Distribution of different fibre types in human skeletal muscles

Journal oJ the Neurological Sciences, 1983, 61:301-314

301

Elsevier

DISTRIBUTION OF DIFFERENT FIBRE TYPES IN HUMAN SKELETAL MUSCLES A Statistical and Computational Model for the Study of Fibre Type Grouping and Early Diagnosis of Skeletal Muscle Fibre Denervation and Reinnervation JAN LEXELL I, DAVID DOWNHAM 2 and MICHAEL SJ()STROM 3

Departments of IAnatomy and 3 Neurology, University of Ume& (Sweden) and Department of 2Statistics and Computational Mathematic University of Liverpool (Great Britain) (Received 9 May, 1983) (Accepted 19 May, 1983)

SUMMARY

To define fibre type grouping in terms of random and non-random arrangements of the two fibre types, type 1 (ST) and type 2 (FT), we adopted the measure of counting the number of "enclosed fibres". The statistical properties of the number of enclosed fibres, and the number and size of groups of enclosed fibres were studied in computersimulated muscle cross-sections, using a model based upon hexagonal-shaped fibres. The effects on the results of differences in the sizes of the muscle fibres were considered. The applicability of the model, and the derived results and methods of analysis were tested on 10 samples from a cross-section of a whole human muscle. The results show that the model can be applied to various shapes and sizes of muscle samples and various sizes of muscle fibres. The number of enclosed fibres within a muscle sample is the best of the three measures of non-randomness considered. A test is also described for assessing whether or not the observed number of enclosed fibres is random at a given significance level.

Key words: Diagnosis, computer a s s i s t e d - H i s t o c y t o c h e m i s t r y - M i c r o c o m p u t e r s - Muscle denervation - Muscles - Needle biopsy - Nerve degeneration - Nerve regeneration

Correspondence and reprint requests: Jan Lexell, MD, Department of Anatomy, University of Ume~, S-901 87 Ume~, Sweden. This work was supported by grants from Gun and Bertil Stohnes Foundation, the Research Council of the Swedish Sports Federation and the Swedish Medical Research Council. 0022-510X/83/$ 03.00 © 1983 Elsevier Science Publishers B.V.

302 INTRODUCTION Human skeletal muscles are characteristically composed of fibres with different structural and functional properties. By staining thin cross-sections of biopsied muscles for myofibrillar adenosine triphosphatase (mATPase), the fibres may at light-microscopic level be classified into at least two main types: type 1 (slow-twitch) and type 2 (fast-twitch). This classification is found valuable and is often used in research and for clinical purposes (Dubowitz and Brooke 1973; Swash and Schwartz 1981). The proportion of the two fibre types varies between different muscles and between different sites within a muscle (Henriksson-Lars6n et al. 1983; Lexell et al. 1983). In crosssections of human muscles, the two fibre types are commonly described as being randomly arranged, and so the term "mosaic pattern" is often used to describe the morphology. A very common change occurring in this mosaic pattern is the formation of groups of many fibres with the same enzyme histochemical properties. Such an arrangement of fibre types is regarded as a pathological process and is referred to as "fibre type grouping". It is thought to be caused by the reinnervation of denervated individual muscle fibres due to collateral sprouting from nearby axons (Jolesz and Sreter 1981). However, the occurrence of fibre type grouping depends very much upon the proportion of the two fibre types and their distribution in the muscle. If, in a random arrangement of fibres, the proportion of a given fibre type is high, then there are also likely to be large groups of this particular fibre type. Thus, in a given cross-section it is difficult to assess whether the grouping has occurred by chance or by a pathological process. Another aspect is that large groups consisting of one fibre type and being the sign of denervation and reinnervation are easy to recognize in a cross-section of a muscle. However, the clinical features are then obvious and often more reliable and interesting. If, instead, small changes in the distribution of the fibre types can be recognized, and if such changes are early signs of a successive denervation and reinnervation process, it would be of great value for the clinician. The main purposes of this study are therefore, to define fibre type grouping on the basis of randomness, and to find an objective measure with which to distinguish between muscles with fibre type grouping and muscles with a random arrangement of the fibres. Thus it might be possible to recognize early signs of denervation and reinnervation in muscles. We adopt the measure of counting the number of "enclosed fibres" (Jennekens et al. 1971a) in a cross-section, i.e. when a fibre is surrounded by fibres of the same histochemicai type (Fig. 1). To study the statistical properties of the number of enclosed fibres in a muscle cross-section, a model, based upon hexagonal-shaped fibres, is used (Johnson et al. 1973) and simulated muscle cross-sections are prepared in a microcomputer. Muscle cross-sections with different shapes and different proportions of the fibre types, and fibres with varying fibre sizes, are considered. The applicability of the model, and the derived results and methods of analysis, are tested on 10 sample areas, each comprising about 300 fibres, from a cross-section of a whole muscle, m. vastus lateralis, prepared by a recently described technique (Henriksson-Lars6n et al. 1983; Lexell et al. 1983).

