Β Β· Stereological formulas . 25 . OPTICAL FRACTIONATOR . Estimate of total number of particles (N)...
Transcript of Β Β· Stereological formulas . 25 . OPTICAL FRACTIONATOR . Estimate of total number of particles (N)...
Stereological formulas
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AREA FRACTION FRACTIONATOR ............................................................................................................................................................................. 3
CAVALIERI ESTIMATOR ............................................................................................................................................................................................. 4
COMBINED POINT INTERCEPT .................................................................................................................................................................................. 6
CONNECTIVITY ASSAY .............................................................................................................................................................................................. 7
CYCLOIDS FOR LV ..................................................................................................................................................................................................... 8
CYCLOIDS FOR SV ................................................................................................................................................................................................... 10
DISCRETE VERTICAL ROTATOR ............................................................................................................................................................................... 12
FRACTIONATOR ...................................................................................................................................................................................................... 13
ISOTROPIC FAKIR .................................................................................................................................................................................................... 17
ISOTROPIC VIRTUAL PLANES .................................................................................................................................................................................. 18
IUR PLANES OPTICAL FRACTIONATOR .................................................................................................................................................................... 21
L-CYCLOID OPTICAL FRACTIONATOR ...................................................................................................................................................................... 22
MERZ ..................................................................................................................................................................................................................... 23
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NUCLEATOR ........................................................................................................................................................................................................... 24
OPTICAL FRACTIONATOR ....................................................................................................................................................................................... 25
OPTICAL ROTATOR ................................................................................................................................................................................................. 29
PETRIMETRICS ........................................................................................................................................................................................................ 31
PHYSICAL FRACTIONATOR ...................................................................................................................................................................................... 32
PLANAR ROTATOR .................................................................................................................................................................................................. 33
POINT SAMPLED INTERCEPT .................................................................................................................................................................................. 35
SIZE DISTRIBUTION ................................................................................................................................................................................................. 36
SPACEBALLS ........................................................................................................................................................................................................... 38
SURFACE-WEIGHTED STAR VOLUME ...................................................................................................................................................................... 40
SURFACTOR ........................................................................................................................................................................................................... 41
SV-CYCLOID FRACTIONATOR .................................................................................................................................................................................. 42
VERTICAL SPATIAL GRID ......................................................................................................................................................................................... 43
WEIBEL................................................................................................................................................................................................................... 44
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AREA FRACTION FRACTIONATOR
Estimated volume fraction (πππ£π£οΏ½ )
πππ£π£οΏ½ (ππ, ππππππ) =β ππ(ππ)ππππππ=1
β ππ(ππππππ)ππππππ=1
P(ref) Points hitting reference volume
Y Sub-region P(Y) Points hitting sub-region
Estimated area (οΏ½ΜοΏ½π΄)
οΏ½ΜοΏ½π΄ =1ππππππ
.ππ(ππ).ππ(ππππ)
asf Area sampling fraction a(p) Area associated with a point
R e f e r e n c e s Howard, C. V., & Reed, M. G. (1998). Unbiased Stereology, Three-Dimensional Measurement in Microscopy (pp. 170β172). Milton Park, England: BIOS Scientific Publishers.
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CAVALIERI ESTIMATOR
Area associated with a point (Ap)
π΄π΄ππ = ππ2 g2 Grid area
Volume associated with a point (VP)
ππππ = ππ2πππ‘π‘Μ
m Section evaluation interval π‘π‘Μ Mean section cut thickness
Estimated volume (π½π½οΏ½) πποΏ½ = π΄π΄ππππβ²π‘π‘Μ οΏ½οΏ½ππππ
ππ
ππ=1
οΏ½ Ap Area associated with a point mβ Section evaluation interval π‘π‘Μ Mean section cut thickness Pi Points counted on grid
Estimated volume
corrected for over-projection ([v])
[π£π£] = π‘π‘.οΏ½ππ.οΏ½ππβ²ππ
ππ
ππ=1
β ππππππ(ππβ²)οΏ½ t Section cut thickness k Correction factor g Grid size aβ Projected area
Coefficient of error
(CE)
πΆπΆπΆπΆ =βπππππ‘π‘ππππππππππβ ππππππππ=1
TotalVar Total variance of the estimated volume n Number of sections Pi Points counted on grid
πππππ‘π‘ππππππππππ = ππ2 + πππ΄π΄ππππππππ
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Cavalieri Estimator (2)
Variance of systematic
random sampling (VARSRS)
πππ΄π΄ππππππππ =3(π΄π΄ β ππ2) β 4π΅π΅ + πΆπΆ
12,ππ = 0
πππ΄π΄ππππππππ =3(π΄π΄ β ππ2) β 4π΅π΅ + πΆπΆ
240,ππ = 1
m Smoothness class of sampled function s2 Variance due to noise π΄π΄ = β ππππ2ππ
ππ=1 , π΅π΅ = β ππππππππ+1ππβ1ππ=1 , πΆπΆ = β ππππππππ+2ππβ2
ππ=1
With:
n : number of sections
ππ2 = 0.0724 οΏ½ ππβπποΏ½οΏ½ππβ ππππππ
ππ=1 ππβππ
Shape factor
R e f e r e n c e s GarcΓa-FiΓ±ana, M., Cruz-Orive, L.M., Mackay, C.E., Pakkenberg, B. & Roberts, N. (2003). Comparison of MR imaging against physical sectioning to estimate the volume of human cerebral compartments. Neuroimage, 18 (2), 505β516. Gundersen, H. J. G., & Jensen, E.B. (1987). The efficiency of systematic sampling in stereology and its prediction. Journal of Microscopy, 147 (3), 229β263. Howard, C. V., & Reed, M.G. (2005). Unbiased Stereology, Three-Dimensional Measurement in Microscopy (Chapter 3). New York: Garland Science/BIOS Scientific Publishers.
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COMBINED POINT INTERCEPT
Profile area (a) ππ = ππ(ππ).οΏ½ππ a(p) Area associated with a point βππ Number of points
Profile boundary (b) ππ =
ππ2ππ.οΏ½πΌπΌ d Distance between points
βπΌπΌ Number of intersections
This method is based on the principles described in the following:
Howard, C.V., Reed, M.G. (2010). Unbiased Stereology (Second Edition). QTP Publications: Coleraine, UK. See equations 2.5 and 3.2
Miles, R.E., Davy, P. (1976). Precise and general conditions for the validity of a comprehensive set of stereological fundamental formulae. Journal of Microscopy, 107 (3), 211β226.
