- Email: [email protected]
3D modeling for sagittal suture
ARTICLE IN PRESS
Physica A 359 (2006) 538–546 www.elsevier.com/locate/physa
3D modeling for sagittal suture Yasuhito Ootaa, Kenzo Onob, Sasuke Miyazimac, a
Department of Computer Science, Chubu University, Kasugai, Aichi 487, Japan b Department of Pathology, Tosei General Hospital, Seto, Aichi 489, Japan c Department of Natural Science and Mathematics, Chubu University, Kasugai, Aichi 487, Japan Received 2 December 2004; received in revised form 26 May 2005 Available online 7 July 2005
Abstract Sagittal suture has a complex pattern which is formed by collision of edges of left and right growing parietal bones. There is fibrous tissue between the left and the right parietal bones, where an osteoblast becomes a basis of ossification. We simulate the growth and the collision of the bone by using the Eden growth model in 3-dimensional space (3D) and obtain a pattern which is similar to the sagittal suture on the inner surface as well as on the outer surface of the calvaria. r 2005 Elsevier B.V. All rights reserved. Keywords: Sagittal suture; The Eden model; The Hurst exponent
1. Introduction A new concept of fractality introduced by Mandelbrot [1] is applied not only to the fields of mathematics and physics but also to economics, biology, medical science and many other fields [2–12]. The fractal geometry treats a structure of non-integer dimension. No matter whether a pattern is ordered or not, the fractal pattern is scaleinvariant, i.e., the character of the construction of the pattern does not change even if the scale is multiplied isotropical. Growing patterns such as DLA and the Eden model are also scale-invariant. This property is observed in the critical phenomena of Corresponding author.
E-mail address: [email protected] (S. Miyazima). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.05.095
ARTICLE IN PRESS Y. Oota et al. / Physica A 359 (2006) 538–546
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statistical physics. Since 1980s many researchers in statistical physics have been interested in fractal [2–12]. Recently, a number of researches of growing surface have been developed in the various fields [13–19]. For example, the pattern of tumor tissue is investigated as an indicator to diagnose malignancy or benignancy of tumor in pathology [9]. Here we investigate what factors are important for formation of complex patterns of ‘‘suture’’ parts on calvaria (Fig. 1). Many cranial bones are combined with each other by fibrous junction that is called suture. The suture has complex pattern, and it is the collision interface where the growing bones collide with each other. Masuda and Yohro discussed the fractality of the suture of cranium [20]. They measured the fractal dimension of suture of children (3 years old) to old persons (72 years old) by using the box counting method. As a result, they obtained the fractal dimension of 1:2oDo1:5. The suture with complex pattern is formed by growth and collision of the left and the right parietal bones. A 2-dimensional (2D) simulation for the sagittal suture is already discussed by the present authors [21]. The result of analysis shows that not only the collision of the growing surface of the two parietal bones but also cells in fibrous tissue between the two bones play an important part for formation of the complex pattern of the suture. Here we have compared the Hurst exponent of simulation with observed value of 0.770–0.950 and found that it changes with concentration of osteoblast in fibrous tissue between two parietal bones. In this paper we try to think of the collision of the left and the right parietal bones in 3-dimensional (3D) space. We found that the Hurst exponent a which characterizes the collision interface changes with concentration of osteoblasts in the same way as 2D and the pattern of the suture on the inner and outer surfaces on the calvaria has a different pattern by changing width of fibrous tissue.
2. The structure and the role of bones A frame of vertebrate such as a human being is composed of bone and cartilage. The bone and the cartilage are living tissues, and have regenerative ability, too. The Front frontal bone coronal suture sagittal suture parietal bone
parietal bone lamboid suture
occipital bone Rear
Fig. 1. Top view of the calvaria.
