Stereology for pores in wheat bread: statistical analyses for the Boolean model by serial sections

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1 Journal of Microscopy, Vol. 162, Pr 2, May 1991, pp ADONIS E Received 18 December 1989; revised 28 September 1990; accepted 5 October 1990 Stereology for pores in wheat bread: statistical analyses for the Boolean model by serial sections by UTE BINDRICH and DIETRICH STOYAN, Znstitut fur Lebensrnitteltechnik, Technische Universitat Dresden, Dresden, 8027 and Fachbereich Mathernatik, Bergakadernie Freiberg, Freiberg, 9200, Germany KEY w ORDS. Boolean model, bread, pair correlation function, pores, serial sections, specific convexity number, stereology. SUMMARY A simple approximative stereological method is proposed for obtaining the fundamental parameters of a three-dimensional Boolean model. It uses serial sections for the determination of N;, the specific convexity number. This procedure was applied to two different examinations of bread, which can be very well described by Boolean models. INTRODUCTION This paper has two purposes. First we will show how the pore structure in bread can be investigated stereologically. Secondly we will point out that the serial section method is also of use in the stereological analysis of the Boolean model. Until now, the geometrical structure of bread (excluding the crust) has been characterized only by very simple parameters such as the specific weight or volume density V, of pores (Wassermann, 1979; Maleki et al., 1980; Stephan, 1980). It is clear that neither V, nor the specific weight will characterize the variability of the bread structure. For example, the same value of V, is possible for quite different structures, e.g. structures with either a few big or many small pores. Visual classification (Dallmann, 1969) relies on subjective factors. It is desirable to evaluate results of process and product developments, and therefore it seems necessary to establish quantitative methods for analysis of bread structures. Many of the qualitative properties of bread (such as softness, elasticity and compressibility) are determined by the state of the pore walls (PW; the solid phase of bread), the water content of the PW, and the structure of the pore system. It is known that the moisture content and the state of the PW are closely related, controlled by the recipe and technology of the manufacturing process. If the properties of the raw material are kept constant, the influence of the PW properties is practically constant because of the uniform distribution of the ingredients. Therefore, because of the complicated and multiple controllable processes of structure formation, the structure of the pores (or the bread) is the dominant characteristic which has to be studied. A well-known model in stereological studies is the Boolean model (see Miles, 1976; Weibel, 1980, pp ; Stoyan et al., 1987, pp ). As we will show, this ( The Royal Microscopical Society 23 1

2 232 U. Bindrich and D. Stoyan model is acceptable as a stochastic model for the system of pores in bread. Therefore, the union of all pores can be approximated by a union of spheres of independent random diameters with centres completely randomly scattered in space. The spheres can be interpreted as primary pores, resulting from a growth process starting with pore germs. If using only single planar or thin sections, the determination of parameters of a Boolean model is a difficult problem, including the solution of ill-conditioned equations and ad hoc assumptions. As we will show, by the means of serial sections, it is very easy to estimate the specific convexity number N,+ directly, without stereological tricks. Once N,+ is determined, the other parameters can be obtained by stable stereological procedures using planar sections. MATERIALS AND METHODS Two samples of wheat bread were produced from 100 parts wheat flour, 1.4 parts dry yeast, 1.7 parts salt and water. The structure of the bread was modified by the effective viscosity of the dough: batch A was a soft dough and batch B was a firm dough. The specific weights are g/ml (sample A) and g/ml (sample B); sample A is of better quality. Only two samples of bread were studied because our aim was to describe the application of stereological methods and not to establish a theory of bread structures. The general requirements of preparation for quantitative structure analysis impose the following particular demands on bread preparations: (i) fixing the state of the structure, (ii) increasing mechanical stability and guaranteeing storage stability, (iii) production of relatively smooth planar sections, and (iv) clear discrimination between pores and pore walls in the section planes. Here it should be noted that bread is mechanically and biochemically unstable. These demands are satisfied by the following preparative steps: (1) minimization of water activity by freeze drying, (2) displacement of the gas phase in the material by paraffin, (3) production of planar sections by a microtome, (4) colouring the PW in the section plane with a J-K-iodide solution. Steps (3) and (4) have to be repeated in order to produce serial sections. Photographs of the planar sections were used for evaluation; the contrast could be increased by UV light. Figure 1 shows two typical structures corresponding to samples from batches A and B. Fig. 1. Planar sections through samples A and B. Black = pore walls, white = pores. Magnification = c. 10.

