A new approach to the quantification of fiber type grouping is presented, in which the distribution of histochemicaltype in a muscle cross section is regarded as a realization of a binary Markov random field (BMRF). Methods for the estimation of the parameters of this model are discussed. The first order BMRF, which is used in this article, contains 2 parameters: a and p. The parameter p is of prime importance, as it is an interaction parameter which governs the degree of type grouping. The value of this parameter is estimated for 9 muscle biopsies. The interpretation of the results is discussed. Key words: fiber type grouping Markov random field spatial distribution denervation reinnervation MUSCLE & NERVE 15:725-732 1992
MODELING FIBER TYPE GROUPING BY A BINARY MARKOV RANDOM FIELD HENK
W.
VENEMA, PhD
T h e quantitative analysis of cross sections of musand p. T h e second parameter (p) is often called cle tissue has made steady progress over the last the interaction parameter as it governs the spatial interactions between fiber types. It is thus related 30 years. In the first years of this period, research to the amount of fiber type grouping. concentrated mainly on the fiber size3 and It is not unreasonable to surmise that a model Since 1973, a number of‘ studies have which allows for interaction between the types of been published on the quantification of the spatial neighboring fibers could fit the data observed in a distribution of different fiber types in cross secmuscle fiber pattern which exhibits fiber type tions of muscle ti~sue.’”’”~Several measures have grouping. One generally assumes that this clusterbeen used to detect departures from a complete ing is brought about by direct interaction between random distribution of fiber type.I3 The most imneighbors, i.e., by denervation of a fiber and sucportant are: the number of enclosed fibers?,’,’’ cessive reinnervation by a collateral sprout from a the codispersion index (CDI),” the mean cluster neuron innervating one of its neighbors. size,7 and the number of neighbor pairs of differis to introduce the The aim of this study ent type.21 Finally, some insight into the mechaBMRF model as a tool for the analysis of fiber nism which presumably underlies fiber type clustype grouping, and to apply it to biopsies taken tering has been obtained by simulation s t u d i e ~ . ~ . * ~ from healthy individuals and from patients with In this study, another approach to the quantifipolyneuropathy, and to discuss the interpretation cation of fiber type grouping is described: the disof the results. tribution of histochemical types in a muscle cross section is considered to be a realization of a spatial stochastic process. The model we have used is the METHODS most simple version of what in the statistical literThe BMRF Model-Some Examples. We present ature is known as a binary Markov random field here a cursory introduction to the binary Markov (BMRF).‘ This model contains 2 parameters, a random field (BMRF) model, as applied to the analysis of fiber type distributions in muscle cross sections. For more details, the reader is referred From the Laboratory of Medical Physics and Informatics, University of Amsterdam, Amsterdam, The Netherlands to the relevant literature.’’6 Acknowledgments: The author is indebted to Prof. Dr. J. Strackee and In the present approach, the muscle fiber patDr. M. de Visser for helpful discussions. tern is regarded to consist of 2 separate entities: Address reprint requests to Dr. Venema. Laboratorium voor Medische the configuration of cross sections of fibers (the Fysica en Informatiekunde. Academisch Medisch Centrum, Meibergdreef “pattern”); and the distribution of fiber types, 15, 1105 A2 Amsterdam, The Netherlands. which is superimposed on this pattern. It is the Accepted for publication September 1, 1991 latter distribution which we consider to be a realCCC 0148-639x1921060725-08$04.00 ization of a BMRF. 0 1992 John Wiley & Sons, Inc.
Modeling Fiber Type Grouping
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FIGURE 1. Examples of realizations of the first-order BMRF model on a simulated muscle fiber pattern with 596 cells. The pattern has periodic boundary conditions, thus the left and right side and the upper and lower side are connected. For all distributions, cx = 0; p = -0.5 (a), -0.2 (b), 0 (c), 0.1 (d), 0.2 (e), and 0.27 (f).
