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3D NMR Imaging of Foam Structures
JOURNAL OF MAGNETIC RESONANCE, ARTICLE NO.
Series A 118, 195–201 (1996)
0027
3D NMR Imaging of Foam Structures KATSUMI KOSE Institute of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Received June 13, 1995; revised October 9, 1995
Three-dimensional foam structures were measured using NMR imaging, and their 3D geometrical properties were analyzed. Eight bubble polyhedra of a polyurethane foam specimen were extracted from 3D NMR image data using a newly developed 3D geometrical structure analysis program, and quantitative geometrical data were measured for the first time for real foam systems. The results agreed well with the study by Matzke with soap bubbles but did not agree with the optimum solution by Weaire and Phelan for 3D space division into equal volume cells with minimum partitional area. The reason for this disagreement is not clear; however, improved foam preparation and more systematic measurements using the method developed here may clarify this difference. q 1996 Academic Press, Inc.
INTRODUCTION
How to divide 3D space into equal volume cells with minimum partitional area is a classical mathematical problem (1). In 1887, Lord Kelvin proposed a solution based on the 14-sided polyhedron (tetrakaidecahedron) whose faces are eight hexagons and six squares (2). He curved its edges slightly to meet Plateau’s rules which require that surfaces of cell polyhedra must meet at 1207 and the edges of cell polyhedra must meet at the tetrahedral angle of 109.477. Although there was no mathematical proof nor experimental verification, his tetrakaidecahedron has been thought of as an optimal solution for a long time. Recently, however, a better solution than Kelvin’s tetrakaidecahedron has been discovered by Weaire and Phelan (3). Their unit cell consists of six 14-sided polyhedra and two 12-sided polyhedra. The 14-sided polyhedron consists of 12 pentagonal faces and two hexagonal faces while the 12-sided polyhedron consists of 12 pentagonal faces. Although they found this structure through a computational search based on some crystal structures, this solution has not been verified experimentally. Since it is considered that foam gives a division of 3D space with minimum partitional area (2), measurements of 3D foam structures are essential to this problem. As for experimental research, in 1946, Matzke, American botanist, published an extensive study of three-dimensional shapes of bubbles in foam (4). His foam consisted of about
2000 equal volume bubbles made one by one with a syringe in a cylindrical dish. He then made observation using a binocular dissecting microscope and recorded the shapes of 600 bubbles in the central region of the dish. He observed many 12-, 13-, 14-, and 15-sided polyhedral bubbles but very few 11-, 16-, and 17-sided ones and he could not find Kelvin’s tetrakaidecahedron. Although Matzke’s paper gives us valuable knowledge on the 3D shapes of bubble polyhedra, even now we cannot obtain any quantitative data from it. Thus, quantitative measurements of 3D foam structures are highly desirable. The purpose of this study is, thus, to visualize 3D foam structures using NMR imaging, to measure 3D geometrical quantities using the image data, and to evaluate the observed structures. METHOD
Two specimens were prepared by immersing two kinds (with different cell sizes) of commercially available polyurethane foam into CuSO4-doped water in 20 mm diameter NMR sample tubes. Polyurethane foams, in which bubbles were produced by chemical reactions, were used because the cell size seemed to be uniform and their foam structures were physically stable. For convenience, we denote the specimen with the smaller cell size as specimen I and that with the larger cell size as specimen II. In order to evaluate geometric distortion possibly present in the NMR images, a phantom consisting of four concentric NMR sample tubes (with the outer diameters 5.0, 10.0, 15.0, and 20.0 mm and the inner diameters 4.2, 9.0, 13.5, and 18.0 mm) filled with CuSO4-doped water was prepared. Three-dimensional microscopic images [FOV, 19.2 mm3 ; image matrix, 128 3 , voxel size, 0.15 mm3 ] were obtained with a homebuilt NMR imaging system using a 4.74 T, 89 mm vertical-bore superconducting magnet (Oxford Instruments) and an actively shielded gradient coil (Doty Scientific). The pulse sequence used was a conventional spinecho 3D imaging sequence (TR Å 200 ms, TE Å 12 ms) to avoid susceptibility artifacts. The phase-encoding directions were the x and y directions and the signal readout direction was the z direction. Since 4 to 16 signal accumulations were
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1064-1858/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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FIG. 1. Two-dimensional cross-sectional images of the concentric sample-tube phantom in a plane parallel to the xz plane (a) and in a plane parallel to the yz plane (b). For the 3D image, FOV, (19.2 mm) 3 ; image matrix, 128 3 ; voxel size, (0.15 mm) 3 .
