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Minimal Surface
Brain Research Bulletin, Vol. 44, No. 2, pp. 117–124, 1997 Copyright © 1997 Elsevier Science Inc. Printed in the USA. All rights reserved 0361-9230/97 $17.00 1 .00
PII S03061-9230(97)00113-5
Minimal Surface: A Useful Paradigm to Describe the Deeper Part of the Corpus Callosum? J.-L. STIEVENART,*†1 M.-T. IBA-ZIZEN,* A. TOURBAH,*‡ A. LOPEZ,* M. THIBIERGE,* A. ABANOU,* AND E. A. CABANIS* *Service de Neuroradiologie, Centre Hospitalier National d’Ophtalmologie des XV-XX, 28 rue de Charenton, Paris, F-75012, France, †Service de Biophysique et de Me´decine Nucle´aire, Hoˆpital Beaujon, Clichy, France; Fe´de´ration de Neurologie and ‡Institut National de la Sante´ et de la Recherche Me´dicale, Unite´ 134, Hoˆpital de la Salpeˆtrie´re, Paris, France [Received 28 November 1996; Revised 28 April 1997; Accepted 1 May 1997] ABSTRACT: The aim of this magnetic resonance imaging study was to find a geometrical characterization of the deeper part of the corpus callosum. Its shape was studied in 12 middle-aged persons free of white matter pathology. Profiles of curvatures were measured showing that this surface was close to a minimal one, especially at the genu and near the splenium. To assess the effect of a white matter pathology on these geometrical features, the same measurements were performed in an extra group of nine patients with definite multiple sclerosis. The hypothesis of curvatures profiles parallelism for the two groups could be rejected at the 0.05 confidence level for the mean curvatures but not for the Gaussian ones. Curvatures profiles may give indications on balance between the cortex and the fiber bundles growth rates during the development and on large scale modifications cooccurring with multilocular white matter pathologies. © 1997 Elsevier Science Inc.
the CC by analyzing the covariability of the CC thickness measured at 100 equally spaced locations along the corpus callosum axis in a population of 104 normal adults and determined seven regions with consistent variations (seven factors). Angle measurements, which give a more intrinsic description, were done between some lines built on characteristic points of the outer surface of the corpus callosum [8]. These measurements are clearly related to the standardized longitudinal planar curvatures. These angles are altered in certain malformative syndromes. However, the sagittal view alone of the corpus callosum is not sufficient to yield a three-dimension geometrical characterization that seems more interesting because of the essential bihemispheric interconnecting function of the CC. The studies of the human brain development show that the corpus callosum begins its growth at the anterior region during the 10th week of gestation, when fibers cross the midline in the massa commissuralis. It then progressively extends backwards so that all its axonal elements are present at the 20th week [19]. Then, due to the global growth of the brain and to the myelination process, the shape of the CC is modified. For example the genu, which has appeared by the 16th week, progressively acquires its final morphology. The program of such a development can be thought of as a maximal extension of the neocortex with the constraint to preserve connections between cortical regions (mainly homotopic, but also heterotopic ones). Because of metabolic limitation, the cost of creating and developing such connections must be kept to a minimum. Let us consider as a surface the layer of fibers passing through the CC and connecting two homotopic longitudinal lines of neurons drawn on the cortex. For a given length of this surface trace on the CC, it can be proved that when this surface has a negative Gaussian curvature everywhere (Appendix 1), the associated layer of fibers connects longer lines of neurons than in any other condition. Among those surfaces, the ones with null mean curvature, the so-called minimal surfaces [2,6], may represent a critical situation. Any local deformation of such a surface produces an area stretching that can generate an unusual stress on some elements of this fibers layer. This could be an explanation of localized weakness of the white matter. We, thus, tried to characterize the closeness of the CC inner surface to a minimal surface in
KEY WORDS: MR imaging, 3D postprocessing, Neuroanatomy, Corpus Callosum, Morphometry, Multiple sclerosis.
INTRODUCTION The corpus callosum (CC) plays a major role in the architecture of the brain and is considered as a sensitive indicator of both the cortical and the white matter state [26]. Numerous studies have analyzed the thickness as well as the areas of different parts of the CC [26]. Some variations were noted in relation with age, gender [11], handedness [27], or with different pathologies [12]: ischemic modifications [29], multiple sclerosis [16,22], schizophrenia [15]. These studies are difficult to reproduce or even can lead to conflicting results [23,28]. It is a well-acknowledged fact that the difficulties stem from the shape variability of the corpus callosum in the midsagittal plane where the longitudinal curvature modifies relationships between length, width, and area of corpus callosum segments. To overcome this purely geometrical difficulty, the between-group comparison can be done on a deformation function used to perform a previous shape normalization of the CC [3]. Moreover, there are some uncertainties in the definition of CC regions as, for example, the “posterior fifth of the CC” for the splenium [4]. A study [5] tried to find a “natural” segmentation of 1
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a population of 12 middle-aged patients or healthy volunteers with no white matter lesions by computing its mean curvatures profiles. We chose to explore the curvatures of the CC inner surface, because it was more easily delineated on the transverse sections than the outer one, which was partially obscured by the indusium griseum. The primary purpose of our study was to present a geometric characterization that could give information on differential growing rates of cortex and fiber bundles. However, to demonstrate its utility in investigating large-scale modification coexisting with multilocular white matter pathologies, we studied in the same way nine patients with definite multiple sclerosis to see whether they could be distinguished from the former group on the basis of the corpus callosum inner surface shape. We thought that multiple sclerosis was a well-suited pathology to test the relevance of the proposed analysis, because the white matter alterations may cause— or may be due to—a modification of its global shape. MATERIALS AND METHODS The studied population was retrospectively selected among patients who underwent a 3D MR examination (GE Medical Systems SIGNA 1.5T) in our institution. These patients were classified into two groups: a reference group and a multiple sclerosis group. The reference group (four men, eight women) is assumed to be representative of the normal middle aged population as far as the CC shape is concerned. It consists in three healthy volunteers and nine patients whose MR examination was normal or demonstrated minute cortical angiomas or hypophisial microadenoma. Their ages ranged from 21 to 41 (mean 30.7). Patients in the MS group (three men, six women) met the Poser’s criteria for definite multiple sclerosis [17]. Their ages ranged from 21 to 48 (mean 32.3). The 3D-Fourier Transform acquisition sequence was T1weighted and performed in the sagittal plane (SPGR flip angle 25°, TE 8 ms, TR 35 ms). The stack of 124 contiguous 1.3 mm-thick slices was reformatted on a workstation (GEMS Advantage Window 1.2) with dedicated software (GEMS Voxtool) to display the true midsagittal plane (Fig. 1a). This plane was manually determined by the operator using the standard orthogonal views display and the oblique plane manipulation of the software. In this midsagittal plane, the deeper part of the CC was delineated by a cubic spline passing through control points given by the operator. The geodesic (curvilinear) length of this curve—the corpus callosum length (CCL)—was measured from the rostrum to a posterior landmark corresponding to the closest point to the posterior commissure (Fig. 1b). This point can be constructed as the tangency point to the splenium of a circle centered by the posterior commissure. The CCL, directly given by the software in millimeters, was used as a standardization variable. Ten points were equally spaced (geodesic distance) on this curve. This determined 11 regular points (C1, C2,. . ., C11—rostrum excluded) where the surface curvatures were estimated (Fig. 1b). Due to symmetry, the principal planes (Appendix 1) of the surface along its intersection with the midsagittal plane are this plane itself and the ones perpendicular to the trace of corpus callosum on it. These planes are automatically given by the Voxtool software by using the “lock cursor to trace” option. The two curvatures c1 and c2 were calculated by means of formula 3 (Appendix 2). The various lengths involved in this formula were measured on calibrated films by using the template of Fig. A4 drawn on graph tracing paper. To assess the accuracy of this method, a 3D sagittal acquisition, with identical parameters to the ones used for patients, was performed on a cylindrical 51-mm diameter phantom. Its axis was placed along the anteroposterior direction. Repeated measurements on
FIG. 1. (a) Midsagittal view of reference patient #3 (FOV: 13 cm); (b) points where curvatures were measured. Line segments figure the transverse planes where the planar curvatures were measured. Intersections of these segments with the inferior trace of the CC are the points C1, C2, C11. The length of the thick line is by definition the length of the corpus callosum (CCL)—see text.
resulting images demonstrated that the measurement was unbiased and provided us with an estimation of the error of about 3 mm on the radius of curvature. For the patients, measures were repeated twice and the resulting curvature radii were averaged. Then the curvature radii were divided by the CCL, yielding thus dimensionless curvatures expressed in (%CCL)21 for the mean curvatures (H-curvatures) and in (%CCL)22 for the Gaussian curvatures (K-curvatures). The reproducibility of this procedure was assessed by a simple linear regression analysis between 22 pairs of measurements done by the same operator 2 weeks apart. This reproducibility was good for the H-curvatures (r2 5 0.906) but rather low for the K-curvatures (r2 5 0.804). All the discrepancies occurred on the more curved regions as, for example, the genu. All the measurements in this study were performed by the same operator.