726
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The random variables of this model take the value -1 (for a type 1 fiber) or + 1 (for a type 2 fiber). In the following, type 1 fibers are assumed to be colored light (“white”)and type 2 fibers dark (“black’), corresponding with the staining for myofibrillar adenosine triphosphatase, pH 9.4, which is used in this study. The most simple version of the BMRF model, the first-order model, contains 2 parameters, a and p. The first parameter governs the relative numbers of type 1 and type 2 fibers. For ci = 0 the odds on a type 1 and type 2 fiber are equal. Thus, a = 0 corresponds with muscle cross sections with (in the mean) equal numbers of type 1 and 2 fibers. For a < 0, more type 1 fibers are present; for ci > 0, more type 2 fibers. T h e second parameter (p) governs the interaction between the fiber types of nearest neighbors. If p is positive, the probability of neighboring fibers of the same type is favored, corresponding with type grouping. For negative values of p, the opposite is the case, i.e., the probability of unlike neighbors is enhanced. For p = 0, a random distribution of fiber type is obtained. More general models can be obtained by introducing additional parameters corresponding with interactions between, e.g., triples of mutually nei hboring fibers, next-nearest neighbors, and so on!‘) We restrict ourselves in this study to the first-order model. A n attractive property of the BMRF model is that one can easily generate (by means of a special computer procedure, often called a Monte Carlo procedure) an arbitrary realization of the model.”.“ This can be done both on regular lattices (consisting of, e.g., hexagonal cells) and on irregular cell patterns. Examples of fiber type distributions of the first order model are shown in Figure 1. These distributions were generated on an artificial cell pattern, containing 576 cells, which closely mimics a muscle fiber pattern. This cell pattern was obtained by constructing the radical axis tessellation2‘ of 576 closely packed circles of varying size. T h e parameter ci = 0 for all 6 distributions, which leads to roughly equal numbers of type 1 and type 2 cells. T h e first 2 distributions (a and b), with negative values of p (P = -0.5 and -0.2), show segregation of like cell types. T h e third pattern (c) has a random distribution o f cell type (p = O), while the remaining three distributions (d, e, and f ) exhibit an increasing amount of type grouping (p = 0.1, 0.2, and 0.27). For comparison, Figure -2 shows the fiber type distributions in 3 fascicles of the cross section of
Modeling Fiber Type Grouping
the quadriceps femoris muscle of a patient with polyneuropathy, which exhibit a moderate amount of type grouping. T h e general resemblance of these fiber type distributions and the BMRF realizations of Figure 1 with p = 0.1 or 0.2 seems to be reasonable. I n the next section, we discuss methods to estimate the parameters (a,p) for an arbitrary fiber type distribution. As a final illustration, Figure 3 shows 2 realizations of the BMRF model which exhibit type predominance. In both patterns, 80% of the cells are type 1. T h e left pattern (a) has a random distribution of cell type ( a = 0.69, p = O), and in the right one (b) type grouping is present ( a = 0.10, p = 0.20). T h e presence of type grouping in the right pattern may readily be unnoticed, because, for example, no enclosed “fibers” of the scarce type are present. This example emphasizes the value of a quantitative analysis of biopsies when fiber type grouping has to be demonstrated or ruled out. The fact that a is different for the 2 realizations in Figure 3 (in order to obtain 80% type I cells both times) shows that there is not an unequivocal relationship between a and the relative numbers of type 1 and 2 fibers, but that there is an influence of f3 as well. Usually, it is more convenient to disregard a altogether, and use the relative numbers of both fiber types and p to describe
FIGURE 2. Three fascicles of a patient with polyneuropathy which exhibit a moderate amount of type grouping.