performed, the total measurement time for one 3D image was between 3.6 and 14.5 hours. To evaluate geometrical image distortion due to static magnetic field inhomogeneity, a pair of 3D images was acquired for specimen II with positive and negative readout gradients. EXPERIMENTAL RESULTS
Figure 1 shows cross-sectional images of the water phantom in a plane parallel to the xz plane ( a ) and in a plane parallel to the yz plane ( b ) . The vertical direction in the images, which is parallel to the readout gradient, is the z direction. Since this phantom is axially symmetric, the axis is parallel to the static magnetic field and the image area is far enough from the tube ends so that the susceptibility-induced magnetic filed in the phantom can be neglected. Thus, the geometric distortion along the x and y directions observed in Fig. 1 is due to the nonlinearity of the gradient field because static field inhomogeneity distorts the images of the sample tubes only along the z direction. The maximum image distortion is about two pixel lengths in the xz plane and about five pixel lengths in the yz plane near the corners of the images. However, within the central spherical region whose diameter is 15 mm, the image distortion is less than one pixel length. Image distortion due to nonlinearity of the z gradient, which cannot be evaluated from images in Fig. 1, has already been confirmed to be less than one pixel length within the spherical region in earlier studies ( 5, 6 ) .
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Image distortion due to static field inhomogeneity can be evaluated using Fig. 2. Figure 2a shows a 2D cross-sectional image cut from 3D image data of specimen II in a plane parallel to the xz plane. Figure 2b shows a difference between two 3D image data sets at the same slice position which were measured with positive and negative readout (z) gradients. Dark regions in Fig. 2a show edges or vertices of bubbles, and dark and bright pair regions in Fig. 2b demonstrate image distortion of the edges or vertices along the z direction due to static field inhomogeneity. Although whether this inhomogeneity is due to the magnet itself or spatial variation of the magnetic susceptibility of the specimen cannot be identified, it is probably due to the magnet itself because the dark and bright pair regions are absent near the center of the image. In any event, this distortion or image shift is also less than one pixel length in the 15 mm diameter central spherical region. From the above results, the geometrical distortions observed in the 3D images are on the order of one pixel length in the central spherical region, though there are two factors which affect the image distortion. Figures 3a and 3c show cross-sectional images of specimens I and II. In these images, only cross sections of edges or vertices of bubble polyhedra are visible and the connections between them are not clear. Thus, to visualize the network of the polyhedral edges, minimum intensity projection (mIP) images were computed from the 3D image data sets. Figures 3b and 3d are mIP images computed from 8 and 24 contiguous 2D slices, respectively, of the 3D image
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FIG. 2. (a) Two-dimensional cross-sectional image of specimen II cut in a plane parallel to the xz plane from the 3D volume data set. For the 3D image, FOV, (19.2 mm) 3 ; image matrix, 128 3 ; voxel size, (0.15 mm) 3 . (b) Difference at the same slice position between two 3D image data sets which were measured under positive and negative readout gradients.
data. These images were made by taking a minimum value along each projection ray perpendicular to the projection plane, which was the xy plane in Fig. 3b and the xz plane in Fig. 3d. Since the thickness of one 2D slice is 0.15 mm, the thickness of the slabs for the mIP calculation is 1.2 and 3.6 mm in Fig. 3b and Fig. 3d, respectively. In these figures, we can clearly see the connections or network of the polyhedral edges, so that we can realize the polyhedral faces. Although these mIP images give no information along the projection direction which is perpendicular to the image plane, we can determine the x , y , and z coordinates of the vertices of the bubble polyhedra by combining three series of mIP images whose projection directions are the x , y , and z directions. In the next section, the methods to measure the x , y , and z coordinates for one polyhedron will be presented. Figure 4 is a surface-rendered 3D display of specimen II. Although we can realize the 3D structure of bubble polyhedra by seeing such displays from various viewing angles, it is difficult to measure the geometrical quantities of the polyhedra from these images. MEASUREMENTS OF 3D GEOMETRICAL QUANTITIES
First, I shall explain why usual 3D volume-extraction techniques cannot be applied to the image data measured in this study. Such volume-extraction methods are based on ‘‘image segmentation’’ in 2D slices: specific regions
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such as brain, heart, or some other organ are cut by selecting appropriate threshold values in 2D cross-sectional slices, and 3D volume regions are defined by sets of the 2D regions. This algorithm is useful only for 3D objects of which exterior and interior regions can be distinguished from each other in 2D slices. In the 3D image data measured in this study, however, only polyhedral edges of bubbles are visualized as seen in Figs. 3a and 3c because their faces were already lost or too thin ( !100 mm) for detection. Thus, it is very difficult to distinguish between interior and exterior regions of bubble polyhedra in the 3D image data. Therefore, some other 3D volume-extraction technique must be developed to extract bubble polyhedra from the 3D image data set. I have developed one technique utilizing series of mIP images as shown in Figs. 3b and 3d. The computer program to measure geometrical quantities of the bubble polyhedra was developed on an X-window system running on a workstation ( HP9000 / 712 / 60 ) , and bubble polyhedra were interactively extracted on the CRT screen. However, geometrical quantities were measured only for specimen II because the contrast-to-noise ratio of the mIP images was not enough for specimen I. In all the mIP images examined, little curvature was observed for the polyhedral edges. Thus, all bubble polyhedra were described as having straight edges. The procedure to measure the x, y, and z coordinates of vertices for one bubble polyhedron is as follows:
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FIG. 3. Single-slice image (a) and minimum-intensity projection (mIP) image (b) of specimen I. Single-slice image (c) and mIP image (d) of specimen II. Images of specimen I are displayed in the xy plane and those of specimen II are displayed in the xz plane.