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bone is regenerated by the following process continuously. The osteoclast which releases the calcium is produced by differentiating from one kind of a hematopoietic cell. The osteoclast is placed on the surface of the bone, and then it transforms into an apocyte. The osteoclast stretches a ruffled border to all directions, and absorbs the calcium to make an absorption fossa to the longitudinal direction. A new bone is formed from the osteoblast following to the osteoclast that destroys the bone. Most of the quantity of the calcium in a human body can be stored in the bone. The bone which consists of osteocytes, osteoblasts, osteoclasts and bone matrix is the very important part of the human body, which absorbs the calcium by osteoblast and releases the calcium by osteoclast as well as hematopoiesis and regulation of minerals. The calcium is the main ingredient which composes the bone, and one can consider in general that the osteoblast makes the bone and the osteoclast destroys the bone. But a recent research presents a new fact that the osteoblast and the osteocyte are deeply concerned with the release of the calcium [22–25].
3. The cranium formation and the suture pattern The cranium is not a piece of bone but it is composed of 23 pieces of cranial bones to form a tightly combined structure. The growth of the cranial bones starts from eminence and the calvalia consists of five cranial bones (the left and the right frontal, the left and the right parietal and occipital bones as shown in Fig. 2). It is well known that the parietal bone region has a gap just after the birth. The cranial bones collide with each other in the center area of the fibrous tissue and form the suture. The cranium formed by such a process plays an important roll in maintaining enough space for the brain. A set of the cranial bones does not have ‘‘articulation’’ point. The suture is a fibrous junction and various names are given to each suture. Fig. 1 is a shape of the calvaria. ‘‘Coronal suture’’ is the suture of a frontal bone and the parietal bones, and ‘‘sagittal suture’’ of the left and the right parietal bones, and ‘‘lambdoid suture’’ of the parietal bones and an occipital bone. The growth of cranium does not finish yet at the birth and the pattern of the suture is not accomplished. After the birth about two years are necessary before the complete growth. The periods for the complete growth of the cranium, however, depend on the individuals. The suture part opens easily under the intracranial pressure in the infancy. It is in the late middle age when complete formation of the cranium is performed [22–25]. The suture formed by the collision of the two parietal bones is a hook engaged with firmly like a coastline which shows a complicated and irregular pattern. We simulate this complex pattern of the sagittal suture in the 3D space, considering that the suture is formed by the collision of the growing front of the left and the right parietal bones, where the Eden growth model is applied.
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Fig. 2. An embryo calvaria picture [26]. A symbol shows point of the eminence which is the starting point of the growth of the cranial bones.
4. Growing surfaces in the Eden growth model Complex patterns such as Koch curve and a coastline are called self-similar fractal. These patterns have an important property that a similar pattern to the original one is obtained, even if the length of vertical (Y-axis) and horizontal (Xaxis) directions is equally multiplied. But in the nature, there are many complex patterns which are obtained by anisotropic multiplication between the vertical and the horizontal directions. These complex patterns are called self-affine fractal. The patterns of the ridge curve of mountains and the surfaces of the growth front of the Eden growth model have the self-affinity. These self-affine patterns cannot be characterized by only one index of the fractal dimension [3,5,9]. An example shown in Fig. 3 requires two indices which characterize the multiplication factors in the X and the Y directions of the Eden growth model, where the seed cells are placed on a straight line. A scaling form is expressed as in the following equation for the pattern in the Eden growth model where the growth starts from the straight line: h a sð‘; hÞ‘ f z , (1) ‘ where sð‘; hÞ is the fluctuation of edge of the growth surface, ‘ the partition width of collision surface, and h the width which is measured from the straight line. Here the
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scaling function is given by ( xb ðxhh1Þ f ðxÞ ¼ const.ð1hhxÞ :
(2)
The exponent b characterizes the increase of fluctuation of the growing surface. The exponent a is the Hurst exponent and z is defined to be a/b [15].
5. Formation of suture The suture formed by the collision of the left and the right parietal bones is selfaffine as mentioned above. From Eqs. (1) and (2), the following relation between the fluctuation of collision interface and the partition width of collision interface ‘ is
Fig. 3. The pattern of the Eden growth model which started from 2000 cells on a straight line.
Fig. 4. The growth model of the parietal bones in a cube of L W D. L, W and D are the length of the wall, the gap and the depth, respectively. The left and right rectangular cubes are the edge of the growth of the left and the right parietal bones, and the square is osteoblast in the surface of the gap. The walls grow up in the direction of arrow.
