Volume estimation of multicellular colon carcinoma spheroids using cavalieri's principle

Path. Res. Pract. 191, 1192-1197 (1995)

Volume Estimation of Multicellular Colon Carcinoma Spheroids Using Cavalieri's Principle~1- ~I- ~IJ. Bauer1,

w.

Gries 2 and F. A. Bahmer3

1Universitats-Hautklinik, Homburg/Saar, 2Fraunhofer Institut fOr Medizintechnik, St. Ingbert, 3Dermatologische Klinik, ZKH, Bremen, Germany

SUMMARY Multicellular tumour-spheroids are regarded as suitable models for cancer research, similar to avascular tumour parts. As a size parameter of the spheroids, usually their maximum diameter is used, estimated on a section presumed to be equatorial or near equatorial. Estimation of the volume of spheroids is of interest for the detection of subtle changes in different kinds of investigations. Since spheroids are often not truly spherical, and because model-based methods for volume determination may be biased, Cavalieri's principle, rediscovered recently for stereo logy, was used to determine the volume of the spheroids. Here we report the results of the volume estimation of colon carcinoma spheroids, together with an outline of the basic stereological principle and formulas used. The spheroids investigated had a volume between 1.4 and 92.3 mm3 (mean 33.9). The volume fraction of necrotic to viable tissue cells was between 0.6:1 and 2.2:1. The coefficient of error (CE) was remarkably low with 3.7% for the volumes. Both inter- and intraobserver-variability were extremely low with correlation coefficients (r2) of 99%. Thus, the high precision of the stereological method, combined with a low workload, make it ideally suitable for routine volume estimation.

Introduction

Tumour cell aggregates, derived from tumour cell explants in a culture medium at continuous movement, known as multicellular tumour spheroids, have been considered as models for early avascular tumour parts or for micrometastases, based on their histological structure and their ability to synthesize extracellular matrix substances14, 16. Furthermore, spheroids retain

* Dedicated to Prof. Dr. H. Zaun (Homburg) on the occasion of his 65th birthday. **Presented in part at the GZG-workshop "Biologische Grundlagen und Anwendungen dreidirnensionaler Zellsysterne", SchontallJagst, Feb. 1994. 0344-0338/95/0191-1192$3.50/0

a certain degree of structural and functional differentiation of the primary tumor with spatial variation in cellular proliferation, reflected by cellular heterogeneity14. Three phases of spheroid growth can be distinguished, leading to spheroids with a maximum diameter of 1-4 mm, depending on the tumor type and culture conditions2, 4,10. At the final stage, the spheroids consist of an outer shell of viable cells and a core of necrotic cells4. These two compartements can easily be distinguished in most spheroid types 14 . Multicellular spheroids are used to study biological, pathophysiological and therapeutical effects, especially those of cytotoxic drugs. In these studies, the maximum diameter of a spheroid from a section regarded as equatorial or near equatorial is used as size parameter. How© 1995 by Gustav Fischer Verlag, Stuttgart

Volume Estimation of Multicellular· 1193

ever, if subtle volume changes have to be detected, the diameter of the sphere is not an adequate measure because the volume of a sphere changes with the third power of its radius. The lack of a simple volume determination has led to the conclusion "that volume regrowth or growth delay is not a very accurate measure for the effectiveness of tumor treatment, even though it may be of high practical relevance in clinical oncology"14. However, with the ancient principle of Cavalieri, it is possible to estimate unbiasedly and efficiently (in the sense of free of systematic error and ease of use, respectively) the volume of a body irrespective of its shapeS. This independence from shape is an advantage because spheroids are not always spherical (Fig. 1). Here we report the results of the volume estimation of colon carcinoma spheroids, together with an outline of the basic stereological principle and formulas used, together with some hints regarding the precision of the method. Results

a) Volume of the Spheroids The volumes of the individual spheroids showed a high variation, ranging from 1.4 to 92.3 mm 3 (mean 33.9; SD 25.3). From the total volume, the inner, necrotic part of the spheroids occupied between 0.8 to 63.0 mm 3 (mean 19.9; SD 17.7), whereas the outer shell of vital cells occupied 0.6 to 29.3 mm 3 (mean 13.9; SD 8.5). The volume of the necrotic cells exceeded that of the vital cells in 6 spheroids. The volume of the necrotic cells was about the same as that of the vital cells in only one spheroid, and in one spheroid, the

vital cell volume exceeded. The values are given Table 1.