1. Compute three series of mIP images whose projection directions are x, y, and z. The computation is repeated for slabs of 24 contiguous slices while the slab volume is incrementally moved from one edge to the other in the field of view to cover the whole image area. Thus, three image series each consisting of 104 [128 0 (12 1 2)] mIP images are obtained. 2. Select a polygon (polyhedral face) in one mIP image,
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for example, parallel to the xy plane. Click the mouse button on a vertex of the polygon to determine the x and y coordinates of the vertex. To measure the z coordinate, a mIP image perpendicular to the xy plane (parallel to the xz or the yz plane) and containing the vertex is used. 3. Click the mouse button on the next vertex which is on the same polygon and has an edge connection with the first
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FIG. 5. Bubble polyhedra extracted from the mIP images of specimen II: (a) 12-sided, (b) 13-sided, (c) 14-sided, and (d) 15-sided polyhedra.
from the mIP images. Although only the vertices and edges of the bubble polyhedra were measured from the images, surface areas, volumes, and flatness of the polyhedral faces were calculated under some approximations. The results are listed in Table 1, and some of them are shown in Fig. 5. In Table 1, surface areas and volumes of the polyhedra were calculated by taking averages over possible divisions of the polyhedral faces into triangles because the faces were not always planar. Flatness, which represents flatness of a polyhedral face, was calculated as follows: a polyhedral face was divided into triangles, vectors normal to the triangles were calculated, angles which the normal vectors made with other normal vectors were calculated and averaged over possible combinations of the normal vectors, and the averaged angles were further averaged over possible divisions of the polygon into triangles. The average
FIG. 4. Surface-rendered 3D display of specimen II.
vertex. Coordinates x, y, and z are determined in the same way as described above. In this way, the edge connecting the vertices is determined. 4. Repeat operation 3 until the measured vertex reaches the initial vertex. In this way, the x, y, and z coordinates of vertices for the polygon are determined. 5. Repeat operations 2–4 until the measured polygons cover a polyhedron. The above operations 2 – 5 were repeated, and eight randomly chosen polyhedra in the central region, where image distortion was on the order of one pixel length, were measured. It took about an hour to extract one polyhedron
TABLE 1 Geometrical Quantities of Polyhedra Extracted from Specimen II Number of faces Volume (mm3)
Average flatness (7)
Isometric quotient
Bubble no.
Total
Quadrilateral
Pentagon
Hexagon
Surface area (mm2)
1 2 3 4 5 6 7 8
12 13 13 13 14 14 15 15
0 1 1 1 0 2 2 3
12 10 10 10 12 8 8 6
0 2 2 2 2 4 5 6
76.7 75.6 75.7 82.5 81.1 76.3 81.9 88.8
50.3 51.6 50.4 56.8 56.8 51.6 58.6 65.0
8.2 9.1 8.5 7.4 9.3 9.1 10.5 7.7
0.703 0.712 0.736 0.726 0.741 0.714 0.731 0.720
13.63
1.25
9.50
2.88
79.8
55.1
8.7
0.723
Average
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FIG. 7. Distribution of intersection angles at the vertices of the bubble polyhedra. The mean value is 109.227, which is close to the tetrahedral angle of 109.477.