W ¼ 198
W ¼ 298
D
p ¼ 0:01
p ¼ 0:001
p ¼ 0:0001
p ¼ 0:01
p ¼ 0:001
p ¼ 0:0001
p ¼ 0:01
p ¼ 0:001
p ¼ 0:0001
Top 5 10 15 20 25 30 35 40 45 Bottom
0.82 0.82 0.79 0.76 0.72 0.69 0.64 0.58 0.51 0.43 0.39
0.76 0.76 0.75 0.73 0.70 0.66 0.59 0.52 0.44 0.37 0.35
0.52 0.52 0.51 0.49 0.46 0.44 0.41 0.39 0.36 0.35 0.36
0.85 0.85 0.83 0.81 0.79 0.78 0.76 0.75 0.73 0.73 0.73
0.84 0.85 0.84 0.83 0.82 0.80 0.79 0.78 0.76 0.74 0.73
0.70 0.71 0.70 0.70 0.69 0.68 0.67 0.66 0.65 0.64 0.62
0.88 0.88 0.86 0.84 0.82 0.81 0.80 0.79 0.78 0.78 0.78
0.86 0.87 0.86 0.85 0.85 0.84 0.83 0.82 0.81 0.81 0.79
0.78 0.79 0.78 0.78 0.78 0.77 0.77 0.76 0.75 0.75 0.74
ARTICLE IN PRESS
W ¼ 98
Y. Oota et al. / Physica A 359 (2006) 538–546
Table 1 The Hurst exponent a depends on osteoblast concentration p
543
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obtained: sð‘Þ‘a .
(3)
At first we prepare a 3D space of L W D with L ¼ 1000, D ¼ 50 and the gap W from 98 to 298 as intervals of the left and the right walls (Fig. 4). Now, the osteoblast (square in the Fig. 4) is arranged at random as a root of the osteocyte on the surface of the gap W, on the depth D ¼ 1 (top) (we suppose that top layer is the fibrous tissue). The osteoblasts of L W p are assumed to distribute in the surface of the gap. Here p is the concentration of the osteoblast, changing from p ¼ 0:01 to p ¼ 0:0001. The seed cells on the left and the right walls which grow in the same way as the Eden growth model can be considered as the growth fronts of the left and the right parietal bones. The growth of the parietal bones on both the sides forms the collision interface when the gap is filled out by the osteocyte. A distance h from a wall on one of the sides to the collision interface is measured and the Hurst exponent a is calculated from Eq. (3) (Table 1). Table 1 shows that in the depth D ¼ 1 (top), no matter whether the gap W is large or not, when the value of the concentration p is large, the Hurst exponent is large. When the value of the concentration p is small, the Hurst exponent is large for the gap W ¼ 298. This fact is corresponding to following discussion of our simulation in 2D: when the gap W is large enough, the Eden growth of osteoblasts at the center area gives more influences to the suture pattern of a cell, because the central osteoblasts can grow for a longer time and occupy a wider space before the left and the right parietal bones.
6. Results and conclusions The cells which are a root of the osteocyte have been grown by using Eden growth. The cells in the fibrous tissue contact with one of the left and the right growing
Fig. 5. Pictures of an adult cranium [26]: (a) is the outer surface of the cranium, and (b) is the inner surface.
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parietal bones. The collision interface of the left and the right parietal bones forms the sagittal suture. The sagittal suture patterns are observed to be different between the outer and the inner surfaces of the calvalia (Fig. 5). From the result in the present simulation, we find that the complexity of the collision interface disappears gradually, as the depth D increases (Fig. 6). This result is matched with the fact shown in anatomy photograph [26]. The Hurst exponent becomes smaller by 10% in the gap of W ¼ 198. When the gap W is 298, we have found almost constant (0.74–0.88) and large the Hurst exponent at any depths and concentrations (Table 1). As the gap between the left and the right parietal bones of a newborn baby is estimated to be about 0.5 cm, the gap W in this simulation is set to be 200–300, since the size of the osteoblast is 20–30 mm. The Hurst exponent in the case of depth
Fig. 6. A stage of ‘‘suture’’ in the present simulation (p ¼ 0:001 and W ¼ 98): (a) is depth D ¼ 1 (top layer), (b) is depth D ¼ 25 (middle layer), (c) is depth D ¼ 50 (bottom layer).
