1 Specific aims

2 Massive new datasets obtained with techniques such as in situ hybridization

3 (ISH) and BAC-transgenics allow the expression levels of many genes at many

4 locations to be compared. Our goal is to develop automated methods to relate

5 spatial variation in gene expression to anatomy. We want to find marker genes

6 for specific anatomical regions, and also to draw new anatomical maps based on

7 gene expression patterns. We have three specific aims:

8 (1) develop an algorithm to screen spatial gene expression data for combi-

9 nations of marker genes which selectively target anatomical regions

10 (2) develop an algorithm to suggest new ways of carving up a structure into

11 anatomical subregions, based on spatial patterns in gene expression

12 (3) create a 2-D “flat map” dataset of the mouse cerebral cortex that con-

13 tains a flattened version of the Allen Mouse Brain Atlas ISH data, as well as

14 the boundaries of cortical anatomical areas. Use this dataset to validate the

15 methods developed in (1) and (2).

16 In addition to validating the usefulness of the algorithms, the application of

17 these methods to cerebral cortex will produce immediate benefits, because there

18 are currently no known genetic markers for many cortical areas. The results

19 of the project will support the development of new ways to selectively target

20 cortical areas, and it will support the development of a method for identifying

21 the cortical areal boundaries present in small tissue samples.

22 All algorithms that we develop will be implemented in an open-source soft-

23 ware toolkit. The toolkit, as well as the machine-readable datasets developed

24 in aim (3), will be published and freely available for others to use.

25 Background and significance

26 Aim 1

27 Machine learning terminology: supervised learning

28 The task of looking for marker genes for anatomical subregions means that

29 one is looking for a set of genes such that, if the expression level of those genes

30 is known, then the locations of the subregions can be inferred.

31 If we define the subregions so that they cover the entire anatomical structure

32 to be divided, then instead of saying that we are using gene expression to find

33 the locations of the subregions, we may say that we are using gene expression to

34 determine to which subregion each voxel within the structure belongs. We call

35 this a classification task, because each voxel is being assigned to a class (namely,

36 its subregion).

37 Therefore, an understanding of the relationship between the combination of

38 1

39

40 their expression levels and the locations of the subregions may be expressed as

41 a function. The input to this function is a voxel, along with the gene expression

42 levels within that voxel; the output is the subregional identity of the target

43 voxel, that is, the subregion to which the target voxel belongs. We call this

44 function a classifier. In general, the input to a classifier is called an instance,

45 and the output is called a label (or a class label).

46 The object of aim 1 is not to produce a single classifier, but rather to develop

47 an automated method for determining a classifier for any known anatomical

48 structure. Therefore, we seek a procedure by which a gene expression dataset

49 may be analyzed in concert with an anatomical atlas in order to produce a

50 classifier. Such a procedure is a type of a machine learning procedure. The

51 construction of the classifier is called training (also learning), and the initial

52 gene expression dataset used in the construction of the classifier is called training

53 data.

54 In the machine learning literature, this sort of procedure may be thought

55 of as a supervised learning task, defined as a task in whcih the goal is to learn

56 a mapping from instances to labels, and the training data consists of a set of

57 instances (voxels) for which the labels (subregions) are known.

58 Each gene expression level is called a feature, and the selection of which

59 genes to include is called feature selection. Feature selection is one component

60 of the task of learning a classifier. Some methods for learning classifiers start

61 out with a separate feature selection phase, whereas other methods combine

62 feature selection with other aspects of training.

63 One class of feature selection methods assigns some sort of score to each

64 candidate gene. The top-ranked genes are then chosen. Some scoring measures

65 can assign a score to a set of selected genes, not just to a single gene; in this

66 case, a dynamic procedure may be used in which features are added and sub-

67 tracted from the selected set depending on how much they raise the score. Such

68 procedures are called “stepwise” or “greedy”.

69 Although the classifier itself may only look at the gene expression data within

70 each voxel before classifying that voxel, the learning algorithm which constructs

71 the classifier may look over the entire dataset. We can categorize score-based

72 feature selection methods depending on how the score of calculated. Often

73 the score calculation consists of assigning a sub-score to each voxel, and then

74 aggregating these sub-scores into a final score (the aggregation is often a sum or

75 a sum of squares). If only information from nearby voxels is used to calculate a

76 voxel’s sub-score, then we say it is a local scoring method. If only information

77 from the voxel itself is used to calculate a voxel’s sub-score, then we say it is a

78 pointwise scoring method.