3 & Stereology of the pore structure in bread 233 SOME FORMULAE FOR THE BOOLEAN MODEL Model description The Boolean model is a spatial random set constructed as follows. There is a spatial system of completely randomly distributed points ( germs ; in our case centres of primary pores) of density 1. In mathematical terms the germs form a Poisson point process of intensity E. Every point is the centre of a random convex and isotropic particle (in our case a pore), where all particles are independent. It is assumed that particles are independent, and thus overlapping of particles is possible such that the union of all particles may be a topologically complicated structure. The planar section of a Boolean model can also be described as a Boolean model. This offers the possibility of (indirectly) testing the model assumption by using planar sections. The fundamental parameters of the elements of a Boolean model are the injensity and the three particle characteristics: V (mean volume); S (mean surface area); b (mean average breadth). They are connected with parameters available from planar sections, i.e. area fraction, specific boundary length, and planar specific convexity number, A, = 1 - exp (- E.V), (1) 7I L - - i.s exp (- i.b), A-4 NAf = lbexp (-2.V). (3) The planar convexity number NA+ can be explained as follows. Consider the profiles of particles in the section plane, which is equipped with an (x, y)-coordinate system. They are all convex and thus each has an upper right point on its boundary. (The upper right point is the point with the maximum y-coordinate; if there is no single point with the maximum y-coordinate, then from these the point is chosen which has the maximum x-coordinate.) A certain fraction of the upper right points lies in the particle profiles (and generally cannot be observed). The intensity of the point process of upper right points which are not overlapped by other particles is denoted by NAf. In other words, N i is the intensity of the point process of upper right points of the union of all particles. It can be determined without identifying the original particles. Parameter estimation If only A,, LA and NAf are available, then the problem becomes the stereological determination of four unknowns: E., V, S, and b; from the three known numbers A,, LA and NA. A trick of classical stereology is to assume that the particles are spheres with normally or lognormally distributed diameters. The model then depends on three parameters only. Let p and o2 be the distribution parameters of radii in the densitiesf(x), f(x) = ~ 1 OX 1 (lognormal).

4 234 U. Bindrich and D. Stoyan Then the following equations lead to 1, p and a : or the lognormal equivalents: These equations are simple consequences of Eqs. (l), (2), and (3), and of the formulae for the first three moments of the normal and lognormal distribution. Clearly, the normal case is useful only if p % a ; otherwise negative radii would appear. The accuracy of the estimates obtained by means of Eqs. (6)-( 11) depends on the accuracy of A,, LA and NL. Their distributional properties are not yet known, and thus it is impossible to give formulae for standard errors. It is very often quite difficult to check the distributional assumptions made above and thus this procedure is not exact. A stable stereological estimation method can be established if the spatial specific convexity number NV+ is also available. The quantity NV+ can be described analogously to NA+ in 3-D space. If the particles are spheres, then their poles with maximal z-coordinates play the role of upper right points. Thus, NV+ = 1 exp (- 1P). (12) The four quantities A,, LA, NL, NV+ can be used to calculate 1, P, s and Fusing Eqs. (11, (21, (31, and (12). Model test The distribution of random sets in general and of the Boolean model in particular may be characterized by so-called contact functions (see Stoyan et al., 1987, pp. 74 and 80). We shall use contact functions for the planar sections, namely the linear and quadratic contact functions HI and H,. Note that H,(r) is the probability that an arbitrarily placed square of side length r intersects the Boolean model, provided that the centre of the square lies outside of the union of the particles (pores); HI is an analogous quantity when the square is replaced by a segment of length r. The functions H, and H, have the form and Hl(r) = 1 - exp (-ar) r20, (13) H,(r) = 1 - exp (-2ar - b?) r >O (14)

