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Moments of distributions of distances to the nearest capillary in tissue
MICROVASCULAR
Moments
RESEARCH
27, 114-116 (1984)
of Distributions
of Distances to the Nearest Capillary in Tissue LOUISD. HOMER
Naval Medical Research Institute, Bethesda, Maryland 20814 Received June 21, 1983
Kayar and co-workers have published methods for estimating the distribution of distances to the nearest capillary (1982a,b) and applied their methods to the distances to be found in the gastrocnemius muscle (1982~). One of these papers (1982a) presents formulas for the mean and variance of the distributions of distances to be expected, assuming the capillaries are arranged in square lattices or hexagonal lattices or are randomly spaced. The formulas they gave for the mean and variance of the square and hexagonal lattice arrangements are approximations, good to only one decimal place. Exact general formulas for the moments of these three distributions are given below. For a square lattice of unit length along a side, the distance to the nearest capillary of a randomly positioned point is r, and the probability density function of r is (1982a,b) f(r) f(r)
= 2m
0 S r S t,
(1)
= 2m - 8r cos-‘(1/2r) f < r 6 m.
(2)
The kth moment about the origin is defined by U, =
I
#f(r)dr,
(3)
and the integration is taken over the domain of J Using Eqs. (1) and (2), integration by parts of Eq. (3) leads to 8 u,
=
k
+
X’t%
rk+l
2 I 1’2 dm
dr.
(4)
Applying a standard integration formula from a CRC handbook to Eq. (4) (Weast et al., 1964) we also have 2
2k
v5ii
uk = (k + 2) (k + 1) 2k/2+ (k + 2) (k + 1) I 1’2 dm
114 0026-2862/84 $3.00 Copyright Q 19&I by Academic Press, Inc. All rights of reproduction in any form reserved, Printed in U.S.A.
).!-1 dr.
(5)
BRIEF
115
COMMUNICATIONS
But using (4), we may make a substitution for the integral in (5) to obtain a recursive expression. 2 k2 U,-, uk = (k + 2)(k + 1)2”* + (k + 2)(k + 1) 4’
(6)
Thus, if U1 and U, are known, higher moments may be calculated recursively. Equation (5) may be integrated for the cases k = 1 and k = 2, again using standard integration formulas.
u-qi 1 1
6 +;log(VY+
1).
u, = Q. A similar approach with the formula for the hexagonal capillary arrangement (Kayer er al., 1982a) leads to moments V,,
”
k* V,-, 2 = (k + l)(k + 2) + 4 (k + I)(k + 2)’
v,=L+
-L log (2 + $,, 6j/3
3
(7)
v, = 4.
For the random distribution of capillaries with capillary density D, the kth moment about the origin is w
=
k
Ul
+
(k/2))
(~0)‘~
’
(8)
The gamma function, T(x), may be found in a number of tables (Weast et al., 1964). Except for the random distribution our formulas apply to normalized distances. To obtain moments in familiar units, the kth moment must be multiplied by Lk. Kayar et al. (1982a) give relationships between the capillary density D and L. For the square capillary array, L = D-l’*. For the hexagonal array, L = 0.87740-‘12. Kayar et al. (1982a) gave the mean and variance for the square array as 0.40241, and 0.0220L2, where L is the length of a side of a square in the array. More exact are 0.3826L and 0.02029L2. For the hexagonal array they gave 0.4387L and 0.0417L2. L is the length of one of the sides of the hexagon. Better approximations for the mean and variance of the distances in the hexagonal array are 0.4601L and 0.03831L2. ACKNOWLEDGMENTS This research was supported in part by Naval Medical Research and Development Command, Research Work Unit MF585271C3.0002. The opinions and assertions contained herein are the private
116
BRIEF COMMUNICATIONS
ones of the writer and are not to be construed as official or as reflecting the views of the Navy Department or the naval service at large.