303

Fig. 1. Small part of the whole cross-section of m. vastus lateralis. The star (*) shows an enclosed type 2 fibre. In the lower right corner is a group of type 1 fibres. The marked area corresponds to the area studied in Table 4. Staining: m A T P a s e at pH 9.4; Magnification x 70.

MATERIALS AND METHODS

Equipment Two microcomputers were used for the computations in this study: a standard Apple II (Apple Computer, Inc., U.S.A.) at Liverpool University and an ABC 800 (Luxor AB, Sweden) at Umeh University.

Definitions of enclosedfibre and group sizes Fibres in a cross-section of a human muscle are seen to be tightly packed in well-defined fascicles, which vary in size from 150 up to 700 fibres. In a cross-section of a muscle specimen a fibre is defined to be an "enclosed fibre" if it is surrounded by fibres of the same histochemical type (Jennekens et al. 197 la). An enclosed fibre with none of its surrounding fibres enclosed forms a group of size 1. A group of size r ( > 1) consists of r enclosed fibres that: (a) are neighbours of one or more enclosed fibres in the group, (b) can be "reached" from each other by at least one path passing from one enclosed fibre to another where each fibre on the path belongs to the group, and (c) have no neighbouring enclosed fibres that are not themselves members of the group. With this definition, only the enclosed fibres are counted in the determination of group sizes. The random arrangement of fibres in a cross-section are so considered in

304 terms o f the number of enclosed fibres and the number, and sizes of groups of enclosed fibres.

The model A model, formulated by J o h n s o n et al. (1973), can be used to study statistically these three counts. The model consists of three assumptions: (i) each fibre is a regular hexagon lying in a bee-hive desgin as illustrated in Fig. 2a, (ii) fibres are of two types, 1 (slow-twitch) and 2 (fast-twitch), and, (iii) a fibre is of type 1 with probability P and type 2 with probability (1 - P) where P is a constant for each section. When simulating a particular region, P is then taken to be the ratio of the number of type 1 fibres to the total number of fibres. Simulation Although some results can be derived with probabilistic arguments, we also have to resort to the numerical technique of computer simulation. The simulations are restricted in this study to cross-sections in which each row contains the same number o f hexagons: Fig. 2a consists of 8 rows each with 4 hexagons. The simulations are described in terms of configurations of points which are the centres of the hexagon. Thus, Fig. 2a becomes Fig. 2b, which also includes a co-ordinate system. For a configuration with m rows each with n points, there are (m - 2) (n - 2) 1,1

) ) )

1,2 2,1

3,1

2,2 3,2

•

5,1

8,1 a

•

4,3 5,3

6,2 7,2

4,4 5,4

6,3 7,3

8,2

2,4 3,4

•

5,2

7,1

2,3

4,2

6,1

1,4

3,3

•

4,1

)

1,3

6,4 7,4

8,3

8,4

b

Fig. 2. a: A schematic illustration of a cross-sectioned skeletal muscle as a hexagonal lattice, consisting of 8 rows each with 4 columns of hexagons = 32 hexagons, of which 12 are internal and the remaining 20 on the boundary, b: Replacing the centre of each hexagon by a point, it gives the representation of the co-ordinate system used.