5 with and Stereology of the pore structure in bread 235. T a = ).a- 71 b = j.a, (16) where Tis the mean perimeter of the particle and j., the corresponding intensity of germs of the planar Boolean model. Thus, and -In [l - H,(r)] = ar (17) -In [l - H,(r)]/r = 2a + br. MEASUREMENT AND RESULTS Basic statistics Initially the planar characteristics AA, LA and NA+ were determined. AA and L A were obtained from samples of total size c. 7 x 10 mm by means of a Quantimet QTM 720. The quantity NA+ was obtained manually, because the QTM could not give an accurate measurement. The results are shown in Table 1. Furthermore, the QTM 720 was used to determine the linear and quadratic contact functions Hi and H, for the same probes. Figure 2 shows these functions transformed according to Eqs. (17) and (18). The deviation from a linear form is small for all four functions, which shows that the assumption of a Boolean model for the set of pores seems to be acceptable. Parameter estimation is the spherical case Using Eqs. (6)-( 11) we tried to estimate I., p and 6 using the values of AA, LA and NA+ obtained above. In the normal case the radicand in (6) is negative; thus the normal model is not applicable. Table 2 shows the results obtained in the lognormal case. Discussion and model tests These values seem reasonable, but as the assumptions (spheres, with lognormal diameters) are not proven, they may not be accurate. Further measurements were carried out below. Using only single planar sections we determined the pair correlation functions of the centres of pore profiles (see Hanisch & Stoyan, 1980; Hanisch et al., 1985; Stoyan, 1987, for an explanation of the pair correlation function of point patterns). When the pore profiles were not circles, the centres were defined as the intersection point of the longest and shortest axes of the pores. Occasionally, for pores very close together, it was difficult to find the centres and to distinguish between pores. By the procedure described in Stoyan (1987) we obtained the pair correlation function in Fig. 3. For samples A and B this is very close to the pair correlation Table 1. Planar characteristics of bread samules A and B. Sample A Sample B , (18)

6 U. Bindrich and D. Stoyan /I Fig. 2. Averaged linear and quadratic contact distributions H, and Hq for samples A and B. (For greater values of r as shown here, the estimated H,(r) and Hq(r) are equal to 1 or very close to 1.) function of a Poisson process, where the pair correlation function is a constant, 1. Fluctuations in the pair correlation function around the value 1 for r > 0.3mm were treated as statistical variation. The small values for r < 0.3mm show that in the patterns there is a weak inhibition which results from the need for pore space. Note that there are several possibilities for deviation from a Boolean model. The point process of germs may not be a Poisson process, but the pores are still independent; or the centre process is Poisson, but the pores are dependent. In both cases the point process of section circle centres is not a Poisson process. An example of the latter type is the Stienen model (see Stoyan, 1990). The pair correlation function of the point process of centres of section circles for this model was calculated by Stoyan. Its form shows a similarity to the curves in Fig. 3. Note also that there is as yet no mathematical theory which estimates the accuracy of pair corrrelation functions. Experiments showed that reasonable results are obtained provided the sample is large enough. Table 2. Parameters estimated in the lognormal case. Sample A Sample B

7 Stereology of the pore structure in bread 237 SW 1!XX7 1SGQ (rm) Fig. 3. Pair correlation function for the pore centres for samples A and B. Parameter estimation by the disector Serial sections 20 pm thick were measured. This thickness was chosen because the minimal pore diameter was estimated to be c. 30 pm. Using the disector principle (see Gundersen, 1986), we estimated the density N,+ of upper right pore ends in space. Note that originally the disector was used for the determination of the particle density N,+ for systems of non-overlapping particles. In this case N, = N,+, if the particles are convex. (0) t / i C Fig. 4. The counting rule of the disector for a Boolean model with spherical particles. (a) Classical disector, for isolated particles. (b) Disector for Boolean model case. c = counting plane, t = test or look-up plane, (0) = particles (or right upper points) counted. A non-overlapping right upper point of the Boolean model may be counted if it lies between the test and counting plane and its planar projection lies in the sampling window in the counting plane. Figure 5 shows which points are counted and which are not.