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CONNECTIVITY ASSAY
Euler number (X3) ππ3 = πΌπΌ + π»π» β π΅π΅ I Total island markers π»π» Total hole markers π΅π΅ Total bridge markers
Number of alveoli (Nalv) πππππππ£π£ = βππ3 X3 Euler number
Sum counting frame volumes (V)
ππ = β.ππ.ππ h Disector height n Number of disectors a Area counting frame
Numerical density of alveoli (Nv)
πππ£π£ =πππππππ£π£ππ
Nalv Number of alveoli V Sum counting frame volumes
R e f e r e n c e s Ochs, M., Nyengaard, J.R., Jung, A., Knudsen, L., Voigt, M., Wahlers, T., Richter, J., & Gundersen, H.J.G. (2004). The number of alveoli in the human lung. American journal of respiratory and critical care medicine, 169 (1), 120β124.
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CYCLOIDS FOR LV
Area associated with a point (Ap)
π΄π΄ππ = ππ2 g2 Grid area
Volume associated with a
point (Vp) ππππ = ππ2πππ‘π‘Μ g2 Grid area
m Section evaluation interval π‘π‘Μ Mean section cut thickness
Length per unit volume (Lv) πΏπΏππ = 2
οΏ½πΌπΌοΏ½Μ οΏ½πΏπΆπΆοΏ½ππππππβ
πΏπΏππ =2β
.οΏ½πΌπΌοΏ½Μ οΏ½ππππππποΏ½ππππππ
πποΏ½. οΏ½πππποΏ½=
2βοΏ½πππποΏ½β πΌπΌππππππ=1
β ππππππππ=1
οΏ½ILΜ CοΏ½prj Number of counting frames β Section cut thickness Ii Intercepts Pi Test points οΏ½ICΜ cycοΏ½
prj Average number of intersections of
projected images pl Test points per unit length of cycloid
Estimated volume (πποΏ½) πποΏ½ = ππβοΏ½
πππποΏ½οΏ½ππππ
ππ
ππ=1
m Sampling fractions β Section cut thickness a Area p Number of test points Pi Test points
Estimated length (πΏπΏοΏ½) πΏπΏοΏ½ = 2 οΏ½
πππποΏ½πποΏ½πΌπΌππ
ππ
ππ=1
a Area l Line length m Sampling fractions Ii Intercepts
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Cycloids for Lv (2)
Coefficient of error for line length πΆπΆπΆπΆοΏ½πΏπΏοΏ½|πΏπΏοΏ½ =
οΏ½πππ΄π΄ππππππππβ πΌπΌππππππ=1
VARSRS Variance of systematic random sampling LοΏ½|L Estimated length per length Ii Intercepts
Variance of systematic
random sampling (VARSRS) πππ΄π΄ππππππππ =
3ππ0 β 4ππ1 + ππ212
ππππ = οΏ½ πΏπΏπππΏπΏππ+ππππβππ
ππ=1
g Grid size Li Line length at section i
Coefficient of error for length
density πΆπΆπΆπΆ(πΏπΏππ) = οΏ½
ππππ β 1οΏ½
β πΌπΌππ2ππππ=1
β πΌπΌππππππ=1 β πΌπΌππππ
ππ=1+
β ππππ2ππππ=1
β ππππππππ=1 β ππππππ
ππ=1β 2
β πΌπΌππππππππππ=1
β πΌπΌππππππ=1 β ππππππ
ππ=1οΏ½
Ii Intercepts Pi Test points n Number of probes
R e f e r e n c e s
ArtachoβPΓ©rula, E., RoldΓ‘nβVillalobos, R. (1995). Estimation of capillary length density in skeletal muscle by unbiased stereological methods: I. Use of vertical slices of known thickness The Anatomical Record, 241 (3), 337-344.
Gokhale, A. M. (1990). Unbiased estimation of curve length in 3βD using vertical slices. Journal of Microscopy, 159 (2), 133β141.
Howard, C. V., Reed, M.G. (1998). Unbiased Stereology, Three-Dimensional Measurement in Microscopy (pp. 170β172). BIOS Scientific Publishers.
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CYCLOIDS FOR SV
Area associated with a point (Ap)
π΄π΄ππ = ππ2 g2 Grid area
Volume associated with a
point (Vp) ππππ = ππ2πππ‘π‘Μ g2 Grid area
m Evaluation interval π‘π‘Μ Section cut thickness
Estimated surface area per unit volume (est Sv) πππππ‘π‘ πππ£π£ = 2 οΏ½
2πππποΏ½β πΌπΌππππππ=1
β ππππππππ=1
p/l Points per unit length of cycloid Ii Intercepts with cycloids Pi Point counts
Estimated volume (πποΏ½) πποΏ½ = πππ‘π‘Μ οΏ½
πππποΏ½οΏ½ππππ
ππ
ππ=1
m Evaluation interval π‘π‘Μ Section cut thickness a/p Area associated with each point Pi Point counts
Estimated surface area (οΏ½ΜοΏ½π) οΏ½ΜοΏ½π = 2 οΏ½πππποΏ½πππ‘π‘Μ οΏ½ πΌπΌππ
ππ
ππ=1 m Evaluation interval
π‘π‘Μ Section cut thickness a/l Area per unit length Ii Intercepts with cycloids C
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Cycloids for Sv (2)
Coefficient of error for
estimated surface (CE)
πΆπΆπΆπΆοΏ½οΏ½ΜοΏ½π|πποΏ½ =οΏ½πππ΄π΄ππππππππβ πΌπΌππππππ=1
VarSRS Variance due to
systematic random sampling
VarSRS =3g0 β 4g1 + g2
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Coefficient of error for surface
density (CE (Sv))
πΆπΆπΆπΆ(πππ£π£) = οΏ½ππ
ππ β 1οΏ½β πΌπΌππ2ππππ=1
β πΌπΌππππππ=1 β πΌπΌππππ
ππ=1+
β ππππ2ππππ=1
β ππππππππ=1 β ππππππ
ππ=1β 2
β πΌπΌππππππ=1 ππππ
β πΌπΌππππππ=1 β ππππππ
ππ=1οΏ½
n Number of measurements Ii Intercepts with cycloids Pi Point counts
References
Baddeley, A. J., Gundersen, H.J.G., & CruzβOrive, L.M. (1998) Estimation of surface area from vertical sections. Journal of Microscopy, 142 (3), 259β276.