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D ¼ 50 (bottom) and the gap W ¼ 98 is about 0.38 and can be compared with the collision interface calculated by the simple collision of Eden growth model without osteoblast. This fact means that the collision occurs on the lower layer before the growth of the osteoblast influences from the top surface. It is shown that the shape of the sagittal suture on the inner surface of the calvalia is produced under the influence of the complexity of the pattern of the top surface. The roughness of the inner surface, however, disappears because the depth eliminates the influence of osteoblast on the top surface as D increases. The Hurst exponent of the collision interface is large at the top and small at the bottom (Table 1) as far as the value of the osteoblast concentration p is large and W is small. Finally, we can say that the complex pattern of the suture is produced by the collision of two parietal bones with many osteoblasts between the two bones using the Eden growth model. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
B.B. Mandelbrot, Fractal Geometry of Nature, Freeman, San Francisco, 1982. H. Takayasu, Fractal, Asakura Publishing Co., Tokyo, 1986 (in Japanese). J. Feder, Fractals, Plenum Press, New York, 1988. M. Matsushita, M. Sano, Y. Hayakawa, H. Honjo, Y. Sawada, Phys. Rev. Lett. 53 (1984) 286. T. Vicsek, F. Family, Phys. Rev. Lett. 52 (1984) 1669. H.E. Stanley, N. Qstrowski, On Growth and Form, Martinus Nijhoff, Dordrecht, 1986. R. Jullien, R. Botet, Aggregation and Fractal Aggregates, World Scientific, Singapore, 1987. H. Fujikawa, M. Matsushita, J. Phys. Soc. Japan 58 (1989) 3875. M. Matsushita (Ed.), Fractal in Medical Science and Biology, Asakura Publishing Co., Tokyo, 1992 (in Japanese). S. Matsuura, S. Miyazima, Fractals 1 (1993) 11. M. Takayasu, H. Takayasu, T. Sato, Physica A 233 (1996) 824. K. Yamamoto, S. Miyazima, R.K. Koshal, Y. Yamada, Physica A 314 (2002) 743. P. Meakin, P. Ramanlal, L.M. Sander, R.C. Ball, Phys. Rev. A 34 (1986) 5091. D.E. Wolf, J. Kerte`sz, Europhysics. Lett. 4 (1987) 651. T. Vicsek, Fractal Growth Phenomena, World Scientific, Singapore, 1989. D.E. Wolf, J. Villain, Europhys. Lett. 13 (5) (1990) 389. A.-L. Baraba´si, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, 1995. T. Nagamine, K. Maruyama, S. Miyazima, Fractals 1 (1993) 998. Y. Ueno, T. Nagamine, K. Maruyama, S. Miyazima, Physica A 233 (1996) 672. Y. Masuda, T. Yohro, Okajimas Folia Anat. Japan 64 (1) (1987) 39. Y. Oota, T. Nagamine, K. Ono, S. Miyazima, Forma 19 (2004) 197. K. Kimura, Human of Anatomy, Taisyukan Publishing Co., Tokyo, 1988 (in Japanese). H. Terada, T. Fujita, Laboratory Manual of Osteology, Nanzando Co., Tokyo, 1992 (in Japanese). T. Matsumoto (Ed.), Bone Signal and Osteoporosis, Yodosha Co., Tokyo, 1997 (in Japanese). L.P. Gartner, J.L. Hiatt, J.M. Strum, Cell Biology and Historogy, third ed., Lippincott Williams & Willkins, Philadelphia, 1998. P.H. Abrahams, R.T. Hutchings, S.C. Marks Jr, McMinn’s Colour Atlas of Human Anatomy, fourth ed., Nankodo Co., Tokyo, 1999 (in Japanese).