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FIGURE 3. Difference between fiber type predominance (a), and fiber type predominance in combination with type grouping (b). In both distributions, 80% of the cells are type 1. For the left distribution, p = 0; for the right one, p = 0.2.
the spatial distribution of fiber type, instead 01 (equivalently) (Y and p. In the previous section, w e have shown that it is possible to generate realizations of the BMKF model by means of a computer procedure. We now want to operate the other way around: given a realization of a BMKF (a muscle fiber pattern with a distribution of fiber types which we consider as such, o r an artificially generated realization), can we estimate the parameters of the corresponding model? As an example w e will estimate (Y and p of the 3 fascicles of Figure 2. Neighboring fibers have been determined with an algori t him of' Vene ma. Parameter estimation can be achieved in 2 ways."' T h e first method is the conditional pseudolikelihood (CPL) technique. This method is rather cumbersome, and it has the drawback that only estimates of the parameters themselves are obtained and not of their standard errors. Details of this technique can be found in the literature.' Parame-
Parameter Estimation.
"'
Table 1. Estimates of the parameters Fascicle
n
n,,
1 2 3 Together
168 256 226 650
452 704 610 1766
" i
55 100 86 241
cy
ter estimates of the 3 fascicles of Figure 2, separately and together, are given in 'Table l . 'The simplest method for parameter estimation is the maximum likelihood (MI,) method. Estimates of both the parameters and of their standard errors are obtained. This method utilizes the fact that, given the number of fibers (n) and of neighbor pairs (n,,), there exists a one-to-one relationship between 0 1 1 the one hand, the number of type 1 o r 2 fibers ( n , o r n2)and the number of unlike neighbor pairs (nI2},and, on the other hand, the parameters a and p of the first-order BMKF model. I n a previous communication," we showed that n12is a sensitive measure to detect deviations from a random distribution of fiber type. Application o f this method is not completely straightforward, as the relationship between n, (or n2)and nI2,and (Y and @, is unknown in analytical form. We have circumvented this problem by approximating this relationship using Monte Carlo simulations (Venema, submitted for publication). Some results are given in 'Table 2a and b. With aid
and p of the first-order model for the 3 fascicles of Figure 2 *
ni2
aPL
PPL
f2
166 248 229 643
0122 0074 0082 0083
0145 0182 0169 0 169
0673 0609 0619 0629
f,,
0367 0352 0375 0364
~
M
0118 0040 0065 0062
L
W ~ M L ) PML 0071 0032 0044 0025
0137 0177 0151 0 160
SE (PMJ 0036 0024 0 029 0016
'Obtained wifh pseudoiikebhood (Pi) estimation jcoiurnns 6 and 71, and maximum kkelihood (Mi) estimation (coi~mffs 70 and 12) Fof the iatter esrimates, !he standard errors are also given (columns I 1 and 73)
728
Modeling Fiber Type Grouping
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~~~
Table 2a. Fractions of unlike neighbor pairs f,, (= n,&~,) for values of p between -0.20 and 0.25 (rows), and percentages of fibers of the most-occurring type between 50 and 80 (columns)
P
50
52
54
56
58
60
62
64
66
68
70
71
72
73
74
75
76
77
78
79
80
-0.200 -0.180 -0.160 -0.140 -0.120
571 566 560 554 548
570 565 559 553 547
567 562 556 551 544
562 557 552 546 540
555 550 545 539 533
546 541 536 531 525
535 530 525 520 514
522 517 512 507 502
507 503 498 493 488
490 486 482 477 472
471 467 463 459 454
461 457 453 449 445
450 447 443 439 435
439 436 432 429 424
428 425 421 418 414
416 413 410 406 402
404 401 398 394 391
391 388 385 382 379
378 375 372 369 366
364 362 359 356 353
350 348 345 343 341
-0.100 -0.080 -0.060 -0.040 -0.020
542 534 527 519 510
541 533 526 518 509
538 531 523 515 506
533 526 519 511 502
527 520 513 505 496
518 512 505 497 489
508 502 495 488 480
496 490 484 476 469
482 477 471 464 457
467 461 456 449 443
449 444 439 433 427