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TABLE 1 DESCRIPTIVE STATISTICS OF STANDARDIZED H AND K CURVATURES H-Curvatures Location, Group
C1, C C2, C C3, C C4, C C5, C C6, C C7, C C8, C C9, C C10, C C11, C C1, MS C2, MS C3, MS C4, MS C5, MS C6, MS C7, MS C8, MS C9, MS C10, MS C11, MS
K-Curvatures
Mean
Std. Dev.
Mean
Std. Dev.
0,06985 20,00323 20,02361 20,02327 20,02332 20,02798 20,03029 20,03049 20,01811 0,00144 20,10044 0,02792 20,01820 20,02652 20,01975 20,02181 20,02221 20,03435 20,06642 20,04029 0,02155 20,14891
0,19253 0,03977 0,00932 0,01127 0,01058 0,00538 0,00550 0,01354 0,01930 0,04250 0,03137 0,06474 0,01797 0,01456 0,00996 0,01003 0,01733 0,01216 0,03648 0,02747 0,05943 0,05430
20,02346 20,00542 20,00125 20,00114 20,00127 20,00084 20,00074 20,00170 20,00195 20,00248 0,00156 20,02563 20,00347 20,00134 20,00180 20,00143 20,00169 20,00150 20,00229 20,00429 20,00619 0,00164
0,03809 0,00772 0,00077 0,00050 0,00100 0,00061 0,00083 0,00145 0,00095 0,00269 0,00238 0,02622 0,00186 0,00174 0,00105 0,00061 0,00249 0,00177 0,00477 0,00448 0,00739 0,00317
C1 and C2 exhibited a large variability and were no further analyzed. On the greatest part of the CC this surface had a negative mean curvature with the chosen orientation (transverse curvature greater than longitudinal one), and a negative Gaussian curvature. Two regions of null mean curvature were identified near the genu and near the splenium (Fig. 2). Analysis of variance for profiles data demonstrated that the parallelism of mean curvatures profiles from C3 to C11 (no curvatures by group interaction) can be rejected at the confidence level of 0.05 (p 5 0.0007 and p 5 0.0123 with the Huynh–Feldt’s correction). This was not true for the Gaussian curvatures profiles. The factor analysis of the mean curvatures was performed on both groups considered as a single one. Three factors were used for this analysis. They explained 68.5% of the variation of the original curvatures (Table 2a). Mean curvatures at C3, C4, and C10 were poorly described by this three-factor model because the corresponding communalities were less than 60%. The first factor was correlated with the posterior mean curvatures (C8 –C9). The second factor was strongly linked with the curvature at C11. The third factor did not receive a clear interpretation. To visualize a possible relationship between mean curvatures and atrophy, symbols corresponding to patients with corpus callosum atrophy were identified. In the space of the two first factors the control patients were gathered near the origin and all patients with multiple sclerosis but one were located at the periphery (Fig. 3). For the factor analysis
9 MS patients. Only curvatures from C3 to C11 were used for the profile analysis. The hypothesis of profiles parallelism in both groups can be rejected (p , 0.05) for the mean curvatures but not for the Gaussian curvatures.
Mean profiles of H- and K-curvatures were first calculated and analyzed with a repeated measures analysis of variance [13]. To explore the covariance structure of the curvatures along the inferior trace, two factor analysis [13] were performed with a standard statistical package (Abacus Concepts, StatView 4.0). The aim of this kind of analysis is to reduce a large number of correlated variables measured on individuals to a smaller set of underlying variables called factors. Each original variable is a linear combination of these factors plus a specific random term. The coefficient by which is multiplied a factor in this linear form is called the factor loading. In our application, the factor loading was the correlation of the factor with the original variable. No oblique transformation was done on these factors, which were the original orthogonal, unrotated ones, extracted by a principal components analysis. The factor score is the value taken by the factor for a given individual. Corpus callosum atrophy was quantified by the ratio of the square root of its sagittal area to CCL yielding a dimensionless index expressing the overall thinness of the CC. In our reference population this ratio average was 0.36, with a standard deviation of 0.035. Assuming a normal distribution of this thinness index, less than 5% of the population has an index below 0.30 (0.36 –1.645 3 0.035). The corpus callosum was thus termed “atrophic” when this ratio was below 0.30. RESULTS Profiles of mean and gaussian curvatures along the deep sagittal strip of the CC were established (Table 1). Curvatures at locations
FIG. 2. Profiles of averaged curvatures along the deeper part of the corpus callosum. Units are (%CCL)21 for the mean curvatures and (%CCL)22 for the Gaussian curvatures. Vertical bars represent the 95% confidence intervals. (a) Mean curvatures (regions with null mean curvature are circled); (b) Gaussian curvatures.
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STIEVENART ET AL. TABLE 2 FACTOR ANALYSIS OF STANDARDIZED CURVATURES Factors Loadings
Location
Communality Summary
Factor 1
Factor 2
Factor 3
SMC
Final Estimate
C3 C4 C5 C6 C7 C8 C9 C10 C11
20,258 20,393 20,608 20,568 0,530 0,849 0,886 20,133 0,244
0,558 0,271 20,015 0,554 0,630 0,022 20,066 20,438 0,832
0,396 0,583 0,502 20,500 0,202 0,324 0,044 0,417 20,052
0,496 0,409 0,500 0,703 0,505 0,736 0,757 0,361 0,586
0,535 0,568 0,622 0,879 0,719 0,826 0,792 0,383 0,755
C3 C4 C5 C6 C7 C8 C9 C10 C11
0,242 20,078 0,382 0,857 0,863 0,865 0,437 0,056 0,585
20,436 0,829 0,118 20,162 20,297 0,055 0,834 0,819 0,108
20,038 0,150 0,741 20,307 20,154 20,135 20,218 20,245 0,616
0,326 0,672 0,394 0,837 0,787 0,875 0,896 0,792 0,640
0,250 0,717 0,710 0,855 0,858 0,769 0,936 0,735 0,734
a
b
The absolute value of a factor loading close to one express a strong contribution of the variable to this factor. Factor loadings greater than 0.8 have been highlighted. The communality is the proportion of variance of the variable, which is explained by the three-factor model. The squared multiple correlation (SMC) of a variable with all the other variables is often considered as an initial estimate of this quantity. (a) Analyis of the H-curvatures; (b) Analysis of the K-curvatures.
of the Gaussian curvatures, three factors explained 72.9% of the variation of the original curvatures (Table 2b). The first factor was linked with curvatures at C6, C7, and C8, whereas the second one was linked with curvatures at C4, C9, and C10. C3 was poorly described by this three-factor model because its communality was 0.25. The repartition of the individuals in the factor space had no particularity related to their group. In the multiple sclerosis group the mean of the previously defined CC atrophy index, 0.30, was significantly smaller than in the reference group (p 5 0.004, Student’s t-test).