Howard, C. V., Reed, M.G. (1998). Unbiased Stereology, Three-Dimensional Measurement in Microscopy(pp.170β172). BIOS Scientific Publishers.
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DISCRETE VERTICAL ROTATOR
Estimated volume (Est v) πππππ‘π‘ π£π£ =
ππππ
.ππππ.οΏ½ππππ
ππ
ππ=1
π·π·ππ
n Number of centriolar sections ap Area associated with each point Pi Number of points in each class Di Distance of class from central axis
References
Mironov, A. A. (1998). Estimation of subcellular organelle volume from ultrathin sections through centrioles with a discretized version of the vertical rotator. Journal of microscopy, 192(1), 29-36.
Stereological formulas
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FRACTIONATOR
Estimate of total number of particles (N) ππ = οΏ½ππβ .
1ππππππ
.1ππππππ
ππβ Particles counted ππππππ Area sampling fraction ππππππ Section sampling fraction
Variance due to
systematic random sampling β Gundersen
(VARSRS)
πππ΄π΄ππππππππ =3(π΄π΄ β ππ2) β 4π΅π΅ + πΆπΆ
12,ππ = 0
πππ΄π΄ππππππππ =3(π΄π΄ β ππ2) β 4π΅π΅ + πΆπΆ
240,ππ = 1
π΄π΄ = β (ππππβ)2ππππ=1
π΅π΅ = β ππππβππππ+1βππβ1ππ=1
πΆπΆ = β ππππβππππ+2βππβ2ππ=1
s2 Variance due to noise
Variance due to noise - Gundersen (s2) ππ2 = οΏ½ππβ
ππ
ππ=1
ππβ Particles counted n Number of sections used
Total variance β Gundersen (TotalVar)
πππππ‘π‘ππππππππππ = ππ2 + πππ΄π΄ππππππππ VARSRS Variance due to SRS s2 Variance due to noise
Coefficient of error β Gundersen (CE) πΆπΆπΆπΆ =
βπππππ‘π‘ππππππππππππ2
TotalVar Total variance s2 Variance due to noise
Number-weighted mean section cut thickness
(πππΈπΈβοΏ½οΏ½οΏ½οΏ½οΏ½)
π‘π‘ππβοΏ½οΏ½οΏ½οΏ½ =β π‘π‘ππππππ=1 ππππβ
β ππππβππππ=1
m Number of sections ti Section thickness at site i Qi Particles counted
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Fractionator (2)
Coefficient of error β Scheaffer (CE) πΆπΆπΆπΆ =
οΏ½ππ2 οΏ½1ππ β
1πΉπΉοΏ½
πποΏ½
f Number of counting frames πΉπΉ Total possible sampling sites ππ2 Estimated variance πποΏ½ Average particles counted
Average number of particles β
Scheaffer (πΈπΈοΏ½) πποΏ½ =β ππππππππ=1ππ
Qi Particles counted f Number of counting frames
Estimated variance - Scheaffer (s2) ππ2 =
β (ππππ β πποΏ½)2ππππ=1ππ β 1
f Number of counting frames Qi Particles counted QοΏ½ Average particles counted
Estimated variance of estimated cell population - Scheaffer
πΆπΆπππππΉπΉ2ππ2
ππ Cfp Finite population correction
ππ2 Estimated variance f Number of counting frames πΉπΉ Total possible sampling sites Estimated variance of mean cell
count - Scheaffer πΆπΆππππππ2
ππ Cfp Finite population correction
ππ2 Estimated variance f Number of counting frames
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Fractionator (3)
Estimated mean coefficient of error β Cruz-Orive (est Mean CE) πππππ‘π‘ ππππππππ πΆπΆπΆπΆ (πππππ‘π‘ ππ) = οΏ½
13ππ
.οΏ½οΏ½ππ1ππ β ππ2ππππ1ππ + ππ2ππ
οΏ½2ππ
ππ=1
οΏ½
12οΏ½
Q1i Counts in sub-sample 1 Q2i Counts in sub-sample 2 n Size of sub-sample
Predicted coefficient of error for estimated population β Schmitz-
Hof (CEpred) πΆπΆπΆπΆππππππππ(πππΉπΉ) = οΏ½
ππππππ(ππππβ)ππ. (ππππβ)2
πΆπΆπΆπΆππππππππ(πππΉπΉ) =1
οΏ½β ππππβππππ=1
=1
οΏ½β πππ π βπππ π =1
R Number of counting spaces S Number of sections Qrβ Counts in the βrβ-th counting space
Qsβ Counts in the βsβ-th section
References Geiser, M., CruzβOrive, L.M., Hof, V.I., & Gehr, P. (1990) Assessment of particle retention and clearance in the intrapulmonary conducting airways of hamster lungs with the fractionator. Journal of Microscopy, 160 (1), 75β88.
Glaser, E. M., Wilson, P.D. (1998). The coefficient of error of optical fractionator population size estimates: a computer simulation comparing three estimators. Journal of Microscopy, 192 (2), 163β171.
Gundersen, H.J.G., Vedel Jensen, E.B., Kieu, K., & Nielsen, J. (1999). The efficiency of systematic sampling in stereologyβreconsidered. Journal of Microscopy, 193 (3), 199β211.
Gundersen, H. J. G., Jensen, E.B. (1987). The efficiency of systematic sampling in stereology and its prediction. Journal of Microscopy, 147 (3), 229β263.
Howard, V., Reed, M. (2005). Unbiased stereology: three-dimensional measurement in microscopy (vol. 4, chapter 12). Garland Science/Bios Scientific Publishers.
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Fractionator (4)
Scheaffer, R.L., Ott, L., & Mendenhall, W. (1996). Elementary survey sampling (chapter 7). Boston: PWS-Kent.
Schmitz, C., Hof, P.R. (2000). Recommendations for straightforward and rigorous methods of counting neurons based on a computer simulation approach. Journal of Chemical Neuroanatomy, 20 (1), 93β114.
