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Stereological formulas
1
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
Stereological formulas
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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
Stereological formulas
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AREA FRACTION FRACTIONATOR
Estimated volume fraction (๐๐๐ฃ๐ฃ๏ฟฝ )
๐๐๐ฃ๐ฃ๏ฟฝ (๐๐, ๐๐๐๐๐๐) =โ ๐๐(๐๐)๐๐๐๐๐๐=1
โ ๐๐(๐๐๐๐๐๐)๐๐๐๐๐๐=1
P(ref) Points hitting reference volume
Y Sub-region P(Y) Points hitting sub-region
Estimated area (๏ฟฝฬ๏ฟฝ๐ด)
๏ฟฝฬ๏ฟฝ๐ด =1๐๐๐๐๐๐
.๐๐(๐๐).๐๐(๐๐๐๐)
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. 170โ172). Milton Park, England: BIOS Scientific Publishers.
Stereological formulas
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CAVALIERI ESTIMATOR
Area associated with a point (Ap)
๐ด๐ด๐๐ = ๐๐2 g2 Grid area
Volume associated with a point (VP)
๐๐๐๐ = ๐๐2๐๐๐ก๐กฬ
m Section evaluation interval ๐ก๐กฬ Mean section cut thickness
Estimated volume (๐ฝ๐ฝ๏ฟฝ) ๐๐๏ฟฝ = ๐ด๐ด๐๐๐๐โฒ๐ก๐กฬ ๏ฟฝ๏ฟฝ๐๐๐๐
๐๐
๐๐=1
๏ฟฝ Ap Area associated with a point mโ Section evaluation interval ๐ก๐กฬ Mean section cut thickness Pi Points counted on grid
Estimated volume
corrected for over-projection ([v])
[๐ฃ๐ฃ] = ๐ก๐ก.๏ฟฝ๐๐.๏ฟฝ๐๐โฒ๐๐
๐๐
๐๐=1
โ ๐๐๐๐๐๐(๐๐โฒ)๏ฟฝ t Section cut thickness k Correction factor g Grid size aโ Projected area
Coefficient of error
(CE)
๐ถ๐ถ๐ถ๐ถ =โ๐๐๐๐๐ก๐ก๐๐๐๐๐๐๐๐๐๐โ ๐๐๐๐๐๐๐๐=1
TotalVar Total variance of the estimated volume n Number of sections Pi Points counted on grid
๐๐๐๐๐ก๐ก๐๐๐๐๐๐๐๐๐๐ = ๐๐2 + ๐๐๐ด๐ด๐๐๐๐๐๐๐๐
Stereological formulas
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Cavalieri Estimator (2)
Variance of systematic
random sampling (VARSRS)
๐๐๐ด๐ด๐๐๐๐๐๐๐๐ =3(๐ด๐ด โ ๐๐2) โ 4๐ต๐ต + ๐ถ๐ถ
12,๐๐ = 0
๐๐๐ด๐ด๐๐๐๐๐๐๐๐ =3(๐ด๐ด โ ๐๐2) โ 4๐ต๐ต + ๐ถ๐ถ
240,๐๐ = 1
m Smoothness class of sampled function s2 Variance due to noise ๐ด๐ด = โ ๐๐๐๐2๐๐
๐๐=1 , ๐ต๐ต = โ ๐๐๐๐๐๐๐๐+1๐๐โ1๐๐=1 , ๐ถ๐ถ = โ ๐๐๐๐๐๐๐๐+2๐๐โ2
๐๐=1
With:
n : number of sections
๐๐2 = 0.0724 ๏ฟฝ ๐๐โ๐๐๏ฟฝ๏ฟฝ๐๐โ ๐๐๐๐๐๐
๐๐=1 ๐๐โ๐๐
Shape factor
R e f e r e n c e s Garcรญa-Fiรฑana, 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), 505โ516. Gundersen, H. J. G., & Jensen, E.B. (1987). The efficiency of systematic sampling in stereology and its prediction. Journal of Microscopy, 147 (3), 229โ263. Howard, C. V., & Reed, M.G. (2005). Unbiased Stereology, Three-Dimensional Measurement in Microscopy (Chapter 3). New York: Garland Science/BIOS Scientific Publishers.
Stereological formulas
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COMBINED POINT INTERCEPT
Profile area (a) ๐๐ = ๐๐(๐๐).๏ฟฝ๐๐ a(p) Area associated with a point โ๐๐ Number of points
Profile boundary (b) ๐๐ =
๐๐2๐๐.๏ฟฝ๐ผ๐ผ d Distance between points
โ๐ผ๐ผ 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), 211โ226.
Stereological formulas
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CONNECTIVITY ASSAY
Euler number (X3) ๐๐3 = ๐ผ๐ผ + ๐ป๐ป โ ๐ต๐ต I Total island markers ๐ป๐ป Total hole markers ๐ต๐ต Total bridge markers
Number of alveoli (Nalv) ๐๐๐๐๐๐๐ฃ๐ฃ = โ๐๐3 X3 Euler number
Sum counting frame volumes (V)
๐๐ = โ.๐๐.๐๐ h Disector height n Number of disectors a Area counting frame
Numerical density of alveoli (Nv)
๐๐๐ฃ๐ฃ =๐๐๐๐๐๐๐ฃ๐ฃ๐๐
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), 120โ124.
