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

Modeling Fiber Type Grouping

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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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727

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

Modeling fiber type grouping by a binary Markov random field.

A new approach to the quantification of fiber type grouping is presented, in which the distribution of histochemical type in a muscle cross section is...
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