phantom study, where the relative error on the radius of curvature measurement was approximately 12% (for a radius of curvature of 25.5 mm.). This could be an explanation of the great variability of H- and K-curvatures at C1 and C2 where the radii of curvature were both smaller than anywhere else (in the range 5–20 mm.). An other explanation could be that the CC shape at the genu is sometimes close to a singularity with a very small radius of curvature and that this near-singularity is variably located with respect to C1 and C2. In spite of those difficulties, the curvatures seem to us relevant parameters because they are presumably related to the growth rates of different regions during the development of the brain. These preliminary results incite to develop a more automatic and a more accurate procedure to build the curvatures profiles, especially on regions with small radius of curvature. One must stress the fact that H- and K-curvatures are different in nature. The mean curvature is an extrinsic property in the sense that it is changed, for example, by pure folding, without stretching or tearing the surface. On the contrary, the Gaussian curvature cannot be changed this way and is, thus, an intrinsic property [9]. The same difference exists between the Euclidian distance, which is extrinsic, and the geodesic distance, which is intrinsic, or between the volume enclosed by a surface and this surface area. The presented reference cases had a remarkable invariance of the standardized mean curvature of the deeper part of the anterior region (C3–C6) of the corpus callosum trunk with an average mean curvature equal to 20.024. The region of C7 corresponds to the level of the pyramidal tract. The H-curvature at this point was the one with the smallest coefficient of variation respectively in both populations. The constraints induced by the pyramidal tract during the growth of the brain could explain the stable extrinsic geometrical properties of this region. Differences in CC H-curvatures profiles between the two groups laid in the posterior part of the corpus callosal trunk (C8 –C9) where large-diameter fibers are more dense [1]. There, the form of the CC in the multiple sclerosis group is farther from a minimal surface than in the reference group. This region covers approximately the isthmus that interconnects the right and left parieto-temporal cortex. Its sagittal area was found to be greater in women [23,28]. It corresponds to the regions W65-74 and W77-85 of the factor analysis of the widths of the CC [5]. These regions as well as the splenium were also found to show important modifi-
DISCUSSION The accuracy of the curvature measurements is a serious problem. That is why we used an averaged value of two independent measurements. Sophisticated methods exist [21]. But those methods themselves could be ineffective for example on the posterior third of the CC where images show a mixing between the CC and the crus of fornix. There, some anatomical knowledge is necessary to correctly detect the relevant surface (Fig. 4). The error calculus outlined in Appendix 2 shows that the smaller the radius of curvature the bigger is the absolute error on H-curvature measurement. On the other hand, the relative error on K-curvature measurement is simply proportional to relative error on radius of curvature measurement. Even if the absolute error on radius of curvature is not strictly constant for the whole range of encountered values, the relative error is clearly a decreasing function of the radius of curvature. A point of this function is given by the
FIG. 3. Repartition of individuals in the space of the two first factors (H-curvatures analysis)—axes correspond to the factor scores. The curve encloses all normal patients plus one patient with multiple sclerosis. Symbols of patients with corpus callosum atrophy are solid. The arrow shows the representative point of patient #3 (Fig. 1a).
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FIG. 4. Detection of the relevant surface on the posterior part of the corpus callosum trunk (points C8 and C9). (a) Sagittal view (FOV: 3 cm., the line is the trace of the transverse view); (b) transverse view orthogonal to the trace of the corpus callosum on the midsagittal plane. The crus of fornix (arrows) is not to be taken into account to determine the curvature at M.
cations in size and shape with age (in the range 4 –18) [18]. In the factor analysis of H-curvatures, the first factor indicates that curvatures at C8 and C9 are strongly correlated and that they are linked with corpus callosum atrophy because most of the representative points of such cases lay on the positive region of the first factor axis. To confirm this point, another factor analysis was conducted, not reproduced here, adding our thinness index into the variables set. It clearly demonstrated the positive link between the
121 H-curvatures at C8 and C9, the thinness index, and the first factor. This method avoids to make assumption on what is called a CC atrophy. There is no obvious geometrical relationship between the CC atrophy and its inferior curvatures. But a parallel outward displacement of the CC inner surface results in a decrease (more negative value) of its mean curvature. The second factor was associated with the mean curvature at C11. Due to the local cylindrical shape of the splenium (null transverse curvature) this curvature is roughly proportional to the inverse of its sagittal radius and is, thus, related to its standardized size. The K-curvatures at C3–C5 are relatively stable with an average standardized value of 20.00122 in the reference population. The region of C6 –C7 is the less intrinsically curved one (excluding a small posterior region between C10 and C11 where the Gaussian curvature is null). Its average standardized Gaussian curvature is equal to 20.00079 in the reference population. We examined with more attention K-curvatures at locations C4, C9, and C10, where differences between groups were the closest to significance with respective p-values of 0.068, 0.092, and 0.12. An estimation of the test power, carried out for these locations, yielded values below 20% to detect a 50% difference on the radii of curvature with the usual type I risk of 5%. Especially for C9 and C10, these low values are attributable to the high variability of the corresponding curvatures measures in the multiple sclerosis group. This high variability is partly related to the more curved aspect of this region, which makes the measurements less accurate, as it was seen before. Nevertheless, K-curvatures at C9, C10, and at the remote location C4 seem to covary consistently because they are linked with the second factor of the factor analysis. Simple inspection suggests that some well-known minimal surfaces, such as the Enneper’s one (Fig. 5), are much alike the deeper part of the corpus callosum. This fact is not so surprising, because fibers layers can be seen as made of roughly orthogonal families of curves, the interhemispheric and the longitudinal intrahemispheric connections. If the brain development is thought of as an energy limited growth of interconnected elements, keeping the connecting paths as short as possible results in the generation of minimal or near-minimal surfaces. The two regions that were identified as the closest to a minimal surface are the ones where the thin fibers are more densely packed in the corpus callosum [1,24]. The posterior one corresponds approximately to the region W89-94 [5]. In the context of a tension-based morphogenesis hypothesis [25], it seems more interesting to analyze the curvatures of the interconnecting bundles or layers rather than the ones of the cortex. This is suggested by gyrations abnormalities coexisting with partial CC development [20]. Cross-species comparisons are another way to approach this question. Most of these studies concentrate on the cortex shape or on the outer subcortical aspect [10]. We underline that the form of scaling used in this study is applicable for cross-species comparisons. Similarly to what happens in diffusion governed dynamic systems, for critical values of some parameters, a specific mode can be captured in random fluctuations and amplified as the system is growing [14]. This kind of phenomenon could participate to the cortex gyration, and a parameter of interest could be the cortex to fiber bundles growth ratio. With this interpretation, a reduced set of geometric variational principles could explain the morphogenesis of brain and its interindividual variability. The development of this hypothesis is far beyond the scope of this presentation. CONCLUSION Studies of the corpus callosum are now to be done in a 3D fashion. Modern workstations make it easy. A collection of linear
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FIG. A1. Positive and negative intrinsic curvatures C1 and C2 are the traces of the surface on two orthogonal planes intersecting along its normal at M. If the concavities of these curves are on the same side of the tangent plane at M the curvature is said positive; if not, it is said negative.
APPENDIX 1 All definitions and properties used here can be found in classical textbooks about differential geometry as for example [7]. Let us consider a surface S, a point M on it and the normal N to it. Two orthogonal planes intersecting along this normal determine two curves C1 and C2 on the surface (Fig. A1). The radius of curvature of C1 at M is the radius of the circle that best fits the curve on a neighborhood of M (Fig. A2). The inverse of the radius of curvaFIG. 5. The Enneper’s surface. (a) Posterosuperior view of the part of this surface that is close to its symmetry plane; (b) sagittal trace of the surface on its symetry plane.
lengths or areas measurements is unable to fully render the complex shape of this anatomical structure. Moreover, mixing size and shape information must be avoided. Standardized curvatures, with their mutual correlations, seem appropriate to describe the CC. Departure from the mean H-curvature profile of the sagittal strip observed in the reference population could have a pathological signification because it may unveil an unusual stress pattern on the white matter fibers bundles at some step of the development or a specific mode of degeneration. There is a balance between the posterior (C8 –C9) mean curvatures of the CC trunk and the splenial (C11) one, which is altered in our group of patients with multiple sclerosis. This alteration is, for a part, independent from global corpus callosum atrophy. The design of this study makes it clearly exploratory, and much confirmatory work remains to be done. Its retrospective nature does not allow us to state whether these shape anomalies were or were not preexistent to the onset of the disease. A future step would be to improve the accuracy of curvatures measurements and, concerning the multiple sclerosis, to correlate the deviation from the mean profiles to the topography of the white matter lesions and to the time course of the disease. ACKNOWLEDGEMENTS
The authors greatly thank Pr. C. MARGERIN for illuminating discussions about surfaces with negative curvature.
FIG. A2. Radius of curvature of a curve. Passing through M one can draw circles which have the same tangent as the curve C1. All these circles have their center on the normal N. Among all of them the one, dashed line, which best fits the curve is called the osculating circle. Its radius is by definition the radius of curvature of C1 at M.
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FIG. A3. The Gaussian curvature of a surface (example with a negative curvature). On the left is the surface where the Gaussian curvature has to be determined at M. A closed path surrounding M is covered clockwise, it encloses on the surface an area A. 1,2,3 are some unit normals encountered. On the right is the unit sphere S where are the extremities of the vectors, parallel to the unit normals, issued from O. The path on S is covered counterclockwise, it encloses an area Au.
ture is simply called the curvature c1. The mean curvature of the surface is the arithmetic mean of c1 and c2. H 5 1/ 2~c1 1 c2!