305 internal, and 2(m + n ) - 4 boundary points: in Fig. 2b there are 12 internal and 20 boundary points. If x(1), x(2) .... x(mn) is a sequence of random numbers from the uniform distribution on the interval (0,1) and ifk = n(i - 1) + j, then the fibre represented by the point (i,j) is of type 1 when x(k) < P and of type 2 ifx(k) > e. (In practice, a random sequence is never available and is replaced by a sequence of pseudo-random numbers which are numbers generated in a deterministic manner displaying "random-like" properties.) Thus, the computer program simulates a random configuration for a particular value of P, and then determines the number of enclosed points, and the number and size of groups. The algorithm for determining the group statistics is the most time consuming part of the program. RESULTS

Cross-sections of simulated muscles In a configuration with M internal points, the number of type 1 points is binomially distributed. The probability that a point is both type 1 and enclosed is p7 and so the expected number of type 1 enclosed points is MP 7. Similarly, the expected number of type 2 enclosed points is M(I - p ) 7 . The expected numbers of type 1 and type 1 enclosed points are given in Table 1 for P taking values 0.3, 0.4 and 0.5 and for 3 sizes of configurations: 12 x 12, 16 x 16 and 20 x 20. Twenty configurations were simulated for each of these 9 possibilities. The mean number of enclosed points for each sample of 20 configurations is given in Table 1 for comparison with the expected value; the variances and ranges are also given. The number and the sizes of groups of enclosed points were computed for each of the 180 simulated configurations. The mean, variance and range of the number of groups of type 1 and type 2 enclosed points are given in Table 2 for each of the 9 possibilities. The distribution of the sizes of groups of enclosed points are represented in Table 3 by 9 frequency tables. Cross-section of a whole muscle (a) The effects of varying fibre sizes In a cross-section of a human or animal muscle specimen, fibres do not always have 6 neighbours, because the shapes and sizes of the fibres vary. If a fibre has j neighbours, then it is a type 1 enclosed fibre with probability PJ + i. Consequently, if there are mj internal fibres with j neighbours so that the number of internal fibres M = m I + m e + .... then the expected number of enclosed fibres is ~ j mj {PJ + 1 + (1 - P)J ÷ 1}. Most, if not all, fibres have 4, 5, 6, 7 or 8 neighbours and the summation effectively has 5 or fewer terms. The expected value is therefore likely to be close to M { e 7 + (1 - p ) 7 } . Counting the number of neighbours of each fibre in part of Fig. 1, no fibre has less than 4 neighbours and no fibre has more than 8 neighbours. The number of neighbours of internal fibres of each type are given in Table 4; the mean numbers of

100

196

324

12 × 12

16 x 16

20 x 20

Expected ~ Mean Variance Range

Expected ~ Mean Variance Range

Expected a Mean Variance Range

Enclosed points

0.07 0.05 0.05 0-1

0.04 0 0 0-0

0.02 0 0 0-0

26.68 23.05 48.47 17-37

16.14 18.45 32.37 5-29

8.24 7.9 19.88 1-19

0.53 0.40 0.36 0-6

0.32 0.30 0.33 0-2

0.16 0.15 0.24 0-2

Type 1

Type 1

Type 2

P=0.4

P=0.3

The expected number of enclosed fibres is MP v for type 1 fibres and M(1 - pT) for type 2 fibres.

Number of internal points (M)

Size of configuration

9.07 9.55 19.13 1-17

5.49 6.35 10.34 0-14

2.80 1.80 3.01 0-6

Type 2

2.53 2.55 3.73 0-6

1.53 1.55 2.16 0-5

0.78 0.80 1.12 0-4

Type 1

P=0.5

2.53 2.60 5.09 0-9

1.53 1.75 1.67 0-4

0.78 0.65 1.08 0-3

Type 2

THE MEAN, VARIANCE AND RANGE OF THE NUMBER OF ENCLOSED POINTS FOR 20 C O N F I G U R A T I O N S W I T H P T A K I N G VALUES 0.3, 0.4 AND 0.5, AND OF SIZE 12 x 12, 16 x 16 AND 20 x 20