Physica A 359 (2006) 538–546 www.elsevier.com/locate/physa
3D modeling for sagittal suture Yasuhito Ootaa, Kenzo Onob, Sasuke Miyazimac, a
Department of Computer Science, Chubu University, Kasugai, Aichi 487, Japan b Department of Pathology, Tosei General Hospital, Seto, Aichi 489, Japan c Department of Natural Science and Mathematics, Chubu University, Kasugai, Aichi 487, Japan Received 2 December 2004; received in revised form 26 May 2005 Available online 7 July 2005
Abstract Sagittal suture has a complex pattern which is formed by collision of edges of left and right growing parietal bones. There is fibrous tissue between the left and the right parietal bones, where an osteoblast becomes a basis of ossification. We simulate the growth and the collision of the bone by using the Eden growth model in 3-dimensional space (3D) and obtain a pattern which is similar to the sagittal suture on the inner surface as well as on the outer surface of the calvaria. r 2005 Elsevier B.V. All rights reserved. Keywords: Sagittal suture; The Eden model; The Hurst exponent
1. Introduction A new concept of fractality introduced by Mandelbrot [1] is applied not only to the fields of mathematics and physics but also to economics, biology, medical science and many other fields [2–12]. The fractal geometry treats a structure of non-integer dimension. No matter whether a pattern is ordered or not, the fractal pattern is scaleinvariant, i.e., the character of the construction of the pattern does not change even if the scale is multiplied isotropical. Growing patterns such as DLA and the Eden model are also scale-invariant. This property is observed in the critical phenomena of Corresponding author.
E-mail address: [email protected] (S. Miyazima). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.05.095
ARTICLE IN PRESS Y. Oota et al. / Physica A 359 (2006) 538–546
539
statistical physics. Since 1980s many researchers in statistical physics have been interested in fractal [2–12]. Recently, a number of researches of growing surface have been developed in the various fields [13–19]. For example, the pattern of tumor tissue is investigated as an indicator to diagnose malignancy or benignancy of tumor in pathology [9]. Here we investigate what factors are important for formation of complex patterns of ‘‘suture’’ parts on calvaria (Fig. 1). Many cranial bones are combined with each other by fibrous junction that is called suture. The suture has complex pattern, and it is the collision interface where the growing bones collide with each other. Masuda and Yohro discussed the fractality of the suture of cranium [20]. They measured the fractal dimension of suture of children (3 years old) to old persons (72 years old) by using the box counting method. As a result, they obtained the fractal dimension of 1:2oDo1:5. The suture with complex pattern is formed by growth and collision of the left and the right parietal bones. A 2-dimensional (2D) simulation for the sagittal suture is already discussed by the present authors [21]. The result of analysis shows that not only the collision of the growing surface of the two parietal bones but also cells in fibrous tissue between the two bones play an important part for formation of the complex pattern of the suture. Here we have compared the Hurst exponent of simulation with observed value of 0.770–0.950 and found that it changes with concentration of osteoblast in fibrous tissue between two parietal bones. In this paper we try to think of the collision of the left and the right parietal bones in 3-dimensional (3D) space. We found that the Hurst exponent a which characterizes the collision interface changes with concentration of osteoblasts in the same way as 2D and the pattern of the suture on the inner and outer surfaces on the calvaria has a different pattern by changing width of fibrous tissue.
2. The structure and the role of bones A frame of vertebrate such as a human being is composed of bone and cartilage. The bone and the cartilage are living tissues, and have regenerative ability, too. The Front frontal bone coronal suture sagittal suture parietal bone
parietal bone lamboid suture
occipital bone Rear
Fig. 1. Top view of the calvaria.
