Image Anal Stereol 2001;20:15-25 Original Research Paper METRIC CHARACTERISTICS OF VARIOUS METHODS FOR NUMERICAL DENSITY ESTIMATION IN TRANSMISSION LIGHT MICROSCOPY – A COMPUTER SIMULATION Miroslav Kališnik1, Andrej Blejec2, Zdenka Pajer1 and Janja Majhenc3 1Institute of Histology and Embryology, Faculty of Medicine, Ljubljana, Slovenia, 2National Institute of Biology, Ljubljana, Slovenia, 3Institute of Biophysics, Faculty of Medicine, Ljubljana, Slovenia e-mails: miroslav.kalisnik@mf.uni-lj.si, andrej.blejec@uni-lj.si, zdenka.pajer@mf.uni-lj.si, janja@biofiz.mf.uni-lj.si (Accepted February 28, 2001) ABSTRACT In the introduction the evolution of methods for numerical density estimation of particles is presented shortly. Three pairs of methods have been analysed and compared: (1) classical methods for particles counting in thin and thick sections, (2) original and modified differential counting methods and (3) physical and optical disector methods. Metric characteristics such as accuracy, efficiency, robustness, and feasibility of methods have been estimated and compared. Logical, geometrical and mathematical analysis as well as computer simulations have been applied. In computer simulations a model of randomly distributed equal spheres with maximal contrast against surroundings has been used. According to our computer simulation all methods give accurate results provided that the sample is representative and sufficiently large. However, there are differences in their efficiency, robustness and feasibility. Efficiency and robustness increase with increasing slice thickness in all three pairs of methods. Robustness is superior in both differential and both disector methods compared to both classical methods. Feasibility can be judged according to the additional equipment as well as to the histotechnical and counting procedures necessary for performing individual counting methods. However, it is evident that not all practical problems can efficiently be solved with models. Keywords: accuracy, efficiency, feasibility, light microscopy, numerical density, robustness, vertical resolution. INTRODUCTION At the beginning let us quote the opinion of Bodziony et al. (1998) that “numerical density stereology should by no means be considered to be a closed area of investigation”. Since 1925 several methods for numerical density (NV) estimation in the light microscope with transmitted light have been developed. For the illustration of various counting methods we have used the same simplified model as Weibel (1979), i.e. particles as equal spheres randomly distributed in 3D with their centres as associated points for counting them. In order for the particles to be seen in the microscope, they must be distinguished from the surroundings by adequate contrast. In our model we have assumed a maximal contrast between the particles and the surroundings. In counting particles with a diameter larger than the lateral resolution (d) of the microscope we take into account the particles with a particle associated point, e.g. with the centres or centroids inside the reference space. The test area (At) defines the 2D reference space. To help at the decision whether the particle with its centre intersecting the limiting lines of the test area is inside or outside it, a rule of two forbidden and two allowed lines has been accepted, a rule known for long time e.g. in haematology. However, the particles have to be counted in 3-D reference space, therefore the two forbidden and the two allowed lines have been extended into two forbidden and two allowed planes. The known demand that the forbidden lines or planes have to be infinite in extent is irrelevant for the model of spheres in our simulation. In transmitted light the particle height (h) (i.e. the dimension in the direction of the optical axis of the 15 KaliŠNIK M et al: Metric characteristics of various methods for NV estimation microscope) should be larger than the vertical resolution of the microscope. From the particle counting theory the term lost polar caps height is known, but its estimation has not been exactly defined. We propose that, instead of the lost polar caps height we calculate and use the vertical resolution (depth of focus = h) for a microscope objective with a defined numerical aperture as an approximation. If the contrast between the particles and the surroundings is not maximal, the height of lost caps is probably larger than the depth of focus, but a correlation between the lost polar caps height and the vertical resolution nevertheless exists (Fig. 1). 