FIG. 6. Distributions of surface areas for (a) quadrilaterals, (b) pentagons, and (c) hexagons in the bubble polyhedra.
flatness for one polyhedron is the average of the flatness of polyhedral faces over the polyhedron. Thus, if a polyhedron consists of planar faces only, the average flatness is 07. The isometric quotient, which is shown in the ninth column of Table 1 and represents a figure of merit for surface area minimization, is defined by 36pV 2 /A 3 (7), where V and A are the volume and surface area of the cell. Its value is 0.5236 for a cube which can completely fill a 3D space and 1.0 for a sphere which cannot fill a 3D space without
voids. To attain a larger isometric quotient, polyhedra which can fill 3D spaces should be as close to a sphere as possible. However, as shown in Fig. 5, the measured polyhedra have structures elongated along the vertical direction in the figure. Since this direction is the same for all polyhedra, this elongation of the structures is presumed to be caused by some tension or other physical mechanism during the formation of the polyurethane foam. Thus, to minimize the surface area and obtain a better isometric quotient, the structures of the measured polyhedra were shrunk by a factor of 0.60, which was optimal for this foam, along the vertical direction in Fig. 5. The isometric quotients are tabulated in Table 1. To the best of the author’s knowledge, this is the first such result for real foams. The average value of the isometric quotients for the eight polyhedra was improved from 0.674 to 0.723 by this correction. We will discuss this quotient in more detail in the next section. Figure 6 shows distributions of surface areas of the faces in the measured polyhedra. There is a definite correlation between surface areas and the number of sides. Figure 7 shows a distribution of intersection angles at the vertices of the polyhedra. These angles were calculated after the correction described above were made. Most of them ranged from 707 to 1507 and their average was 109.227, which was very close to the tetrahedral angle of 109.477. This result
TABLE 2 Parameters of Space-Filling Polyhedra
Kelvin (2) Weaire and Phelan (3) Matzke (4) This study
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Ratio of numbers of polyhedral faces (4:5:6)
Ratio of numbers of polyhedra (12:13:14:15)
Average number of faces per polyhedron
Isometric quotient
43:0:57 0:89:11 11:67:22 9:70:21
0:0:100:0 25:0:75:0 12:30:36:18 13:38:25:25
14 13.5 13.70 13.63
0.757 0.764 — 0.723
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shows that the intersection angles at vertices are randomly distributed around the tetrahedral angle: the distribution is nearly Gaussian and its standard deviation is about 157. To summarize, the measured results are as follows: Only 12-, 13-, 14-, and 15-sided polyhedra were observed. Polyhedra with less than 12 faces and more than 15 faces were not observed. The average number of faces per one polyhedron was 13.63. Only quadrilateral, pentagonal, and hexagonal faces were observed and triangular faces and faces with more than six edges were not observed. The ratio of numbers of quadrilaterals, pentagons, and hexagons was 9.2:69.7:21.1. The cell size was nearly uniform: most cells fell within {10% of the mean value. The average of isometric quotients for polyhedra was improved to 0.723 by shrinking them along one direction. There was a positive correlation between the surface area and the number of sides of the polyhedral faces. Intersection angles at vertices of the bubble polyhedra were distributed around the tetrahedral angle. DISCUSSION
Table 2 summarizes parameters of space-filling polyhedra by theoretical studies of Kelvin (2) and Weaire and Phelan (3) and experimental studies by Matzke (4) and in the present paper. The results by Matzke and those presented here gave good agreement. Therefore, the same physical mechanism must determine the structure of those foams, although Matzke made soap bubbles one by one and the polyurethane foam used in this study was made through some chemical reactions which were accompanied by gas production. Although the agreement between experimental results is satisfactory, as described above, there is a significant difference between experimental and theoretical studies. The dis-
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agreement between the experiments and Kelvin’s tetrakaidecahedron is, however, not important because a more optimal solution has been already proposed by Weaire and Phelan. A remarkable difference between the experiments and the solution is that the latter does not contain any quadrilateral faces nor any 13-sided and 15-sided polyhedra. In other words, the unit cell of the theory contains only eight polyhedra but the unit cell in real foam, if it exists, seems to contain a much larger number of polyhedra. However, since the isometric quotient obtained in this experiment is much smaller than that of the theory and the experimental precision is limited, we cannot explain this disagreement quantitatively. The method developed here is, however, very powerful for studying 3D structure of foams, so more careful foam preparation and more systematic research using this technique may clarify this contradiction. ACKNOWLEDGMENTS The author thanks Dr. E. Fukushima for critical reading of the manuscript. He also thanks Dr. T. Hashimoto for cooperation at the early stage of this study, Professor D. Weaire for useful comments on this study, and Professor T. Ogawa for kind guidance to this field and encouragement. This work was supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture in Japan.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
J. Gray, Nature 367, 598 (1994). W. Thomson (Lord Kelvin), Philos. Mag. 24, 503 (1887). D. Weaire and R. Phelan, Philos. Mag. Lett. 69, 107 (1994). E. Matzke, Am. J. Botany 32, 58 (1946). K. Kose, Phys. Rev. Lett. 72, 1467 (1994). K. Kose, Phys. Fluids 6, S4 (1994). S. Ross, Am. J. Phys. 46, 513 (1978).