79 Key questions when choosing a learning method are: What are the instances?

80 What are the features? How are the features chosen? Here are four principles

81 that outline our answers to these questions.

82 Principle 1: Combinatorial gene expression

83 Above, we defined an “instance” as the combination of a voxel with the

84 “associated gene expression data”. In our case this refers to the expression level

85 of genes within the voxel, but should we include the expression levels of all

86 2

87

88 genes, or only a few of them?

89 It is too much to hope that every anatomical region of interest will be iden-

90 tified by a single gene. For example, in the cortex, there are some areas which

91 are not clearly delineated by any gene included in the Allen Brain Atlas (ABA)

92 dataset. However, at least some of these areas can be delineated by looking

93 at combinations of genes (an example of an area for which multiple genes are

94 necessary and sufficient is provided in Preliminary Results).

95 Principle 2: Only look at combinations of small numbers of genes

96 When the classifier classifies a voxel, it is only allowed to look at the expres-

97 sion of the genes which have been selected as features. The more data that is

98 available to a classifier, the better that it can do. For example, perhaps there

99 are weak correlations over many genes that add up to a strong signal. So, why

100 not include every gene as a feature? The reason is that we wish to employ

101 the classifier in situations in which it is not feasible to gather data about every

102 gene. For example, if we want to use the expression of marker genes as a trigger

103 for some regionally-targeted intervention, then our intervention must contain a

104 molecular mechanism to check the expression level of each marker gene before

105 it triggers. It is currently infeasible to design a molecular trigger that checks

106 the level of more than a handful of genes. Similarly, if the goal is to develop a

107 procedure to do ISH on tissue samples in order to label their anatomy, then it

108 is infeasible to label more than a few genes. Therefore, we must select only a

109 few genes as features.

110 Principle 3: Use geometry in feature selection

111 When doing feature selection with score-based methods, the simplest thing

112 to do would be to score the performance of each voxel by itself and then com-

113 bine these scores (pointwise scoring). A more powerful approach is to also use

114 information about the geometric relations between each voxel and its neighbors;

115 this requires non-pointwise, local scoring methods. See Preliminary Results for

116 evidence of the complementary nature of pointwise and local scoring methods.

117 Principle 4: Work in 2-D whenever possible

118 There are many anatomical structures which are commonly characterized in

119 terms of a two-dimensional manifold. When it is known that the structure that

120 one is looking for is two-dimensional, the results may be improved by allowing

121 the analysis algorithm to take advantage of this prior knowledge. In addition,

122 it is easier for humans to visualize and work with 2-D data.

123 Therefore, when possible, the instances should represent pixels, not voxels.

124 Aim 2

125 Machine learning terminology: clustering

126 If one is given a dataset consisting merely of instances, with no class labels,

127 then analysis of the dataset is referred to as unsupervised learning in the jargon

128 of machine learning. One thing that you can do with such a dataset is to group

129 instances together. A set of similar instances is called a cluster, and the activity

130 of finding grouping the data into clusters is called clustering or cluster analysis.

131 3

132

133 The task of deciding how to carve up a structure into anatomical subregions

134 can be put into these terms. The instances are once again voxels (or pixels)

135 along with their associated gene expression profiles. We make the assumption

136 that voxels from the same subregion have similar gene expression profiles, at

137 least compared to the other subregions. This means that clustering voxels is

138 the same as finding potential subregions; we seek a partitioning of the voxels

139 into subregions, that is, into clusters of voxels with similar gene expression.

140 It is desirable to determine not just one set of subregions, but also how

141 these subregions relate to each other, if at all; perhaps some of the subregions

142 are more similar to each other than to the rest, suggesting that, although at a

143 fine spatial scale they could be considered separate, on a coarser spatial scale

144 they could be grouped together into one large subregion. This suggests the

145 outcome of clustering may be a hierarchial tree of clusters, rather than a single

146 set of clusters which partition the voxels. This is called hierarchial clustering.

147 Similarity scores

148 A crucial choice when designing a clustering method is how to measure

149 similarity, across either pairs of instances, or clusters, or both. There is much

150 overlap between scoring methods for feature selection (discussed above under

151 Aim 1) and scoring methods for similarity.

152 Spatially contiguous clusters; image segmentation

153 We have shown that aim 2 is a type of clustering task. In fact, it is a

154 special type of clustering task because we have an additional constraint on

155 clusters; voxels grouped together into a cluster must be spatially contiguous.

