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Optical fractionator formulas

All stereological formulas

ClosedEstimate of total number of particles (N)

  •  : Particles counted
  • t: Section mounted thickness
  • h: Counting frame height
  • : Area sampling fraction
  • : Section sampling fraction
ClosedVariance due to systematic random sampling – Gundersen  (VARSRS)

or

, ,

  • s2; Variance due to noise
ClosedVariance due to noise - Gundersen (s2)

  • : Particles counted
  • n: Number of sections used
ClosedTotal variance – Gundersen (TotalVar)

  • VARSRS: Variance due to SRS
  • s2: Variance due to noise
ClosedCoefficient of error – Gundersen (CE)

  • TotalVar:Total variance
  • s2: Variance due to noise
ClosedNumber-weighted mean section cut thickness ()

  • m: Number of sections
  • ti : Section thickness at site i
  • Qi: Particles counted
ClosedCoefficient of error – Scheaffer (CE)

  • f: Number of counting frames
  • : Total possible sampling sites
  • : Estimated variance
  • : Average particles counted
ClosedAverage number of particles – Scheaffer ()

  • Qi: Particles counted
  • f: Number of counting frames
ClosedEstimated variance - Scheaffer (s2)

  • f: Number of counting frames
  • Qi: Particles counted
  • : Average particles counted
ClosedEstimated variance of estimated cell population - Scheaffer

  • Cfp: Finite population correction
  • : Estimated variance
  • f: Number of counting frames
  • : Total possible sampling sites
ClosedEstimated variance of mean cell count - Scheaffer

  • Cfp: Finite population correction
  • : Estimated variance
  • f: Number of counting frames
ClosedEstimated mean coefficient of error – Cruz-Orive (est Mean CE)

  • Q1i: Counts in sub-sample 1
  • Q2i: Counts in sub-sample 2
  • n: Size of sub-sample
ClosedPredicted coefficient of error for estimated population – Schmitz-Hof (CEpred)

and

  • R: Number of counting spaces
  • S: Number of sections
  • : Counts in the “r”-th counting space
  • : Counts in the “s”-th section

 

References

Geiser, M., Cruz‐Orive, 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), 75–88.

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), 163–171.

Gundersen, H.J.G., Vedel Jensen, E.B., Kieu, K., & Nielsen, J. (1999). The efficiency of systematic sampling in stereology—reconsidered. Journal of Microscopy, 193 (3), 199–211.

Gundersen, H. J. G., Jensen, E.B. (1987). The efficiency of systematic sampling in stereology and its prediction. Journal of Microscopy, 147 (3), 229–263.

Howard, V., Reed, M. (2005). Unbiased stereology: three-dimensional measurement in microscopy (vol. 4, chapter 12). Garland Science/Bios Scientific Publishers.

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), 93–114.

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), 482–497.

 

See Optical Fractionator workflow