AP: Mag Res
Series A 118, 195–201 (1996)
0027
3D NMR Imaging of Foam Structures KATSUMI KOSE Institute of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Received June 13, 1995; revised October 9, 1995
Three-dimensional foam structures were measured using NMR imaging, and their 3D geometrical properties were analyzed. Eight bubble polyhedra of a polyurethane foam specimen were extracted from 3D NMR image data using a newly developed 3D geometrical structure analysis program, and quantitative geometrical data were measured for the first time for real foam systems. The results agreed well with the study by Matzke with soap bubbles but did not agree with the optimum solution by Weaire and Phelan for 3D space division into equal volume cells with minimum partitional area. The reason for this disagreement is not clear; however, improved foam preparation and more systematic measurements using the method developed here may clarify this difference. q 1996 Academic Press, Inc.
INTRODUCTION
How to divide 3D space into equal volume cells with minimum partitional area is a classical mathematical problem (1). In 1887, Lord Kelvin proposed a solution based on the 14-sided polyhedron (tetrakaidecahedron) whose faces are eight hexagons and six squares (2). He curved its edges slightly to meet Plateau’s rules which require that surfaces of cell polyhedra must meet at 1207 and the edges of cell polyhedra must meet at the tetrahedral angle of 109.477. Although there was no mathematical proof nor experimental verification, his tetrakaidecahedron has been thought of as an optimal solution for a long time. Recently, however, a better solution than Kelvin’s tetrakaidecahedron has been discovered by Weaire and Phelan (3). Their unit cell consists of six 14-sided polyhedra and two 12-sided polyhedra. The 14-sided polyhedron consists of 12 pentagonal faces and two hexagonal faces while the 12-sided polyhedron consists of 12 pentagonal faces. Although they found this structure through a computational search based on some crystal structures, this solution has not been verified experimentally. Since it is considered that foam gives a division of 3D space with minimum partitional area (2), measurements of 3D foam structures are essential to this problem. As for experimental research, in 1946, Matzke, American botanist, published an extensive study of three-dimensional shapes of bubbles in foam (4). His foam consisted of about
2000 equal volume bubbles made one by one with a syringe in a cylindrical dish. He then made observation using a binocular dissecting microscope and recorded the shapes of 600 bubbles in the central region of the dish. He observed many 12-, 13-, 14-, and 15-sided polyhedral bubbles but very few 11-, 16-, and 17-sided ones and he could not find Kelvin’s tetrakaidecahedron. Although Matzke’s paper gives us valuable knowledge on the 3D shapes of bubble polyhedra, even now we cannot obtain any quantitative data from it. Thus, quantitative measurements of 3D foam structures are highly desirable. The purpose of this study is, thus, to visualize 3D foam structures using NMR imaging, to measure 3D geometrical quantities using the image data, and to evaluate the observed structures. METHOD
Two specimens were prepared by immersing two kinds (with different cell sizes) of commercially available polyurethane foam into CuSO4-doped water in 20 mm diameter NMR sample tubes. Polyurethane foams, in which bubbles were produced by chemical reactions, were used because the cell size seemed to be uniform and their foam structures were physically stable. For convenience, we denote the specimen with the smaller cell size as specimen I and that with the larger cell size as specimen II. In order to evaluate geometric distortion possibly present in the NMR images, a phantom consisting of four concentric NMR sample tubes (with the outer diameters 5.0, 10.0, 15.0, and 20.0 mm and the inner diameters 4.2, 9.0, 13.5, and 18.0 mm) filled with CuSO4-doped water was prepared. Three-dimensional microscopic images [FOV, 19.2 mm3 ; image matrix, 128 3 , voxel size, 0.15 mm3 ] were obtained with a homebuilt NMR imaging system using a 4.74 T, 89 mm vertical-bore superconducting magnet (Oxford Instruments) and an actively shielded gradient coil (Doty Scientific). The pulse sequence used was a conventional spinecho 3D imaging sequence (TR Å 200 ms, TE Å 12 ms) to avoid susceptibility artifacts. The phase-encoding directions were the x and y directions and the signal readout direction was the z direction. Since 4 to 16 signal accumulations were
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1064-1858/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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FIG. 1. Two-dimensional cross-sectional images of the concentric sample-tube phantom in a plane parallel to the xz plane (a) and in a plane parallel to the yz plane (b). For the 3D image, FOV, (19.2 mm) 3 ; image matrix, 128 3 ; voxel size, (0.15 mm) 3 .