In

b) Coefficient of Error The CErn! (V) for the volume determination from all slabs was low with values between 1.26% and 2.65%, with a mean total CE of 1.92% (Table 2). The estimation of CE for the subset of the slabs with odd numbers yielded values between 1.53% and 6.10%. Here, the total CE was 3.7% (Table 2). Table 3 shows the number of points of the Merz-grid counted on the spheroid sections with their respective CEo The last column in Table 3 gives the minimal number of points on all slabs necessary to achieve a CE of 2.5 %. Figs. 3 and 4 show the inter- and intra-observer variability, which was very low with less than 1%. Discussion By the use of the simple principle of Cavalieri, adopted for stereological purposes, we have estimated the volume of multicellular colon carcinoma spheroids. The spheroids are cut into slabs of known thickness. The area of the slabs, multiplied by their thickness, yields an estimate of their volume. It is not necessary to know the exact thickness of each slab, as for practical purposes their average thickness suffices 6 • A typical phenomenon of stereo logical volume determination is confirmed: the high variability of the volume of the spheroids, with rather low error variances on the level of the slabs of the individual spheroid used for area determination. This phenomenon is not due to estimation errors but due to the fact that the total var-

Table 1. Volume and volume fractions of the spheroids (Vvitl = vital cells; Vnee = necrotic cells; Veot ! = total volume 1st determination; Vlot2 = total volume 2nd determination; Vtot3 = total volume 2nd investigator) Volume [mm 3] Vnec1

Vvirl

Veol !

VlOt2

Vlol3

VnecrlNvill

Nr.1

8.4

8.4

16.8

15.5

15.8

1.0

Nr.2

63.0

29.3

92.3

90.5

91.0

2.2

Nr.3

19.9

10.7

30.6

31.4

31.4

1.9

Nr.4

18.5

13.3

31.8

32.5

32.7

1.4

Nr.5

24.4

21.6

46.0

45.1

45.5

1.1

Nr.6

12.4

19.3

31.7

26.5

30.2

0.6

Nr.7

12.1

7.9

20.0

20.0

18.9

1.5

Nr.8

0.8

0.6

1.4

1.3

1.3

1.3

Mean

19.9

13.9

33.9

32.9

33.4

(J

17.7

8.5

25.3

24.9

26.8

Spheroid

Inter-Observer Variability (Colon Carcinoma Spheroids) V 100 0

I 80

v e

R =0,99

60

40

20

40

20

60

80

Volume Determination 1st Investigator (mm 3) A

Fig. 3. Interobserver variability for volume determination of colon carcinoma spheroids. Intra-Observer Variability

Fig. 1. 3-D reconstruction of spheroid number 1 (courtesy of Dr. El Gammel, Bochum).

iance is a combination of the variances obtained on each sampling level; here the level of the slabs and the level of the spheroids. It has been demonstrated that between 53% and 78% of the total variance is due to the biological variance, i.e. between spheroids 3, 8, 9. Thus, lowering the error variance requires more objects and not a higher precision of "measuring," summarized by Gundersen and Osterby into the formula "Do more less well"8. The method described allows the unbiased volume determination of spheroids. This, in turn, allows the enumeration of the total number of cells (numerical density) contained in a spheroid, e.g. by the "disector" method 15 . This might also be of use for determining the numerical density of certain cell subsets such as those proliferating.