TABLE 1

Number of internal points (M)

100

196

324

Size of configuration

12 x 12

16 x 16

20 × 20

Mean Variance Range

Mean Variance Range

Mean Variance Range

Groups of enclosed points

0.05 0.05 0-1

0 0 0-0

0 0 0-0

9.15 6.66 4-15

5.55 3.42 3-10

3.4 1.94 1-6

0.4 0.36 0-2

0.25 0.20 0-1

0.1 0.09 0-1

Type 1

Type 1

Type 2

P = 0.4

P = 0.3

4.85 3.71 1-8

3.4 3.32 0-7

1.4 1.41 0-4

Type 2

2 1.89 0-5

1.2 1.11 0-3

0.55 0.37 0-2

Type 1

P = 0.5

2 1.89 0-5

1.35 0.98 0-3

0.45 0.47 0-2

Type 2

T H E MEAN, V A R I A N C E A N D R A N G E OF N U M B E R OF G R O U P S O F EACH TYPE F O R 20 C O N F I G U R A T I O N S W I T H P T A K I N G VALUES 0.3, 0.4 A N D 0.5, A N D OF SIZE 12 x 12, 16 x 16 A N D 20 x 20

TABLE 2

308 TABLE 3 THE FREQUENCY DISTRIBUTION OF THE SIZE OF GROUPS OF ENCLOSED POINTS IN 20 CONFIGURATIONS WITH P TAKING VALUES 0.3, 0.4 AND 0.5, AND OF SIZE 12 x 12, 16 x 16 AND 20x 20 Size of configuration 12 x 12

16 x 16

Number of points in a group

P = 0.4

P = 1).5

2 3 4 5+

33 15 11 3 6

24 4 1 1 0

14 4 1 1 0

Total Mean

68 2.32

30 1.29

20 1.45

1

39 20 16 8 6 22

47 11 5 5 2 3

40 8 2 1 0 0

73 1.87

51 1.29

68 17 9 2 4 5

63 12 4 1 0 0

1

2 3 4 5 6+ Total Mean 20 x 20

P = (1.3

1

2 3 4 5 6+ Total Mean

111 3.31 92 35 11 14 14 17 183 2.52

1115 1.97

80 1.29

s u r r o u n d i n g fibres are 5.6 and 6.1 for type 1 and type 2 fibres, respectively. T h e p r o p o r t i o n o f type 1 fibres is 57~o w h e n c o n s i d e r i n g only the internal fibres and 54~o w h e n c o n s i d e r i n g all fibres. T h e values o f the e x p e c t e d n u m b e r o f e n c l o s e d fibres w e r e c a l c u l a t e d for b o t h values o f P and are given in T a b l e 4 for e a c h fibre type. T a b l e 4 also i n c l u d e s the o b s e r v e d n u m b e r o f e n c l o s e d fibres in the c r o s s - s e c t i o n .

(b) Numbers of enclosedfibres, and numbers and sizes of groups F i g u r e 1 is part o f a c r o s s - s e c t i o n f r o m a w h o l e muscle, m. v a s t u s lateralis (31-year-old m a l e w h o h a d d i e d accidentally). T e n different s a m p l e regions f r o m this c r o s s - s e c t i o n w e r e studied. T h e values o f P lay in the r a n g e 0 . 4 4 - 0 . 7 5 . T h e n u m b e r o f internal fibres w e r e e s t i m a t e d f r o m the total n u m b e r o f f b r e s , by a s s u m i n g a circular shape. T h e total n u m b e r o f fibres, the p r o p o r t i o n o f type 1 fibres, the n u m b e r o f e n c l o s e d fibres and the n u m b e r o f g r o u p s w e r e d e t e r m i n e d in e a c h s a m p l e region a n d are given in T a b l e 5. T h e e x p e c t e d n u m b e r o f e n c l o s e d fibres was c a l c u l a t e d for e a c h region on