10X 20X 0.25 0.45 40X 0.65 60X 0.75 Francon Michel Stevens et al. Wicksell (1925, 1926) expressed the classical stereological principle that the number of particle profiles in the test area (NA) is proportional to the numerical density (NV) of the particles and their average tangent diameter (D), from which we deduce the following Eq. 1 Nv = NA/D. (1) This principle could be used for particle counting in reflected light (Fig. 2). But in biological objects we usually observe and count particles in slices of thickness (t) in transmitted light. The thickness of physical slices (called also sections) is usually 0.05 -0.08 µm for ultrathin sections, 0.5 - 2.0 µm for semithin sections, 5 - 10 µm for normally thick sections and several tens to over 200 µm for thick sections. The section can be thick or thin. In thick sections we can count particles in the whole slice thickness by successively turning the micrometer screw from top to down in order to see clearly the particles at all depths (Fig. 3). In both thin and thick sections, we have to take into account, besides the Wicksell's principle, corrections for slice thickness (t) and lost polar caps (h), according to Eq. 2 developed by Agduhr (1941), Floderus (1944) and Abercrombie (1946): M 10X 20X NA 0.250.45 40X 0.65 60X 0.75 NV = NA/(t + D - 2h). (2) Fig. 1. The relation between the magnification of the objectives 10x to 100x and (a) lateral resolution (d) in µm, calculated according to the equation of Abbe (Boyd, 1995) and (b) vertical resolution (depth of focus) of the microscope (h) in µm, calculated according to equations from Francon (1961) and Michel (1981) for the light microscope or Stevens et al. (1994) for the confocal microscope. Possible additional effect of the physiological eye accommodation has not been taken into account. M: objective magnification, NA: numerical aperture. inzga Fig. 2. In the reference space all the particles with centres dislocated from the section plane for < D/2 are included. For particles in reflected light the equation (1) NV = NA/D is used. Fig. 3. Particles inside the height (t+ D - 2h) belong to the reference space of a thick section with the height t and NV = NA/(t + D - 2h). Consequently, we count the particles in a »superslice«, containing the common space of the real (physical or optical) section and of both virtual spaces (above and under the real section), containing the centres of the particles, which are visible by their parts in the real section. 16 Image Anal Stereol 2001;20:15-25 The smallest usable slice thickness is the optical section thickness with height h (Fig. 4). In this case we count the particles at the level of the optical section without moving the micrometer screw. 'i""D'/2-hl ..*...D/2-hT Fig. 4. The particles with the centres inside space with the height (h + D - 2h) = D - h belong to the reference space of the thinnest optical section with the vertical resolution h. Using the method for differential counting according to Ebbeson and Tang (1965) we take two physical sections of differing thicknesses (t1 > t2), count the particles per test area in each of them (NA1 and NA2) and calculate the numerical density according to Eq. 3 NV = (NA1 - NA2)/(t1 - t2), (3) for which knowledge of the particle diameter is obviously not necessary. Fig. 5. Using the method for modified differential counting we count first the particles seen in one thick section (NAt) and then in one thin optical section of thickness t = h inside the same physical section (NAh). Equation NV = (NAt - NAh)/(t - h) is The above procedure can be simplified by counting particles within a thick slice, and by taking an optical section inside of the thick physical section instead of a thinner physical section (Pajer and Kališnik, 1984; Kališnik and Pajer, 1985) (Fig. 5), leading to the Eq. 4 NV = (NAt - NAh)/(t - h). (4) In all cases above the reference space is limited by two planes but is open above and under the real space, extended in both directions by a virtual space. Limiting the reference space also from above by a third forbidden plane (Fig. 6) we obtain two disector methods, for physical sections (Sterio, 1984) and for optical sections (Howard et al, 1985). In the physical disector method we cut the object into pairs of sections, the one being the forbidden or control section, and the other used as counting or reference section. We do not count particles in the forbidden section (look-up section). Additionally, in the counting section we do not count particles, which are seen also in the forbidden section. We count particles, which are seen in the counting section only. But since the reference space of the second, counting section, is reduced from above for the thickness of the »shadow« from the first, control section, (i.e. the lower virtual space of the first section), it is augmented under the real space for the same thickness (i.e. the lower virtual space of the second section). The final slice thickness is thus equal to the physical section thickness t (Fig. 7a, b). The Eq. 5 for this method is simply NV = NAt / t, as has been published by Collan (1991). (5) Fig. 6. Delimitation of the reference space with three forbidden planes AEHD, DCGH and EFGH is characteristic for both disector methods. 