Stereological formulas
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CYCLOIDS FOR LV
Area associated with a point (Ap)
๐ด๐ด๐๐ = ๐๐2 g2 Grid area
Volume associated with a
point (Vp) ๐๐๐๐ = ๐๐2๐๐๐ก๐กฬ g2 Grid area
m Section evaluation interval ๐ก๐กฬ Mean section cut thickness
Length per unit volume (Lv) ๐ฟ๐ฟ๐๐ = 2
๏ฟฝ๐ผ๐ผ๏ฟฝฬ ๏ฟฝ๐ฟ๐ถ๐ถ๏ฟฝ๐๐๐๐๐๐โ
๐ฟ๐ฟ๐๐ =2โ
.๏ฟฝ๐ผ๐ผ๏ฟฝฬ ๏ฟฝ๐๐๐๐๐๐๐๏ฟฝ๐๐๐๐๐๐
๐๐๏ฟฝ. ๏ฟฝ๐๐๐๐๏ฟฝ=
2โ๏ฟฝ๐๐๐๐๏ฟฝโ ๐ผ๐ผ๐๐๐๐๐๐=1
โ ๐๐๐๐๐๐๐๐=1
๏ฟฝILฬ C๏ฟฝprj Number of counting frames โ Section cut thickness Ii Intercepts Pi Test points ๏ฟฝICฬ cyc๏ฟฝ
prj Average number of intersections of
projected images pl Test points per unit length of cycloid
Estimated volume (๐๐๏ฟฝ) ๐๐๏ฟฝ = ๐๐โ๏ฟฝ
๐๐๐๐๏ฟฝ๏ฟฝ๐๐๐๐
๐๐
๐๐=1
m Sampling fractions โ Section cut thickness a Area p Number of test points Pi Test points
Estimated length (๐ฟ๐ฟ๏ฟฝ) ๐ฟ๐ฟ๏ฟฝ = 2 ๏ฟฝ
๐๐๐๐๏ฟฝ๐๐๏ฟฝ๐ผ๐ผ๐๐
๐๐
๐๐=1
a Area l Line length m Sampling fractions Ii Intercepts
Stereological formulas
9
Cycloids for Lv (2)
Coefficient of error for line length ๐ถ๐ถ๐ถ๐ถ๏ฟฝ๐ฟ๐ฟ๏ฟฝ|๐ฟ๐ฟ๏ฟฝ =
๏ฟฝ๐๐๐ด๐ด๐๐๐๐๐๐๐๐โ ๐ผ๐ผ๐๐๐๐๐๐=1
VARSRS Variance of systematic random sampling L๏ฟฝ|L Estimated length per length Ii Intercepts
Variance of systematic
random sampling (VARSRS) ๐๐๐ด๐ด๐๐๐๐๐๐๐๐ =
3๐๐0 โ 4๐๐1 + ๐๐212
๐๐๐๐ = ๏ฟฝ ๐ฟ๐ฟ๐๐๐ฟ๐ฟ๐๐+๐๐๐๐โ๐๐
๐๐=1
g Grid size Li Line length at section i
Coefficient of error for length
density ๐ถ๐ถ๐ถ๐ถ(๐ฟ๐ฟ๐๐) = ๏ฟฝ
๐๐๐๐ โ 1๏ฟฝ
โ ๐ผ๐ผ๐๐2๐๐๐๐=1
โ ๐ผ๐ผ๐๐๐๐๐๐=1 โ ๐ผ๐ผ๐๐๐๐
๐๐=1+
โ ๐๐๐๐2๐๐๐๐=1
โ ๐๐๐๐๐๐๐๐=1 โ ๐๐๐๐๐๐
๐๐=1โ 2
โ ๐ผ๐ผ๐๐๐๐๐๐๐๐๐๐=1
โ ๐ผ๐ผ๐๐๐๐๐๐=1 โ ๐๐๐๐๐๐
๐๐=1๏ฟฝ
Ii Intercepts Pi Test points n Number of probes
R e f e r e n c e s
ArtachoโPรฉrula, E., RoldรกnโVillalobos, 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 3โD using vertical slices. Journal of Microscopy, 159 (2), 133โ141.
Howard, C. V., Reed, M.G. (1998). Unbiased Stereology, Three-Dimensional Measurement in Microscopy (pp. 170โ172). BIOS Scientific Publishers.
Stereological formulas
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CYCLOIDS FOR SV
Area associated with a point (Ap)
๐ด๐ด๐๐ = ๐๐2 g2 Grid area
Volume associated with a
point (Vp) ๐๐๐๐ = ๐๐2๐๐๐ก๐กฬ g2 Grid area
m Evaluation interval ๐ก๐กฬ Section cut thickness
Estimated surface area per unit volume (est Sv) ๐๐๐๐๐ก๐ก ๐๐๐ฃ๐ฃ = 2 ๏ฟฝ
2๐๐๐๐๏ฟฝโ ๐ผ๐ผ๐๐๐๐๐๐=1
โ ๐๐๐๐๐๐๐๐=1
p/l Points per unit length of cycloid Ii Intercepts with cycloids Pi Point counts
Estimated volume (๐๐๏ฟฝ) ๐๐๏ฟฝ = ๐๐๐ก๐กฬ ๏ฟฝ
๐๐๐๐๏ฟฝ๏ฟฝ๐๐๐๐
๐๐
๐๐=1
m Evaluation interval ๐ก๐กฬ Section cut thickness a/p Area associated with each point Pi Point counts
Estimated surface area (๏ฟฝฬ๏ฟฝ๐) ๏ฟฝฬ๏ฟฝ๐ = 2 ๏ฟฝ๐๐๐๐๏ฟฝ๐๐๐ก๐กฬ ๏ฟฝ ๐ผ๐ผ๐๐
๐๐
๐๐=1 m Evaluation interval
๐ก๐กฬ Section cut thickness a/l Area per unit length Ii Intercepts with cycloids C
Stereological formulas
11
Cycloids for Sv (2)
Coefficient of error for
estimated surface (CE)
๐ถ๐ถ๐ถ๐ถ๏ฟฝ๏ฟฝฬ๏ฟฝ๐|๐๐๏ฟฝ =๏ฟฝ๐๐๐ด๐ด๐๐๐๐๐๐๐๐โ ๐ผ๐ผ๐๐๐๐๐๐=1
VarSRS Variance due to
systematic random sampling
VarSRS =3g0 โ 4g1 + g2
12
Coefficient of error for surface
density (CE (Sv))
๐ถ๐ถ๐ถ๐ถ(๐๐๐ฃ๐ฃ) = ๏ฟฝ๐๐
๐๐ โ 1๏ฟฝโ ๐ผ๐ผ๐๐2๐๐๐๐=1
โ ๐ผ๐ผ๐๐๐๐๐๐=1 โ ๐ผ๐ผ๐๐๐๐
๐๐=1+
โ ๐๐๐๐2๐๐๐๐=1
โ ๐๐๐๐๐๐๐๐=1 โ ๐๐๐๐๐๐
๐๐=1โ 2
โ ๐ผ๐ผ๐๐๐๐๐๐=1 ๐๐๐๐
โ ๐ผ๐ผ๐๐๐๐๐๐=1 โ ๐๐๐๐๐๐
๐๐=1๏ฟฝ
n Number of measurements Ii Intercepts with cycloids Pi Point counts
References
Baddeley, A. J., Gundersen, H.J.G., & CruzโOrive, L.M. (1998) Estimation of surface area from vertical sections. Journal of Microscopy, 142 (3), 259โ276.
Howard, C. V., Reed, M.G. (1998). Unbiased Stereology, Three-Dimensional Measurement in Microscopy(pp.170โ172). BIOS Scientific Publishers.
Stereological formulas
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DISCRETE VERTICAL ROTATOR
Estimated volume (Est v) ๐๐๐๐๐ก๐ก ๐ฃ๐ฃ =
๐๐๐๐
.๐๐๐๐.๏ฟฝ๐๐๐๐
๐๐
๐๐=1
๐ท๐ท๐๐
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) ๐๐ = ๏ฟฝ๐๐โ .