309 TABLE 4 DETAILS OF THE CROSS-SECTIONED MUSCLE SPECIMEN IN FIG. 1

Number of neighbours of internal fibres

34 5 6 7 8 9+

Total number of internal fibres Proportions Number of fibres on the fascicle border Proportions Total number of fibres Proportions Expected number of enclosed fibres

Observed number of enclosed fibres

Type 1

Type 2

Total

0 4 28 30 7 1 0

0 0 8 29 15 0 0

0 4 36 59 22 1 0

70

52

122

57 % 19

43 ~o 23

100 ~/o 42

45 ~/o 89 54 ~o

55 ~o 75 46 ~o

164 100 ~o

P = 0.57

(a) (b)

2.38 2.88

0.33 0.47

2.71 3.35

P = 0.54

(a) (b)

1.63 2.02

0.53 0.73

2.16 2.75

3

1

4

(a), The expected values are evaluated from the formula based on all fibres having 6 neighbours (Mp7), and (b) from the more general form ~ mj × PJ ÷ t.

the basis of the model (Johnson et al. 1973), and the mean number of groups of enclosed fibres was determined from 50 simulated configurations for each sample region. These computed values are also included in Table 5. The shapes of the simulated configurations were the same as that shown in Fig. 2b. The number and size of the rows were chosen so that the number of points was close to the number of fibres in the region under consideration: for example, configurations with 18 rows each with 18 points were used to resemble sample region 1. To obtain the significance level for the number of enclosed fibres in a sample region, 200 configurations with an appropriate size were simulated and the number of enclosed points computed for each configuration (Monte Carlo technique). Rough 5 ~o and 1 ~o significance limits were then available. For 8 of the 10 sample regions, the observed numbers of enclosed points were clearly far from these limits, and the significance levels are given in Table 5. Where the observed number of enclosed fibres were close to a significance limit, refined significance limits were determined from 500 simulated configurations. Computing the group sizes in the simulated configuration is time-consuming on a micro-computer. However, for P in the interval (0.4, 0.6), the Poisson distribution fits

262 243 263 288 266 276 274 272 293 235

Estimated no. of internal fibres a (M)

75 9o 64°~, 44~,, 559o 539~, 56 ~o 46 9o 44 9o 44~o 66~o

Proportion of type 1 fibres (P)

39 10 1 28 4 3 2 1 0 15

35 10.7 0.84 4.38 3.12 4.77 1.19 0.87 0.93 12.8

0 1 5 4 0 2 3 6 3 2

0.02 0.19 4.54 1.08 1.35 0.88 3.67 4.70 5.06 0.12

n.s. c n.s. n.s. < 0.01 n.s. n.s. n.s. n.s. n.s. < 0.05

5 4 1 3 3 3 1 1 0 9

Observed 8.54 d 4.92 0.70 2.52 2.28 2.94 0.96 0.54 0.66 5.4

Estimate&

Observed

Observed

Expected b

Type 1

Type 2

Type 1 Expected b

Number of groups

Number of enclosed fibres

Estimated by assuming a circular shape; thus, M is the largest integer less than n - 2 ~/'ztn. The expected number of enclosed type 1 fibres is MP 7 and of enclosed type 2 fibres is M(1 - p)7. Computed from the simulated configuration. The values in these columns are means of 50 configurations.

326 305 327 355 330 342 399 337 360 296

1 2 3 4 5 6 7 8 9 10

" b ~ d

No. of fibres (n)

Sample region

0 1 2 3 0 2 2 4 3 1

Observed

Type 2

0.06 d 0.12 2.82 0.92 1.2 0.88 2.48 2.72 2.76 0.16

Estimated ~

THE OBSERVED NUMBER OF ENCLOSED FIBRES A N D NUMBER OF GROUPS OF FIBRES IN TEN D I F F E R E N T SAMPLE R E G I O N S FROM A W H O L E CROSS-SECTION OF M. VASTUS LATERALIS

TABLE 5

< 0.05

n.s.

n.S.

n.s.

n.s.

n.s.

< 0.05

n.s.

l'l.S.

n.s.