17 Kališnik M et al: Metric characteristics of various methods for NV estimation a b D/2-h jjjj' D/2-h Fig. 7 a, b. Physical disector: The particles with centres in the lower virtual space of the forbidden paired section (a) with height (D/2 – h) are not counted in the reference section (b). In the reference section we do not count the particles with the centres inside the part of the upper virtual space which is in the »shadow« of the forbidden paired section. The final reference space of the counting section has the height (t + D/2 – h) – (D/2 – h) = t. Theoretically this method would be ideal and superior to all former methods. In reality its application is much more demanding and the results are less reliable than those of the older methods. It is necessary to cut exact series of pairs of sections and to mount each first section on one slide and each second on the other slide, marking them to indicate they are a pair, in order to match them perfectly. Further, additional equipment is necessary in form of a tandem projection microscope, or two microscopes each with its own digital camera and own videomonitor or one microscope with motor driven stage, one camera and a monitor. Ignoring the additional costs for equipment and tedious preparation of pairs of sections, the results of this method are uncertain because of possible mistakes in identification of particles, occurring in a D/2-h both sections (look-up and counting), this error being additive in each pair of sections. Some of these problems in using the physical disector method are avoided by the use of the so-called optical disector method. This method is actually an improved method for thick sections, with the reference space delimited with three forbidden planes (Fig. 8a, b). This method can also be performed, under favourable conditions, in the common optical microscope, though the use of a confocal microscope is recommended. But, one constraint remains: in counting particles, not only those seen in the forbidden optical section have to be ignored but also those seen in the first counting optical section, if they have been seen in the forbidden section. Otherwise this method becomes simply the method for thick sections. b U w D/2-h Fig. 8 a, b. Optical disector: The particles with centres inside the lower virtual space of the forbidden upper optical section (a) are not counted in the first counting optical section (b). The particles with their centres in the space, delimited upwards with the lower limit of the shadow of the forbidden section and downwards with the bottom of the lower virtual space of the last optical section are counted in all optical counting sections with the total height T. The total reference space has the height T. T 18 Image Anal Stereol 2001;20:15-25 The aim of the present study is to verify the hypothesis that all enumerated methods for numerical density estimation give accurate results if all necessary conditions are assumed and properly respected and if the sample is representative and large enough. In addition, we shall compare the efficiency and robustness of all the methods using the computer simulation as well as geometrical and mathematical analysis. We shall conclude by summarizing the feasibility of some of the above methods. COMPUTER SIMULATION AND MATHEMATICAL ANALYSIS For each type of methods for estimation of particle number we have used a computer simulated counting process. Particles with selected diameter D = 7 µm were counted in a model object of nearly 4000 nonoverlapping particles in a cube of 300 µm edge with a particle density NV = 145296 mm3. Particles were treated as spheres with diameter D, with minimal distance between the particle centres being 2D = 14 µm. The model object was generated in two phases. First, a large number uniformly distributed points in 3-dimensional space was generated. In the next step, points with minimal distance to other points smaller than 2D were eliminated. The remaining points represent particle centres of nonoverlapping spheres. To avoid problems with border space, the actual model space was enlarged in each dimension by D. To select a slice, the vertical position of a slice was selected at random. Slices with thicknesses 0, 0.5, 1, 2.5, 5, 10, 25, 50, and 100 µm were generated. On each slice, the random sample of n = 10 test areas was taken. For each area the particles meeting the criteria of the particular estimation method were counted and used in the appropriate formula for particle density estimation. The counting process was