8 238 U. Bindrich and D. Stoyan Lo) Fig. 5. (a) These right upper points cannot be distinguished by the disector (because on the counting plane only one object is seen, the union of the planar sections of the two particles). (b) These points can be distinguished. Table 3. Estimates of the Boolean model parameters. Sample A Sample B L (mm-3) t7 (nun3) ;(m2) b (mm) It is obvious that the disector also yields a lower bound for the number of upper right points in a Boolean model, as shown schematically in Figs. 4 and 5. If n subsequent parallel section planes of distance d are used, an estimator of N, is Here N, is the number of upper right points between section planes i and i + 1, and A is the area of the window in each plane. By this method we obtained the values NV+ = 13-2mm-3 (sample A) and NV+ = 10.9mm-3 (sample B). These and A,, LA and NL lead to the estimates of the Boolean model parameters shown in Table 3. DISCUSSION The results shown in Table 3 are very close to those for spheres with lognormal diameters. This was an unexpected result. This seems to indicate the lognormal assumption is reasonable, and confirms the belief that the Boolean model parameters are correctly estimated. Another possible interpretation is a robustness property of the Boolean model which ensures that calculations by means of the formulae for the Boolean model lead to quite realistic results even if the structure deviates from a Boolean model. The effect of underestimating N,+ is probably only small. (If the true Nv is x times greater than the value obtained, then 1 has to be multiplied by x and b', s and by 1 /x.) Comparison of both samples shows significant differences in their structure which perhaps could not be expected using the information on the specific weights only.

9 Stereology of the pore structure in bread 239 Sample A has a greater pore volume fraction V, and much more pore germs (% z for sample A and A z 23mm-3 for sample B), but the pores in sample A are slightly smaller than in sample B. This may be the reason for its improved taste and quality. ACKNOWLEDGMENT We would like to thank the referees for the suggestions and criticisms which have improved the paper. REFERENCES Dallmann, H. (1969) Porenrabelle. Moritz Schafer, Detrnold. Gundersen, H.J.G. (1986) Stereology of arbitrary particles. A review of unbiased number and size estimators and the presentation of some new ones, in memory of William R. Thompson. 3. Microsc. 143, Hanisch, K.-H., Konig, D. & Stoyan, D. (1985) The pair correlation function for point and fibre systems and its stereological determination by planar sections. 3. Microsc. 140, Hanisch, K.-H. & Stoyan, D. (1980) Stereological estimation of the radial distribution function of centres of spheres. 3. Microsc. 122, Maleki, M., Hoseney, R.C. & Mattern, P.J. (1980) Effects of loaf volume, moisture content and protein quality on the softness and staling rate of bread. Cereal Chem. 57, Miles, R.E. (1976) Estimating aggregate and overall characteristics from thick sections by transmission microscopy. 3. Microsc. 107, Stephan, H. (1980) Die Beduetung der Weizenbackmittel fur die Gebackherstellung. Brot Backwaren, 28, Stoyan, D. (1987) Statistical analysis of spatial point processes: a soft-core model and cross-correlations of marks. Biom , Stoyan, D. (1990) Stereological formulae for a random system of non-intersecting spheres. Statistics, 21, Stoyan, D., Kendall, W.S. & Mecke, J. (1987) Stochasric Geometry and Its Applications. J. Wiley and Sons, Chichester. Wassermann, L. (1979) Relation between structure and rheological properties of bread crumb. Food Texture and Rheology (ed. by P. Sherman), pp Academic Press, London. Weibel, E. (1980) Stereological Methods, Vol. 2. Academic Press, London.