1๐๐๐๐๐๐
.1๐๐๐๐๐๐
๐๐โ Particles counted ๐๐๐๐๐๐ Area sampling fraction ๐๐๐๐๐๐ Section sampling fraction
Variance due to
systematic random sampling โ Gundersen
(VARSRS)
๐๐๐ด๐ด๐๐๐๐๐๐๐๐ =3(๐ด๐ด โ ๐๐2) โ 4๐ต๐ต + ๐ถ๐ถ
12,๐๐ = 0
๐๐๐ด๐ด๐๐๐๐๐๐๐๐ =3(๐ด๐ด โ ๐๐2) โ 4๐ต๐ต + ๐ถ๐ถ
240,๐๐ = 1
๐ด๐ด = โ (๐๐๐๐โ)2๐๐๐๐=1
๐ต๐ต = โ ๐๐๐๐โ๐๐๐๐+1โ๐๐โ1๐๐=1
๐ถ๐ถ = โ ๐๐๐๐โ๐๐๐๐+2โ๐๐โ2๐๐=1
s2 Variance due to noise
Variance due to noise - Gundersen (s2) ๐๐2 = ๏ฟฝ๐๐โ
๐๐
๐๐=1
๐๐โ Particles counted n Number of sections used
Total variance โ Gundersen (TotalVar)
๐๐๐๐๐ก๐ก๐๐๐๐๐๐๐๐๐๐ = ๐๐2 + ๐๐๐ด๐ด๐๐๐๐๐๐๐๐ VARSRS Variance due to SRS s2 Variance due to noise
Coefficient of error โ Gundersen (CE) ๐ถ๐ถ๐ถ๐ถ =
โ๐๐๐๐๐ก๐ก๐๐๐๐๐๐๐๐๐๐๐๐2
TotalVar Total variance s2 Variance due to noise
Number-weighted mean section cut thickness
(๐๐๐ธ๐ธโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ)
๐ก๐ก๐๐โ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ =โ ๐ก๐ก๐๐๐๐๐๐=1 ๐๐๐๐โ
โ ๐๐๐๐โ๐๐๐๐=1
m Number of sections ti Section thickness at site i Qi Particles counted
Stereological formulas
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Fractionator (2)
Coefficient of error โ Scheaffer (CE) ๐ถ๐ถ๐ถ๐ถ =
๏ฟฝ๐๐2 ๏ฟฝ1๐๐ โ
1๐น๐น๏ฟฝ
๐๐๏ฟฝ
f Number of counting frames ๐น๐น Total possible sampling sites ๐๐2 Estimated variance ๐๐๏ฟฝ Average particles counted
Average number of particles โ
Scheaffer (๐ธ๐ธ๏ฟฝ) ๐๐๏ฟฝ =โ ๐๐๐๐๐๐๐๐=1๐๐
Qi Particles counted f Number of counting frames
Estimated variance - Scheaffer (s2) ๐๐2 =
โ (๐๐๐๐ โ ๐๐๏ฟฝ)2๐๐๐๐=1๐๐ โ 1
f Number of counting frames Qi Particles counted Q๏ฟฝ Average particles counted
Estimated variance of estimated cell population - Scheaffer
๐ถ๐ถ๐๐๐๐๐น๐น2๐๐2
๐๐ Cfp Finite population correction
๐๐2 Estimated variance f Number of counting frames ๐น๐น Total possible sampling sites Estimated variance of mean cell
count - Scheaffer ๐ถ๐ถ๐๐๐๐๐๐2
๐๐ Cfp Finite population correction
๐๐2 Estimated variance f Number of counting frames
Stereological formulas
15
Fractionator (3)
Estimated mean coefficient of error โ Cruz-Orive (est Mean CE) ๐๐๐๐๐ก๐ก ๐๐๐๐๐๐๐๐ ๐ถ๐ถ๐ถ๐ถ (๐๐๐๐๐ก๐ก ๐๐) = ๏ฟฝ
13๐๐
.๏ฟฝ๏ฟฝ๐๐1๐๐ โ ๐๐2๐๐๐๐1๐๐ + ๐๐2๐๐
๏ฟฝ2๐๐
๐๐=1
๏ฟฝ
12๏ฟฝ
Q1i Counts in sub-sample 1 Q2i Counts in sub-sample 2 n Size of sub-sample
Predicted coefficient of error for estimated population โ Schmitz-
Hof (CEpred) ๐ถ๐ถ๐ถ๐ถ๐๐๐๐๐๐๐๐(๐๐๐น๐น) = ๏ฟฝ
๐๐๐๐๐๐(๐๐๐๐โ)๐๐. (๐๐๐๐โ)2
๐ถ๐ถ๐ถ๐ถ๐๐๐๐๐๐๐๐(๐๐๐น๐น) =1
๏ฟฝโ ๐๐๐๐โ๐๐๐๐=1
=1
๏ฟฝโ ๐๐๐ ๐ โ๐๐๐ ๐ =1
R Number of counting spaces S Number of sections Qrโ Counts in the โrโ-th counting space
Qsโ Counts in the โsโ-th section
References Geiser, M., CruzโOrive, 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), 75โ88.
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), 163โ171.
Gundersen, H.J.G., Vedel Jensen, E.B., Kieu, K., & Nielsen, J. (1999). The efficiency of systematic sampling in stereologyโreconsidered. Journal of Microscopy, 193 (3), 199โ211.
Gundersen, H. J. G., Jensen, E.B. (1987). The efficiency of systematic sampling in stereology and its prediction. Journal of Microscopy, 147 (3), 229โ263.
Howard, V., Reed, M. (2005). Unbiased stereology: three-dimensional measurement in microscopy (vol. 4, chapter 12). Garland Science/Bios Scientific Publishers.
Stereological formulas
16
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), 93โ114.
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), 482โ497.
Stereological formulas
17
ISOTROPIC FAKIR
Estimated total surface area ๐๐๐๐๐ก๐ก๐๐ = 2
1๐๐
.๏ฟฝ๐ฃ๐ฃ๐๐๐๐
. ๐ผ๐ผ๐๐
๐๐
๐๐=1
n Number of line sets (always set to 3) ๐ฃ๐ฃ๐๐๐๐ Inverse of the probe per unit volume
Ii Intercepts with test lines
References
Kubรญnovรก, 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), 201โ211.
Stereological formulas
18
ISOTROPIC VIRTUAL PLANES
Length per unit volume ๐ฟ๐ฟ๐๐ =2๐๐(๐๐๐๐๐๐)๐๐(๐๐๐๐๐๐๐๐๐๐) .