performed, the total measurement time for one 3D image was between 3.6 and 14.5 hours. To evaluate geometrical image distortion due to static magnetic field inhomogeneity, a pair of 3D images was acquired for specimen II with positive and negative readout gradients. EXPERIMENTAL RESULTS
Figure 1 shows cross-sectional images of the water phantom in a plane parallel to the xz plane ( a ) and in a plane parallel to the yz plane ( b ) . The vertical direction in the images, which is parallel to the readout gradient, is the z direction. Since this phantom is axially symmetric, the axis is parallel to the static magnetic field and the image area is far enough from the tube ends so that the susceptibility-induced magnetic filed in the phantom can be neglected. Thus, the geometric distortion along the x and y directions observed in Fig. 1 is due to the nonlinearity of the gradient field because static field inhomogeneity distorts the images of the sample tubes only along the z direction. The maximum image distortion is about two pixel lengths in the xz plane and about five pixel lengths in the yz plane near the corners of the images. However, within the central spherical region whose diameter is 15 mm, the image distortion is less than one pixel length. Image distortion due to nonlinearity of the z gradient, which cannot be evaluated from images in Fig. 1, has already been confirmed to be less than one pixel length within the spherical region in earlier studies ( 5, 6 ) .
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Image distortion due to static field inhomogeneity can be evaluated using Fig. 2. Figure 2a shows a 2D cross-sectional image cut from 3D image data of specimen II in a plane parallel to the xz plane. Figure 2b shows a difference between two 3D image data sets at the same slice position which were measured with positive and negative readout (z) gradients. Dark regions in Fig. 2a show edges or vertices of bubbles, and dark and bright pair regions in Fig. 2b demonstrate image distortion of the edges or vertices along the z direction due to static field inhomogeneity. Although whether this inhomogeneity is due to the magnet itself or spatial variation of the magnetic susceptibility of the specimen cannot be identified, it is probably due to the magnet itself because the dark and bright pair regions are absent near the center of the image. In any event, this distortion or image shift is also less than one pixel length in the 15 mm diameter central spherical region. From the above results, the geometrical distortions observed in the 3D images are on the order of one pixel length in the central spherical region, though there are two factors which affect the image distortion. Figures 3a and 3c show cross-sectional images of specimens I and II. In these images, only cross sections of edges or vertices of bubble polyhedra are visible and the connections between them are not clear. Thus, to visualize the network of the polyhedral edges, minimum intensity projection (mIP) images were computed from the 3D image data sets. Figures 3b and 3d are mIP images computed from 8 and 24 contiguous 2D slices, respectively, of the 3D image
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FIG. 2. (a) Two-dimensional cross-sectional image of specimen II cut in a plane parallel to the xz plane from the 3D volume data set. For the 3D image, FOV, (19.2 mm) 3 ; image matrix, 128 3 ; voxel size, (0.15 mm) 3 . (b) Difference at the same slice position between two 3D image data sets which were measured under positive and negative readout gradients.
data. These images were made by taking a minimum value along each projection ray perpendicular to the projection plane, which was the xy plane in Fig. 3b and the xz plane in Fig. 3d. Since the thickness of one 2D slice is 0.15 mm, the thickness of the slabs for the mIP calculation is 1.2 and 3.6 mm in Fig. 3b and Fig. 3d, respectively. In these figures, we can clearly see the connections or network of the polyhedral edges, so that we can realize the polyhedral faces. Although these mIP images give no information along the projection direction which is perpendicular to the image plane, we can determine the x , y , and z coordinates of the vertices of the bubble polyhedra by combining three series of mIP images whose projection directions are the x , y , and z directions. In the next section, the methods to measure the x , y , and z coordinates for one polyhedron will be presented. Figure 4 is a surface-rendered 3D display of specimen II. Although we can realize the 3D structure of bubble polyhedra by seeing such displays from various viewing angles, it is difficult to measure the geometrical quantities of the polyhedra from these images. MEASUREMENTS OF 3D GEOMETRICAL QUANTITIES
First, I shall explain why usual 3D volume-extraction techniques cannot be applied to the image data measured in this study. Such volume-extraction methods are based on ‘‘image segmentation’’ in 2D slices: specific regions
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such as brain, heart, or some other organ are cut by selecting appropriate threshold values in 2D cross-sectional slices, and 3D volume regions are defined by sets of the 2D regions. This algorithm is useful only for 3D objects of which exterior and interior regions can be distinguished from each other in 2D slices. In the 3D image data measured in this study, however, only polyhedral edges of bubbles are visualized as seen in Figs. 3a and 3c because their faces were already lost or too thin ( !100 mm) for detection. Thus, it is very difficult to distinguish between interior and exterior regions of bubble polyhedra in the 3D image data. Therefore, some other 3D volume-extraction technique must be developed to extract bubble polyhedra from the 3D image data set. I have developed one technique utilizing series of mIP images as shown in Figs. 3b and 3d. The computer program to measure geometrical quantities of the bubble polyhedra was developed on an X-window system running on a workstation ( HP9000 / 712 / 60 ) , and bubble polyhedra were interactively extracted on the CRT screen. However, geometrical quantities were measured only for specimen II because the contrast-to-noise ratio of the mIP images was not enough for specimen I. In all the mIP images examined, little curvature was observed for the polyhedral edges. Thus, all bubble polyhedra were described as having straight edges. The procedure to measure the x, y, and z coordinates of vertices for one bubble polyhedron is as follows:
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FIG. 3. Single-slice image (a) and minimum-intensity projection (mIP) image (b) of specimen I. Single-slice image (c) and mIP image (d) of specimen II. Images of specimen I are displayed in the xy plane and those of specimen II are displayed in the xz plane.