(1)
It is remarkable that this quantity does not vary with the position of the two orthogonal planes about the normal. Its sign depends on the orientation chosen on the normal. Let us consider now a closed path on the surface determining an area A around the point M. Let U be a unit vector parallel to N and originating from an arbitrary fixed point O. When a point X and its associate normal vector N run along the path, the extremity of U describes a curve on the unit sphere S centered by O. This curve encloses an area Au on the sphere (Fig. A3). If the initial path shrinks towards M, the ratio Au/A has a limit that is by definition the Gaussian curvature at M. If corresponding points travel along the two paths on the surface and on the sphere in the same direction the gaussian curvature K is positive, if not, it is negative. In fact, there is a relation between K and the curvatures c1 and c2 precedently seen. For a particular position of the orthogonal planes that makes the curvatures extremal, K 5 c1 z c2
(2)
These specific planes are called the principal planes. A sphere of radius R has constant mean and Gaussian curvatures, respectively 1/R (if the normal is positively oriented towards the outside) and 1/R2. To show how our standardization method operates, let us consider a great circle of the sphere and its corresponding strip. Its length is 2p z R. The standardized radius of curvature with our unity is thus 100/2p. The values of mean and Gaussian curvatures are then respectively 0.0628 (2p/100) and 0.00395 (4p2/10000). A plane has null mean and Gaussian curvatures. It has the property that any local deformation results in an area stretching. Every surface with a null mean curvature everywhere has this property and is thus called a minimal surface.
FIG. A4. Measurement of the radius of curvature at M of a planar curve. The axes are the normal at M and a perpendicular to it.
distance d from the origin. The Y-axis intersects the curve at altitude b. If the curve is approximated with a polynomial of degree 2, its equation is y 5 Ax2 1 Bx 1 C. The three unknowns A, B, C are determined by writing down the three equations corresponding to the three measured heights. Resolving this system yields A5
a 1 c 22b c 2a B5 C 5 b. 2d2 2d
One knows that the radius of curvature is given by uRu 5
~1 1 y9 2! 3/ 2 uy0u
where y9 and y0, respectively, stand for the first and the second derivative of the y function. At the point x 5 0 the radius of curvature is thus uRu 5
~1 1 B2! 3/ 2 ; u2Au
making the substitutions for B and A gives the formula uRu 5
~4 z d2 1 ~c 2 a! 2! 3/ 2 u8 z d z ~a 1 c 2 2zb)u
(3)
The approximation of a smooth curve by a polynomial of degree 2 is not too restrictive because formula (3) applied, for example, to a circular arc results in a systematic relative error on the radius of curvature that is less than 0.8%. Formula (3) was applied at each point for sagittal and transverse curvatures and the surface curvatures were computed as: H5
S D
1 1 1 1 2 and K 5 2 2 r1 r2 r1r2
For a near-minimal surface (where r1 ' r2 5 r) the errors on H and K can be approximated by DH 5
Î2
Dr DK and u u 5 2r r K z
Dr
Î2 r .
APPENDIX 2
REFERENCES
To measure the curvature at point M of a planar curve, two orthogonal reference axes are placed so that the Y-axis coincides with the normal in M (Fig. A4). Two symmetric parallels intersect the curve at altitudes a and c. Those parallels are located at a
1. Aboitiz, F.; Scheibel, A. B.; Fisher, R. S.; Zaidel, E. Fiber composition of the human corpus callosum. Brain Res. 598:143–153; 1992. 2. Bourguignon, J. P.; Lawson, H. B.; Margerin, C. Les surfaces minimales. Pour la Science 99:90 –101; 1986.
124 3. Davatzikos, C.; Vaillant, M.; Resnick, S. M.; Prince, J. L.; Letovsky, S.; Bryan, R. N. A computerized approach for morphological analysis of the corpus callosum. J. Comput. Assist. Tomogr. 20:88 –97; 1996. 4. Demeter, S.; Ringo, J. L.; Doty, R. W. Morphometric analysis of the human corpus callosum and anterior commissure. Hum. Neurobiol. 6:219 –226; 1988. 5. Denenberg, V. H.; Kertesz, A.; Cowell, P. E. A factor analysis of the human’s corpus callosum. Brain Res. 548:126 –132; 1991. 6. Dierkes, U.; Hildebrandt, S.; Ku¨ster, A.; Wohlrab, O. Minimal surfaces—I. Berlin: Springer Verlag; 1992. 7. Do Carmo, M. Differential geometry of curves and surfaces. Englewoods Cliffs, NJ: Prentice Hall; 1976. 8. Gabrielli, O.; Salvolini, U.; Bonifazi, V.; Ciferri, L.; Lanza, R.; Rossi, R.; Coppa, G. V.; Giorgi, P. L. Morphological studies of the corpus callosum by MRI in children with malformative syndromes. Neuroradiology 35:109 –112; 1993. 9. Griffin, L. D. The intrinsic geometry of the cerebral cortex. J. Theor. Biol. 166:261–273; 1994. 10. Hofman, M. A. On the evolution and geometry of the brain in mammals. Prog. Neurobiol. 32:137–158; 1989. 11. Johnson, S. C.; Farnworth, T.; Pinkston, J. B.; Bigler, E. D.; Blatter, D. D. Corpus callosum surface area across the human adult life span: Effect of age and gender. Brain Res. Bull. 35:373–377; 1994. 12. McLeod, N. A.; Williams, J. P.; Machen, B.; Lum, G. B. Normal and abnormal morphology of the corpus callosum. Neurology 37:1240 – 1242; 1987. 13. Morrison, D. F. Multivariate statistical methods, 2nd ed. Singapore: McGraw-Hill; 1976:153–160; 302–332. 14. Murray, J. D. Spatial pattern formation with reaction/population interaction diffusion mechanisms/(ch 14); Mechanical models for generating pattern and form in development. (ch 17). In: Mathematical biology, 2nd ed. Berlin: Springer Verlag; 1993. 15. Nasrallah, H. A.; Andreasen, N. C.; Coffman, J. A.; Olson, S. C.; Dunn, V. D.; Ehrhardt, J. C.; Chapman, S. M. A controlled magnetic resonance imaging study of corpus callosum thickness in schizophrenia. Biol. Psychiatry 21:274 –282; 1986. 16. Pelletier, J.; Habib, M.; Lyon-Caen, O.; Salamon, G.; Poncet, M.; Khallil, R. Functional and magnetic resonance imaging correlates of
STIEVENART ET AL.
17.
18.
19. 20. 21.
22.
23.
24. 25.
26.
27. 28. 29.
callosal involvement in multiple sclerosis. Arch Neurol. 50:1077– 1082; 1993. Poser, M. C.; Paty, D. W.; Scheinberg, L. New diagnostic criteria for multiple sclerosis: Guidelines for research protocols. Ann Neurol. 3:227–231; 1983. Rajapakse, J. C.; Giedd, J. N.; Rumsey, J. M.; Vaituzis, A. C.; Hamburger, S. D.; Rapoport, J. L. Regional MRI measurements of the corpus callosum: A methodological and developmental study. Brain Dev. 18:379 –388; 1996. Rakic, P.; Yakovlev, P. I. Development of the corpus callosum and cavum septi in man. J. Comp. Neurol. 32:45–72; 1968. Rubinstein, D.; Youngman, V.; Hise, J. H.; Damiano, T. R. Partial development of the corpus callosum. AJNR 15:869 – 875; 1994. Sander, P. T.; Zu¨cker, S. W. Inferring surface trace and differential structure from 3-D images. IEEE Transact. Pattern Anal. Machine Intell. 12:833– 854; 1990. Simon, J. H.; Schiffer, R. B.; Rudick, R. A.; Hernonn, R. M. Quantitative determination of MS induced corpus callosum atrophy in vivo using MRI imaging. AJNR 8:599 – 604; 1987. Steinmetz, H.; Ja¨ncke, L.; Kleinschmidt, A.; Schlaug, G.; Volkman, J.; Huang, Y. Sex but no hand difference in the isthmus of the corpus callosum. Neurology 42:749 –752; 1992. Tomasch, J. S. Size, distribution and number of fibres in the human corpus callosum. Anat. Rec. 119:119 –135; 1954. Van Essen, D. C. A tension-based theory of morphogenesis and compact wiring in the central nervous system. Nature 385:313–318; 1997. Weiss, S.; Kimbacher, M.; Wenger, E.; Neuhold, A. Morphometric analysis of the corpus callosum using MR: Correlation of measurements with aging in healthy individuals. AJNR 14:637– 645; 1993. Witelson, S. F. The brain connection: The corpus callosum is larger in left-handers. Science 229:665– 668; 1985. Witelson, S. F. Cognitive neuroanatomy: A new era. Neurology 42: 709 –713; 1992. Yamauchi, H.; Fukuyama, H.; Nabatame, H.; Kiyoshi, H.; Jun, K. Callosal atrophy with reduced cortical oxygen metabolism in carotid artery disease. Stroke 24:88 –93; 1993.