โ๐๐โ๐๐(๐๐๐๐๐๐) p(box) Number of corners considered
a(plane) Exact sampling area p(ref) Number of corners in region โ๐๐ Total number of transects
Estimated total length ๐ฟ๐ฟ =1๐๐๐๐๐๐
.1๐๐๐๐๐๐
.1โ๐๐๐๐
.1๐๐๐๐๐๐
. 2๏ฟฝ๐๐
Or ๐ฟ๐ฟ = 1๐ ๐ ๐ ๐ ๐๐
. ๐๐๐๐.๐๐๐๐๐๐(๐๐๐๐๐๐)
. ๐ก๐กฬ โ(๐๐๐๐๐๐) .๐๐. 2โ๐๐
๐๐๐๐๐๐ =๐๐(๐๐๐๐๐๐)๐๐๐๐.๐๐๐๐
โ๐๐๐๐ =โ(๐๐๐๐๐๐)
๐ก๐กฬ
๐๐๐๐๐๐ =๐ถ๐ถ[๐๐(๐๐๐๐๐๐๐๐๐๐)]๐ฃ๐ฃ(๐๐๐๐๐๐) =
1๐๐
ssf Section sampling fraction asf Area sampling fraction hsf Height sampling fraction psd Probe sampling density โ๐๐ 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 ๐ก๐กฬ Average section thickness a(plane) Sampling plane area E Expected value v(box) Volume of sampling box
Total plane area ๐ด๐ด = ๏ฟฝ๏ฟฝ๐ด๐ด๐๐,๐๐
๐ ๐
๐๐=1
๐๐
๐๐=1
l Number of layouts s Number of sampling sites Ai,j Plane area inside of each sampling box for
each layout
Stereological formulas
19
Isotropic Virtual Planes (2)
Coefficient of error
(Gundersen) ๐ถ๐ถ๐ถ๐ถ =๏ฟฝ3(๐ด๐ด โ ๐๐๐๐๐๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐)
12 + ๐๐๐๐๐๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐
โ๐๐,๐๐ = 0
๐ถ๐ถ๐ถ๐ถ =๏ฟฝ3(๐ด๐ด โ ๐๐๐๐๐๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐)โ 4๐ต๐ต + ๐ถ๐ถ
240 + ๐๐๐๐๐๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐
โ๐๐,๐๐ = 1
Poisson noise is โQi
A = ๏ฟฝ(Qi. Qi)
B = ๏ฟฝ(Qi. Qi+1)
C = ๏ฟฝ(Qi. Qi+2)
A,B,C Covariogram values Qi Particles counted
Plane areas ๐๐๏ฟฝโ = (๐ด๐ด,๐ต๐ต,๐ถ๐ถ)
๐๐๐๐๐๐๐๐๐๐๐๐:๐ด๐ด๐๐ + ๐ต๐ต๐๐ + ๐ถ๐ถ๐ถ๐ถ + ๐ท๐ท๐๐ = 0, ๐ท๐ท๐๐ = ๐ท๐ท + ๐๐. ๐๐
๐๐๐๐๐๐: ๏ฟฝ๏ฟฝ(๐๐,๐๐, ๐ถ๐ถ)๏ฟฝ๐๐0 โค ๐๐ โค ๐๐0 + ๐๐๐๐๐๐0 โค ๐๐ โค ๐๐0 + ๐๐๐๐๐ถ๐ถ0 โค ๐ถ๐ถ โค ๐ถ๐ถ0 + ๐๐๐ง๐ง
๏ฟฝ๏ฟฝ
๐๐๐๐๐๐๐๐: (๐๐๐๐๐๐ โฉ ๐๐๐๐๐๐๐๐๐๐๐๐)
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
Stereological formulas
20
Isotropic Virtual Planes (3)
Average number of counts ๐๐๏ฟฝ =
โ โ ๐๐๐๐ ๐๐๐๐๐๐๐๐=1
๐๐๐๐=1
โ ๐๐๐๐๐๐๐๐=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
๐ถ๐ถ = ๏ฟฝ๐ถ๐ถ๐๐
๐๐
๐๐=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, 238โ248.
Stereological formulas
21
IUR PLANES OPTICAL FRACTIONATOR
Estimated length ๐๐๐๐๐ก๐ก ๐ฟ๐ฟ = 2.๐๐๐๐๏ฟฝ๐ผ๐ผ .
1๐๐๐๐๐๐
.1๐๐๐๐๐๐
.๐ก๐กโ
a/l Area per unit length of test line โ๐ผ๐ผ Number of intersections ssf Section sampling fraction asf Area sampling fraction t Section cut thickness h Height of counting frame
Area sampling fraction ๐๐๐๐๐๐ =
๐๐๐๐๐๐๐๐(๐น๐น๐๐๐๐๐๐๐๐)๐๐๐๐๐๐๐๐(๐๐,๐๐ ๐๐๐ก๐ก๐๐๐๐) =
๐๐๐ถ๐ถ๐น๐น .๐๐๐ถ๐ถ๐น๐น๐๐๐ ๐ ๐ก๐ก๐๐๐๐.๐๐๐ ๐ ๐ก๐ก๐๐๐๐
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
Stereological formulas
22
L-CYCLOID OPTICAL FRACTIONATOR
Estimated length of lineal structure ๐๐๐๐๐ก๐ก ๐ฟ๐ฟ = 2.
๐๐๐๐๏ฟฝ๐ผ๐ผ .
1๐๐๐๐๐๐
.1๐๐๐๐๐๐
.๐ก๐กโ
a/l Area per unit cycloid length โ๐ผ๐ผ Number of intercepts ssf Section sampling fraction asf Area sampling fraction t Section cut thickness h Height of counting frame
Area sampling fraction ๐๐๐๐๐๐ =
๐๐๐๐๐๐๐๐(๐น๐น๐๐๐๐๐๐๐๐)๐๐๐๐๐๐๐๐(๐๐,๐๐ ๐๐๐ก๐ก๐๐๐๐) =
๐๐๐ถ๐ถ๐น๐น .๐๐๐ถ๐ถ๐น๐น๐๐๐ ๐ ๐ก๐ก๐๐๐๐.๐๐๐ ๐ ๐ก๐ก๐๐๐๐
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), 637โ644.
Stereological formulas
23
MERZ
Length of semi-circle (L) ๐ฟ๐ฟ =12๐๐๐๐
d Circle diameter
Surface area per unit volume (Sv) ๐๐๐ฃ๐ฃ =2โ ๐ผ๐ผ๐๐๐๐โ๐๐
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 ๐๐ = ๐๐๐๐2๏ฟฝ l Length of rays
Volume estimate ๐ฃ๐ฃ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ =
4๐๐3๐๐๐๐3๏ฟฝ l Length of rays
Estimated coefficient of error
๐๐๐๐๐ก๐ก ๐ถ๐ถ๐๐(๐๐) =๏ฟฝ 1๐๐ โ 1โ (๐๐๐๐ โ ๐๐๏ฟฝ)2๐๐
๐๐=1
๐๐๏ฟฝ
n Number of nucleator estimates Ri Area/volume estimate for each sampling site
Average area/volume estimate ๐๐๏ฟฝ =
1๐๐๏ฟฝ๐๐๐๐
๐๐
๐๐=1
n Number of nucleator estimates Ri Area/volume estimate for each sampling site
Relative efficiency ๐ถ๐ถ๐ถ๐ถ๐๐(๐๐) =๐ถ๐ถ๐๐(๐๐)โ๐๐
n Number of nucleator estimates CV (R) Estimated coefficient of variation
Geometric mean of area/volume estimate ๐๐๏ฟฝ
1๐๐โ ๐๐๐๐๐๐๐๐๐๐
๐๐=1 ๏ฟฝ 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), 3โ21.