IColon Carcinoma Spheroids; n-8) V o

100,----------------------0--1

I

80

e t

60

40

•

20I I

0 0

20

40

60

80

100

Volume (mm 3) 2nd determination A

Fig. 4. Intraobserver variability for volume determination of colon carcinoma spheroids.

Fig. 2. Near equatorial 3 11m section of a spheroid with superimposed Merz-grid (final magnification x 100).

Volume Estimation of Multicellular· 1195 Table 2. CE's for the total number of slabs (ml) and the subset of every 2nd slab (ml) investigated

is very important that the starting point for the slabs in the interval of has to be random 7,12. Characteristic for stereological methods is the low inter- and intra-observer variability. We have chosen not to estimate the sections directly in the microscope, but to use colour slides which facilitates the same investigation to be performed again or by another investigator. In our opinion, the high reproducibility makes the method suitable for routine work. We feel that the accuracy of stereological methods provide for quantification which might substantially contribute to the analysis of structure-function relationships, as pointed out convincingly by WeibeP7 more than ten years ago.

Spheroid

ml

CE(V) [%]

ml

CE(V) [%]

Nr.1

21

1.73

11

2.41

Nr.2

32

1.26

16

1.53

Nr.3

22

1.75

11

2.44

Nr.4

21

1.56

11

2.85

Nr.5

32

1.50

16

2.89

Nr.6

33

2.12

17

6.10

Nr.7

21

2.36

11

3.47

Nr.8

10

2.65

5

5.44

Material and Methods

Mean

24

1.92

3.70

a) Spheroids and Preparation

12.25

Table 3. Number of points counted on each spheroid (Sum P), corresponding CE, as well as optimal number of points (P opt)

a =2,5)

Spheroid

SumP

CE(A) [%]

P (

Nr.1

159

1.30

67

Nr.2

213

0.96

60

Nr.3

124

1.31

53

Nr.4

134

1.31

57

Nr.5

181

1.33

78

Nr.6

112

2.06

87

Nr.7

171

0.97

49

Nr.8

48

2.17

40

The Cavalieri-method of volume determination is both very simple and rapid. Only about 100 points or intersections, according to our investigations even somewhat less, need to be counted for a remarkably high precision. In addition, no expensive automatic or semiautomatic image analyzer is required. Our results also confirm that only about 10 to 15 slabs are needed for a rather precise volume determination 7. With an average of 12.3 slabs, we obtained a total error variance of only 3.7%. If the spheroid number 8 with only 5 slabs, and spheroid number 6 with a defect are omitted, the total variance drops to 2.7%. In contrast, if 24 slabs are used for estimation, the total variance drops only an additional 0.8%. Thus, it is not necessary to investigate more than 10 to 15 slabs. Instead of the Merz-grid, any other square lattice grid could be used, making the estimation procedure even more simple. From the investigation of the subsets of odd and even numbered slabs, it is clear that the CE's are influenced by the position of the first slab. This emphasizes that it

For quantification, 8 spheroids from the colon carcinoma line WIDR, kindly supplied by H. Degani (Weizman Institute, Tel Aviv, Israel), were used. The spheroids were cultivated in Dulbecco's Minimal Eagle's Medium (DMEM, Biochrom KG, Berlin, Germany). After fixation in 3% Glutaraldehyde, cacodylate buffered solution at room temperature, they were dehydrated in ethanol. Embedding of all spheroids was performed in acrylate plastic (Technovit 7100, Kulzer GmbH, Wehrheim, Germany). Since all spheroids are processed in the same manner, shrinkage is not a problem. Measured on sections regarded as equatorial or near equatorial, the spheroids had a diameter between 0.6 and 2.9 mm. Most of them were not truly spherical visually; some appeared to be rather complex bodies (Fig. 1). One cross section near the equator is shown in Fig. 2, superimposed by a Merz-grid 13 . Sections were cut on a microtome (Reichert-Jung, Heidelberg, Germany) with an average thickness of 3 /lm at an average distance of 27 /lm, and stained with hematoxylin-eosin for 120 minutes at room temperature. Thus, the average thickness t of the slabs was 30 (27 + 3) /lm. Measurements with an interferometer-based metering device in a previous study had shown that the actual section thickness did not differ more than 20% of the true thickness. For stereo logical determinations, the average section thickness is sufficient for the estimation procedure 1.