440 435 430 424 418
430 426 421 415 410
420 416 411 406 400
409 405 401 396 391
399 395 390 385 380
387 383 379 375 370
375 372 368 364 359
363 360 356 352 348
350 347 345 340 336
337 334 331 328 324
0.000 0.010 0.020 0.030 0.040
500 495 490 484 478
499 494 489 483 478
497 492 486 481 475
493 487 480 471 488 482 475 467 483 477 470 462 477 472 465 457 472 467 460 452
461 456 452 447 442
449 445 440 436 431
435 431 427 423 419
420 416 413 409 405
412 408 405 401 397
403 400 396 393 389
394 391 388 384 381
385 382 379 375 372
375 372 369 366 363
365 362 359 356 353
354 351 349 346 343
343 341 338 336 333
331 329 327 325 322
320 318 316 313 311
0.050 0.060 0.070 0.080 0.090
472 466 459 452 445
472 465 459 452 445
469 463 457 450 443
466 460 453 447 440
454 449 442 436 430
447 441 435 429 423
437 432 426 421 414
426 421 416 411 405
414 409 404 399 394
400 396 391 386 381
393 389 384 379 374
385 381 377 372 367
377 373 369 364 360
368 365 361 356 352
359 356 352 348 344
350 347 343 339 335
340 337 334 330 326
330 327 324 321 317
320 317 314 311 307
309 306 303 300 297
0.100 0.110 0.120 0.130 0.140
438 430 421 413 403
437 435 432 428 422 429 427 425 420 415 421 419 416 413 407 412 411 408 404 399 403 401 399 395 391
416 409 401 393 385
408 401 394 386 378
398 392 385 378 370
387 381 375 368 361
376 370 364 358 351
369 364 358 352 345
362 357 351 346 339
355 350 345 339 332
348 343 337 332 326
340 335 330 325 320
331 327 322 317 312
323 318 313 309 304
314 310 305 301 296
304 301 296 292 288
294 290 287 283 279
0.150 0.160 0.170 0.180 0,190
394 383 372 361 348
393 383 372 360 348
392 382 371 359 347
389 379 369 357 345
386 376 366 354 342
382 372 362 351 339
376 367 357 346 335
370 361 351 341 330
362 353 344 335 324
353 345 337 327 317
343 336 328 319 309
338 331 323 314 305
332 325 318 310 300
326 319 312 304 296
320 313 307 299 291
313 307 301 293 285
306 300 294 287 280
299 293 288 281 274
391 286 281 274 268
283 279 273 267 261
275 270 265 260 254
0.200 0.205 0.210 0.215 0.220
335 328 321 313 305
335 328 320 313 305
334 327 320 312 304
332 325 318 310 303
330 323 316 308 301
327 320 313 306 298
323 316 310 302 295
318 312 305 298 291
313 307 300 294 287
307 301 295 288 282
299 294 288 282 276
296 290 285 279 273
291 286 281 276 270
287 282 277 272 266
282 278 272 268 262
277 273 268 263 258
272 268 263 258 254
266 262 258 254 249
260 256 253 248 244
254 250 247 243 239
247 244 241 237 233
0225 0230 0235 0240 0245 0250
297 288 279 269 259 248
297 288 279 269 259 247
296 287 278 268 258 247
295 286 277 268 257 746
293 284 275 266 256 245
291 282 274 265 255 244
288 280 271 262 252 242
284 276 268 259 250 239
280 273 264 256 247 237
276 268 260 252 244 234
270 263 255 248 240 231
267 260 253 246 238 229
264 257 250 243 236 227
260 254 247 241 233 224
257 250 244 238 230 222
253 247 241 234 227 220
249 243 237 231 224 217
244 239 233 227 221 214
239 234 229 223 218 211
234 229 224 219 214 207
229 225 220 215 210 204
461 455 449 442 435
The tabulated values are the decimal fractions off,,
Example If 60% of the fibers are type 1 or type 2 and p = 0.100, then f,,
of these tables, estimates of P and its standard error can be obtained when P is in the range (-0.20 to 0.25). Equivalent tables for CY are not given here (see also the comments at the end of the previous section). Upon request, these tables are available from the author. Parameter estimates of a and p and their stan-
Modeling Fiber Type Grouping
=
0 422
dard errors for the 3 fascicles of Figure 2 and the 3 fascicles together, obtained by ML estimation using the approximations mentioned above, are listed in Table 1. The reader may check that estimates of P and SE (P) can be obtained by linear interpolation of the data in Table 2. Example. For the 3 fascicles together, 409 of
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729
~
Table 2b. Standard deviation (SD) of p for n = 1 for values of p between -0.20 and 0.25 (rows), and percentages of fibers of the most-occurring type between 50 and 80 (columns).
