, y
y
for 0 ≤ n1 < N1 and 0 ≤ n2 < N2 . The pixel at (n1 , n2 ) is a point outside of GBM if g (n1 , n2 ) is true or 1. Converted from the gray-level image in Fig. 2(d), Fig. 2(e) shows the resulting binary image in which black is for zero and white is for one. It is observed from g that white pixels are located y in some distances from the centerline of n2 = (N2 − 1)/2. We + − compose two functions, f (n1 ) and f (n1 ), for the two distances to the central line with respect to the horizontal axis of n1 , such as f + (n1 ) = min−1 n −(Ny −1)/2 {(g (n1 , n2 ) = 1)&(n2 >
y (N2 y (N2
− 1)/2)} and
f − (n1 )
= min
−1
2
2
y {(g (n1 , n2 ) (N2 −1)/2−n2
= 1)&(n2 <
− 1)/2)}. The function is the smallest positive integer y of n2 − (N2 − 1)/2 for which the image pixel g (n1 , n2 ) is white, y and f− (n1 ) is the smallest positive integer of (N2 − 1)/2 − n2 for which the image pixel g (n1 , n2 ) is white. Fig. 2(f) shows y the two functions where pixels at (n1 , f + (n1 ) + (N2 − 1)/2) and y − (n1 , (N2 − 1)/2 − f (n1 )) are marked with white dots. Fig. 3(a) shows, in dashed line, the function f+ (n1 ) above the central line y of n2 = (N2 − 1)/2 = 50, while Fig. 3(b) shows, also in dashed line, the function f− (n1 ) below the central line. With the inverse −1 , the functions of f+ (n1 ) and f− (n1 ) will be mapping of ϕ/4 f+ (n1 )
mapped to the two lines on both sides of the line in Fig. 1(c), respectively. It is assumed that any pixel at (n1 , n2 ) for which g (n1 , n2 ) is white is not a pixel in the GBM. Since the GBM has smooth boundaries on both sides, we extract a smooth envelope line for each side of boundaries such that each envelope line is in tangent touch to the peaks of the waveform it covers. If e+ (n) is a lower envelope for f+ (n), it has to satisfy the following three conditions: (1) e+ (n) ≤ f+ for 0 ≤ n < M; (2) e+ (n) is a smooth function of n, meaning that |e+ (n + 1) − e+ (n)| for 0 ≤ n < M − 1 are relatively small; and (3) e+ (n) line is in tangent with f+ (n) line at some lowest points, i.e., e+ (n) = f+ (n) for some points among the lowest trough of f+ (n) line. We present an iterative procedure to obtain, from the function f+ (n), its envelope e+ (n). The other envelope function e− (n) can be obtained from the function f− (n) in a similar manner. Let h(n) be the finite impulse response (FIR) of a low-pass filter. If the input is fk (n), the output of the filter will be k (n) = h(n) ∗ fk (n), where the sign, ‘*’, represents the discrete convolution operation. Compare k (n) with fk (n) and select the smaller one as the system output fk+1 (n), i.e., fk+1 (n) = min{k (n), fk (n)}.
(2)
This output, fk+1 (n), is fed back to the same system as the new input if the termination condition is not satisfied. The initial iteration number is set as k = 0 and the upper distance function, f+ (n), is set to be the initial image, such as of iterations, the diff0 (n) = f+ (n). With the increasing number y ference εk+1 = max{ |fk (n) − fk+1 (n)| 0 ≤ n < N1 } will decrease and finally converge to zero if iteration number goes to infinite high. When the difference is small enough, say εk+1 < 0.1, the iterations can be terminated and the system produces the envelope function e+ (n) = fk+1 (n). Fig. 3(a) shows the distance function of f+ (n) in the dashed line with its envelope, e+ (n), in the solid line. In a similar procedure except replacing the min{·, ·} operation with the max{·, ·} operation in Eq. (2), we obtain the other envelope function, e− (n), in the solid line in Fig. 3(b), from the distance function, f− (n), in the dashed line. −1 , the solid lines of e+ (n) Taking the inverse mapping of ϕ/4 − and e (n) are mapped to the smooth lines along the GBM boundaries on both sides of the GBM. Filling the two ends of the curves of e+ and e− with two straight lines, we have an enclosed belt region as shown in Fig. 1(d). The length of the belt can be estimated by
L=
K−2
2
2 1/2
((c1 (k + 1) − c1 (k)) + (c2 (k + 1) − c2 (k)) )
.