b) Stereological Measurements The specimens were photographed using a macrophoto lens on an Olympus SLR OM3 camera at a magnification of 2 fold on an ordinary Agfachrome slide film (Agfa, Leverkusen, Germany). The slides were projected with a slide projector to yield a final magnification of 28.4 directly onto the grids on plain white paper. This method is convenient with bright pictures allowing unanimous identification of the structures to be measured. The rather abrupt transition from the outer rim of stained viable cells, to the inner unstained necrotic cells is clearly visible even in H&E stained sections. The use of slides allows repeated evaluation or evaluation by different investigators, thus facilitating studies on inter- and intra-observer variability. Based on the principle of the Italian mathematician Cavalieri, it can be shown that the volume of a three-dimensional object can be determined if the sum of the areas of equidistant parallel slabs is multiplied by the distance t between the slabs (formula 1, appendix), provided that the first section is placed

1196 .

J.

Bauer, W. Gries and F. A. Bahmer

into the object between . This principle is completely general, valid for three-dimensional bodies regardless of their shape, and not influenced by the direction of cutting the object, which can be selected at will!, 5, 6. The area of the viable and necrotic compartments of the spheroids was determined simply by superimposing a Merz-Grid 13 onto the spheroid specimens and counting the cross-points on the respective fractions (formula 2, appendix). Combined with formula 1, an estimate of the spheroid volume was obtained (formula 3, appendix).

1 A(spb) = a(p) . M' EP

a(p) area associated with each test point of the test sys-

tem, M linear factor of magnification and P number of points hitting a spheroid in a section. Formula 3: Estimate of the spheroid volume by combining Formula 1 and Formula 2:

c) Estimation of Error Systematic sampling designs which are often used in stereology do not allow the estimation of error variance in the usual way7. Instead, other methods have been proposed to determine the coefficient of error (CE), e.g. by the quadratic approximation to the covariogram7,11 (formula 4, appendix). Total CE, based on all cases, was estimated using formula 5 (appendix). To learn something on the magnitude of the CE of the volume estimation in relation to the number of slabs, the CE was estimated in a subset of cases consisting of every second slab. Another source of error stems from the point counting procedure for area determination. Clearly, the precision of the method depends on the number of points falling onto the structure, as well as on the complexity of the structure. The CE for the area determination might then be calculated according to formula 6 (appendix). An easy way, which avoids any calculation, is the use of the nomogram given by Gundersen and Jensen 7. To assess the reproducibility of the method, the total volume of the spheroids was estimated twice by the same investigator, as well as by a student not involved in the study.

1 V = a(p) . - . t· EP M

Formula 4: Estimation of the Coefficient of Error (CE) for volume determination, using the quadratic ap-

proximation to the covariogram?' 11, 12:

CE(V) = J(3A

'LPi

with m

A=

d) Calculations

L

m-l

Pi . Pi

B=

L

Formula 2: Area determination A(spb) of the spheroid section with a Merz grid 13 • Here,

L

Pi . P i+2

i=1

Formula 5: Estimation of the total CE of k spher-

!. LCE2(V) k

=

k

i=1

Formula 6: CE for area determination 7 :

CE(a) = 0.269·

Appendix

V= t· EA

C=

oids?:

Acknowledgements

The following formulas were used (see text): Formula 1: Volume determination, were V is the volume of the spheroid, t the average thickness of the slabs and La the combined area of the slabs 6 :

m-2

Pi . P i+ 1

i=1

CE(V)tot Thanks are due to Dr. H. Leinebach (Saarbriicken) for cultivation of the spheroids, A. Kerber (Homburg) for skillful technical assistance, Dr. El Gammal (Bochum) for 3D-reconstruction of one spheroid and Prof. T. Mattfeldt (Ulm) for valuable suggestions.