P
50
52
54
56
58
60
62
64
66
68
70
71
72
73
74
75
76
77
78
79
80
-0.200 -0,180 -0,160 -0,140 -0,120
82 79 77 74 72
82 79 77 75 72
82 80 77 75 73
83 80 78 75 73
84 81 78 76 74
84 81 79 77 75
85 82 80 78 76
86 84 82 79 77
88 86 83 81 79
91 89 86 84 81
95 92 89 86 84
96 93 91 88 85
98 95 92 90 87
100 97 94 91 89
102 99 96 94 91
106 102 98 95 94
108 104 100 97 95
113 107 103 99 98
116 110 106 102 100
120 114 109 106 103
123 118 113 109 107
-0,100 -0.080 -0.060 -0040 -0.020
70 67 65 62 60
70 67 65 62 60
70 68 65 63 60
71 68 66 63 61
72 69 67 64 61
73 70 68 65 62
74 71 69 66 64
75 73 70 68 65
77 74 72 70 67
79 77 75 72 69
82 80 78 75 71
83 81 79 76 73
85 83 81 78 74
87 85 83 79 76
89 87 85 82 78
92 90 87 83 80
94 92 89 86 82
97 95 92 89 85
99 98 95 92 88
102 100 98 95 91
105 103 101 98 94
0.000 0.010 0020 0.030 0.040
58 57 55 54 53
58 57 56 54 53
58 57 56 55 53
58 57 56 55 54
59 58 57 56 55
60 59 58 57 56
61 60 59 58 56
62 61 60 59 58
64 63 62 61 59
66 65 64 62 61
68 67 66 65 64
69 68 67 66 65
70 69 69 67 66
71 71 70 69 68
73 72 72 70 69
76 75 74 72 71
78 77 76 75 73
81 80 79 77 75
84 83 81 80 78
88 86 84 82 80
91 90 88 86
0.050 52
83
0.060 0.070 0.080 0.090
51 50 49 48
52 51 50 49 48
52 51 50 49 48
53 51 50 49 48
53 52 51 50 49
54 53 52 51 49
55 54 53 51 50
57 55 54 52 51
58 57 55 54 52
60 58 57 55 54
62 61 59 57 56
63 62 60 59 57
65 63 61 60 58
66 64 63 61 60
68 66 64 63 61
69 68 66 64 63
71 70 68 66 64
73 71 70 68 66
76 74 72 71 68
78 76 74 73 71
81 79 77 76 74
0.100 0.110 0 120 0 130 0 140
46 45 44 43 42
47 45 44 43 42
47 46 45 43 42
47 46 45 44 42
48 46 45 44 43
48 47 46 45 43
49 48 47 45 44
50 49 47 46 45
51 50 49 47 46
53 52 50 49 47
55 53 52 50 49
56 54 53 51 50
57 55 53 52 51
58 57 55 53 52
60 58 56 54 53
61 59 57 55 54
63 60 58 57 56
64 62 59 58 57
66 64 62 60 59
69 67 64 62 61
72 70 67 64 63
0.150 0.160 0.170 0.180 0.190
41 40 38 37 36
41 40 39 37 36
41 40 39 38 36
41 40 39 38 36
42 40 39 38 37
42 41 40 38 37
43 42 40 39 38
44 42 41 40 38
45 43 42 41 39
46 45 43 42
48 47 45 44 42
49 47 46 44 42
50 49 47 45 44
52 50 48 47 45
53 51 49 48 46
54 52 50 48 46
55 54 52
40
47 46 45 43 41
48
57 55 53 51 49
59 57 55 53 50
62 60 57 55 52
0.200 0.205 0.210 0.215 0.220
35 34 33 32 32
35 34 33 33 32
35 34 34 33 32
35 34 34 33 32
35 35 34 33 33
36 35 34 34 33
36 36 35 34 33
37 36 36 35 34
37 37 36 35 34
38 38 37 36 35
40 39 38 37 36
40 40 39 38 37
41 40 39 38 37
42 41 40 39 38
42 42 41 40 38
44 43 42 41 39
45 44 43 42 40
46 45 44 43 41
47 46 45 44 43
48 48 47 46 44
50 49 48 47 46
0.225 0.230 0.235 0.240 0.245 0.250
31 30 30 29 28 27
31 31 30 29 28 27
31 31 30 29
32 31 30 29 28 27
32 31 30 29 28 27
32 32 30 29 28 27
33 32 31 30 29 28