1. Compute three series of mIP images whose projection directions are x, y, and z. The computation is repeated for slabs of 24 contiguous slices while the slab volume is incrementally moved from one edge to the other in the field of view to cover the whole image area. Thus, three image series each consisting of 104 [128 0 (12 1 2)] mIP images are obtained. 2. Select a polygon (polyhedral face) in one mIP image,
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for example, parallel to the xy plane. Click the mouse button on a vertex of the polygon to determine the x and y coordinates of the vertex. To measure the z coordinate, a mIP image perpendicular to the xy plane (parallel to the xz or the yz plane) and containing the vertex is used. 3. Click the mouse button on the next vertex which is on the same polygon and has an edge connection with the first
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FIG. 5. Bubble polyhedra extracted from the mIP images of specimen II: (a) 12-sided, (b) 13-sided, (c) 14-sided, and (d) 15-sided polyhedra.
from the mIP images. Although only the vertices and edges of the bubble polyhedra were measured from the images, surface areas, volumes, and flatness of the polyhedral faces were calculated under some approximations. The results are listed in Table 1, and some of them are shown in Fig. 5. In Table 1, surface areas and volumes of the polyhedra were calculated by taking averages over possible divisions of the polyhedral faces into triangles because the faces were not always planar. Flatness, which represents flatness of a polyhedral face, was calculated as follows: a polyhedral face was divided into triangles, vectors normal to the triangles were calculated, angles which the normal vectors made with other normal vectors were calculated and averaged over possible combinations of the normal vectors, and the averaged angles were further averaged over possible divisions of the polygon into triangles. The average
FIG. 4. Surface-rendered 3D display of specimen II.
vertex. Coordinates x, y, and z are determined in the same way as described above. In this way, the edge connecting the vertices is determined. 4. Repeat operation 3 until the measured vertex reaches the initial vertex. In this way, the x, y, and z coordinates of vertices for the polygon are determined. 5. Repeat operations 2–4 until the measured polygons cover a polyhedron. The above operations 2 – 5 were repeated, and eight randomly chosen polyhedra in the central region, where image distortion was on the order of one pixel length, were measured. It took about an hour to extract one polyhedron
TABLE 1 Geometrical Quantities of Polyhedra Extracted from Specimen II Number of faces Volume (mm3)
Average flatness (7)
Isometric quotient
Bubble no.
Total
Quadrilateral
Pentagon
Hexagon
Surface area (mm2)
1 2 3 4 5 6 7 8
12 13 13 13 14 14 15 15
0 1 1 1 0 2 2 3
12 10 10 10 12 8 8 6
0 2 2 2 2 4 5 6
76.7 75.6 75.7 82.5 81.1 76.3 81.9 88.8
50.3 51.6 50.4 56.8 56.8 51.6 58.6 65.0
8.2 9.1 8.5 7.4 9.3 9.1 10.5 7.7
0.703 0.712 0.736 0.726 0.741 0.714 0.731 0.720
13.63
1.25
9.50
2.88
79.8
55.1
8.7
0.723
Average
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FIG. 7. Distribution of intersection angles at the vertices of the bubble polyhedra. The mean value is 109.227, which is close to the tetrahedral angle of 109.477.