Stereological formulas
25
OPTICAL FRACTIONATOR
Estimate of total number of particles (N) ๐๐ = ๏ฟฝ๐๐โ.
๐ก๐กโ
.1๐๐๐๐๐๐
.1๐๐๐๐๐๐
๐๐โ Particles counted t Section mounted thickness h Counting frame height ๐๐๐๐๐๐ Area sampling fraction ๐๐๐๐๐๐ Section sampling fraction
Variance due to
systematic random sampling โ Gundersen
(VARSRS)
๐๐๐ด๐ด๐๐๐๐๐๐๐๐ =3(๐ด๐ด โ ๐๐2) โ 4๐ต๐ต + ๐ถ๐ถ
12,๐๐ = 0
๐๐๐ด๐ด๐๐๐๐๐๐๐๐ =3(๐ด๐ด โ ๐๐2) โ 4๐ต๐ต + ๐ถ๐ถ
240,๐๐ = 1
๐ด๐ด = โ (๐๐๐๐โ)2๐๐๐๐=1
๐ต๐ต = โ ๐๐๐๐โ๐๐๐๐+1โ๐๐โ1๐๐=1
๐ถ๐ถ = โ ๐๐๐๐โ๐๐๐๐+2โ๐๐โ2๐๐=1
s2 Variance due to noise
Variance due to noise - Gundersen (s2) ๐๐2 = ๏ฟฝ๐๐โ
๐๐
๐๐=1
๐๐โ Particles counted n Number of sections used
Total variance โ Gundersen (TotalVar)
๐๐๐๐๐ก๐ก๐๐๐๐๐๐๐๐๐๐ = ๐๐2 + ๐๐๐ด๐ด๐๐๐๐๐๐๐๐ VARSRS Variance due to SRS s2 Variance due to noise
Coefficient of error โ Gundersen (CE) ๐ถ๐ถ๐ถ๐ถ =
โ๐๐๐๐๐ก๐ก๐๐๐๐๐๐๐๐๐๐๐๐2
TotalVar Total variance s2 Variance due to noise
Number-weighted mean section cut thickness
(๐๐๐ธ๐ธโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ)
๐ก๐ก๐๐โ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ =โ ๐ก๐ก๐๐๐๐๐๐=1 ๐๐๐๐โ
โ ๐๐๐๐โ๐๐๐๐=1
m Number of sections ti Section thickness at site i Qi Particles counted
Stereological formulas
26
Optical Fractionator (2)
Coefficient of error โ Scheaffer (CE) ๐ถ๐ถ๐ถ๐ถ =
๏ฟฝ๐๐2 ๏ฟฝ1๐๐ โ
1๐น๐น๏ฟฝ
๐๐๏ฟฝ
f Number of counting frames ๐น๐น Total possible sampling sites ๐๐2 Estimated variance ๐๐๏ฟฝ Average particles counted
Average number of particles โ
Scheaffer (๐ธ๐ธ๏ฟฝ) ๐๐๏ฟฝ =โ ๐๐๐๐๐๐๐๐=1๐๐
Qi Particles counted f Number of counting frames
Estimated variance - Scheaffer (s2) ๐๐2 =
โ (๐๐๐๐ โ ๐๐๏ฟฝ)2๐๐๐๐=1๐๐ โ 1
f Number of counting frames Qi Particles counted Q๏ฟฝ Average particles counted
Estimated variance of estimated cell population - Scheaffer
๐ถ๐ถ๐๐๐๐๐น๐น2๐๐2
๐๐ Cfp Finite population correction
๐๐2 Estimated variance f Number of counting frames ๐น๐น Total possible sampling sites Estimated variance of mean cell
count - Scheaffer ๐ถ๐ถ๐๐๐๐๐๐2
๐๐ Cfp Finite population correction
๐๐2 Estimated variance f Number of counting frames
Stereological formulas
27
Optical Fractionator (3)
Estimated mean coefficient of error โ Cruz-Orive (est Mean CE) ๐๐๐๐๐ก๐ก ๐๐๐๐๐๐๐๐ ๐ถ๐ถ๐ถ๐ถ (๐๐๐๐๐ก๐ก ๐๐) = ๏ฟฝ
13๐๐
.๏ฟฝ๏ฟฝ๐๐1๐๐ โ ๐๐2๐๐๐๐1๐๐ + ๐๐2๐๐
๏ฟฝ2๐๐
๐๐=1
๏ฟฝ
12๏ฟฝ
Q1i Counts in sub-sample 1 Q2i Counts in sub-sample 2 n Size of sub-sample
Predicted coefficient of error for estimated population โ Schmitz-
Hof (CEpred) ๐ถ๐ถ๐ถ๐ถ๐๐๐๐๐๐๐๐(๐๐๐น๐น) = ๏ฟฝ
๐๐๐๐๐๐(๐๐๐๐โ)๐๐. (๐๐๐๐โ)2
๐ถ๐ถ๐ถ๐ถ๐๐๐๐๐๐๐๐(๐๐๐น๐น) =1
๏ฟฝโ ๐๐๐๐โ๐๐๐๐=1
=1
๏ฟฝโ ๐๐๐ ๐ โ๐๐๐ ๐ =1
R Number of counting spaces S Number of sections Qrโ Counts in the โrโ-th counting space
Qsโ Counts in the โsโ-th section
References Geiser, M., CruzโOrive, 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), 75โ88.
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), 163โ171.
Gundersen, H.J.G., Vedel Jensen, E.B., Kieu, K., & Nielsen, J. (1999). The efficiency of systematic sampling in stereologyโreconsidered. Journal of Microscopy, 193 (3), 199โ211.
Gundersen, H. J. G., Jensen, E.B. (1987). The efficiency of systematic sampling in stereology and its prediction. Journal of Microscopy, 147 (3), 229โ263.
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), 93โ114.
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), 482โ497.