(3)
k=0
Let A represent the area in the shape drawn by the two boundary lines on both sides of and the two straight end lines as shown in Fig. 1(d). The area, A, can be estimated by simply counting the number of pixels inside the enclosure. The average thickness, , can then be estimated by = A/L. The unit of is the size of a single pixel. The actual GBM thickness is obtained by applying on the factor of the metric width of a single pixel. We summarize the algorithm as the following. At the beginning, the user specifies a sequence of scattered points (cross
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marks in Fig. 1(b)) along the central line of the GBM segment. The gap between the two neighboring user-specified points is filled by a linear interpolation. Thus, a consecutive sequence of locations is obtained as the central line (Fig. 1(c)). Mapping the central line into a straight horizontal line, we have a smaller local image vertically centered at the central line (Fig. 2(b)). Evaluating the average intensity in the narrow region along the central line, we then calculate the absolute difference (Fig. 2(c)) between the image (Fig. 2(b)) and the average intensity of the
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GBM segment. To reduce the effects of local spikes, a low-pass filter is applied to obtain a smoother image (Fig. 2(d)). The intensity range in the narrow region along the central line is evaluated. All pixels whose intensities are out of this intensity range are considered outside of GBM and indicated by white color in Fig. 2(e). We find the two sequences of points in the white regions vertically closest to the central line on both the upper and lower sides of the central line as shown the white dots in Fig. 2(f). Two 1-D functions for the upper and lower
Fig. 4 – (a) Original image acquired at magnification of 2k, (b) user specified sequence shown by the cross signs along the segment of BM to be measured, (c) two segmented sub-BM segments indicated by the shapes.
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dots are plotted in Fig. 3 as the dashed lines. Their respective smooth envelopes are plotted as the solid lines in Fig. 3. The inverse mappings convert the envelope lines back to the original image space as the GBM edges in Fig. 1(d). The GBM segment region is obtained after adding the two straight end lines. The GBM thickness is estimated as the ratio of the area of the extracted GBM region to the length of the central line.
3.
GBM with a wide-angled sequence
If a straight line determined by two separate points (c1 (k), c2 (k)) and (c1 (k + 1), c2 (k + 1)), for 0 ≤ k < K − 2, the slope of the line is (k) = (c2 (k + 1) − c2 (k))/(c1 (k + 1) − c1 (k)). The angle between the line and the horizontal axis is k = tan−1 ( (k)) if c1 (k + 1) − c1 (k) > 0, and k = tan−1 ( (k)) + if c1 (k + 1) − c1 (k) < 0. The angle is between −/2 and −3/2. To have the angle range of [0, 2), we increase k by 2 when c1 (k + 1) − c1 (k) > 0 and c2 (k + 1) − c2 (k) < 0. When c1 (k + 1) − c1 (k) = 0, k is either /2 or 3/2 depending on the sign of c2 (k + 1) − c2 (k). The angle range, [min{ k |0 ≤ k < K − 2}, max{ k |0 ≤ k < K − 2}], for the GBM segment in Fig. 1(c) is relatively narrow. A single mapping works well since the inverse mapping is unique. However, if the angle range is very wide, one mapping may not be enough. Fig. 4(a) shows an original image with a long segment of GBM. In Fig. 4(b), the GBM segment is manually marked with a sequence of locations, c, whose angle ranges from just above /4 to just below 5/4 as shown in Fig. 5. Since the angle ranges close to , we have to partition the GBM segment into two or more sub-segments so that each GBM sub-segment has a narrower angle range to have a unique inverse mapping. Let = min{ k |0 ≤ k < K − 2}, the minimum angle, and = min{ k |0 ≤ k < K − 2} the maximum angle. Both and are inside [0, 2). The total range of [0, 2) is partitioned into eight periods, each of which covers /4. Thus, we have a discrete sequence { k |0 ≤ k < K − 2} whose component k = [4 k /], val-
Fig. 5 – k , the angle between the central line and the horizontal axis vs. k, the index of the input points, for 0 ≤ k < K − 1, where K is the total number of central points inputted manually.