repeated m = 10 times, giving m estimates for the particle density. Standard deviation of m estimates was used as standard error (SE) and mean value of m estimates was taken as the particle density estimate (NV) for a particular method. The standard deviation s for a sample of test areas was estimated from SE as s = SE ¦ *Jn . Accuracy Accuracy was described by the relative standard error RSE = SE / NV , and relative error of estimate RE = (NV — NV ) / NV (6) (7) Efficiency To measure efficiency, the required number of test areas (nreq) and total number of counted particles (Creq), required for RSEreq = 5% were estimated from simulated estimates: n req 2 NV ¦ RSEreq J Creq=nreq-v (8) (9) where s is the standard deviation and V the average number of particles per area counted in the simulation. Table 1. Overall relative error e for various counting methods. Counting method Wicksell Thick slice Number of Estimated assumptions parameters1 "1 8 2 T, 5 Original differential 2 Modified differential 1 T, T-, 1 , 2 Disector 1 Overall relative error e S/D T + S-2h t + D-2h (T1 —T2)/(t1—t2) T1 / (t1 - h) T/t Ö difference between false and true particle diameter; T difference between false and true slice thickness s T 19 Kališnik M et al: Metric characteristics of various methods for NV estimation Robustness RESULTS Robustness was described by the relative error of estimate. It depends on the number of assumed parameters and the relative error of estimate of each parameter. In general, the overall overestimate of parameters (e.g. particle diameter D and slice thickness t ) leads to an underestimate of particle number and vice versa. The relative error of particle density RE’ = – e / (1 + e) where e is the overall relative error of the parameter estimate given in Table 1. Feasibility Feasibility was evaluated qualitatively. Results are presented under Accuracy, Efficiency and Robustness. Feasibility is considered in the Discussion. Accuracy The values for relative standard error RSE and relative error of estimate RE, as criteria of accuracy, are presented in Table 2 and Fig. 9. In all methods, with increasing slice thickness, both parameters decrease and the estimates approach the true value NV. Table 2. Results of computer simulation presenting relative standard error (RSE), relative error (RE), number of areas and number of counted particles required for RSE = 5%, and the corresponding number of particles per area for various counting methods and slice thicknesses (t). Counting methods Slice thickness t [µm] Slices Wicksell Thin Thick Differential counting Physical section t2 = 5 µm 0 23.7 0.5 18.3 1 14.8 2.5 12.8 5 10.1 10 7.0 25 4.3 50 2.6 100 2.0 Physical section t2 = 3 µm Optical section t2 = h = 1 µm 10 23.8 25 13.7 50 5.4 100 3.4 10 22.4 25 8.7 50 4.7 100 2.5 10 17.0 25 7.8 50 3.1 100 1.9 Disector Physical Optical 0.5 62.3 1 42.2 2.5 32.7 5 14.5 10 13.1 25 4.9 50 3.7 100 2.2 Number of particles per area -1.0 225.2 5.3 134.3 4.4 88.1 -2.9 66.3 1.3 41.0 0.7 20.1 -0.1 7.6 0.4 2.8 1.1 1.7 583 2.6 406 3.0 385 4.4 251 3.8 217 5.3 158 7.9 119 15.7 80 28.9 93 55.5 12.4 226.4 -1.7 74.8 -0.8 11.7 1.4 4.7 -5.0 201.0 2.8 30.5 -0.7 8.7 0.8 2.6 -8.2 115.6 2.2 24.0 -1.6 3.9 0.0 1.4 2939 13.0 1573 21.0 391 33.6 283 60.5 2500 12.4 625 20.5 286 32.8 155 59.7 1237 10.7 443 18.4 123 31.6 83 58.2 3.2 1554.9 -4.4 711.1 -9.8 426.4 0.6 83.6 -6.1 68.7 0.7 9.5 0.4 5.5 1.4 2.0 513 0.3 356 0.5 503 1.2 220 2.6 338 4.9 125 13.2 145 26.3 104 53.1 20 Image Anal Stereol 2001;20:15-25 010 25 50 Slice thickness t [µm] a +10% -1> Nv -10% 100 b --*- 010 25 50 Slice thickness t [µm] +10% -4> Nv -10% 100 C -.:$: +10% -€> Nv -10% 010 25 50 100 Slice thickness t [µm] d iL +10% Nv -10% 135 Thin slice thickness t` [µm] Fig. 9. Accuracy of different particles counting methods estimated by the relative number of particles indicated by 95% confidence intervals (vertical bars) and averages (?) at various slice thicknesses t for a classical thin and thick slices, b physical and optical disector, c modified differential counting (t2 = h), and d differential counting with t1 = 100 µm and different t2 (1, 3, 5 µm). Efficiency The values for the required number of areas nreq and required number of counted particles Creq in order to obtain RSE = 5% as criteria for efficiency a are presented in Table 2 and Fig. 10. In general, with increasing slice thickness both parameters decrease and converge to similar values regardless of the counting method. b Slice ¦-¦-¦ Disector 0--0--0 °-. Differential counting: ''¦-. t' = 1u.m A--A--A t'=3u.m ¦-¦-¦ """ °. t'=5u.m T-T-T m^, """ ¦------- -m^^^ 'o..... A.X^VVV .....O, "¦v. \ "ˇ. ~~~~~~H / ¦ Slice ¦-¦-¦ Disector o-- 0-- 0 Differential I - counting: w t =1u.m A -A -A \ "¦-. t'=3u.m ¦ -¦-¦ t =5u.m T-T-T A \ w \ v-v \ o., ..O, V \ \ ¦—^8-