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bone is regenerated by the following process continuously. The osteoclast which releases the calcium is produced by differentiating from one kind of a hematopoietic cell. The osteoclast is placed on the surface of the bone, and then it transforms into an apocyte. The osteoclast stretches a ruffled border to all directions, and absorbs the calcium to make an absorption fossa to the longitudinal direction. A new bone is formed from the osteoblast following to the osteoclast that destroys the bone. Most of the quantity of the calcium in a human body can be stored in the bone. The bone which consists of osteocytes, osteoblasts, osteoclasts and bone matrix is the very important part of the human body, which absorbs the calcium by osteoblast and releases the calcium by osteoclast as well as hematopoiesis and regulation of minerals. The calcium is the main ingredient which composes the bone, and one can consider in general that the osteoblast makes the bone and the osteoclast destroys the bone. But a recent research presents a new fact that the osteoblast and the osteocyte are deeply concerned with the release of the calcium [22–25].
3. The cranium formation and the suture pattern The cranium is not a piece of bone but it is composed of 23 pieces of cranial bones to form a tightly combined structure. The growth of the cranial bones starts from eminence and the calvalia consists of five cranial bones (the left and the right frontal, the left and the right parietal and occipital bones as shown in Fig. 2). It is well known that the parietal bone region has a gap just after the birth. The cranial bones collide with each other in the center area of the fibrous tissue and form the suture. The cranium formed by such a process plays an important roll in maintaining enough space for the brain. A set of the cranial bones does not have ‘‘articulation’’ point. The suture is a fibrous junction and various names are given to each suture. Fig. 1 is a shape of the calvaria. ‘‘Coronal suture’’ is the suture of a frontal bone and the parietal bones, and ‘‘sagittal suture’’ of the left and the right parietal bones, and ‘‘lambdoid suture’’ of the parietal bones and an occipital bone. The growth of cranium does not finish yet at the birth and the pattern of the suture is not accomplished. After the birth about two years are necessary before the complete growth. The periods for the complete growth of the cranium, however, depend on the individuals. The suture part opens easily under the intracranial pressure in the infancy. It is in the late middle age when complete formation of the cranium is performed [22–25]. The suture formed by the collision of the two parietal bones is a hook engaged with firmly like a coastline which shows a complicated and irregular pattern. We simulate this complex pattern of the sagittal suture in the 3D space, considering that the suture is formed by the collision of the growing front of the left and the right parietal bones, where the Eden growth model is applied.
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Fig. 2. An embryo calvaria picture [26]. A symbol shows point of the eminence which is the starting point of the growth of the cranial bones.
4. Growing surfaces in the Eden growth model Complex patterns such as Koch curve and a coastline are called self-similar fractal. These patterns have an important property that a similar pattern to the original one is obtained, even if the length of vertical (Y-axis) and horizontal (Xaxis) directions is equally multiplied. But in the nature, there are many complex patterns which are obtained by anisotropic multiplication between the vertical and the horizontal directions. These complex patterns are called self-affine fractal. The patterns of the ridge curve of mountains and the surfaces of the growth front of the Eden growth model have the self-affinity. These self-affine patterns cannot be characterized by only one index of the fractal dimension [3,5,9]. An example shown in Fig. 3 requires two indices which characterize the multiplication factors in the X and the Y directions of the Eden growth model, where the seed cells are placed on a straight line. A scaling form is expressed as in the following equation for the pattern in the Eden growth model where the growth starts from the straight line: h a sð‘; hÞ‘ f z , (1) ‘ where sð‘; hÞ is the fluctuation of edge of the growth surface, ‘ the partition width of collision surface, and h the width which is measured from the straight line. Here the
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scaling function is given by ( xb ðxhh1Þ f ðxÞ ¼ const.ð1hhxÞ :
(2)
The exponent b characterizes the increase of fluctuation of the growing surface. The exponent a is the Hurst exponent and z is defined to be a/b [15].
5. Formation of suture The suture formed by the collision of the left and the right parietal bones is selfaffine as mentioned above. From Eqs. (1) and (2), the following relation between the fluctuation of collision interface and the partition width of collision interface ‘ is
Fig. 3. The pattern of the Eden growth model which started from 2000 cells on a straight line.
Fig. 4. The growth model of the parietal bones in a cube of L W D. L, W and D are the length of the wall, the gap and the depth, respectively. The left and right rectangular cubes are the edge of the growth of the left and the right parietal bones, and the square is osteoblast in the surface of the gap. The walls grow up in the direction of arrow.