4B)j12

i=1

i=1

Simple calculations were done on a hand-held calculator, calculations based on a formula were done in a spreadsheet program (Lotus 1-2-3, Lotus, Munich, Germany) on a personal computer equipped with a 486 processor. Regression analysis was performed by the appropriate module in SPSSWIN 5.0.1 (SPSS Inc., Chicago, Ill., USA).

+: -

~j(EP)i Yv'A(Siih)

with 7r

B = a(l) . - . EI 2

with A(spb) total area of structure; a(l) area of the grid associated with test line length, LP total number of points hitting a spheroid, and LI number of intersections of the grid lines with the boundaries of the structure.

Volume Estimation of Multicellular . 1197

References 1 Cruz-Orive LM, Weibel ER (1990) Recent stereological methods for cell biology: a brief survey. Am J Physiol 258 (Lung Cell Mol Physiol 2): L148-L156 2 Durand RE (1975) Cure, regression and cell survival: a comparison of common radiobiological endpoints using an in vitro tumour model. Br J Radiol 48: 556-571 3 Ebbesson SOE, Tang DB (1967) A comparison of sampling procedures in a structured cell population. Proceedings of the Second International Congress for Stereologie. Ed Elias H, Springer, Berlin 4 Folkman J, Hochberg M (1973) Self-regulation of growth in three dimensions. J Exp Med 138: 745 - 753 5 Gundersen HJG (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, J Microscopy 143: 3-45 6 Gundersen HJG, Bendtsen L, Korbo N et al. (1988) Some new, simple and efficient stereological methods and their use in fathological research and diagnosis. APMIS 96: 379 -394 Gundersen HJG, Jensen EB (1987) The efficiency of systematic sampling in stereo logy and its prediction. J Microscopy 147 (3): 229-263 8 Gundersen HJG, Osterby R (1981) Optimizing sampling efficiency of stereological studies in biology: or 'Do more less well'. J Microscopy 121: 65 -73

9 Gupta M, Mayhew TM, Bedi KS et al. (1983) Inter-animal variation and its influence on the overall precision of morphometric estimates based on nested sampling designs. J Microscopy 131: 147-154 10 Landry J, Freyer J, Sutherland RM (1982) A model for the growth of multicellular spheroids. Cell Tiss Kinet 15: 585-594 11 Mattfeldt T (1989) The accuracy of one-dimensional systematic sampling. J Microscopy 153 (3): 301-313 12 Mattfeldt T (1990) Stereologische Methoden in der Pathologie, Thieme, Stuttgart 13 Merz WA (1967) Die Streckenmessung an gerichteten Strukturen im Mikroskop und ihre Anwendung zur Bestimmung von Oberflachen-Volumen-Relationen im Knochengewebe. Mikroskopie 22: 132 -142 14 Miiller-Klieser W (1987) Multicellular spheroids - A review on cellular aggregates in cancer research. J Cancer Res Clin Oncol113: 101-122 15 Sterio DC (1984) The unbiased estimation of number and sizes of arbitrary particles using the disector. J Microse 134: 127-136 16 Sutherland RM, McCredieJA, Inch WR (1971) Growth of multicell spheroids in tissue culture as a model of nodular carcinomas. J Nat Cancer Inst 46: 113-120 17 Weibel ER (1981) Stereological Methods in Cell Biology: Where are we - where are we going? J Histochem Cytochern 29: 1043-1052

Received January 25, 1995 . Accepted in revised form September 12, 1995

Key words: Cavalieri principle - Colon carcinoma spheroids - Morphometry - Stereology - Volume Prof. Dr. med. F. Bahmer, Oirektor der Hautklinik, Zentralkrankenhaus St.-Jiirgen-Strage, 0-28205 Bremen Germany Phone 0421-497-5321,Fax-3316 ' ,