311 the data closely: one aspect of this is seen in Table 2 where the means and variances are close for P in this interval. Thus, for P in the interval (0.4, 0.6) approximate significance levels are immediately available: for example, in Table 5 the significance levels for the number of groups in region 6 is 1 - e- 3.82 ~4= o (3.82)J/J ! - 0.336. In regions 1, 2, 4 and 10, the significance levels of the number of groups of enclosed fibres were obtained by simulating 500 configurations. The significance levels of the number of groups are given in Table 5. DISCUSSION

Fibre type grouping is known to occur in chronic peripheral neuropathies and in motor neuron disease in the advanced stage (Swash and Schwartz 1981). It has been studied in several ways: for example, in clinical material (Morris 1969; Engel 1970; Jennekens et al. 197 lb), experimentally (Karpati and Engel 1968; Kugelberg et al. 1969) and quantitatively (James 1972). Fibre type grouping has so been considered to be present when around 25 fibres of the same histochemical type form a group (Swash and Schwartz 1981). This is a quite rough defmition, as the observed grouping depends upon the relative proportion of the two fibre types. A model, such as the one used in the present study, has many advantages in the definition of fibre type grouping. However, the applicability of the described model depends upon how close its assumptions resemble the muscle sections studied. In particular, there are two aspects that could invalidate the results: (i) one fibre type is consistently differing in size from the other type, (ii) the proportion of the two fibre types varies systematically within a fascicle. The sizes of the two fibre types are known to vary according to e.g. the type of muscle, the sex, age, and level of activity of the individual. If then a fibre type in a muscle is smaller than the other type, it will have fewer surrounding fibres, and consequently the observed number of enclosed fibres are likely to be larger than the expected number derived from the model. The expected number, and the observed number: of enclosed fibres for the region marked in Fig. 1 are seen in Table 4 to be close, even though the mean sizes of the two fibre types differ, so that type 1 fibres are in general smaller. This suggests that it is unnecessary to change the model to take account of different fibre sizes, although such changes in the model and the computer program could be made with envisaged difficulty. Furthermore, large altertions in the general morphology leading to e.g, small, hypotrophic fibres and groups of such fibres are themselves strong indications of a pathological process in the muscle. The proportion of type 1 and type 2 fibres on the border of the fascicle is seen in Table 4 to differ from the proportion of the internal fibres. If this occurs regularly, it might be necessary to change the model with a consequent difficulty in deriving an expression for the expected number of enclosed fibres. However, the computer program is easily modified and an estimate of the expected number of enclosed fibres can be computed. If fibres in a sample of a muscle are assumed to be hexagons and if each fibre

312 is assigned a type according to assumptions (i) and (ii) of the model, then the number of one of the fibre types is binomially distributed. However, the number of enclosed fibres of either type is not binomially distributed as asserted by Johnson et al. (1973), because the probability that a fibre is enclosed is dependent upon a surrounding fibre being enclosed or not. If their assertion were correct, then the variance of the number of enclosed fibres should be slightly less than the mean. In Table 1 the variances of the numbers of enclosed fibres in simulated configureations are seen, for P :/: 0.5, to be consistently greater than the means and the differences increase the further P is from 0.5. The known expected numbers of enclosed fibres are included in Table 1 to show their closeness with the sample means. To test then, whether or not a cross-sectioned muscle specimen displays fibre type grouping, one needs to assess whether the observed number of enclosed fibres is too great; the possibility that there are too few enclosed fibres ("ungrouping") is not considered here. A one-tailed significance test is appropriate. Unfortunately, the probability distributions of the number of enclosed fibres is not known, nor has a satisfactory approximation been found and so we resort to Monte Carlo significance tests (cf. RESULTS)*. No results have been derived for the distribution of the number of groups of enclosed fibres. However, for P in the range 0.4-0.6 the Poisson distribution provided a close fit to the data from the simulated configurations. No details of the fits are given here but the means and variances of the numbers of groups for the simulated configurations are seen to be close in the columns headed P = 0.4 and P = 0.5 of Table 2. Thus, when the observed proportion of type 1 fibres is in the range 0.4-0.6, the significance tests can be based on the Poisson distribution. If the proportion is outside this range, or if the number of groups in the section is close to the limit for the chosen significance level, then we resort to a Monte Carlo significance test, which unfortunately is timeconsuming. In Table 5 the significance levels of sections 1, 2, 4 and 10 were obtained using the Monte Carlo method. The sizes of the groups in the simulated configurations are presented in Table 3 in the form of 9 frequency tables. The group sizes display great variability and so the number of fibres in the largest group is a poor measure of non-randomness. The whole muscle cross-section in Table 5 was chosen not only for the test of the applicability of the model, but also because it comprises large areas with excessive grouping of both fibre types; some fascicles consist of more than a hundred fibres of the same histochemical type. Because of this, it was of interest to study whether any other area in the muscle might show signs of a non-random distribution. However, only two of the examined areas show non-randomness for both the number of enclosed fibres and the number of groups but not for both fibre types. Sample region 1 comprises 5 groups with a total of 39 enclosed fibres. Neither the total number of enclosed fibres, nor the number of groups are significantly non-random at the 5 ~o level. However, the largest group consisting of 10 ,enclosed fibres would, according to previous definitions (Swash and Schwartz 1981), have been described as * We have had no success in fitting various functions of M and P to the variances of the number of enclosed fibres of the simulated configurations.