33 32 31 30 29 28
33 33 32 31 30 28
34 33 32 31 30 29
35 34 33 32 31 30
36 35 34 33 32 30
36 36 35 34 32 31
37 36 35 34 33 31
38 37 36 35 33 32
38 37 36 35 34 32
39 38 37 36 35 33
40 39 38 37 35 34
41 41 39 38 36 34
43 42 41 39 37 36
44 43 42 41 39 37
28
27
The tabulated values are the decimal fractions. The standard error of
p for a fascicle with n fibers is obtained by dividing the SD by $.
650 fibers are type 2 (62.9%) and 643 of the 1766 neighbor pairs are unlike: fi2 (=n12/nn)= 0.364. For 62% type 2 fibers (Table 2a, column 62), this fraction of unlike neighbor pairs corresponds with p = 0.163, for 64% type 2 with p = 0.157. Rounded to 3 decimals, 62.9% type 2 corresponds with p = 0.160. The SD for n = I for p = 0.160 is
730
Modeling Fiber Type Grouping
50
0.42 for both 62 and 64% type 2 fibers (Table 2b). The SE for n = 650 is obtained by division by the square root of 650, and is 0.016. When one compares the estimates of a and p obtained with the CPL method and ML method (Table I ) the overall agreement is satisfactory. In a previous study, w e demonstrated that the
MUSCLE & NERVE
June 1992
fraction of unlike neighbor pairs f 1 2 is slightly influenced by the presence of type-specific size differences." As the ML estimates of a and p are derived from f I 2 , we can expect a slight size dependency for these parameters as well. After adjustment of f I 2 for size differences," we determined new estimates of (Y and p. It appears that, for the fascicles of Figure 2, the correction is almost negligible. The adjusted estimates of p are 0.002 higher than the original ones, while the new estimates of a are 0.003 lower. The estimates of the SEs remain unchanged. A correction procedure for systematic size differences that can be used with the CPL estimation method seems to be, at present, not possible. RESULTS
these corrections was small. At left, the values for the reference group are shown as a function of age. All these biopsies have negative values of p. For this small group, no dependency of p on age could be demonstrated. The mean value of P is -0.09. For the polyneuropathy patterns with slight, moderate, and severe type grouping, p values of 0.07, 0.16, and 0.21 were found. By means of additional Monte Carlo simulations, it is possible to establish the goodness-of-fit of the first-order model.20 It appears that, for all biopsies but one, the fit of the first-order model is excellent. T h e one exception is the fiber type distribution with the most severe type grouping (P = 0.2 1); a slightly better fit is obtained with a higherorder model with an additional interaction term between triples of mutually neighboring fibers.