156 In Preliminary Results, we show that one can get reasonable results without

157 enforcing this constraint, however, we plan to compare these results against

158 other methods which guarantee contiguous clusters.

159 Perhaps the biggest source of continguous clustering algorithms is the field

160 of computer vision, which has produced a variety of image segmentation algo-

161 rithms. Image segmentation is the task of partitioning the pixels in a digital

162 image into clusters, usually contiguous clusters. Aim 2 is similar to an image

163 segmentation task. There are two main differences; in our task, there are thou-

164 sands of color channels (one for each gene), rather than just three. There are

165 imaging tasks which use more than three colors, however, for example multispec-

166 tral imaging and hyperspectral imaging, which are often used to process satellite

167 imagery. A more crucial difference is that there are various cues which are ap-

168 propriate for detecting sharp object boundaries in a visual scene but which are

169 not appropriate for segmenting abstract spatial data such as gene expression.

170 Although many image segmentation algorithms can be expected to work well

171 for segmenting other sorts of spatially arranged data, some of these algorithms

172 are specialized for visual images.

173 Dimensionality reduction

174 Unlike aim 1, there is no externally-imposed need to select only a handful

175 of informative genes for inclusion in the instances. However, some clustering

176 algorithms perform better on small numbers of features. There are techniques

177 which “summarize” a larger number of features using a smaller number of fea-

178 tures; these techniques go by the name of feature extraction or dimensionality

179 4

180

181 reduction. The small set of features that such a technique yields is called the

182 reduced feature set. After the reduced feature set is created, the instances may

183 be replaced by reduced instances, which have as their features the reduced fea-

184 ture set rather than the original feature set of all gene expression levels. Note

185 that the features in the reduced feature set do not necessarily correspond to

186 genes; each feature in the reduced set may be any function of the set of gene

187 expression levels.

188 Another use for dimensionality reduction is to visualize the relationships

189 between subregions. For example, one might want to make a 2-D plot upon

190 which each subregion is represented by a single point, and with the property

191 that subregions with similar gene expression profiles should be nearby on the

192 plot (that is, the property that distance between pairs of points in the plot

193 should be proportional to some measure of dissimilarity in gene expression). It

194 is likely that no arrangement of the points on a 2-D plan will exactly satisfy

195 this property – however, dimensionality reduction techniques allow one to find

196 arrangements of points that approximately satisfy that property. Note that

197 in this application, dimensionality reduction is being applied after clustering;

198 whereas in the previous paragraph, we were talking about using dimensionality

199 reduction before clustering.

200 Clustering genes rather than voxels

201 Although the ultimate goal is to cluster the instances (voxels or pixels), one

202 strategy to achieve this goal is to first cluster the features (genes). There are

203 two ways that clusters of genes could be used.

204 Gene clusters could be used as part of dimensionality reduction: rather than

205 have one feature for each gene, we could have one reduced feature for each gene

206 cluster.

207 Gene clusters could also be used to directly yield a clustering on instances.

208 This is because many genes have an expression pattern which seems to pick

209 out a single, spatially continguous subregion. Therefore, it seems likely that an

210 anatomically interesting subregion will have multiple genes which each individ-

211 ually pick it out1. This suggests the following procedure: cluster together genes

212 which pick out similar subregions, and then to use the more popular common

213 subregions as the final clusters. In the Preliminary Data we show that a num-

214 ber of anatomically recognized cortical regions, as well as some “superregions”

215 formed by lumping together a few regions, are associated with gene clusters in

216 this fashion.

217 Aim 3

218 Background

219 _______________

220 1This would seem to contradict our finding in aim 1 that some cortical areas are combina-

221 torially coded by multiple genes. However, it is possible that the currently accepted cortical

222 maps divide the cortex into subregions which are unnatural from the point of view of gene

223 expression; perhaps there is some other way to map the cortex for which each subregion can

224 be identified by single genes.

225 5

226

227 The cortex is divided into areas and layers. To a first approximation, the

228 parcellation of the cortex into areas can be drawn as a 2-D map on the surface of

229 the cortex. In the third dimension, the boundaries between the areas continue

230 downwards into the cortical depth, perpendicular to the surface. The layer

231 boundaries run parallel to the surface. One can picture an area of the cortex as

232 a slice of many-layered cake.