West, M. J., Slomianka, L., & Gundersen, H.J.G. (1991). Unbiased stereological estimation of the total number of neurons in the subdivisions of the rat hippocampus using the optical fractionator. The Anatomical Record, 231 (4), 482β497.
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ISOTROPIC FAKIR
Estimated total surface area πππππ‘π‘ππ = 2
1ππ
.οΏ½π£π£ππππ
. πΌπΌππ
ππ
ππ=1
n Number of line sets (always set to 3) π£π£ππππ Inverse of the probe per unit volume
Ii Intercepts with test lines
References
KubΓnovΓ‘, L., Janacek, J. (1998). Estimating surface area by the isotropic fakir method from thick slices cut in an arbitrary direction. Journal of Microscopy, 191 (2), 201β211.
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ISOTROPIC VIRTUAL PLANES
Length per unit volume πΏπΏππ =2ππ(ππππππ)ππ(ππππππππππ) .
βππβππ(ππππππ) p(box) Number of corners considered
a(plane) Exact sampling area p(ref) Number of corners in region βππ Total number of transects
Estimated total length πΏπΏ =1ππππππ
.1ππππππ
.1βππππ
.1ππππππ
. 2οΏ½ππ
Or πΏπΏ = 1π π π π ππ
. ππππ.ππππππ(ππππππ)
. π‘π‘Μ β(ππππππ) .ππ. 2βππ
ππππππ =ππ(ππππππ)ππππ.ππππ
βππππ =β(ππππππ)
π‘π‘Μ
ππππππ =πΆπΆ[ππ(ππππππππππ)]π£π£(ππππππ) =
1ππ
ssf Section sampling fraction asf Area sampling fraction hsf Height sampling fraction psd Probe sampling density βππ Total number of transects a(box) Area of sampling box h(box) Depth of sampling box d Sampling plane separation dx, dy Distances in XY π‘π‘Μ Average section thickness a(plane) Sampling plane area E Expected value v(box) Volume of sampling box
Total plane area π΄π΄ = οΏ½οΏ½π΄π΄ππ,ππ
π π
ππ=1
ππ
ππ=1
l Number of layouts s Number of sampling sites Ai,j Plane area inside of each sampling box for
each layout
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Isotropic Virtual Planes (2)
Coefficient of error
(Gundersen) πΆπΆπΆπΆ =οΏ½3(π΄π΄ β ππππππππππππππ ππππππππππ)
12 + ππππππππππππππ ππππππππππ
βππ,ππ = 0
πΆπΆπΆπΆ =οΏ½3(π΄π΄ β ππππππππππππππ ππππππππππ)β 4π΅π΅ + πΆπΆ
240 + ππππππππππππππ ππππππππππ
βππ,ππ = 1
Poisson noise is βQi
A = οΏ½(Qi. Qi)
B = οΏ½(Qi. Qi+1)
C = οΏ½(Qi. Qi+2)
A,B,C Covariogram values Qi Particles counted
Plane areas πποΏ½β = (π΄π΄,π΅π΅,πΆπΆ)
ππππππππππππ:π΄π΄ππ + π΅π΅ππ + πΆπΆπΆπΆ + π·π·ππ = 0, π·π·ππ = π·π· + ππ. ππ
ππππππ: οΏ½οΏ½(ππ,ππ, πΆπΆ)οΏ½ππ0 β€ ππ β€ ππ0 + ππππππ0 β€ ππ β€ ππ0 + πππππΆπΆ0 β€ πΆπΆ β€ πΆπΆ0 + πππ§π§
οΏ½οΏ½
ππππππππ: (ππππππ β© ππππππππππππ)
A,B,C,D Given constants d Distance between planes i Integer x0,y0,z0 Vertex of a sampling box bx,by,bz Dimensions of a sampling
box
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Isotropic Virtual Planes (3)
Average number of counts πποΏ½ =
β β ππππ ππππππππ=1
ππππ=1
β ππππππππ=1
p Number of probes Lj Number of layouts in each probe Qi j Number of counts in each probe and layout
Total corners of sampling boxes inside the region of interest
πΆπΆ = οΏ½πΆπΆππ
ππ
ππ=1
p Number of probes Ci Number of sampling boxes inside region of interest
References
Larsen, J. O., Gundersen, H.J.G., & Nielsen, J. (1998). Global spatial sampling with isotropic virtual planes: estimators of length density and total length in thick, arbitrarily orientated sections. Journal of Microscopy, 191, 238β248.
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IUR PLANES OPTICAL FRACTIONATOR
Estimated length πππππ‘π‘ πΏπΏ = 2.πππποΏ½πΌπΌ .
1ππππππ
.1ππππππ
.π‘π‘β
a/l Area per unit length of test line βπΌπΌ Number of intersections ssf Section sampling fraction asf Area sampling fraction t Section cut thickness h Height of counting frame
Area sampling fraction ππππππ =
ππππππππ(πΉπΉππππππππ)ππππππππ(ππ,ππ πππ‘π‘ππππ) =
πππΆπΆπΉπΉ .πππΆπΆπΉπΉπππ π π‘π‘ππππ.πππ π π‘π‘ππππ
xCF,yCF XY dimensions of counting frame xstep, ystep Dimensions of grid area(Frame) Area of counting frame area(x,y step) Area of grid
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L-CYCLOID OPTICAL FRACTIONATOR
Estimated length of lineal structure πππππ‘π‘ πΏπΏ = 2.
πππποΏ½πΌπΌ .
1ππππππ
.1ππππππ
.π‘π‘β
a/l Area per unit cycloid length βπΌπΌ Number of intercepts ssf Section sampling fraction asf Area sampling fraction t Section cut thickness h Height of counting frame
Area sampling fraction ππππππ =
ππππππππ(πΉπΉππππππππ)ππππππππ(ππ,ππ πππ‘π‘ππππ) =
πππΆπΆπΉπΉ .πππΆπΆπΉπΉπππ π π‘π‘ππππ.πππ π π‘π‘ππππ
xCF,yCF XY dimensions of counting frame xstep, ystep Dimensions of grid area(Frame) Area of counting frame area(x,y step) Area of grid
References Stocks, E. A., McArthur, J.C., Griffen, J.W., & Mouton, P.R. (1996). An unbiased method for estimation of total epidermal nerve fiber length. Journal of Neurocytology, 25 (1), 637β644.