PII S03061-9230(97)00113-5
Minimal Surface: A Useful Paradigm to Describe the Deeper Part of the Corpus Callosum? J.-L. STIEVENART,*†1 M.-T. IBA-ZIZEN,* A. TOURBAH,*‡ A. LOPEZ,* M. THIBIERGE,* A. ABANOU,* AND E. A. CABANIS* *Service de Neuroradiologie, Centre Hospitalier National d’Ophtalmologie des XV-XX, 28 rue de Charenton, Paris, F-75012, France, †Service de Biophysique et de Me´decine Nucle´aire, Hoˆpital Beaujon, Clichy, France; Fe´de´ration de Neurologie and ‡Institut National de la Sante´ et de la Recherche Me´dicale, Unite´ 134, Hoˆpital de la Salpeˆtrie´re, Paris, France [Received 28 November 1996; Revised 28 April 1997; Accepted 1 May 1997] ABSTRACT: The aim of this magnetic resonance imaging study was to find a geometrical characterization of the deeper part of the corpus callosum. Its shape was studied in 12 middle-aged persons free of white matter pathology. Profiles of curvatures were measured showing that this surface was close to a minimal one, especially at the genu and near the splenium. To assess the effect of a white matter pathology on these geometrical features, the same measurements were performed in an extra group of nine patients with definite multiple sclerosis. The hypothesis of curvatures profiles parallelism for the two groups could be rejected at the 0.05 confidence level for the mean curvatures but not for the Gaussian ones. Curvatures profiles may give indications on balance between the cortex and the fiber bundles growth rates during the development and on large scale modifications cooccurring with multilocular white matter pathologies. © 1997 Elsevier Science Inc.
the CC by analyzing the covariability of the CC thickness measured at 100 equally spaced locations along the corpus callosum axis in a population of 104 normal adults and determined seven regions with consistent variations (seven factors). Angle measurements, which give a more intrinsic description, were done between some lines built on characteristic points of the outer surface of the corpus callosum [8]. These measurements are clearly related to the standardized longitudinal planar curvatures. These angles are altered in certain malformative syndromes. However, the sagittal view alone of the corpus callosum is not sufficient to yield a three-dimension geometrical characterization that seems more interesting because of the essential bihemispheric interconnecting function of the CC. The studies of the human brain development show that the corpus callosum begins its growth at the anterior region during the 10th week of gestation, when fibers cross the midline in the massa commissuralis. It then progressively extends backwards so that all its axonal elements are present at the 20th week [19]. Then, due to the global growth of the brain and to the myelination process, the shape of the CC is modified. For example the genu, which has appeared by the 16th week, progressively acquires its final morphology. The program of such a development can be thought of as a maximal extension of the neocortex with the constraint to preserve connections between cortical regions (mainly homotopic, but also heterotopic ones). Because of metabolic limitation, the cost of creating and developing such connections must be kept to a minimum. Let us consider as a surface the layer of fibers passing through the CC and connecting two homotopic longitudinal lines of neurons drawn on the cortex. For a given length of this surface trace on the CC, it can be proved that when this surface has a negative Gaussian curvature everywhere (Appendix 1), the associated layer of fibers connects longer lines of neurons than in any other condition. Among those surfaces, the ones with null mean curvature, the so-called minimal surfaces [2,6], may represent a critical situation. Any local deformation of such a surface produces an area stretching that can generate an unusual stress on some elements of this fibers layer. This could be an explanation of localized weakness of the white matter. We, thus, tried to characterize the closeness of the CC inner surface to a minimal surface in
KEY WORDS: MR imaging, 3D postprocessing, Neuroanatomy, Corpus Callosum, Morphometry, Multiple sclerosis.
INTRODUCTION The corpus callosum (CC) plays a major role in the architecture of the brain and is considered as a sensitive indicator of both the cortical and the white matter state [26]. Numerous studies have analyzed the thickness as well as the areas of different parts of the CC [26]. Some variations were noted in relation with age, gender [11], handedness [27], or with different pathologies [12]: ischemic modifications [29], multiple sclerosis [16,22], schizophrenia [15]. These studies are difficult to reproduce or even can lead to conflicting results [23,28]. It is a well-acknowledged fact that the difficulties stem from the shape variability of the corpus callosum in the midsagittal plane where the longitudinal curvature modifies relationships between length, width, and area of corpus callosum segments. To overcome this purely geometrical difficulty, the between-group comparison can be done on a deformation function used to perform a previous shape normalization of the CC [3]. Moreover, there are some uncertainties in the definition of CC regions as, for example, the “posterior fifth of the CC” for the splenium [4]. A study [5] tried to find a “natural” segmentation of 1
To whom requests for reprints should be addressed.
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a population of 12 middle-aged patients or healthy volunteers with no white matter lesions by computing its mean curvatures profiles. We chose to explore the curvatures of the CC inner surface, because it was more easily delineated on the transverse sections than the outer one, which was partially obscured by the indusium griseum. The primary purpose of our study was to present a geometric characterization that could give information on differential growing rates of cortex and fiber bundles. However, to demonstrate its utility in investigating large-scale modification coexisting with multilocular white matter pathologies, we studied in the same way nine patients with definite multiple sclerosis to see whether they could be distinguished from the former group on the basis of the corpus callosum inner surface shape. We thought that multiple sclerosis was a well-suited pathology to test the relevance of the proposed analysis, because the white matter alterations may cause— or may be due to—a modification of its global shape. MATERIALS AND METHODS The studied population was retrospectively selected among patients who underwent a 3D MR examination (GE Medical Systems SIGNA 1.5T) in our institution. These patients were classified into two groups: a reference group and a multiple sclerosis group. The reference group (four men, eight women) is assumed to be representative of the normal middle aged population as far as the CC shape is concerned. It consists in three healthy volunteers and nine patients whose MR examination was normal or demonstrated minute cortical angiomas or hypophisial microadenoma. Their ages ranged from 21 to 41 (mean 30.7). Patients in the MS group (three men, six women) met the Poser’s criteria for definite multiple sclerosis [17]. Their ages ranged from 21 to 48 (mean 32.3). The 3D-Fourier Transform acquisition sequence was T1weighted and performed in the sagittal plane (SPGR flip angle 25°, TE 8 ms, TR 35 ms). The stack of 124 contiguous 1.3 mm-thick slices was reformatted on a workstation (GEMS Advantage Window 1.2) with dedicated software (GEMS Voxtool) to display the true midsagittal plane (Fig. 1a). This plane was manually determined by the operator using the standard orthogonal views display and the oblique plane manipulation of the software. In this midsagittal plane, the deeper part of the CC was delineated by a cubic spline passing through control points given by the operator. The geodesic (curvilinear) length of this curve—the corpus callosum length (CCL)—was measured from the rostrum to a posterior landmark corresponding to the closest point to the posterior commissure (Fig. 1b). This point can be constructed as the tangency point to the splenium of a circle centered by the posterior commissure. The CCL, directly given by the software in millimeters, was used as a standardization variable. Ten points were equally spaced (geodesic distance) on this curve. This determined 11 regular points (C1, C2,. . ., C11—rostrum excluded) where the surface curvatures were estimated (Fig. 1b). Due to symmetry, the principal planes (Appendix 1) of the surface along its intersection with the midsagittal plane are this plane itself and the ones perpendicular to the trace of corpus callosum on it. These planes are automatically given by the Voxtool software by using the “lock cursor to trace” option. The two curvatures c1 and c2 were calculated by means of formula 3 (Appendix 2). The various lengths involved in this formula were measured on calibrated films by using the template of Fig. A4 drawn on graph tracing paper. To assess the accuracy of this method, a 3D sagittal acquisition, with identical parameters to the ones used for patients, was performed on a cylindrical 51-mm diameter phantom. Its axis was placed along the anteroposterior direction. Repeated measurements on
FIG. 1. (a) Midsagittal view of reference patient #3 (FOV: 13 cm); (b) points where curvatures were measured. Line segments figure the transverse planes where the planar curvatures were measured. Intersections of these segments with the inferior trace of the CC are the points C1, C2, C11. The length of the thick line is by definition the length of the corpus callosum (CCL)—see text.