Parameter Estimates of 9 Muscle Cross Sections.
Cross sections of quadriceps femoris muscle of 9 persons, obtained at biopsy, were evaluated. Six cross sections originated from children and young adults without neuromuscular disorder (reference group). Three cross sections were analyzed of patients with disorders of the peripheral motor nerves (polyneuropathy), which exhibited, on visual inspection, slight, moderate, and severe type grouping. T h e biopsy with moderate type grouping is shown in Figure 2. More details on the biopsy p c e d u r e can be found in a previous study. Figure 4 shows the estimated values of the interaction parameter p and the 95% confidence intervals for these 9 biopsies obtained with the ML method. All estimates were corrected for type-specific size differences, if present. The magnitude of
1 -0.3
0
reference group I
I
I
5
10
15
1
20 25 age ( y )
30
1 2 3 p a t i e n t number
FIGURE 4. Estimates of p (? 2 SE) of 6 biopsies of healthy children and young adults as a function of age (left), and of 3 biopsies of polyneuropathy patients with slight ( l ) , moderate (2),and severe (3) type grouping (right). The mean value of p for the reference group (-0.09) is indicated with a dashed line.
Modeling Fiber Type Grouping
DISCUSSION
In this article, an approach is presented in which the distribution of fiber type over a cross section of muscle tissue is considered as a realization of a binary Markov random field (BMRF) containing 2 parameters, (Y and p. Of these, p is of prime importance, as it is a sensitive measure of the depee of fiber type grouping. With the aid of the model, it becomes possible to go further than just accepting or rejecting a random distribution of fiber type; it is now possible to quantify fiber type grouping accurately and sensitively. The need to quantify clearly exists. T h e CDI,'" for example, was developed for this purpose, but it is an ad hoc measure with some drawbacks, which limits its applicability." The advantages of the present approach is that when it can be shown that the fit of the model t o the data is adequate, an exhaustive description of the spatial distribution of the fiber types in the cross section is obtained by the specification of the parameters of the model, i.e., a and p or, equivalently, f I (nl/n) or f2 (n2/n)and p. If the fit is not adequate, the parameter p still is useful as a measure of the amount of fiber type grouping. Moreover, a straightforward extension of the model is possible by incorporating more interaction terms with parameters to match, which allow for interaction between, e.g., combinations of 3 mutually neighboring fibers, or between second-order neighbors. These extra parameters may provide additional information on details of the process of denervation and reinnervation, which underlie the development of fiber type grouping.'8 We have shown that, in 6 biopsies of muscle
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731
tissue of healthy persons, negative values of (3 OCcur, with a mean value of -0.09. A negative value of p corresponds to a slight repulsion of like fiber types, a findin which was already known from earlier studies.‘s14’21,’24 In 3 biopsies of patients with polyneuropathy, which exhibit slight to severe type grouping, positive values of (3 were found (p = 0.07 to 0.21). We have not looked into possible differences in fiber type proportions on the boundary of a fascicle and internally.’”.’” We showed previously that the number of unlike neighbor pairs in a fascicle is hardly influenced by the presence of these differences.“ As a consequence, the ML estimate of p, which is based on the number of unlike neighbor pairs, will neither be influenced in any significant amount. For all biopsies studied here, except one, the first order model describes the data adequately. The one exception is the pattern with the highest value of p, for which a model with an additional interaction term provides a slightly better fit. More research is needed into the information that is provided by these higher-order terms.