W ¼ 198
W ¼ 298
D
p ¼ 0:01
p ¼ 0:001
p ¼ 0:0001
p ¼ 0:01
p ¼ 0:001
p ¼ 0:0001
p ¼ 0:01
p ¼ 0:001
p ¼ 0:0001
Top 5 10 15 20 25 30 35 40 45 Bottom
0.82 0.82 0.79 0.76 0.72 0.69 0.64 0.58 0.51 0.43 0.39
0.76 0.76 0.75 0.73 0.70 0.66 0.59 0.52 0.44 0.37 0.35
0.52 0.52 0.51 0.49 0.46 0.44 0.41 0.39 0.36 0.35 0.36
0.85 0.85 0.83 0.81 0.79 0.78 0.76 0.75 0.73 0.73 0.73
0.84 0.85 0.84 0.83 0.82 0.80 0.79 0.78 0.76 0.74 0.73
0.70 0.71 0.70 0.70 0.69 0.68 0.67 0.66 0.65 0.64 0.62
0.88 0.88 0.86 0.84 0.82 0.81 0.80 0.79 0.78 0.78 0.78
0.86 0.87 0.86 0.85 0.85 0.84 0.83 0.82 0.81 0.81 0.79
0.78 0.79 0.78 0.78 0.78 0.77 0.77 0.76 0.75 0.75 0.74
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W ¼ 98
Y. Oota et al. / Physica A 359 (2006) 538–546
Table 1 The Hurst exponent a depends on osteoblast concentration p
543
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obtained: sð‘Þ‘a .
(3)
At first we prepare a 3D space of L W D with L ¼ 1000, D ¼ 50 and the gap W from 98 to 298 as intervals of the left and the right walls (Fig. 4). Now, the osteoblast (square in the Fig. 4) is arranged at random as a root of the osteocyte on the surface of the gap W, on the depth D ¼ 1 (top) (we suppose that top layer is the fibrous tissue). The osteoblasts of L W p are assumed to distribute in the surface of the gap. Here p is the concentration of the osteoblast, changing from p ¼ 0:01 to p ¼ 0:0001. The seed cells on the left and the right walls which grow in the same way as the Eden growth model can be considered as the growth fronts of the left and the right parietal bones. The growth of the parietal bones on both the sides forms the collision interface when the gap is filled out by the osteocyte. A distance h from a wall on one of the sides to the collision interface is measured and the Hurst exponent a is calculated from Eq. (3) (Table 1). Table 1 shows that in the depth D ¼ 1 (top), no matter whether the gap W is large or not, when the value of the concentration p is large, the Hurst exponent is large. When the value of the concentration p is small, the Hurst exponent is large for the gap W ¼ 298. This fact is corresponding to following discussion of our simulation in 2D: when the gap W is large enough, the Eden growth of osteoblasts at the center area gives more influences to the suture pattern of a cell, because the central osteoblasts can grow for a longer time and occupy a wider space before the left and the right parietal bones.
6. Results and conclusions The cells which are a root of the osteocyte have been grown by using Eden growth. The cells in the fibrous tissue contact with one of the left and the right growing
Fig. 5. Pictures of an adult cranium [26]: (a) is the outer surface of the cranium, and (b) is the inner surface.
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parietal bones. The collision interface of the left and the right parietal bones forms the sagittal suture. The sagittal suture patterns are observed to be different between the outer and the inner surfaces of the calvalia (Fig. 5). From the result in the present simulation, we find that the complexity of the collision interface disappears gradually, as the depth D increases (Fig. 6). This result is matched with the fact shown in anatomy photograph [26]. The Hurst exponent becomes smaller by 10% in the gap of W ¼ 198. When the gap W is 298, we have found almost constant (0.74–0.88) and large the Hurst exponent at any depths and concentrations (Table 1). As the gap between the left and the right parietal bones of a newborn baby is estimated to be about 0.5 cm, the gap W in this simulation is set to be 200–300, since the size of the osteoblast is 20–30 mm. The Hurst exponent in the case of depth
Fig. 6. A stage of ‘‘suture’’ in the present simulation (p ¼ 0:001 and W ¼ 98): (a) is depth D ¼ 1 (top layer), (b) is depth D ¼ 25 (middle layer), (c) is depth D ¼ 50 (bottom layer).