233 Although it is known that different cortical areas have distinct roles in both

234 normal functioning and in disease processes, there are no known marker genes

235 for many cortical areas. When it is necessary to divide a tissue sample into

236 cortical areas, this is a manual process that requires a skilled human to combine

237 multiple visual cues and interpret them in the context of their approximate

238 location upon the cortical surface.

239 Even the questions of how many areas should be recognized in cortex, and

240 what their arrangement is, are still not completely settled. A proposed division

241 of the cortex into areas is called a cortical map. In the rodent, the lack of a

242 single agreed-upon map can be seen by contrasting the recent maps given by

243 Swanson?? on the one hand, and Paxinos and Franklin?? on the other. While

244 the maps are certainly very similar in their general arrangement, significant

245 differences remain in the details.

246 Significance

247 The method developed in aim (1) will be applied to each cortical area to find

248 a set of marker genes such that the combinatorial expression pattern of those

249 genes uniquely picks out the target area. Finding marker genes will be useful

250 for drug discovery as well as for experimentation because marker genes can be

251 used to design interventions which selectively target individual cortical areas.

252 The application of the marker gene finding algorithm to the cortex will

253 also support the development of new neuroanatomical methods. In addition to

254 finding markers for each individual cortical areas, we will find a small panel

255 of genes that can find many of the areal boundaries at once. This panel of

256 marker genes will allow the development of an ISH protocol that will allow

257 experimenters to more easily identify which anatomical areas are present in

258 small samples of cortex.

259 The method developed in aim (3) will provide a genoarchitectonic viewpoint

260 that will contribute to the creation of a better map. The development of present-

261 day cortical maps was driven by the application of histological stains. It is

262 conceivable that if a different set of stains had been available which identified

263 a different set of features, then the today’s cortical maps would have come out

264 differently. Since the number of classes of stains is small compared to the number

265 of genes, it is likely that there are many repeated, salient spatial patterns in

266 the gene expression which have not yet been captured by any stain. Therefore,

267 current ideas about cortical anatomy need to incorporate what we can learn

268 from looking at the patterns of gene expression.

269 While we do not here propose to analyze human gene expression data, it is

270 conceivable that the methods we propose to develop could be used to suggest

271 modifications to the human cortical map as well.

272 6

273

274 Related work

275 There does not appear to be much work on the automated analysis of spatial

276 gene expression data.

277 There is a substantial body of work on the analysis of gene expression data,

278 however, most of this concerns gene expression data which is not fundamentally

279 spatial, for example, microarray datasets. In some cases, a few locations have

280 been sampled, but such a dataset is still of a fundamentally different character

281 than a dataset containing a large grid of sampling points distributed over space.

282 In relating gene expression to anatomy, it is the spatial aspects of the problem

283 which are the most important.

284 As noted above, there has been much work on both supervised learning and

285 clustering, and there are many available algorithms for each. Many of these

286 algorithms are flexible enough to accomodate new scoring measures; and the

287 performance of most of the algorithms is greatly affected by preprocessing and

288 by the choice of which representation to use for feature values. We think it likely

289 that for this application, the development of domain-specific scoring measures

290 (such as gradient similarity, which is discussed in Preliminary Work) will be

291 necessary in order to achieve the best results. In essence, the machine learning

292 community has provided algorithms, but the scientist must provide a framework

293 for representing the problem domain, and the way that this framework is set

294 up has a large impact on performance. Creating a good framework can require

295 creatively reconceptualizing the problem domain, and is not merely a mechanical

296 “fine-tuning” of numerical parameters. Therefore, the completion of Aims 1

297 and 2 involves more than just reimplementing an existing algorithm, and more

298 than just choosing between a set of existing algorithms, and will constitute a

299 substantial contribution to biology.

300 We are aware of one other effort to computationally analyze spatial gene

301 expression data.

