# Table 3 Examples of stereological calculations

Estimation (formula) Counts (example)a Results (example)a
Cavalieri (Fig. 4a) $$\begin{array}{ll}\sum P\hfill & =125\hfill \\ {}a/p\hfill & ={\left(3.5\ mm\right)}^2\hfill \\ {}d=\hfill & 3\ mm\hfill \end{array}$$ $$\begin{array}{ll}V(lung)\hfill & =4594\ m{m}^3\hfill \end{array}$$
V(lung) = ∑P x a/p x d
Parenchymal Volume (Fig. 4b) $$\begin{array}{ll}\sum P(par)\hfill & =218\times 4=872\hfill \\ {}\sum P(nonpar)\hfill & =107\hfill \\ {}\sum P(lung)\hfill & =\sum \left(P(par)+P(nonpar)\right)\hfill \\ {}\hfill & =979\hfill \end{array}$$ $$\begin{array}{ll}V\left(par, lung\right)\hfill & =4092\ m{m}^3\hfill \\ {}V\left( nonpar, lung\right)\hfill & =502\ m{m}^3\hfill \end{array}$$
$$\begin{array}{ll}V\left(par, lung\right)\hfill & =\sum P(par)/\sum P(lung)\times V(lung)\hfill \\ {}V\left( nonpar, lung\right)\hfill & =\sum P(nonpar)/\sum P(lung)\times V(lung)\hfill \end{array}$$
Alveolar volume, surface area and septal thickness (Fig. 4c) $$\begin{array}{ll}\sum I\hfill & =394\hfill \\ {}\sum P(sept)\hfill & =94\hfill \\ {}\sum P(alv)\hfill & =333\hfill \\ {}\sum P(par)\hfill & =94+333=427\hfill \\ {}l/p\hfill & =35\ \mu m\hfill \end{array}$$ $$\begin{array}{ll}{V}_V\left( sept/par\right)\hfill & =0.22\hfill \\ {}{V}_V\left(alv/par\right)\hfill & =0.78\hfill \\ {}{S}_V\left( sept/par\right)\hfill & =527.3\ c{m}^{-1}\hfill \end{array}$$
$$\begin{array}{ll}{V}_V\left( sept/par\right)\hfill & =\sum P(sept)/\sum P(par)\hfill \\ {}{V}_V\left(alv/par\right)\hfill & =\sum P(alv)/\sum P(par)\hfill \\ {}{S}_V\left( sept/par\right)\hfill & =\left(2\times \sum I\right)/\left(\sum P(par)\times l/p\right)\hfill \\ {}S\left( sept, lung\right)\hfill & ={S}_V\left( sept/par\right)\times V(par)\hfill \\ {}V\left(alv, lung\right)\hfill & ={V}_V\left(alv/par\right)\times V(par)\hfill \\ {}\tau (sept)\hfill & =2\times {V}_V\left( sept/par\right)/{S}_V\left( sept/par\right)\hfill \end{array}$$ $$\begin{array}{ll}V\left(alv, lung\right)\hfill & =3191.8\ m{m}^3\hfill \\ {}S\left( sept, lung\right)\hfill & =2158\ c{m}^2\hfill \\ {}\tau (sept)\hfill & =8.3\ \mu m\hfill \end{array}$$
Epithelial mucous cell metaplasia (Fig. 4d) $$\begin{array}{ll}\sum I\hfill & =294\hfill \\ {}\sum P(muc)\hfill & =89\hfill \\ {}\sum P(epi)=\hfill & 245\hfill \\ {}\sum P(lung)\hfill & =396\times 50=19800\hfill \\ {}l/p\hfill & =35\ \mu m\hfill \end{array}$$ $$\begin{array}{ll}{V}_V\left(muc/ lung\right)\hfill & =0.0045\hfill \\ {}{V}_V\left(epi/ lung\right)\hfill & =0.012\hfill \\ {}{S}_V\left(bm/ lung\right)\hfill & =0.85\ m{m}^{-1}\hfill \end{array}$$
$$\begin{array}{ll}{V}_V\left(muc/ lung\right)\hfill & =\sum P(muc)/\sum P(lung)\hfill \\ {}{S}_V\left(bm/ lung\right)\hfill & =\left(2\times \sum I\right)/\left(\sum P(lung)\times l/p\right)\hfill \\ {}V/S\left(muc/bm\right)\hfill & =\left(\sum P(muc)\times l/p\right)/\left(2\times \sum I\right)\hfill \\ {}\tau (epi)\hfill & ={V}_V\left(epi/ lung\right)/{S}_V\left(bm/ lung\right)\hfill \end{array}$$ $$\begin{array}{ll}V/S\left(muc/bm\right)\hfill & =5.3\ \mu {m}^3/\mu {m}^2\hfill \\ {}\tau (epi)\hfill & =14.1\ \mu m\hfill \end{array}$$
Cell numbers (Fig. 5a) $$\begin{array}{ll}\sum {Q}^{-}\hfill & =107\ \left( both\ ways\right)\hfill \\ {}n\hfill & =53\hfill \\ {}A\hfill & =200\times 250\ \mu {m}^2\hfill \\ {}h\hfill & =5\ \mu m\hfill \end{array}$$ $$\begin{array}{ll}{N}_V\left( cell/ lung\right)\hfill & =4038\ m{m}^{-3}\hfill \\ {}N\left( cell, lung\right)\hfill & =18.5\times {10}^6\hfill \end{array}$$
$$\begin{array}{ll}{N}_V\left( cell/ lung\right)\hfill & =\sum {Q}^{-}/\left(2\times n\times A\times h\right)\hfill \\ {}N\left( cell, lung\right)\hfill & =NV\left( cell/ lung\right)\times V(lung)\hfill \end{array}$$
Alveolar number (Fig. 5b)
$$\begin{array}{ll}N\left(alv, lung\right)\hfill & =\left(\sum B\right./\left.\left(2\times n\times A\times h\right)\right)\times V(lung)\hfill \end{array}$$
$$\begin{array}{ll}\sum B\hfill & =112\ \left( both\ ways\right)\hfill \\ {}n\hfill & =90\hfill \\ {}A\hfill & =200\times 250\ \mu {m}^2\hfill \\ {}h\hfill & =5\ \mu m\hfill \end{array}$$ $$\begin{array}{ll}{N}_V\left(alv/ lung\right)\hfill & =2489\ m{m}^{-3}\hfill \\ {}N\left(alv, lung\right)\hfill & =11.4\times {10}^6\hfill \end{array}$$
1. aThe values presented in the following examples are related to Figs. 4 and 5. All density calculations are multiplied with the lung volume as estimated with the Cavalieri method in Fig. 4a to obtain estimates of volume, surface area and number of structure of interest. Note that the calculations are based on examples from rat lungs, but results might dissent from expected values and no shrinking corrections were applied in the formulas