Stereological formulas
29
OPTICAL ROTATOR
Volume of particle ๐ฃ๐ฃ๏ฟฝ = ๐๐๏ฟฝ๐๐(๐๐๐๐)
+/โ
๐๐
a Reciprocal line density a=k.h k Length of slice h Systematic spacing
For vertical slabs and lines parallel to vertical
axis
๐๐(๐๐) = ๐๐1, ๐๐๐๐ ๐๐2 < ๐ก๐ก
๐๐(๐๐) =๐๐2 ๐๐1
๐๐๐๐๐๐๐๐๐๐๐๐ ๏ฟฝ ๐ก๐ก๐๐2๏ฟฝ
, ๐๐๐๐ ๐ก๐ก โค ๐๐2
d1 Distance along test line d2 Distance from origin to test line t ยฝ thickness of optical slice
For vertical slabs and lines perpendicular to
vertical axis
๐๐(๐๐) = ๐๐1, ๐๐๐๐ ๏ฟฝ๐๐12 + ๐ถ๐ถ2 < ๐ก๐ก
๐๐(๐๐) = ๐๐ ๏ฟฝ๏ฟฝ๐ก๐ก2 โ ๐ถ๐ถ2๏ฟฝ , ๐๐๐๐ |๐ถ๐ถ| < ๐ก๐ก โค ๏ฟฝ๐๐12 + ๐ถ๐ถ2
๐๐(๐๐) = ๐๐(0), ๐๐๐๐ ๐ก๐ก โค |๐ถ๐ถ|
๐๐(๐๐) = ๐๐ +๐๐2๏ฟฝ
1
๐๐๐๐๐๐๐๐๐๐๐๐ ๏ฟฝ ๐ก๐กโ๐ข๐ข2 + ๐ถ๐ถ2
๏ฟฝ๐๐๐ข๐ข
๐๐1
๐๐
d1 Distance along test line t ยฝ thickness of optical slice z Distance in z from intercept to origin
Stereological formulas
30
Optical Rotator (2)
For isotropic
slabs
๐๐(๐๐) = ๐๐1, ๐๐๐๐ ๐๐3 < ๐ก๐ก
๐๐(๐๐) =12๐ก๐ก
[โ(๐ก๐ก,๐๐2) + ๐๐(๐ก๐ก,๐๐1,๐๐2,๐๐3)], ๐๐๐๐ ๐๐2 < ๐ก๐กโค ๐๐3
โ(๐ก๐ก,๐๐) = ๐ก๐ก2๏ฟฝ1 โ๐๐2
๐ก๐ก2
๐๐(๐ก๐ก,๐๐1,๐๐2,๐๐3) = ๐๐1๐๐3 + ๐๐22๐๐๐๐๐๐ ๏ฟฝ๐๐1 + ๐๐3
๐ก๐ก + ๏ฟฝ๐ก๐ก2 โ ๐๐22๏ฟฝ
d1 Distance along test line d2 Distance from origin to test line d3 Distance from intercept to origin t ยฝ thickness of optical slice
Estimated surface area
๏ฟฝฬ๏ฟฝ๐ = ๐๐๏ฟฝ๐๐๐๐๐๐๏ฟฝ๐๐๐๐๏ฟฝ๐๐
๐๐(๐๐) = 2, ๐๐๐๐ ๐๐2 < ๐ก๐ก
๐๐(๐๐) = ๐๐.1
๐๐๐๐๐๐๐๐๐๐๐๐ ๏ฟฝ ๐ก๐ก๐๐2๏ฟฝ
, ๐๐๐๐ ๐ก๐ก โค ๐๐2
a Reciprocal line density lj Number of intersections between grid line and cell boundary d2 Distance from origin to test line t ยฝ thickness of optical slice
References Tandrup, T., Gundersen, H.J.G., & Vedel Jensen, E.B. (1997). The optical rotator Journal of microscopy, 186 (2), 108โ120.
Stereological formulas
31
PETRIMETRICS
Total length (๐ณ๐ณ๏ฟฝ) ๐ฟ๐ฟ๏ฟฝ =๐๐2
.๐๐๐๐
.1๐๐๐๐๐๐
.๏ฟฝ๐ผ๐ผ
๐ฟ๐ฟ๏ฟฝ = ๐๐.1๐๐๐๐๐๐
.๏ฟฝ๐ผ๐ผ
a/l = 2d/ฯ Grid constant (2d/ฯ 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) ๐๐ = ๏ฟฝ๐๐โ .1๐๐๐๐๐๐
.1๐๐๐๐๐๐
๐๐โ Particles counted ๐๐๐๐๐๐ Area sampling fraction ๐๐๐๐๐๐ Section sampling fraction
Variance due to noise (s2) ๐๐2 = ๏ฟฝ๐๐โ
๐๐
๐๐=1
๐๐โ Particles counted n Number of sections used
Variance due to systematic random sampling (VARSRS) ๐๐๐ด๐ด๐๐๐๐๐๐๐๐ =
3(๐ด๐ด โ ๐๐2) โ 4๐ต๐ต + ๐ถ๐ถ12
,๐๐ = 0
๐๐๐ด๐ด๐๐๐๐๐๐๐๐ =3(๐ด๐ด โ ๐๐2) โ 4๐ต๐ต + ๐ถ๐ถ
240,๐๐ = 1
๐ด๐ด = โ (๐๐๐๐โ)2๐๐๐๐=1
๐ต๐ต = โ ๐๐๐๐โ๐๐๐๐+1โ๐๐โ1๐๐=1
๐ถ๐ถ = โ ๐๐๐๐โ๐๐๐๐+2โ๐๐โ2๐๐=1
s2 Variance due to noise
Total variance (TotalVar) ๐๐๐๐๐ก๐ก๐๐๐๐๐๐๐๐๐๐ = ๐๐2 + ๐๐๐ด๐ด๐๐๐๐๐๐๐๐ VARSRS Variance due to SRS s2 Variance due to noise
Coefficient of error (CE) ๐ถ๐ถ๐ถ๐ถ =
โ๐๐๐๐๐ก๐ก๐๐๐๐๐๐๐๐๐๐๐๐2
TotalVar Total variance s2 Variance due to noise
References Gundersen, Hans-Jรธrgen 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
๐๐ = 2๐ก๐ก๏ฟฝ๐๐๐๐๐๐
t Separation between test lines gi Isotropic planar rotator function
Volume for vertical planar rotator
๐๐ = ๐๐๐ก๐ก๏ฟฝ๐๐๐๐2
๐๐
t Separation between test lines li Intercept length along a test line
Isotropic planar rotator function ๐๐๐๐(๐๐) = ๐๐๏ฟฝ๐๐2 + ๐๐๐๐2 + ๐๐๐๐2๐๐๐๐ ๏ฟฝ
๐๐๐๐๐๐
+ ๏ฟฝ๏ฟฝ๐๐๐๐๐๐๏ฟฝ2
+ 1๏ฟฝ
๐๐๐๐+ = ๏ฟฝ ๐๐๐๐๏ฟฝ๐๐๐๐ ๐๐+๏ฟฝ โ ๏ฟฝ ๐๐๐๐๏ฟฝ๐๐๐๐ ๐๐+๏ฟฝ๐๐ ๐๐๐๐๐๐๐๐ ๐๐๐ฃ๐ฃ๐๐๐๐
๐๐๐๐โ = ๏ฟฝ ๐๐๐๐๏ฟฝ๐๐๐๐ ๐๐โ๏ฟฝ โ ๏ฟฝ ๐๐๐๐๏ฟฝ๐๐๐๐ ๐๐โ๏ฟฝ๐๐ ๐๐๐๐๐๐๐๐ ๐๐๐ฃ๐ฃ๐๐๐๐
๐๐๐๐ =12
(๐๐๐๐+ + ๐๐๐๐โ)
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 (contโd) ๐๐๐๐+2 = ๏ฟฝ ๐๐๐๐ ๐๐+2
๐๐ ๐๐๐ฃ๐ฃ๐๐๐๐
โ ๏ฟฝ ๐๐๐๐ ๐๐+2
๐๐ ๐๐๐๐๐๐
๐๐๐๐โ2 = ๏ฟฝ ๐๐๐๐ ๐๐โ2
๐๐ ๐๐๐ฃ๐ฃ๐๐๐๐
โ ๏ฟฝ ๐๐๐๐ ๐๐โ2
๐๐ ๐๐๐๐๐๐
๐๐๐๐2 =12 ๏ฟฝ๐๐๐๐+
2 + ๐๐๐๐โ2 ๏ฟฝ
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), 35โ44.