Stereological formulas
23
MERZ
Length of semi-circle (L) πΏπΏ =12ππππ
d Circle diameter
Surface area per unit volume (Sv) πππ£π£ =2β πΌπΌππππβππ
I Number of intercepts l/p Length of semi-circle per point P Number of points
References
Howard, C. V., Reed, M. G. (2010). Unbiased stereology. Liverpool, UK: QTP Publications. {See equation 6.4}
Weibel, E.R. (1979). Stereological Methods. Vol. 1: Practical methods for biological morphometry. London, UK: Academic Press.
Stereological formulas
24
NUCLEATOR
Area estimate ππ = ππππ2οΏ½ l Length of rays
Volume estimate π£π£πποΏ½οΏ½οΏ½οΏ½ =
4ππ3ππππ3οΏ½ l Length of rays
Estimated coefficient of error
πππππ‘π‘ πΆπΆππ(ππ) =οΏ½ 1ππ β 1β (ππππ β πποΏ½)2ππ
ππ=1
πποΏ½
n Number of nucleator estimates Ri Area/volume estimate for each sampling site
Average area/volume estimate πποΏ½ =
1πποΏ½ππππ
ππ
ππ=1
n Number of nucleator estimates Ri Area/volume estimate for each sampling site
Relative efficiency πΆπΆπΆπΆππ(ππ) =πΆπΆππ(ππ)βππ
n Number of nucleator estimates CV (R) Estimated coefficient of variation
Geometric mean of area/volume estimate πποΏ½
1ππβ ππππππππππ
ππ=1 οΏ½ n Number of nucleator estimates Ri Area/volume estimate for each sampling site
References Gundersen, H.J.G. (1988). The nucleator. Journal of Microscopy, 151 (1), 3β21.
Stereological formulas
25
OPTICAL FRACTIONATOR
Estimate of total number of particles (N) ππ = οΏ½ππβ.
π‘π‘β
.1ππππππ
.1ππππππ
ππβ Particles counted t Section mounted thickness h Counting frame height ππππππ Area sampling fraction ππππππ Section sampling fraction
Variance due to
systematic random sampling β Gundersen
(VARSRS)
πππ΄π΄ππππππππ =3(π΄π΄ β ππ2) β 4π΅π΅ + πΆπΆ
12,ππ = 0
πππ΄π΄ππππππππ =3(π΄π΄ β ππ2) β 4π΅π΅ + πΆπΆ
240,ππ = 1
π΄π΄ = β (ππππβ)2ππππ=1
π΅π΅ = β ππππβππππ+1βππβ1ππ=1
πΆπΆ = β ππππβππππ+2βππβ2ππ=1
s2 Variance due to noise
Variance due to noise - Gundersen (s2) ππ2 = οΏ½ππβ
ππ
ππ=1
ππβ Particles counted n Number of sections used
Total variance β Gundersen (TotalVar)
πππππ‘π‘ππππππππππ = ππ2 + πππ΄π΄ππππππππ VARSRS Variance due to SRS s2 Variance due to noise
Coefficient of error β Gundersen (CE) πΆπΆπΆπΆ =
βπππππ‘π‘ππππππππππππ2
TotalVar Total variance s2 Variance due to noise
Number-weighted mean section cut thickness
(πππΈπΈβοΏ½οΏ½οΏ½οΏ½οΏ½)
π‘π‘ππβοΏ½οΏ½οΏ½οΏ½ =β π‘π‘ππππππ=1 ππππβ
β ππππβππππ=1
m Number of sections ti Section thickness at site i Qi Particles counted
Stereological formulas
26
Optical Fractionator (2)
Coefficient of error β Scheaffer (CE) πΆπΆπΆπΆ =
οΏ½ππ2 οΏ½1ππ β
1πΉπΉοΏ½
πποΏ½
f Number of counting frames πΉπΉ Total possible sampling sites ππ2 Estimated variance πποΏ½ Average particles counted
Average number of particles β
Scheaffer (πΈπΈοΏ½) πποΏ½ =β ππππππππ=1ππ
Qi Particles counted f Number of counting frames
Estimated variance - Scheaffer (s2) ππ2 =
β (ππππ β πποΏ½)2ππππ=1ππ β 1
f Number of counting frames Qi Particles counted QοΏ½ Average particles counted
Estimated variance of estimated cell population - Scheaffer
πΆπΆπππππΉπΉ2ππ2
ππ Cfp Finite population correction
ππ2 Estimated variance f Number of counting frames πΉπΉ Total possible sampling sites Estimated variance of mean cell
count - Scheaffer πΆπΆππππππ2
ππ Cfp Finite population correction
ππ2 Estimated variance f Number of counting frames
Stereological formulas
27
Optical Fractionator (3)
Estimated mean coefficient of error β Cruz-Orive (est Mean CE) πππππ‘π‘ ππππππππ πΆπΆπΆπΆ (πππππ‘π‘ ππ) = οΏ½
13ππ
.οΏ½οΏ½ππ1ππ β ππ2ππππ1ππ + ππ2ππ
οΏ½2ππ
ππ=1
οΏ½
12οΏ½
Q1i Counts in sub-sample 1 Q2i Counts in sub-sample 2 n Size of sub-sample
Predicted coefficient of error for estimated population β Schmitz-
Hof (CEpred) πΆπΆπΆπΆππππππππ(πππΉπΉ) = οΏ½
ππππππ(ππππβ)ππ. (ππππβ)2
πΆπΆπΆπΆππππππππ(πππΉπΉ) =1
οΏ½β ππππβππππ=1
=1
οΏ½β πππ π βπππ π =1
R Number of counting spaces S Number of sections Qrβ Counts in the βrβ-th counting space
Qsβ Counts in the βsβ-th section
References Geiser, M., CruzβOrive, L.M., Hof, V.I., & Gehr, P. (1990) Assessment of particle retention and clearance in the intrapulmonary conducting airways of hamster lungs with the fractionator. Journal of Microscopy, 160 (1), 75β88.