resulting images demonstrated that the measurement was unbiased and provided us with an estimation of the error of about 3 mm on the radius of curvature. For the patients, measures were repeated twice and the resulting curvature radii were averaged. Then the curvature radii were divided by the CCL, yielding thus dimensionless curvatures expressed in (%CCL)21 for the mean curvatures (H-curvatures) and in (%CCL)22 for the Gaussian curvatures (K-curvatures). The reproducibility of this procedure was assessed by a simple linear regression analysis between 22 pairs of measurements done by the same operator 2 weeks apart. This reproducibility was good for the H-curvatures (r2 5 0.906) but rather low for the K-curvatures (r2 5 0.804). All the discrepancies occurred on the more curved regions as, for example, the genu. All the measurements in this study were performed by the same operator.
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TABLE 1 DESCRIPTIVE STATISTICS OF STANDARDIZED H AND K CURVATURES H-Curvatures Location, Group
C1, C C2, C C3, C C4, C C5, C C6, C C7, C C8, C C9, C C10, C C11, C C1, MS C2, MS C3, MS C4, MS C5, MS C6, MS C7, MS C8, MS C9, MS C10, MS C11, MS
K-Curvatures
Mean
Std. Dev.
Mean
Std. Dev.
0,06985 20,00323 20,02361 20,02327 20,02332 20,02798 20,03029 20,03049 20,01811 0,00144 20,10044 0,02792 20,01820 20,02652 20,01975 20,02181 20,02221 20,03435 20,06642 20,04029 0,02155 20,14891
0,19253 0,03977 0,00932 0,01127 0,01058 0,00538 0,00550 0,01354 0,01930 0,04250 0,03137 0,06474 0,01797 0,01456 0,00996 0,01003 0,01733 0,01216 0,03648 0,02747 0,05943 0,05430
20,02346 20,00542 20,00125 20,00114 20,00127 20,00084 20,00074 20,00170 20,00195 20,00248 0,00156 20,02563 20,00347 20,00134 20,00180 20,00143 20,00169 20,00150 20,00229 20,00429 20,00619 0,00164
0,03809 0,00772 0,00077 0,00050 0,00100 0,00061 0,00083 0,00145 0,00095 0,00269 0,00238 0,02622 0,00186 0,00174 0,00105 0,00061 0,00249 0,00177 0,00477 0,00448 0,00739 0,00317
C1 and C2 exhibited a large variability and were no further analyzed. On the greatest part of the CC this surface had a negative mean curvature with the chosen orientation (transverse curvature greater than longitudinal one), and a negative Gaussian curvature. Two regions of null mean curvature were identified near the genu and near the splenium (Fig. 2). Analysis of variance for profiles data demonstrated that the parallelism of mean curvatures profiles from C3 to C11 (no curvatures by group interaction) can be rejected at the confidence level of 0.05 (p 5 0.0007 and p 5 0.0123 with the Huynh–Feldt’s correction). This was not true for the Gaussian curvatures profiles. The factor analysis of the mean curvatures was performed on both groups considered as a single one. Three factors were used for this analysis. They explained 68.5% of the variation of the original curvatures (Table 2a). Mean curvatures at C3, C4, and C10 were poorly described by this three-factor model because the corresponding communalities were less than 60%. The first factor was correlated with the posterior mean curvatures (C8 –C9). The second factor was strongly linked with the curvature at C11. The third factor did not receive a clear interpretation. To visualize a possible relationship between mean curvatures and atrophy, symbols corresponding to patients with corpus callosum atrophy were identified. In the space of the two first factors the control patients were gathered near the origin and all patients with multiple sclerosis but one were located at the periphery (Fig. 3). For the factor analysis
9 MS patients. Only curvatures from C3 to C11 were used for the profile analysis. The hypothesis of profiles parallelism in both groups can be rejected (p , 0.05) for the mean curvatures but not for the Gaussian curvatures.
Mean profiles of H- and K-curvatures were first calculated and analyzed with a repeated measures analysis of variance [13]. To explore the covariance structure of the curvatures along the inferior trace, two factor analysis [13] were performed with a standard statistical package (Abacus Concepts, StatView 4.0). The aim of this kind of analysis is to reduce a large number of correlated variables measured on individuals to a smaller set of underlying variables called factors. Each original variable is a linear combination of these factors plus a specific random term. The coefficient by which is multiplied a factor in this linear form is called the factor loading. In our application, the factor loading was the correlation of the factor with the original variable. No oblique transformation was done on these factors, which were the original orthogonal, unrotated ones, extracted by a principal components analysis. The factor score is the value taken by the factor for a given individual. Corpus callosum atrophy was quantified by the ratio of the square root of its sagittal area to CCL yielding a dimensionless index expressing the overall thinness of the CC. In our reference population this ratio average was 0.36, with a standard deviation of 0.035. Assuming a normal distribution of this thinness index, less than 5% of the population has an index below 0.30 (0.36 –1.645 3 0.035). The corpus callosum was thus termed “atrophic” when this ratio was below 0.30. RESULTS Profiles of mean and gaussian curvatures along the deep sagittal strip of the CC were established (Table 1). Curvatures at locations
FIG. 2. Profiles of averaged curvatures along the deeper part of the corpus callosum. Units are (%CCL)21 for the mean curvatures and (%CCL)22 for the Gaussian curvatures. Vertical bars represent the 95% confidence intervals. (a) Mean curvatures (regions with null mean curvature are circled); (b) Gaussian curvatures.
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STIEVENART ET AL. TABLE 2 FACTOR ANALYSIS OF STANDARDIZED CURVATURES Factors Loadings
Location
Communality Summary
Factor 1
Factor 2
Factor 3
SMC
Final Estimate
C3 C4 C5 C6 C7 C8 C9 C10 C11
20,258 20,393 20,608 20,568 0,530 0,849 0,886 20,133 0,244
0,558 0,271 20,015 0,554 0,630 0,022 20,066 20,438 0,832
0,396 0,583 0,502 20,500 0,202 0,324 0,044 0,417 20,052
0,496 0,409 0,500 0,703 0,505 0,736 0,757 0,361 0,586
0,535 0,568 0,622 0,879 0,719 0,826 0,792 0,383 0,755
C3 C4 C5 C6 C7 C8 C9 C10 C11
0,242 20,078 0,382 0,857 0,863 0,865 0,437 0,056 0,585
20,436 0,829 0,118 20,162 20,297 0,055 0,834 0,819 0,108
20,038 0,150 0,741 20,307 20,154 20,135 20,218 20,245 0,616
0,326 0,672 0,394 0,837 0,787 0,875 0,896 0,792 0,640
0,250 0,717 0,710 0,855 0,858 0,769 0,936 0,735 0,734
a
b
The absolute value of a factor loading close to one express a strong contribution of the variable to this factor. Factor loadings greater than 0.8 have been highlighted. The communality is the proportion of variance of the variable, which is explained by the three-factor model. The squared multiple correlation (SMC) of a variable with all the other variables is often considered as an initial estimate of this quantity. (a) Analyis of the H-curvatures; (b) Analysis of the K-curvatures.
of the Gaussian curvatures, three factors explained 72.9% of the variation of the original curvatures (Table 2b). The first factor was linked with curvatures at C6, C7, and C8, whereas the second one was linked with curvatures at C4, C9, and C10. C3 was poorly described by this three-factor model because its communality was 0.25. The repartition of the individuals in the factor space had no particularity related to their group. In the multiple sclerosis group the mean of the previously defined CC atrophy index, 0.30, was significantly smaller than in the reference group (p 5 0.004, Student’s t-test).