REFERENCES S. R., ARCHER, P. G., LECHNER, A. J., AND BANCHERO, N. (1982a). The closest-individual method in the analysis of the distribution of capillaries. Microvasc. Res. 24, 326-341. KAYAR, S. R., ARCHER, P. G., LECHNER, A. J., AND BANCHERO, N. (1982b). Evaluation of the concentric-circles method for estimating capillary-tissue diffusion distances. Microvasc. Res. 24, 342-353. KAYAR, S. R., LECHNER, A. J., AND BANCHERO, N. (1982~). The distribution of diffusion distances in the gastrocnemius muscle of various mammals during maturation. Pluegers Arch. 394, 124129. WEAST, R. C., SELBY, S. M., AND HODGMAN, C. D. (Eds.) (1964). In “Handbook of Mathematical Tables,” 2nd ed., Chem. Rubber Co., Cleveland. KAYAR,
Moments
RESEARCH
27, 114-116 (1984)
of Distributions
of Distances to the Nearest Capillary in Tissue LOUISD. HOMER
Naval Medical Research Institute, Bethesda, Maryland 20814 Received June 21, 1983
Kayar and co-workers have published methods for estimating the distribution of distances to the nearest capillary (1982a,b) and applied their methods to the distances to be found in the gastrocnemius muscle (1982~). One of these papers (1982a) presents formulas for the mean and variance of the distributions of distances to be expected, assuming the capillaries are arranged in square lattices or hexagonal lattices or are randomly spaced. The formulas they gave for the mean and variance of the square and hexagonal lattice arrangements are approximations, good to only one decimal place. Exact general formulas for the moments of these three distributions are given below. For a square lattice of unit length along a side, the distance to the nearest capillary of a randomly positioned point is r, and the probability density function of r is (1982a,b) f(r) f(r)
= 2m
0 S r S t,
(1)
= 2m - 8r cos-‘(1/2r) f < r 6 m.
(2)
The kth moment about the origin is defined by U, =
I
#f(r)dr,
(3)
and the integration is taken over the domain of J Using Eqs. (1) and (2), integration by parts of Eq. (3) leads to 8 u,
=
k
+
X’t%
rk+l
2 I 1’2 dm
dr.
(4)
Applying a standard integration formula from a CRC handbook to Eq. (4) (Weast et al., 1964) we also have 2
2k
v5ii
uk = (k + 2) (k + 1) 2k/2+ (k + 2) (k + 1) I 1’2 dm
114 0026-2862/84 $3.00 Copyright Q 19&I by Academic Press, Inc. All rights of reproduction in any form reserved, Printed in U.S.A.
).!-1 dr.
(5)
BRIEF
115
COMMUNICATIONS
But using (4), we may make a substitution for the integral in (5) to obtain a recursive expression. 2 k2 U,-, uk = (k + 2)(k + 1)2”* + (k + 2)(k + 1) 4’
(6)
Thus, if U1 and U, are known, higher moments may be calculated recursively. Equation (5) may be integrated for the cases k = 1 and k = 2, again using standard integration formulas.
u-qi 1 1
6 +;log(VY+
1).
u, = Q. A similar approach with the formula for the hexagonal capillary arrangement (Kayer er al., 1982a) leads to moments V,,
”
k* V,-, 2 = (k + l)(k + 2) + 4 (k + I)(k + 2)’
v,=L+
-L log (2 + $,, 6j/3
3
(7)
v, = 4.
For the random distribution of capillaries with capillary density D, the kth moment about the origin is w
=
k
Ul
+
(k/2))
(~0)‘~
’
(8)
The gamma function, T(x), may be found in a number of tables (Weast et al., 1964). Except for the random distribution our formulas apply to normalized distances. To obtain moments in familiar units, the kth moment must be multiplied by Lk. Kayar et al. (1982a) give relationships between the capillary density D and L. For the square capillary array, L = D-l’*. For the hexagonal array, L = 0.87740-‘12. Kayar et al. (1982a) gave the mean and variance for the square array as 0.40241, and 0.0220L2, where L is the length of a side of a square in the array. More exact are 0.3826L and 0.02029L2. For the hexagonal array they gave 0.4387L and 0.0417L2. L is the length of one of the sides of the hexagon. Better approximations for the mean and variance of the distances in the hexagonal array are 0.4601L and 0.03831L2. ACKNOWLEDGMENTS This research was supported in part by Naval Medical Research and Development Command, Research Work Unit MF585271C3.0002. The opinions and assertions contained herein are the private
116
BRIEF COMMUNICATIONS
ones of the writer and are not to be construed as official or as reflecting the views of the Navy Department or the naval service at large.
REFERENCES S. R., ARCHER, P. G., LECHNER, A. J., AND BANCHERO, N. (1982a). The closest-individual method in the analysis of the distribution of capillaries. Microvasc. Res. 24, 326-341. KAYAR, S. R., ARCHER, P. G., LECHNER, A. J., AND BANCHERO, N. (1982b). Evaluation of the concentric-circles method for estimating capillary-tissue diffusion distances. Microvasc. Res. 24, 342-353. KAYAR, S. R., LECHNER, A. J., AND BANCHERO, N. (1982~). The distribution of diffusion distances in the gastrocnemius muscle of various mammals during maturation. Pluegers Arch. 394, 124129. WEAST, R. C., SELBY, S. M., AND HODGMAN, C. D. (Eds.) (1964). In “Handbook of Mathematical Tables,” 2nd ed., Chem. Rubber Co., Cleveland. KAYAR,