Glaser, E. M., Wilson, P.D. (1998). The coefficient of error of optical fractionator population size estimates: a computer simulation comparing three estimators. Journal of Microscopy, 192 (2), 163β171.
Gundersen, H.J.G., Vedel Jensen, E.B., Kieu, K., & Nielsen, J. (1999). The efficiency of systematic sampling in stereologyβreconsidered. Journal of Microscopy, 193 (3), 199β211.
Gundersen, H. J. G., Jensen, E.B. (1987). The efficiency of systematic sampling in stereology and its prediction. Journal of Microscopy, 147 (3), 229β263.
Howard, V., Reed, M. (2005). Unbiased stereology: three-dimensional measurement in microscopy (vol. 4, chapter 12). Garland Science/Bios Scientific Publishers.
Stereological formulas
28
Optical Fractionator (4)
Scheaffer, R.L., Ott, L., & Mendenhall, W. (1996). Elementary survey sampling (chapter 7). Boston: PWS-Kent.
Schmitz, C., Hof, P.R. (2000). Recommendations for straightforward and rigorous methods of counting neurons based on a computer simulation approach. Journal of Chemical Neuroanatomy, 20 (1), 93β114.
West, M. J., Slomianka, L., & Gundersen, H.J.G. (1991). Unbiased stereological estimation of the total number of neurons in the subdivisions of the rat hippocampus using the optical fractionator. The Anatomical Record, 231 (4), 482β497.
Stereological formulas
29
OPTICAL ROTATOR
Volume of particle π£π£οΏ½ = πποΏ½ππ(ππππ)
+/β
ππ
a Reciprocal line density a=k.h k Length of slice h Systematic spacing
For vertical slabs and lines parallel to vertical
axis
ππ(ππ) = ππ1, ππππ ππ2 < π‘π‘
ππ(ππ) =ππ2 ππ1
ππππππππππππ οΏ½ π‘π‘ππ2οΏ½
, ππππ π‘π‘ β€ ππ2
d1 Distance along test line d2 Distance from origin to test line t Β½ thickness of optical slice
For vertical slabs and lines perpendicular to
vertical axis
ππ(ππ) = ππ1, ππππ οΏ½ππ12 + πΆπΆ2 < π‘π‘
ππ(ππ) = ππ οΏ½οΏ½π‘π‘2 β πΆπΆ2οΏ½ , ππππ |πΆπΆ| < π‘π‘ β€ οΏ½ππ12 + πΆπΆ2
ππ(ππ) = ππ(0), ππππ π‘π‘ β€ |πΆπΆ|
ππ(ππ) = ππ +ππ2οΏ½
1
ππππππππππππ οΏ½ π‘π‘βπ’π’2 + πΆπΆ2
οΏ½πππ’π’
ππ1
ππ
d1 Distance along test line t Β½ thickness of optical slice z Distance in z from intercept to origin
Stereological formulas
30
Optical Rotator (2)
For isotropic
slabs
ππ(ππ) = ππ1, ππππ ππ3 < π‘π‘
ππ(ππ) =12π‘π‘
[β(π‘π‘,ππ2) + ππ(π‘π‘,ππ1,ππ2,ππ3)], ππππ ππ2 < π‘π‘β€ ππ3
β(π‘π‘,ππ) = π‘π‘2οΏ½1 βππ2
π‘π‘2
ππ(π‘π‘,ππ1,ππ2,ππ3) = ππ1ππ3 + ππ22ππππππ οΏ½ππ1 + ππ3
π‘π‘ + οΏ½π‘π‘2 β ππ22οΏ½
d1 Distance along test line d2 Distance from origin to test line d3 Distance from intercept to origin t Β½ thickness of optical slice
Estimated surface area
οΏ½ΜοΏ½π = πποΏ½πππππποΏ½πππποΏ½ππ
ππ(ππ) = 2, ππππ ππ2 < π‘π‘
ππ(ππ) = ππ.1
ππππππππππππ οΏ½ π‘π‘ππ2οΏ½
, ππππ π‘π‘ β€ ππ2
a Reciprocal line density lj Number of intersections between grid line and cell boundary d2 Distance from origin to test line t Β½ thickness of optical slice
References Tandrup, T., Gundersen, H.J.G., & Vedel Jensen, E.B. (1997). The optical rotator Journal of microscopy, 186 (2), 108β120.
Stereological formulas
31
PETRIMETRICS
Total length (π³π³οΏ½) πΏπΏοΏ½ =ππ2
.ππππ
.1ππππππ
.οΏ½πΌπΌ
πΏπΏοΏ½ = ππ.1ππππππ
.οΏ½πΌπΌ
a/l = 2d/Ο Grid constant (2d/Ο units or ratio of area to length of
semi-circle probe) asf Area fraction (ratio of area of counting frame to grid-
step) I Number of intersections counted d = 2*Merz-radius where the Merz-radius refers to the radius of the semi-circle used to probe.
References Howard, C. V., & Reed, M. G. (2005). Unbiased stereology. New York: Garland Science (prev. BIOS Scientific Publishers).
Stereological formulas
32
PHYSICAL FRACTIONATOR
Total number of particles (N) ππ = οΏ½ππβ .1ππππππ
.1ππππππ
ππβ Particles counted ππππππ Area sampling fraction ππππππ Section sampling fraction
Variance due to noise (s2) ππ2 = οΏ½ππβ
ππ
ππ=1
ππβ Particles counted n Number of sections used
Variance due to systematic random sampling (VARSRS) πππ΄π΄ππππππππ =
3(π΄π΄ β ππ2) β 4π΅π΅ + πΆπΆ12
,ππ = 0
πππ΄π΄ππππππππ =3(π΄π΄ β ππ2) β 4π΅π΅ + πΆπΆ
240,ππ = 1
π΄π΄ = β (ππππβ)2ππππ=1
π΅π΅ = β ππππβππππ+1βππβ1ππ=1
πΆπΆ = β ππππβππππ+2βππβ2ππ=1
s2 Variance due to noise
Total variance (TotalVar) πππππ‘π‘ππππππππππ = ππ2 + πππ΄π΄ππππππππ VARSRS Variance due to SRS s2 Variance due to noise
Coefficient of error (CE) πΆπΆπΆπΆ =
βπππππ‘π‘ππππππππππππ2
TotalVar Total variance s2 Variance due to noise
References Gundersen, Hans-JΓΈrgen G. "Stereology of arbitrary particles*." Journal of Microscopy 143, no. 1 (1986): 3-45. Sterio, D. C. "The unbiased estimation of number and sizes of arbitrary particles using the disector." Journal of Microscopy 134, no. 2 (1984): 127-136.