phantom study, where the relative error on the radius of curvature measurement was approximately 12% (for a radius of curvature of 25.5 mm.). This could be an explanation of the great variability of H- and K-curvatures at C1 and C2 where the radii of curvature were both smaller than anywhere else (in the range 5–20 mm.). An other explanation could be that the CC shape at the genu is sometimes close to a singularity with a very small radius of curvature and that this near-singularity is variably located with respect to C1 and C2. In spite of those difficulties, the curvatures seem to us relevant parameters because they are presumably related to the growth rates of different regions during the development of the brain. These preliminary results incite to develop a more automatic and a more accurate procedure to build the curvatures profiles, especially on regions with small radius of curvature. One must stress the fact that H- and K-curvatures are different in nature. The mean curvature is an extrinsic property in the sense that it is changed, for example, by pure folding, without stretching or tearing the surface. On the contrary, the Gaussian curvature cannot be changed this way and is, thus, an intrinsic property [9]. The same difference exists between the Euclidian distance, which is extrinsic, and the geodesic distance, which is intrinsic, or between the volume enclosed by a surface and this surface area. The presented reference cases had a remarkable invariance of the standardized mean curvature of the deeper part of the anterior region (C3–C6) of the corpus callosum trunk with an average mean curvature equal to 20.024. The region of C7 corresponds to the level of the pyramidal tract. The H-curvature at this point was the one with the smallest coefficient of variation respectively in both populations. The constraints induced by the pyramidal tract during the growth of the brain could explain the stable extrinsic geometrical properties of this region. Differences in CC H-curvatures profiles between the two groups laid in the posterior part of the corpus callosal trunk (C8 –C9) where large-diameter fibers are more dense [1]. There, the form of the CC in the multiple sclerosis group is farther from a minimal surface than in the reference group. This region covers approximately the isthmus that interconnects the right and left parieto-temporal cortex. Its sagittal area was found to be greater in women [23,28]. It corresponds to the regions W65-74 and W77-85 of the factor analysis of the widths of the CC [5]. These regions as well as the splenium were also found to show important modifi-
DISCUSSION The accuracy of the curvature measurements is a serious problem. That is why we used an averaged value of two independent measurements. Sophisticated methods exist [21]. But those methods themselves could be ineffective for example on the posterior third of the CC where images show a mixing between the CC and the crus of fornix. There, some anatomical knowledge is necessary to correctly detect the relevant surface (Fig. 4). The error calculus outlined in Appendix 2 shows that the smaller the radius of curvature the bigger is the absolute error on H-curvature measurement. On the other hand, the relative error on K-curvature measurement is simply proportional to relative error on radius of curvature measurement. Even if the absolute error on radius of curvature is not strictly constant for the whole range of encountered values, the relative error is clearly a decreasing function of the radius of curvature. A point of this function is given by the
FIG. 3. Repartition of individuals in the space of the two first factors (H-curvatures analysis)—axes correspond to the factor scores. The curve encloses all normal patients plus one patient with multiple sclerosis. Symbols of patients with corpus callosum atrophy are solid. The arrow shows the representative point of patient #3 (Fig. 1a).
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FIG. 4. Detection of the relevant surface on the posterior part of the corpus callosum trunk (points C8 and C9). (a) Sagittal view (FOV: 3 cm., the line is the trace of the transverse view); (b) transverse view orthogonal to the trace of the corpus callosum on the midsagittal plane. The crus of fornix (arrows) is not to be taken into account to determine the curvature at M.
cations in size and shape with age (in the range 4 –18) [18]. In the factor analysis of H-curvatures, the first factor indicates that curvatures at C8 and C9 are strongly correlated and that they are linked with corpus callosum atrophy because most of the representative points of such cases lay on the positive region of the first factor axis. To confirm this point, another factor analysis was conducted, not reproduced here, adding our thinness index into the variables set. It clearly demonstrated the positive link between the
121 H-curvatures at C8 and C9, the thinness index, and the first factor. This method avoids to make assumption on what is called a CC atrophy. There is no obvious geometrical relationship between the CC atrophy and its inferior curvatures. But a parallel outward displacement of the CC inner surface results in a decrease (more negative value) of its mean curvature. The second factor was associated with the mean curvature at C11. Due to the local cylindrical shape of the splenium (null transverse curvature) this curvature is roughly proportional to the inverse of its sagittal radius and is, thus, related to its standardized size. The K-curvatures at C3–C5 are relatively stable with an average standardized value of 20.00122 in the reference population. The region of C6 –C7 is the less intrinsically curved one (excluding a small posterior region between C10 and C11 where the Gaussian curvature is null). Its average standardized Gaussian curvature is equal to 20.00079 in the reference population. We examined with more attention K-curvatures at locations C4, C9, and C10, where differences between groups were the closest to significance with respective p-values of 0.068, 0.092, and 0.12. An estimation of the test power, carried out for these locations, yielded values below 20% to detect a 50% difference on the radii of curvature with the usual type I risk of 5%. Especially for C9 and C10, these low values are attributable to the high variability of the corresponding curvatures measures in the multiple sclerosis group. This high variability is partly related to the more curved aspect of this region, which makes the measurements less accurate, as it was seen before. Nevertheless, K-curvatures at C9, C10, and at the remote location C4 seem to covary consistently because they are linked with the second factor of the factor analysis. Simple inspection suggests that some well-known minimal surfaces, such as the Enneper’s one (Fig. 5), are much alike the deeper part of the corpus callosum. This fact is not so surprising, because fibers layers can be seen as made of roughly orthogonal families of curves, the interhemispheric and the longitudinal intrahemispheric connections. If the brain development is thought of as an energy limited growth of interconnected elements, keeping the connecting paths as short as possible results in the generation of minimal or near-minimal surfaces. The two regions that were identified as the closest to a minimal surface are the ones where the thin fibers are more densely packed in the corpus callosum [1,24]. The posterior one corresponds approximately to the region W89-94 [5]. In the context of a tension-based morphogenesis hypothesis [25], it seems more interesting to analyze the curvatures of the interconnecting bundles or layers rather than the ones of the cortex. This is suggested by gyrations abnormalities coexisting with partial CC development [20]. Cross-species comparisons are another way to approach this question. Most of these studies concentrate on the cortex shape or on the outer subcortical aspect [10]. We underline that the form of scaling used in this study is applicable for cross-species comparisons. Similarly to what happens in diffusion governed dynamic systems, for critical values of some parameters, a specific mode can be captured in random fluctuations and amplified as the system is growing [14]. This kind of phenomenon could participate to the cortex gyration, and a parameter of interest could be the cortex to fiber bundles growth ratio. With this interpretation, a reduced set of geometric variational principles could explain the morphogenesis of brain and its interindividual variability. The development of this hypothesis is far beyond the scope of this presentation. CONCLUSION Studies of the corpus callosum are now to be done in a 3D fashion. Modern workstations make it easy. A collection of linear
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FIG. A1. Positive and negative intrinsic curvatures C1 and C2 are the traces of the surface on two orthogonal planes intersecting along its normal at M. If the concavities of these curves are on the same side of the tangent plane at M the curvature is said positive; if not, it is said negative.
APPENDIX 1 All definitions and properties used here can be found in classical textbooks about differential geometry as for example [7]. Let us consider a surface S, a point M on it and the normal N to it. Two orthogonal planes intersecting along this normal determine two curves C1 and C2 on the surface (Fig. A1). The radius of curvature of C1 at M is the radius of the circle that best fits the curve on a neighborhood of M (Fig. A2). The inverse of the radius of curvaFIG. 5. The Enneper’s surface. (a) Posterosuperior view of the part of this surface that is close to its symmetry plane; (b) sagittal trace of the surface on its symetry plane.
lengths or areas measurements is unable to fully render the complex shape of this anatomical structure. Moreover, mixing size and shape information must be avoided. Standardized curvatures, with their mutual correlations, seem appropriate to describe the CC. Departure from the mean H-curvature profile of the sagittal strip observed in the reference population could have a pathological signification because it may unveil an unusual stress pattern on the white matter fibers bundles at some step of the development or a specific mode of degeneration. There is a balance between the posterior (C8 –C9) mean curvatures of the CC trunk and the splenial (C11) one, which is altered in our group of patients with multiple sclerosis. This alteration is, for a part, independent from global corpus callosum atrophy. The design of this study makes it clearly exploratory, and much confirmatory work remains to be done. Its retrospective nature does not allow us to state whether these shape anomalies were or were not preexistent to the onset of the disease. A future step would be to improve the accuracy of curvatures measurements and, concerning the multiple sclerosis, to correlate the deviation from the mean profiles to the topography of the white matter lesions and to the time course of the disease. ACKNOWLEDGEMENTS
The authors greatly thank Pr. C. MARGERIN for illuminating discussions about surfaces with negative curvature.