Stereological formulas
35
POINT SAMPLED INTERCEPT
Volume based on intercept length (๐ฝ๐ฝ๐๐๏ฟฝ)
๐๐๏ฟฝ๏ฟฝ๐๐ =๐๐3๐๐0ฬ 3 =
๐๐3๐๐
๏ฟฝ ๐๐0,๐๐3
๐๐
๐๐=1
n Number of intercepts l Intercept length
Volume-weighted mean volume (๐๐๏ฟฝ๐ฝ๐ฝ) ๐๐๏ฟฝ๐ฝ๐ฝ =
โ ๏ฟฝฬ ๏ฟฝ๐๐๐๐๐๐๐๐๐=๐๐
๐๐.๐ ๐ ๐๐
n Number of intercepts l Intercept length
Coefficient of error (CE)
๐ถ๐ถ๐ถ๐ถ๏ฟฝ๐๐0ฬ 3๏ฟฝ = ๏ฟฝโ ๏ฟฝ๐๐0ฬ 3๏ฟฝ
2๐๐๐๐=1
๏ฟฝโ ๐๐0ฬ 3๐๐๐๐=1 ๏ฟฝ
2 โ1๐๐
n Number of intercepts l Intercept length
Coefficient of variance (CV) ๐ถ๐ถ๐ถ๐ถ๏ฟฝ๐๐0ฬ 3๏ฟฝ = ๐ถ๐ถ๐ถ๐ถ(๏ฟฝฬ ๏ฟฝ๐ฃ๐๐).โ๐๐ n Number of intercepts l Intercept length ๐๐๏ฟฝ๐ฝ๐ฝ Volume-weighted mean volume
Variance (VarianceV) ๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐(๐ฃ๐ฃ) = ๏ฟฝ๐๐3
. ๐๐๐ท๐ท๏ฟฝ๐๐0ฬ 3๏ฟฝ๏ฟฝ = ๏ฟฝ๐ถ๐ถ๐๐๏ฟฝ๐๐0ฬ 3๏ฟฝ. ๏ฟฝฬ ๏ฟฝ๐ฃ๐๐๏ฟฝ L Intercept length ๐๐๏ฟฝ๐ฝ๐ฝ 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, 127โ142.
Sรธrensen, F.B. (1991). Stereological estimation of the mean and variance of nuclear volume from vertical sections. Journal of Microscopy, 162 (2), 203โ229.
Stereological formulas
36
SIZE DISTRIBUTION
Volume-weighted mean particle volume
๏ฟฝฬ ๏ฟฝ๐ฃ๐๐ = ๏ฟฝฬ ๏ฟฝ๐ฃ๐๐. [1 + ๐ถ๐ถ๐๐๐๐2(๐ฃ๐ฃ)] v๏ฟฝN Number-weighted mean volume
CVN(v) Coefficient of variation
Number-weighted mean volume ๏ฟฝฬ ๏ฟฝ๐ฃ๐๐ =
โ๐๐โ๐๐
๐๐ = ๐๐.๐๐
R Number of contours Q Number of points per contour
Variance
๐๐๐๐๐๐๐๐(๐ฃ๐ฃ) =๏ฟฝโ๐๐ โ (โ๐๐)2
โ๐๐ ๏ฟฝ . ๐ฃ๐ฃ(๐๐)2
โ๐๐ โ 1
๐๐ = ๐๐2.๐๐
R Number of contours Q Number of points per contour v(p) Volume associated with a point
Standard deviation ๐๐๐ท๐ท๐๐(๐ฃ๐ฃ) = ๏ฟฝ๐๐๐๐๐๐๐๐(๐ฃ๐ฃ)
๐๐๐ท๐ท๐๐(๐ฃ๐ฃ) = ๏ฟฝ๏ฟฝฬ ๏ฟฝ๐ฃ๐๐ . (๏ฟฝฬ ๏ฟฝ๐ฃ๐๐ โ ๏ฟฝฬ ๏ฟฝ๐ฃ๐๐)
v๏ฟฝN Number-weighted mean volume v๏ฟฝV Volume-weighted particle volume VarN Variance
Coefficient of variation ๐ถ๐ถ๐๐๐๐(๐ฃ๐ฃ) =
๐๐๐ท๐ท๐๐(๐ฃ๐ฃ)๏ฟฝฬ ๏ฟฝ๐ฃ๐๐
๐ถ๐ถ๐๐๐๐(๐ฃ๐ฃ) = ๏ฟฝ๏ฟฝฬ ๏ฟฝ๐ฃ๐๐ โ ๏ฟฝฬ ๏ฟฝ๐ฃ๐๐๏ฟฝฬ ๏ฟฝ๐ฃ๐๐
v๏ฟฝN Number-weighted mean volume v๏ฟฝV Volume-weighted particle volume SDN Standard deviation
Stereological formulas
37
Size distribution (2)
Coefficient of error ๐ถ๐ถ๐ถ๐ถ๐๐(๐ฃ๐ฃ) =
๐ถ๐ถ๐๐๐๐(๐ฃ๐ฃ)โ๐๐
CVN Coefficient of variation R Number of contours
References Sรธrensen, F.B. (1991). Stereological estimation of the mean and variance of nuclear volume from vertical sections. Journal of Microscopy, 162 (2), 203โ229.