FIG. 6. Distributions of surface areas for (a) quadrilaterals, (b) pentagons, and (c) hexagons in the bubble polyhedra.
flatness for one polyhedron is the average of the flatness of polyhedral faces over the polyhedron. Thus, if a polyhedron consists of planar faces only, the average flatness is 07. The isometric quotient, which is shown in the ninth column of Table 1 and represents a figure of merit for surface area minimization, is defined by 36pV 2 /A 3 (7), where V and A are the volume and surface area of the cell. Its value is 0.5236 for a cube which can completely fill a 3D space and 1.0 for a sphere which cannot fill a 3D space without
voids. To attain a larger isometric quotient, polyhedra which can fill 3D spaces should be as close to a sphere as possible. However, as shown in Fig. 5, the measured polyhedra have structures elongated along the vertical direction in the figure. Since this direction is the same for all polyhedra, this elongation of the structures is presumed to be caused by some tension or other physical mechanism during the formation of the polyurethane foam. Thus, to minimize the surface area and obtain a better isometric quotient, the structures of the measured polyhedra were shrunk by a factor of 0.60, which was optimal for this foam, along the vertical direction in Fig. 5. The isometric quotients are tabulated in Table 1. To the best of the author’s knowledge, this is the first such result for real foams. The average value of the isometric quotients for the eight polyhedra was improved from 0.674 to 0.723 by this correction. We will discuss this quotient in more detail in the next section. Figure 6 shows distributions of surface areas of the faces in the measured polyhedra. There is a definite correlation between surface areas and the number of sides. Figure 7 shows a distribution of intersection angles at the vertices of the polyhedra. These angles were calculated after the correction described above were made. Most of them ranged from 707 to 1507 and their average was 109.227, which was very close to the tetrahedral angle of 109.477. This result
TABLE 2 Parameters of Space-Filling Polyhedra
Kelvin (2) Weaire and Phelan (3) Matzke (4) This study
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Ratio of numbers of polyhedral faces (4:5:6)
Ratio of numbers of polyhedra (12:13:14:15)
Average number of faces per polyhedron
Isometric quotient
43:0:57 0:89:11 11:67:22 9:70:21
0:0:100:0 25:0:75:0 12:30:36:18 13:38:25:25
14 13.5 13.70 13.63
0.757 0.764 — 0.723
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shows that the intersection angles at vertices are randomly distributed around the tetrahedral angle: the distribution is nearly Gaussian and its standard deviation is about 157. To summarize, the measured results are as follows: Only 12-, 13-, 14-, and 15-sided polyhedra were observed. Polyhedra with less than 12 faces and more than 15 faces were not observed. The average number of faces per one polyhedron was 13.63. Only quadrilateral, pentagonal, and hexagonal faces were observed and triangular faces and faces with more than six edges were not observed. The ratio of numbers of quadrilaterals, pentagons, and hexagons was 9.2:69.7:21.1. The cell size was nearly uniform: most cells fell within {10% of the mean value. The average of isometric quotients for polyhedra was improved to 0.723 by shrinking them along one direction. There was a positive correlation between the surface area and the number of sides of the polyhedral faces. Intersection angles at vertices of the bubble polyhedra were distributed around the tetrahedral angle. DISCUSSION
Table 2 summarizes parameters of space-filling polyhedra by theoretical studies of Kelvin (2) and Weaire and Phelan (3) and experimental studies by Matzke (4) and in the present paper. The results by Matzke and those presented here gave good agreement. Therefore, the same physical mechanism must determine the structure of those foams, although Matzke made soap bubbles one by one and the polyurethane foam used in this study was made through some chemical reactions which were accompanied by gas production. Although the agreement between experimental results is satisfactory, as described above, there is a significant difference between experimental and theoretical studies. The dis-
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agreement between the experiments and Kelvin’s tetrakaidecahedron is, however, not important because a more optimal solution has been already proposed by Weaire and Phelan. A remarkable difference between the experiments and the solution is that the latter does not contain any quadrilateral faces nor any 13-sided and 15-sided polyhedra. In other words, the unit cell of the theory contains only eight polyhedra but the unit cell in real foam, if it exists, seems to contain a much larger number of polyhedra. However, since the isometric quotient obtained in this experiment is much smaller than that of the theory and the experimental precision is limited, we cannot explain this disagreement quantitatively. The method developed here is, however, very powerful for studying 3D structure of foams, so more careful foam preparation and more systematic research using this technique may clarify this contradiction. ACKNOWLEDGMENTS The author thanks Dr. E. Fukushima for critical reading of the manuscript. He also thanks Dr. T. Hashimoto for cooperation at the early stage of this study, Professor D. Weaire for useful comments on this study, and Professor T. Ogawa for kind guidance to this field and encouragement. This work was supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture in Japan.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
J. Gray, Nature 367, 598 (1994). W. Thomson (Lord Kelvin), Philos. Mag. 24, 503 (1887). D. Weaire and R. Phelan, Philos. Mag. Lett. 69, 107 (1994). E. Matzke, Am. J. Botany 32, 58 (1946). K. Kose, Phys. Rev. Lett. 72, 1467 (1994). K. Kose, Phys. Fluids 6, S4 (1994). S. Ross, Am. J. Phys. 46, 513 (1978).
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