ued as an integer between 0 and 7, where the floor function [·] yields the highest integer below the float variable. Since each integer in k represents /4 in angle, the sequence in a single GBM sub-segment should have a range narrower than 4 or an angle range less than or equal to 3/4. If the total angle range is wider than 3/4, we have to partition the whole sequence of the input central points into multi sub-sequences so that each sub-sequence has an angle range narrow enough. We start with k = 0 and then search through the { k } sequence until a given angle range is about to be exceeded the limit. The procedure for the first GBM sub-segment is as the following: (1) Initialization step: Let k = 0. (2) Iteration step: If max{ m |0 ≤ m ≤ k} − min{ m |0 ≤ m ≤ k} ≥ 2, terminate and output k, otherwise let k = k + 1 and repeat this iteration step. The above procedure separates the first GBM sub-segment from the whole segment. The procedure can be repetitively applied to separate the subsequent sub-segments, one by one, until the whole GBM segment is exhausted. The angle sequence for the GBM segment in Fig. 4(b) is depicted in Fig. 5 with a range of [/4 + 0.0238, 5/4 − 0.0052]. With the above procedure, the GBM segment is separated into two sub-segments with cut-off angle at 3/4. The middle solid horizontal lines for the two separate angle ranges are /2 and , respectively. Thus, the two mappings are ϕ/2 and ϕ . Each GBM sub-segment is processed in a similar way as that in the previous section where an individual GBM segment is segmented. The average GBM thickness is then calculated as the ratio of the total area enclosed in the multiple shapes and the estimated total length of the GBM segment.
4.
Results
The studies detailed below regarding normal glomerular basement membranes utilized a core needle biopsy from a 63year-old man who had a liver transplant 4 years prior to the biopsy and recently developed proteinuria. The biopsy showed focal segmental glomerulosclerosis and the basement membranes of the non-sclerotic glomeruli were normal by light and electron microscopy. Fig. 1(a) shows an original image of a human renal biopsy specimen fixed in glutaraldehyde acquired by a transmission electron microscope (Hitachi H700) at magnification of 2k. The digital dimension of the image is 469 × 432. The component of GBM evaluated is the lamina densa which appears as a central broader zone of gray intensities with limited fluctuations and sandwiched by subepithelial and subendothelial thin lighter regions (lamina rara interna and lamina rara externa) situated beneath the endothelial cell and epithelial cell (podocyte). Along the GBM central line, we manually specified a sequence of sparsely spaced points as shown in Fig. 1(b) with the plus signs. There are totally 17 manually inputted points in this case. Sparser points can be used at the smooth portion of the GBM. However, in the places of the more curved GBM, denser points are needed so that the central line, , drawn along the specified points, is laid approximately in the middle of the GBM as shown in Fig. 1(c). Fig. 1(c) also shows the two edge
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lines located at both sides of from the inverse mapping of the f+ (n) and f− (n) functions, respectively. The GBM segmentation based on the input central points in Fig. 1(b) is illustrated in Fig. 1(d) as the shape enclosed by the two curved side lines from the inverse mappings of the envelope functions and the two straight end lines. The area, which is the total number of pixels inside the enclosed shape, can be counted automatically, while the length of the GBM segment can be estimated by L in Eq. (3) based on the input central points. The ratio of the area to the length L gives the estimate of the average GBM thickness. The total number of pixels in the enclosure in Fig. 1(d) is counted to be 8140 pixels, while the length of the central line is estimated as 457.24. The average GBM thickness is then calculated as 8140/457.24 = 17.803 pixel units. Applying the factor of 0.020838 m/pixel, we have the average metric GBM thickness of 17.803 × 0.020838 = 0.37098 m or 371 nm. Fig. 4(a) shows another digital GBM image of size 350 × 579. A series of 23 central points are manually inputted as shown
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in Fig. 4(b) marked by the 23 crosses. The angle of the line segments angels from /4 to 5/4. The GBM is partitioned into two sub-segments. Starting from the first point of the lowest cross in Fig. 4(b), moving up along the central line and stopping just before the angle reaches 3/4. The first sub-segment includes the lower 13 points, and the second one includes the upper 11 points. The 13th point is the last in the first subsegment and also the first in the second sub-segment. The central line of the first sub-segment by linearly connecting the lower 13 points is 436.67 pixels long, while the central line of the second sub-segment by linearly connecting the lower 11 upper points is 218.89 pixels long. The complete length of the GBM segment is then 436.67 + 218.89 = 655.56 pixels long. Fig. 4(c) shows the two shapes for the two subsegments. The numbers of the pixels in the two shapes are counted as 7600 and 4499, respectively. Thus, the average GBM thickness measured by the ratio of the total area to the length of central line is 7600 + 4499/655.56 = 18.456 pixel
Fig. 6 – (a) An original GBM image acquired at magnification of 5k, (b) the original image in (a) superimposed with the crosses for the input central points and the shape for the resulting GBM segmentation, (c) another original GBM image acquired at magnification of 3k, and (d) the image in (c) superimposed with the crosses for the input central points and the shape for the resulting GBM segmentation.