Stereological formulas
38
SPACEBALLS
Length estimate ๐ฟ๐ฟ = 2.๏ฟฝ๏ฟฝ๐๐๐๐
๐๐
๐๐=1
๏ฟฝ .๐ฃ๐ฃ๐๐
.1๐๐๐๐๐๐
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
๐๐2 = ๏ฟฝ๐๐๐๐
๐๐
๐๐=1
Qi Intersection counted
Variance due to systematic random
sampling
๐๐๐ด๐ด๐๐๐๐๐๐๐๐ =3(๐ด๐ด โ ๐๐2)โ 4๐ต๐ต + ๐ถ๐ถ
12,๐๐ = 0
๐๐๐ด๐ด๐๐๐๐๐๐๐๐ =3(๐ด๐ด โ ๐๐2)โ 4๐ต๐ต + ๐ถ๐ถ
240,๐๐ = 1
๐ด๐ด = โ (๐๐๐๐โ)2๐๐๐๐=1
๐ต๐ต = โ ๐๐๐๐โ๐๐๐๐+1โ๐๐โ1๐๐=1
๐ถ๐ถ = โ ๐๐๐๐โ๐๐๐๐+2โ๐๐โ2๐๐=1
s2 Variance due to noise m Smoothness class of sampled function
Total variance ๐๐๐๐๐ก๐ก๐๐๐๐๐๐๐๐๐๐ = ๐๐2 + ๐๐๐ด๐ด๐๐๐๐๐๐๐๐ VARSRS Variance due to SRS s2 Variance due to noise
Stereological formulas
39
Spaceballs (2)
Coefficient of error ๐ถ๐ถ๐ถ๐ถ =
โ๐๐๐๐๐ก๐ก๐๐๐๐๐๐๐๐๐๐๐๐2
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), 54โ64.
Stereological formulas
40
SURFACE-WEIGHTED STAR VOLUME
Surface-weighted star volume (๐๐๏ฟฝ๏ฟฝ๐๐โ) ๐๐๏ฟฝ๏ฟฝ๐๐โ =
๐๐๐ ๐ ๐๐
. ๏ฟฝฬ ๏ฟฝ๐๐๐๐๐
๐๐๏ฟฝ๏ฟฝ๐๐โ =๐๐๐ ๐ ๐๐ .
โ โ ๐๐๐๐,(๐๐,๐๐)๐๐๐๐๐๐
๐๐=๐๐๐๐๐๐=๐๐
โ ๐๐๐๐๐๐๐๐=๐๐
n Number of probes l Intercept length mi Number of intercepts
Sum of cubed intercepts in probe (yi) ๐๐๐๐ = ๏ฟฝ๐๐1,(๐๐,๐๐)
3
๐๐๐๐
๐๐=1
mi Number of intercepts l Intercept length
Coefficient of error (CE) ๐ถ๐ถ๐ถ๐ถ๏ฟฝ๏ฟฝฬ ๏ฟฝ๐ฃ๐ ๐ โ๏ฟฝ๏ฟฝ = ๏ฟฝ
๐๐๐๐ โ 1 ๏ฟฝ
โ๐๐๐๐2
โ๐๐๐๐ โ๐๐๐๐+
โ๐๐๐๐2
โ๐๐๐๐ โ๐๐๐๐โ 2.
โ๐๐๐๐๐๐๐๐โ๐๐๐๐ โ๐๐๐๐
๏ฟฝ๏ฟฝ12๏ฟฝ
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), 350โ356.
Stereological formulas
41
SURFACTOR
Surface area for single-ray designs
๏ฟฝฬ๏ฟฝ๐ = 4๐๐๐๐02 + ๐๐(๐ฝ๐ฝ) l Length of intercept ร Angle between test line and surface c(ร) Function of the planar angle
Surface area for multi-ray
designs ๐บ๐บ๏ฟฝ = ๐๐๐ ๐ ๏ฟฝ๐๐๐๐๐๐. ๐๐(๐ท๐ท)๐๐๐๐
๐๐=๐๐
l Length of intercept ร Angle between test line and surface c(ร) Function of the planar angle r Number of test lines
Function of the planar angle ๐๐(๐ฝ๐ฝ) = 1 + ๏ฟฝ
12๐๐๐๐๐ก๐ก๐ฝ๐ฝ๏ฟฝ . ๏ฟฝ
๐๐2โ ๐๐๐๐๐๐โ1
1 โ ๐๐๐๐๐ก๐ก2๐ฝ๐ฝ1 + ๐๐๐๐๐ก๐ก2๐ฝ๐ฝ
๏ฟฝ ร 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 ๐๐๐๐ = 2 ๏ฟฝ
๐๐๐๐๏ฟฝโ ๐ผ๐ผ๐๐๐๐๐๐=1
โ ๐๐๐๐๐๐๐๐=1
p/l Ratio of test points to curve length n Number of micrographs โ Ii Total intercept points on curve โ P๐๐ Total test points
References Baddeley, A. J., Gundersen, H.J.G., & CruzโOrive, L.M. (1986). Estimation of surface area from vertical sections. Journal of Microscopy, 142 (3), 259โ276.
Stereological formulas
43
VERTICAL SPATIAL GRID
Estimated volume ๐๐๏ฟฝ = ๐๐.๐๐๐๐.๏ฟฝ๐๐๐๐
๐๐
๐๐=1
a Area associated with point dz Distance between planes m Number of scanning planes โ P Intersections with points
Area associated with point ๐๐ =
๐ค๐ค2
2๐๐ w Horizontal width
Estimated surface area ๏ฟฝฬ๏ฟฝ๐ = 2.๐๐.๐๐๐๐
๐๐ + 4๐๐ .๐๐๐๐
. ๏ฟฝ๐๐๐๐๐๐ + ๐ผ๐ผ๐๐๐ง๐ง๏ฟฝ a Area associated with point dz Distance between planes l Length of cycloid Ixy X,Y intersections Ixz X,Z intersections Length of cycloid ๐๐ =
2๐ค๐ค๐๐
w Horizontal width
X,Y intersections ๐ผ๐ผ๐๐๐๐ = ๏ฟฝ๐ผ๐ผ๐๐๐๐,๐๐
๐๐
๐๐=1
m Number of scanning planes
X,Z intersections ๐ผ๐ผ๐๐๐ง๐ง = 2.๏ฟฝ๏ฟฝ๐๐๐๐
๐๐
๐๐=1
โ ๏ฟฝ ๐๐๐๐,๐๐+1
๐๐โ1
๐๐=1
๏ฟฝ m Number of scanning planes โ๐๐ 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), 146โ151.
Stereological formulas
44
WEIBEL
Surface area per unit volume (Sv) ๐๐๐ฃ๐ฃ = 2.๏ฟฝ
๐ผ๐ผ๐๐2 .๐๐
๏ฟฝ
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), 23โ38.