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D ¼ 50 (bottom) and the gap W ¼ 98 is about 0.38 and can be compared with the collision interface calculated by the simple collision of Eden growth model without osteoblast. This fact means that the collision occurs on the lower layer before the growth of the osteoblast influences from the top surface. It is shown that the shape of the sagittal suture on the inner surface of the calvalia is produced under the influence of the complexity of the pattern of the top surface. The roughness of the inner surface, however, disappears because the depth eliminates the influence of osteoblast on the top surface as D increases. The Hurst exponent of the collision interface is large at the top and small at the bottom (Table 1) as far as the value of the osteoblast concentration p is large and W is small. Finally, we can say that the complex pattern of the suture is produced by the collision of two parietal bones with many osteoblasts between the two bones using the Eden growth model. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
B.B. Mandelbrot, Fractal Geometry of Nature, Freeman, San Francisco, 1982. H. Takayasu, Fractal, Asakura Publishing Co., Tokyo, 1986 (in Japanese). J. Feder, Fractals, Plenum Press, New York, 1988. M. Matsushita, M. Sano, Y. Hayakawa, H. Honjo, Y. Sawada, Phys. Rev. Lett. 53 (1984) 286. T. Vicsek, F. Family, Phys. Rev. Lett. 52 (1984) 1669. H.E. Stanley, N. Qstrowski, On Growth and Form, Martinus Nijhoff, Dordrecht, 1986. R. Jullien, R. Botet, Aggregation and Fractal Aggregates, World Scientific, Singapore, 1987. H. Fujikawa, M. Matsushita, J. Phys. Soc. Japan 58 (1989) 3875. M. Matsushita (Ed.), Fractal in Medical Science and Biology, Asakura Publishing Co., Tokyo, 1992 (in Japanese). S. Matsuura, S. Miyazima, Fractals 1 (1993) 11. M. Takayasu, H. Takayasu, T. Sato, Physica A 233 (1996) 824. K. Yamamoto, S. Miyazima, R.K. Koshal, Y. Yamada, Physica A 314 (2002) 743. P. Meakin, P. Ramanlal, L.M. Sander, R.C. Ball, Phys. Rev. A 34 (1986) 5091. D.E. Wolf, J. Kerte`sz, Europhysics. Lett. 4 (1987) 651. T. Vicsek, Fractal Growth Phenomena, World Scientific, Singapore, 1989. D.E. Wolf, J. Villain, Europhys. Lett. 13 (5) (1990) 389. A.-L. Baraba´si, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, 1995. T. Nagamine, K. Maruyama, S. Miyazima, Fractals 1 (1993) 998. Y. Ueno, T. Nagamine, K. Maruyama, S. Miyazima, Physica A 233 (1996) 672. Y. Masuda, T. Yohro, Okajimas Folia Anat. Japan 64 (1) (1987) 39. Y. Oota, T. Nagamine, K. Ono, S. Miyazima, Forma 19 (2004) 197. K. Kimura, Human of Anatomy, Taisyukan Publishing Co., Tokyo, 1988 (in Japanese). H. Terada, T. Fujita, Laboratory Manual of Osteology, Nanzando Co., Tokyo, 1992 (in Japanese). T. Matsumoto (Ed.), Bone Signal and Osteoporosis, Yodosha Co., Tokyo, 1997 (in Japanese). L.P. Gartner, J.L. Hiatt, J.M. Strum, Cell Biology and Historogy, third ed., Lippincott Williams & Willkins, Philadelphia, 1998. P.H. Abrahams, R.T. Hutchings, S.C. Marks Jr, McMinn’s Colour Atlas of Human Anatomy, fourth ed., Nankodo Co., Tokyo, 1999 (in Japanese).