Stereological formulas
33
PLANAR ROTATOR
Volume for isotropic planar rotator
ππ = 2π‘π‘οΏ½ππππππ
t Separation between test lines gi Isotropic planar rotator function
Volume for vertical planar rotator
ππ = πππ‘π‘οΏ½ππππ2
ππ
t Separation between test lines li Intercept length along a test line
Isotropic planar rotator function ππππ(ππ) = πποΏ½ππ2 + ππππ2 + ππππ2ππππ οΏ½
ππππππ
+ οΏ½οΏ½πππππποΏ½2
+ 1οΏ½
ππππ+ = οΏ½ πππποΏ½ππππ ππ+οΏ½ β οΏ½ πππποΏ½ππππ ππ+οΏ½ππ ππππππππ πππ£π£ππππ
ππππβ = οΏ½ πππποΏ½ππππ ππβοΏ½ β οΏ½ πππποΏ½ππππ ππβοΏ½ππ ππππππππ πππ£π£ππππ
ππππ =12
(ππππ+ + ππππβ)
l Intercept length along a test line ai Distance from origin to test line j Number of grid lines lij Number of intersections between the j-th
grid line and the cell boundary
Stereological formulas
34
Planar Rotator (2)
Isotropic planar rotator function (contβd) ππππ+2 = οΏ½ ππππ ππ+2
ππ πππ£π£ππππ
β οΏ½ ππππ ππ+2
ππ ππππππ
ππππβ2 = οΏ½ ππππ ππβ2
ππ πππ£π£ππππ
β οΏ½ ππππ ππβ2
ππ ππππππ
ππππ2 =12 οΏ½ππππ+
2 + ππππβ2 οΏ½
l Intercept length along a test line ai Distance from origin to test line j Number of grid lines lij Number of intersections between the j-th
grid line and the cell boundary
References Jensen Vedel, E.B., Gundersen, H.J.G. (1993). The rotator Journal of Microscopy, 170 (1), 35β44.
Stereological formulas
35
POINT SAMPLED INTERCEPT
Volume based on intercept length (π½π½πποΏ½)
πποΏ½οΏ½ππ =ππ3ππ0Μ 3 =
ππ3ππ
οΏ½ ππ0,ππ3
ππ
ππ=1
n Number of intercepts l Intercept length
Volume-weighted mean volume (πποΏ½π½π½) πποΏ½π½π½ =
β οΏ½Μ οΏ½πππππππππ=ππ
ππ.π π ππ
n Number of intercepts l Intercept length
Coefficient of error (CE)
πΆπΆπΆπΆοΏ½ππ0Μ 3οΏ½ = οΏ½β οΏ½ππ0Μ 3οΏ½
2ππππ=1
οΏ½β ππ0Μ 3ππππ=1 οΏ½
2 β1ππ
n Number of intercepts l Intercept length
Coefficient of variance (CV) πΆπΆπΆπΆοΏ½ππ0Μ 3οΏ½ = πΆπΆπΆπΆ(οΏ½Μ οΏ½π£ππ).βππ n Number of intercepts l Intercept length πποΏ½π½π½ Volume-weighted mean volume
Variance (VarianceV) ππππππππππππππππππ(π£π£) = οΏ½ππ3
. πππ·π·οΏ½ππ0Μ 3οΏ½οΏ½ = οΏ½πΆπΆπποΏ½ππ0Μ 3οΏ½. οΏ½Μ οΏ½π£πποΏ½ L Intercept length πποΏ½π½π½ Volume-weighted mean volume CV Coefficient of variance
References
Gundersen, H.J.G., Jensen. E.B. (1985). Stereological Estimation of the Volume-Weighted Mean Volume of Arbitrary Particles Observed on Random Sections. Journal of Microscopy, 138, 127β142.
SΓΈrensen, F.B. (1991). Stereological estimation of the mean and variance of nuclear volume from vertical sections. Journal of Microscopy, 162 (2), 203β229.
Stereological formulas
36
SIZE DISTRIBUTION
Volume-weighted mean particle volume
οΏ½Μ οΏ½π£ππ = οΏ½Μ οΏ½π£ππ. [1 + πΆπΆππππ2(π£π£)] vοΏ½N Number-weighted mean volume
CVN(v) Coefficient of variation
Number-weighted mean volume οΏ½Μ οΏ½π£ππ =
βππβππ
ππ = ππ.ππ
R Number of contours Q Number of points per contour
Variance
ππππππππ(π£π£) =οΏ½βππ β (βππ)2
βππ οΏ½ . π£π£(ππ)2
βππ β 1
ππ = ππ2.ππ
R Number of contours Q Number of points per contour v(p) Volume associated with a point
Standard deviation πππ·π·ππ(π£π£) = οΏ½ππππππππ(π£π£)
πππ·π·ππ(π£π£) = οΏ½οΏ½Μ οΏ½π£ππ . (οΏ½Μ οΏ½π£ππ β οΏ½Μ οΏ½π£ππ)
vοΏ½N Number-weighted mean volume vοΏ½V Volume-weighted particle volume VarN Variance
Coefficient of variation πΆπΆππππ(π£π£) =
πππ·π·ππ(π£π£)οΏ½Μ οΏ½π£ππ
πΆπΆππππ(π£π£) = οΏ½οΏ½Μ οΏ½π£ππ β οΏ½Μ οΏ½π£πποΏ½Μ οΏ½π£ππ
vοΏ½N Number-weighted mean volume vοΏ½V Volume-weighted particle volume SDN Standard deviation
Stereological formulas
37
Size distribution (2)
Coefficient of error πΆπΆπΆπΆππ(π£π£) =
πΆπΆππππ(π£π£)βππ
CVN Coefficient of variation R Number of contours
References SΓΈrensen, F.B. (1991). Stereological estimation of the mean and variance of nuclear volume from vertical sections. Journal of Microscopy, 162 (2), 203β229.