302 In the Preliminary Work, we show that

303 The creation of a domain-specific scoring measure may be required in order

304 to achieve good performance, and it is not impossible that the algorithms them-

305 selves will have to be extended. We plan to test out existing algorithms and

306 scoring measures,

307 Therefore, we anticipate

308 Therefore, it is unclear which of the

309 todo

310 vs. AGEA – i wrote something on this but i’m going to rewrite it

311 Preliminary work

312 Format conversion between SEV, MATLAB, NIFTI

313 todo

314 7

315

316 Flatmap of cortex

317 todo

318 Using combinations of multiple genes is necessary and sufficient to

319 delineate some cortical areas

320 Here we give an example of a cortical area which is not marked by any

321 single gene, but which can be identified combinatorially. according to logistic

322 regression, gene wwc12 is the best fit single gene for predicting whether or not a

323 pixel on the cortical surface belongs to the motor area (area MO). The upper-left

324 picture in Figure shows wwc1’s spatial expression pattern over the cortex. The

325 lower-right boundary of MO is represented reasonably well by this gene, however

326 the gene overshoots the upper-left boundary. This flattened 2-D representation

327 does not show it, but the area corresponding to the overshoot is the medial

328 surface of the cortex. MO is only found on the lateral surface (todo).

329 Gnee mtif23 is shown in figure the upper-right of Fig. . Mtif2 captures MO’s

330 upper-left boundary, but not its lower-right boundary. Mtif2 does not express

331 very much on the medial surface. By adding together the values at each pixel

332 in these two figures, we get the lower-left of Figure . This combination captures

333 area MO much better than any single gene.

334 Correlation todo

335 Conditional entropy todo

336 Gradient similarity todo

337 Geometric and pointwise scoring methods provide complementary

338 information

339 To show that local geometry can provide useful information that cannot be

340 detected via pointwise analyses, consider Fig. . The top row of Fig. displays the

341 3 genes which most match area AUD, according to a pointwise method4. The

342 bottom row displays the 3 genes which most match AUD according to a method

343 which considers local geometry5 The pointwise method in the top row identifies

344 genes which express more strongly in AUD than outside of it; its weakness is that

345 this includes many areas which don’t have a salient border matching the areal

346 border. The geometric method identifies genes whose salient expression border

347 seems to partially line up with the border of AUD; its weakness is that this

348 includes genes which don’t express over the entire area. Genes which have high

349 rankings using both pointwise and border criteria, such as Aph1a in the example,

350 may be particularly good markers. None of these genes are, individually, a

351 perfect marker for AUD; we deliberately chose a “difficult” area in order to

352 better contrast pointwise with geometric methods.

353 __________________________

354 2“WW, C2 and coiled-coil domain containing 1”; EntrezGene ID 211652

355 3“mitochondrial translational initiation factor 2”; EntrezGene ID 76784

356 4For each gene, a logistic regression in which the response variable was whether or not a

357 surface pixel was within area AUD, and the predictor variable was the value of the expression

358 of the gene underneath that pixel. The resulting scores were used to rank the genes in terms

359 of how well they predict area AUD.

360 5For each gene the gradient similarity (see section ??) between (a) a map of the expression

361 of each gene on the cortical surface and (b) the shape of area AUD, was calculated, and this

362 was used to rank the genes.

363 8

364

365

366

367 Figure 1: Upper left: wwc1. Upper right: mtif2. Lower left: wwc1 + mtif2

368 (each pixel’s value on the lower left is the sum of the corresponding pixels in

369 the upper row). Within each picture, the vertical axis roughly corresponds to

370 anterior at the top and posterior at the bottom, and the horizontal axis roughly

371 corresponds to medial at the left and lateral at the right. The red outline is

372 the boundary of region MO. Pixels are colored approximately according to the

373 density of expressing cells underneath each pixel, with red meaning a lot of

374 expression and blue meaning little.

375

376

377 Figure 2: The top row shows the three genes which (individually) best predict

378 area AUD, according to logistic regression. The bottom row shows the three

379 genes which (individually) best match area AUD, according to gradient similar-

380 ity. From left to right and top to bottom, the genes are Ssr1, Efcbp1, Aph1a,

381 Ptk7, Aph1a again, and Lepr

382 9

383

384 Areas which can be identified by single genes

385 todo

386 Specific to Aim 1 (and Aim 3)

387 Forward stepwise logistic regression todo

388 SVM on all genes at once

389 In order to see how well one can do when looking at all genes at once, we

390 ran a support vector machine to classify cortical surface pixels based on their

391 gene expression profiles. We achieved classification accuracy of about 81%6.

392 As noted above, however, a classifier that looks at all the genes at once isn’t

393 practically useful.