313 showing fibre type grouping. In this study, the term "fibre type grouping" is exclusively used if, for an observed proportion, the arrangement of fibre types in a cross-section is non-random at a given significance level. But if the proportion of a fibre type is high, then large groups could of course occur, but might not be defined as fibre type grouping in our sense. This high proportion might actually itself be indicative of a successive denervation and reinnervation process, but could also be due to a natural physiological adaptation in the muscle, e.g. training or normal ageing. To assess more closely the significance of fibre type grouping we must have available data on the extent to which grouping and high fibre type proportions may appear normally and in muscles from athletes and in muscles at various age-groups. The two methods used for sampling muscles, the needle technique (BergstrOm 1962, 1975) and the open surgical technique (Dubowitz and Brooke 1973) are inadequate for this sort of study, as the size of the sample is small. The availability of cryomicrotomes and histochemical techniques for the preparation and study of thin cross-sections of whole muscles will here prove to be useful (Henriksson-Lars6n et al. 1983; Lexell et al. 1983). Finally, the conclusions from this introductory work is that the model of Johnson et al. (1973) is seen to be suitable for the assessing of non-random grouping in muscles, irrespective of the size of cross-sections and the proportion of fibre types. In the simulation study, the number of enclosed fibres in a section is seen to be more indicative of non-random grouping than the number of groups of enclosed fibres. The sizes of the groups are shown to be a poor indicator. Despite the sizes of the fibres being different in human muscles, the model is sufficiently robust to yield useful results. Monte Carlo significance tests (based on the model) have to be used to recognize non-random grouping. The programs for this study were written in BASIC, which is available on many microcomputers. Few changes are therefore necessary when implementing such programs on other types of microcomputers. By linking an image analyser with a microcomputer and by modifying the programs accordingly, the differences between the muscles and the model will be smaller. The study of fibre type grouping and the comparison of results, and the early diagnosis of muscle fibre denervation and reinnervation can thus be made more precisely. ACKNOWLEDGEMENT We wish to thank Professor S. James Taylor, Department of Pure Mathematics, University of Liverpool, England, for valuable advices. REFERENCES Bergstr6m, J. (1962) Muscle electrolytesin man, Scand. J. Clin. Lab. Invest., 14, Suppl. 68. BergstrOm,J. (1975) Percutaneous needle biopsy of skeletal muscle in physiologicaland clinical research, Scand. J. Clin. Lab. lnvest., 35: 609-616. Dubowitz, V. and M. H. Brooke (1973)Muscle Biopsy - - A Modern Approach, WB Saunders CompanyLtd, London. Engel, W.K. (1970) Selectiveand nonselectivesusceptibilityof muscle fiber types -- A new approach to human neuromusculardiseases, Arch. Neurol. (Chic.), 22:97-117.

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