Stereological formulas
38
SPACEBALLS
Length estimate πΏπΏ = 2.οΏ½οΏ½ππππ
ππ
ππ=1
οΏ½ .π£π£ππ
.1ππππππ
This equation does not include the terms F2 (area-fraction) and F3 (thickness-fraction) used by Mouton et al. (equation 2, 2002), but includes that information in v (volume sampled).
n Number of sections used Qi Intersection counted v Volume (grid X * grid Y* section
thickness) a Surface area of the sphere ssf Section sampling fraction
Variance due to noise
ππ2 = οΏ½ππππ
ππ
ππ=1
Qi Intersection counted
Variance due to systematic random
sampling
πππ΄π΄ππππππππ =3(π΄π΄ β ππ2)β 4π΅π΅ + πΆπΆ
12,ππ = 0
πππ΄π΄ππππππππ =3(π΄π΄ β ππ2)β 4π΅π΅ + πΆπΆ
240,ππ = 1
π΄π΄ = β (ππππβ)2ππππ=1
π΅π΅ = β ππππβππππ+1βππβ1ππ=1
πΆπΆ = β ππππβππππ+2βππβ2ππ=1
s2 Variance due to noise m Smoothness class of sampled function
Total variance πππππ‘π‘ππππππππππ = ππ2 + πππ΄π΄ππππππππ VARSRS Variance due to SRS s2 Variance due to noise
Stereological formulas
39
Spaceballs (2)
Coefficient of error πΆπΆπΆπΆ =
βπππππ‘π‘ππππππππππππ2
TotalVar Total variance s2 Variance due to noise
References
Mouton, P. R., Gokhale, A.M., Ward, N.L., & West, M.J. (2002). Stereological length estimation using spherical probes. Journal of Microscopy, 206 (1), 54β64.
Stereological formulas
40
SURFACE-WEIGHTED STAR VOLUME
Surface-weighted star volume (πποΏ½οΏ½ππβ) πποΏ½οΏ½ππβ =
πππ π ππ
. οΏ½Μ οΏ½πππππ
πποΏ½οΏ½ππβ =πππ π ππ .
β β ππππ,(ππ,ππ)ππππππ
ππ=ππππππ=ππ
β ππππππππ=ππ
n Number of probes l Intercept length mi Number of intercepts
Sum of cubed intercepts in probe (yi) ππππ = οΏ½ππ1,(ππ,ππ)
3
ππππ
ππ=1
mi Number of intercepts l Intercept length
Coefficient of error (CE) πΆπΆπΆπΆοΏ½οΏ½Μ οΏ½π£π π βοΏ½οΏ½ = οΏ½
ππππ β 1 οΏ½
βππππ2
βππππ βππππ+
βππππ2
βππππ βππππβ 2.
βππππππππβππππ βππππ
οΏ½οΏ½12οΏ½
n Number of probes yi Sum of cubed intercepts in probe mi Number of intercepts
References Reed, M. G., Howard, C.V. (1998). Surface-weighted star volume: concept and estimation. Journal of Microscopy, 190 (3), 350β356.
Stereological formulas
41
SURFACTOR
Surface area for single-ray designs
οΏ½ΜοΏ½π = 4ππππ02 + ππ(π½π½) l Length of intercept Γ Angle between test line and surface c(Γ) Function of the planar angle
Surface area for multi-ray
designs πΊπΊοΏ½ = πππ π οΏ½ππππππ. ππ(π·π·)ππππ
ππ=ππ
l Length of intercept Γ Angle between test line and surface c(Γ) Function of the planar angle r Number of test lines
Function of the planar angle ππ(π½π½) = 1 + οΏ½
12πππππ‘π‘π½π½οΏ½ . οΏ½
ππ2β ππππππβ1
1 β πππππ‘π‘2π½π½1 + πππππ‘π‘2π½π½
οΏ½ Γ Angle between test line and surface
References Jensen, E.B., Gundersen, H.J.G. (1987). Stereological estimation of surface area of arbitrary particles. Acta Stereologica, 6 (3).
Stereological formulas
42
SV-CYCLOID FRACTIONATOR
Estimated surface area per unit volume ππππ = 2 οΏ½
πππποΏ½β πΌπΌππππππ=1
β ππππππππ=1
p/l Ratio of test points to curve length n Number of micrographs β Ii Total intercept points on curve β Pππ Total test points
References Baddeley, A. J., Gundersen, H.J.G., & CruzβOrive, L.M. (1986). Estimation of surface area from vertical sections. Journal of Microscopy, 142 (3), 259β276.
Stereological formulas
43
VERTICAL SPATIAL GRID
Estimated volume πποΏ½ = ππ.ππππ.οΏ½ππππ
ππ
ππ=1
a Area associated with point dz Distance between planes m Number of scanning planes β P Intersections with points
Area associated with point ππ =
π€π€2
2ππ w Horizontal width
Estimated surface area οΏ½ΜοΏ½π = 2.ππ.ππππ
ππ + 4ππ .ππππ
. οΏ½ππππππ + πΌπΌπππ§π§οΏ½ a Area associated with point dz Distance between planes l Length of cycloid Ixy X,Y intersections Ixz X,Z intersections Length of cycloid ππ =
2π€π€ππ
w Horizontal width
X,Y intersections πΌπΌππππ = οΏ½πΌπΌππππ,ππ
ππ
ππ=1
m Number of scanning planes
X,Z intersections πΌπΌπππ§π§ = 2.οΏ½οΏ½ππππ
ππ
ππ=1
β οΏ½ ππππ,ππ+1
ππβ1
ππ=1
οΏ½ m Number of scanning planes βππ Intersections with points
References Cruz-Orive, L. M., Howard, C.V. (1995). Estimation of individual feature surface area with the vertical spatial grid. Journal of Microscopy, 178 (2), 146β151.
Stereological formulas
44
WEIBEL
Surface area per unit volume (Sv) πππ£π£ = 2.οΏ½
πΌπΌππ2 .ππ
οΏ½
I Intersections (triangular markers) P Points (end points circular markers) l Length of each line
Note: We use l/2 for the length represented at each point since there are two end points per line.
References Weibel, E.R., Kistler, G.S., & Scherle, W.F. (1966). Practical stereological methods for morphometric cytology. The Journal of cell biology, 30 (1), 23β38.