394 The requirement to find combinations of only a small number of genes limits

395 us from straightforwardly applying many of the most simple techniques from

396 the field of supervised machine learning. In the parlance of machine learning,

397 our task combines feature selection with supervised learning.

398 Decision trees

399 todo

400 Specific to Aim 2 (and Aim 3)

401 Raw dimensionality reduction results

402 Dimensionality reduction plus K-means or spectral clustering

403 Many areas are captured by clusters of genes

404 todo

405 todo

406 Research plan

407 todo amongst other things:

408 Develop algorithms that find genetic markers for anatomical re-

409 gions

410 1. Develop scoring measures for evaluating how good individual genes are at

411 marking areas: we will compare pointwise, geometric, and information-

412 theoretic measures.

413 2. Develop a procedure to find single marker genes for anatomical regions: for

414 each cortical area, by using or combining the scoring measures developed,

415 we will rank the genes by their ability to delineate each area.

416 3. Extend the procedure to handle difficult areas by using combinatorial cod-

417 ing: for areas that cannot be identified by any single gene, identify them

418 with a handful of genes. We will consider both (a) algorithms that incre-

419 mentally/greedily combine single gene markers into sets, such as forward

420 __________________________

421 65-fold cross-validation.

422 10

423

424 stepwise regression and decision trees, and also (b) supervised learning

425 techniques which use soft constraints to minimize the number of features,

426 such as sparse support vector machines.

427 4. Extend the procedure to handle difficult areas by combining or redrawing

428 the boundaries: An area may be difficult to identify because the bound-

429 aries are misdrawn, or because it does not “really” exist as a single area,

430 at least on the genetic level. We will develop extensions to our procedure

431 which (a) detect when a difficult area could be fit if its boundary were

432 redrawn slightly, and (b) detect when a difficult area could be combined

433 with adjacent areas to create a larger area which can be fit.

434 Apply these algorithms to the cortex

435 1. Create open source format conversion tools: we will create tools to bulk

436 download the ABA dataset and to convert between SEV, NIFTI and MAT-

437 LAB formats.

438 2. Flatmap the ABA cortex data: map the ABA data onto a plane and draw

439 the cortical area boundaries onto it.

440 3. Find layer boundaries: cluster similar voxels together in order to auto-

441 matically find the cortical layer boundaries.

442 4. Run the procedures that we developed on the cortex: we will present, for

443 each area, a short list of markers to identify that area; and we will also

444 present lists of “panels” of genes that can be used to delineate many areas

445 at once.

446 Develop algorithms to suggest a division of a structure into anatom-

447 ical parts

448 1. Explore dimensionality reduction algorithms applied to pixels: including

449 TODO

450 2. Explore dimensionality reduction algorithms applied to genes: including

451 TODO

452 3. Explore clustering algorithms applied to pixels: including TODO

453 4. Explore clustering algorithms applied to genes: including gene shaving,

454 TODO

455 5. Develop an algorithm to use dimensionality reduction and/or hierarchial

456 clustering to create anatomical maps

457 6. Run this algorithm on the cortex: present a hierarchial, genoarchitectonic

458 map of the cortex

459 11

460

461 _______________________________________________________________________________________________________ stuff i dunno where to put yet (there is more scattered through grant-

462 oldtext):

463 Principle 4: Work in 2-D whenever possible

464 In anatomy, the manifold of interest is usually either defined by a combina-

465 tion of two relevant anatomical axes (todo), or by the surface of the structure

466 (as is the case with the cortex). In the former case, the manifold of interest is

467 a plane, but in the latter case it is curved. If the manifold is curved, there are

468 various methods for mapping the manifold into a plane.

469 The method that we will develop will begin by mapping the data into a

470 2-D plane. Although the manifold that characterized cortical areas is known

471 to be the cortical surface, it remains to be seen which method of mapping the

472 manifold into a plane is optimal for this application. We will compare mappings

473 which attempt to preserve size (such as the one used by Caret??) with mappings

474 which preserve angle (conformal maps).

475 Although there is much 2-D organization in anatomy, there are also struc-

476 tures whose shape is fundamentally 3-dimensional. If possible, we would like

477 the method we develop to include a statistical test that warns the user if the

478 assumption of 2-D structure seems to be wrong.

479 todo: replace aim # bullet pts with #s

480 12

481

482