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units. The average metric GBM thickness is calculated as 18.456 × 0.020838 = 0.384586 m or 384.6 nm. Fig. 6 shows two more GBM images in (a) and (c) and their respective resulting segmentations in (b) and (d). In Fig. 6, (a) displays an original digital image, size of 563 × 765, acquired at magnification of 5k, while (c) shows another original digital GBM image, size of 698 × 742, acquired at magnification of 3k. In Fig. 6(b), the central line is calculated as 434.74, and the shape contains 21,646 pixels. The GBM thickness is then 21,646/434.74 = 49.79 in pixel units. At the magnification of 5k the scale is 0.0083352 m/pixel. Thus, the average metric GBM thickness is calculated as 49.79 × 0.0083352 = 0.41501 m or 415 nm. In Fig. 6(d), the central line is 832.20 pixels long, and the closed shape contains 22,351 pixels. The average GBM thickness is then calculated as 22,351/832.20 = 26.86 pixels thick. At the magnification of 3k the scale is 0.013892 m/pixel. Thus, the average metric GBM thickness is calculated as 26.86 × 0.013892 = 0.373139 m or 373.1 nm. Fig. 7 shows the application to the case of thin basement disease. The patient studied was from a 39-year-old woman
with a family history indicating that both her mother and sister had hematuria. The patient had microscopic hematuria and mild proteinuria. Electron microscopy demonstrated thin lamina densa of the glomerular basement membranes. An evaluation of the patient’s biopsy tissue for abnormalities of type IV collagen to evaluate for Alport’s syndrome utilized monoclonal antibodies against alpha 1, 3 and 5 chains of type IV collagen (Wieslab reagents: normal human kidney and skin utilized as controls) indicated no abnormality. The image was taken with the magnification of 7k×. The digital dimension is 712 × 1170 and the metric dimension is 5.8361 m × 9.5902 m. The scattered cross marks along the central line in the GBM segment are the locations manually specified by a user. The two curved lines along the GBM boundaries are the derived edges. The estimated GBM thickness, the ratio of the area enclosed by the surrounding boundaries to the length of the interpolated central line, is 193.6 nm.
5.
Conclusions
In this paper, we have described an approach for estimation of average GBM thickness. Contrary to the usual manual measurements whose reliability is largely dependent on the skills of users, the proposed algorithm only requires minimum user inputs whose positions are much less critical and determines the GBM thickness based on the local contents in the images. In the case of a narrow angle range, an appropriate mapping of the local image into a smaller rectangular image converts the usually curved GBM segment into a straight one. Two distance functions of the non-GBM pixels to the straight central line on the two sides are evaluated and their smooth envelope functions are derived. The inverse mappings of the two envelopes produce the boundaries of the GBM segment. The average GBM thickness is estimated as the ratio of the area inside the recognized GBM segment to the length of the central line. Long GBM segments with wide-angled vector sequence of manual inputs are partitioned into multiple narrow-angled sub-segments prior to the segmentation procedures. The results on images of human renal biopsy specimen fixed in glutaraldehyde show the effectiveness of the proposed algorithm.
Conflict of interest statement There is no conflict of interest. We do not have any actual or potential conflict of interest including any financial, personal or other relationships with other people or organizations within that could inappropriately influence (bias) this work.
references
Fig. 7 – Image of thin basement membrane disease, magnification 7k×. Digital dimension 712 × 1170, metric dimension 5.8361 m × 9.5902 m. Estimated thickness is 193.6 nm.
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