REFERENCES I . Besag J E : Spatial interaction and the statistical analysis ot lattice systems (with discussion). J Royal Stat Soc B1974;36: 192- 236. 2. Besag JE: Statistical analysis of non-lattice data. The Statistician 1975;24:179- 195. 3. Brooke MH, Engel WK: ‘I’he histological analysis of muscle biopsies with regard to fibre types. 1. Adult male and female. Neurology 1969;19:22 1-233. 4. Cohen MH, Lester JM, Bradley WG, Brenner JF, Hirsch RP, Silber DI, Ziegelmiller D: A computer model of denervation-reinnervation in skeletal muscle. Muscle N m i e 1987; 10:826-836. 5. Cunningham GW, Meijer PHE: A comparison of two Monte Carlo methods for coniputations in statistical mechanics. J Comp Phys 1976;20:50-63. 6. Dubes RC, Jain AK: Random field models in image analysis. J A p p l S t a l 1989;16:131-164. 7. Howel D, Brunsdon C: A simple test for the random arrangement of muscle fibres. J Neurol Sci 1987;77:49-57.
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Modeling Fiber Type Grouping
8. Jennekens FGI, Tomlinson BE, Walton J N : Data on the distribution of fibre types in five human limb muscles. An autopsy study. J Neurol Sci 197 1;14:245-257. 9. Johnson MA, Polgar ,I, Weightman D, Appleton D: Data on the distribution of fibre types in thirty-six human muscles. An autopsy study. ,J Neurol Sti 1973; 18: 11 1- 129. 10. Lester J M , Silber DI, Cohen MH, Hirsch RP, Bradley WG, Brenner JF: ‘I‘he co-dispersion index for the measurement of fiber type distribution patterns. Muscle Nerve 1983;6: 58 1-587. 1I . I,exell J , Downham D, Sjiistriim M: Distribution of different fibre type in human skeletal muscles. A statistical and computational model for the study of fibre type grouping and early diagnosis of skeletal muscle fibre denervatiori and reinnervation.j Neurol Sci 1983;61:301-314. 12. Lexell J , Downham D, Sjostrom M: Distribution of different fibre types in human skeletal muscles. A statistical and computational study of the fibre type arrangement in M. Vastus Lateralis of young healthy males. J Neurol Scz 1984;65:353- 365. 3. Lexell J, Downham D, Sjostrijm M: Morphological detection of neurogenic muscle disorders: how can statistical methods aid diagnosis? Acta NeuropathologicrL (Herl) 1987;75:109-115. 4. Lexell J, Wilson C, Downham D: Detection of fiber type grouping: further improvements to the enclosed fiber method. Muscle Nerve 1989; 12: 1024- 1025. 5. PernuS F, Erzen 1: Arrangement of fiber types within fascicles of human vastus lateralis muscle. Muscle Nerue 1991;14:304- 309. 16. Sjostrom M, Downham DY, Lexell J: Distribution of different fiber types in human skeletal muscles: Why is there a difference within a fascicle? Muscle Nerve 1986;9:30- 36. 17. Song SK, Shimada N, Anderson PJ: Orthogonal diameters in the analysis of muscle fibre size and form. Nature 1963;200:1220- 1221. 18. Sdlberg E: Use of single fiber EMG and macro EMG in study of reinnervation. Muscle Nerue 1990;13:804-813. 19. Venema HW, Overweg J: Analysis of the size and shape of cross-sections of muscle fibres. Med Bzol Eng 1974; 12:681692. 20. Venema HW: Spatial correlation in muscle fibre patterns. PhD Thesis, University of Amsterdam, 1982. 21. Venema HW: Spatial distribution of fiber types in skeletal muscle: Test for a random distribution. Muscle Nerve 1988;11:30l-311. 22. Venema HW: Spatial distribution of fiber types in skeletal muscle: Test for a random distribution-A reply. Mwcle Nerve 1989;12:697-698. 23. Venema HW: Determination of nearest neighbors in niuscle fibre patterns using a generalized version of the Dirichlet tessellation. Putt Recog Lett 1991;12:445-449. 24. Willison RG: Arrangement of muscle fibers of a single motor unit in mammalian muscles. Muscle Nenre 1980;3:360361.
MUSCLE & NERVE
June 1992