Validation of a stereological method for estimating particle size and density from 2D projections with high accuracy

Stereological methods for estimating the 3D particle size and density from 2D projections are essential to many research fields. These methods are, however, prone to errors arising from undetected particle profiles due to sectioning and limited resolution, known as ‘lost caps’. A potential solution by Keiding et al. (1972) accounts for lost caps by quantifying the smallest detectable profiles in terms of their limiting section angle (ϕ). However, this simple solution has not been widely adopted nor validated. Rather, model-independent design-based stereological methods, which do not explicitly account for lost caps, have come to the fore. Here, we provide the first experimental validation of the Keiding model by quantifying ϕ of synaptic vesicles using electron-tomography 3D reconstructions. This analysis reveals a Gaussian distribution for ϕ rather than a single value. Nevertheless, curve fits of the Keiding model to the 2D diameter distribution accurately estimate the mean ϕ and 3D diameter distribution. While systematic testing using Monte Carlo simulations revealed an upper limit to determining ϕ, our analysis shows that mean ϕ can be used to estimate the 3D particle density from the 2D density under a wide range of conditions, and this method is potentially more accurate than minimum-size-based lost-cap corrections and disector methods. We applied the Keiding model to estimate the size and density of somata, nuclei and vesicles in rodent cerebella, where high packing density can be problematic for design-based methods. ### Competing Interest Statement The authors have declared no competing interest. * T : thickness of tissue section (transmission microscopy) or focal plane (ρz) D : 3D diameter of a particle μD ± σD : mean and standard deviation of 3D particle diameters CVD : σD / μD u.d. : unit diameter, length normalised to μD (e.g. T/μD) planar : T < 0.1 u.d. thin : T ≈ 0.3 u.d. thick : T ≥ 1 u.d. d : observed 2D diameter of a particle dmin : minimum 2D diameter of a sample of particles[47][1] hmin : minimum penetration depth of a sample of particles[42][2] δmin : minimum 2D diameter of a given particle (z-stack analysis) darea : equivalent-area 2D diameter: darea = 2(area/π)½ dshort, dlong : short and long-axis 2D diameter dgeometric : (dshort·dlong)½ davg : ½(dshort + dlong) μd ± σd : mean and standard deviation of 2D diameters F(d) : probability density of 3D diameters G(d) : probability density of 2D diameters L(d) probability density of lost caps θ : particle cap angle from section surface: sinθ = d/D where 0 ≤ θ ≤ 90 θmin : equivalent cap angle of dmin: sinθmin = dmin/D ϕ : lower limit of θ where 0 ≤ ϕ ≤ 90 ϕcutoff : upper cutoff limit of when ϕ is determinable μϕ ± σϕ : mean and standard deviation of ϕ CVϕ : σϕ / μϕ dϕ: : equivalent 2D diameter of ϕ dϕ = μD·sinϕ ζ : section z-depth over which particle center points are sampled Areaxy : ROI xy-area over which particles are counted VF : Particle volume fraction within a volume of interest AF : Particle area fraction within a ROI N3D : Particle count within a volume of interest N2D : Particle count within a ROI λ3D : 3D particle density, λ3D = N3D / Volumexyz λ2D : 2D particle density, λ2D = N2D / Areaxy Ω : sum of projection overlaps for a given particle where Ω ≥ 0 ψ : upper limit of Ω, i.e. 0 ≤ Ω ≤ ψ χ2 : sum of squared differences between data and fits (or simulations) Δ : Parameter estimation error: % difference or difference from true value μΔ ± σΔ : bias and (68%) confidence interval of a parameter’s estimation error ρxyz : Microscope resolution Rxyz : Image/z-stack resolution [1]: #ref-47 [2]: #ref-42