FIG. A2. Radius of curvature of a curve. Passing through M one can draw circles which have the same tangent as the curve C1. All these circles have their center on the normal N. Among all of them the one, dashed line, which best fits the curve is called the osculating circle. Its radius is by definition the radius of curvature of C1 at M.
SHAPE OF THE CORPUS CALLOSUM
123
FIG. A3. The Gaussian curvature of a surface (example with a negative curvature). On the left is the surface where the Gaussian curvature has to be determined at M. A closed path surrounding M is covered clockwise, it encloses on the surface an area A. 1,2,3 are some unit normals encountered. On the right is the unit sphere S where are the extremities of the vectors, parallel to the unit normals, issued from O. The path on S is covered counterclockwise, it encloses an area Au.
ture is simply called the curvature c1. The mean curvature of the surface is the arithmetic mean of c1 and c2. H 5 1/ 2~c1 1 c2!
(1)
It is remarkable that this quantity does not vary with the position of the two orthogonal planes about the normal. Its sign depends on the orientation chosen on the normal. Let us consider now a closed path on the surface determining an area A around the point M. Let U be a unit vector parallel to N and originating from an arbitrary fixed point O. When a point X and its associate normal vector N run along the path, the extremity of U describes a curve on the unit sphere S centered by O. This curve encloses an area Au on the sphere (Fig. A3). If the initial path shrinks towards M, the ratio Au/A has a limit that is by definition the Gaussian curvature at M. If corresponding points travel along the two paths on the surface and on the sphere in the same direction the gaussian curvature K is positive, if not, it is negative. In fact, there is a relation between K and the curvatures c1 and c2 precedently seen. For a particular position of the orthogonal planes that makes the curvatures extremal, K 5 c1 z c2
(2)
These specific planes are called the principal planes. A sphere of radius R has constant mean and Gaussian curvatures, respectively 1/R (if the normal is positively oriented towards the outside) and 1/R2. To show how our standardization method operates, let us consider a great circle of the sphere and its corresponding strip. Its length is 2p z R. The standardized radius of curvature with our unity is thus 100/2p. The values of mean and Gaussian curvatures are then respectively 0.0628 (2p/100) and 0.00395 (4p2/10000). A plane has null mean and Gaussian curvatures. It has the property that any local deformation results in an area stretching. Every surface with a null mean curvature everywhere has this property and is thus called a minimal surface.
FIG. A4. Measurement of the radius of curvature at M of a planar curve. The axes are the normal at M and a perpendicular to it.
distance d from the origin. The Y-axis intersects the curve at altitude b. If the curve is approximated with a polynomial of degree 2, its equation is y 5 Ax2 1 Bx 1 C. The three unknowns A, B, C are determined by writing down the three equations corresponding to the three measured heights. Resolving this system yields A5
a 1 c 22b c 2a B5 C 5 b. 2d2 2d
One knows that the radius of curvature is given by uRu 5
~1 1 y9 2! 3/ 2 uy0u
where y9 and y0, respectively, stand for the first and the second derivative of the y function. At the point x 5 0 the radius of curvature is thus uRu 5
~1 1 B2! 3/ 2 ; u2Au
making the substitutions for B and A gives the formula uRu 5
~4 z d2 1 ~c 2 a! 2! 3/ 2 u8 z d z ~a 1 c 2 2zb)u
(3)
The approximation of a smooth curve by a polynomial of degree 2 is not too restrictive because formula (3) applied, for example, to a circular arc results in a systematic relative error on the radius of curvature that is less than 0.8%. Formula (3) was applied at each point for sagittal and transverse curvatures and the surface curvatures were computed as: H5
S D
1 1 1 1 2 and K 5 2 2 r1 r2 r1r2
For a near-minimal surface (where r1 ' r2 5 r) the errors on H and K can be approximated by DH 5
Î2
Dr DK and u u 5 2r r K z
Dr
Î2 r .
APPENDIX 2
REFERENCES
To measure the curvature at point M of a planar curve, two orthogonal reference axes are placed so that the Y-axis coincides with the normal in M (Fig. A4). Two symmetric parallels intersect the curve at altitudes a and c. Those parallels are located at a
1. Aboitiz, F.; Scheibel, A. B.; Fisher, R. S.; Zaidel, E. Fiber composition of the human corpus callosum. Brain Res. 598:143–153; 1992. 2. Bourguignon, J. P.; Lawson, H. B.; Margerin, C. Les surfaces minimales. Pour la Science 99:90 –101; 1986.
124 3. Davatzikos, C.; Vaillant, M.; Resnick, S. M.; Prince, J. L.; Letovsky, S.; Bryan, R. N. A computerized approach for morphological analysis of the corpus callosum. J. Comput. Assist. Tomogr. 20:88 –97; 1996. 4. Demeter, S.; Ringo, J. L.; Doty, R. W. Morphometric analysis of the human corpus callosum and anterior commissure. Hum. Neurobiol. 6:219 –226; 1988. 5. Denenberg, V. H.; Kertesz, A.; Cowell, P. E. A factor analysis of the human’s corpus callosum. Brain Res. 548:126 –132; 1991. 6. Dierkes, U.; Hildebrandt, S.; Ku¨ster, A.; Wohlrab, O. Minimal surfaces—I. Berlin: Springer Verlag; 1992. 7. Do Carmo, M. Differential geometry of curves and surfaces. Englewoods Cliffs, NJ: Prentice Hall; 1976. 8. Gabrielli, O.; Salvolini, U.; Bonifazi, V.; Ciferri, L.; Lanza, R.; Rossi, R.; Coppa, G. V.; Giorgi, P. L. Morphological studies of the corpus callosum by MRI in children with malformative syndromes. Neuroradiology 35:109 –112; 1993. 9. Griffin, L. D. The intrinsic geometry of the cerebral cortex. J. Theor. Biol. 166:261–273; 1994. 10. Hofman, M. A. On the evolution and geometry of the brain in mammals. Prog. Neurobiol. 32:137–158; 1989. 11. Johnson, S. C.; Farnworth, T.; Pinkston, J. B.; Bigler, E. D.; Blatter, D. D. Corpus callosum surface area across the human adult life span: Effect of age and gender. Brain Res. Bull. 35:373–377; 1994. 12. McLeod, N. A.; Williams, J. P.; Machen, B.; Lum, G. B. Normal and abnormal morphology of the corpus callosum. Neurology 37:1240 – 1242; 1987. 13. Morrison, D. F. Multivariate statistical methods, 2nd ed. Singapore: McGraw-Hill; 1976:153–160; 302–332. 14. Murray, J. D. Spatial pattern formation with reaction/population interaction diffusion mechanisms/(ch 14); Mechanical models for generating pattern and form in development. (ch 17). In: Mathematical biology, 2nd ed. Berlin: Springer Verlag; 1993. 15. Nasrallah, H. A.; Andreasen, N. C.; Coffman, J. A.; Olson, S. C.; Dunn, V. D.; Ehrhardt, J. C.; Chapman, S. M. A controlled magnetic resonance imaging study of corpus callosum thickness in schizophrenia. Biol. Psychiatry 21:274 –282; 1986. 16. Pelletier, J.; Habib, M.; Lyon-Caen, O.; Salamon, G.; Poncet, M.; Khallil, R. Functional and magnetic resonance imaging correlates of
STIEVENART ET AL.
17.
18.
19. 20. 21.
22.
23.
24. 25.
26.
27. 28. 29.
callosal involvement in multiple sclerosis. Arch Neurol. 50:1077– 1082; 1993. Poser, M. C.; Paty, D. W.; Scheinberg, L. New diagnostic criteria for multiple sclerosis: Guidelines for research protocols. Ann Neurol. 3:227–231; 1983. Rajapakse, J. C.; Giedd, J. N.; Rumsey, J. M.; Vaituzis, A. C.; Hamburger, S. D.; Rapoport, J. L. Regional MRI measurements of the corpus callosum: A methodological and developmental study. Brain Dev. 18:379 –388; 1996. Rakic, P.; Yakovlev, P. I. Development of the corpus callosum and cavum septi in man. J. Comp. Neurol. 32:45–72; 1968. Rubinstein, D.; Youngman, V.; Hise, J. H.; Damiano, T. R. Partial development of the corpus callosum. AJNR 15:869 – 875; 1994. Sander, P. T.; Zu¨cker, S. W. Inferring surface trace and differential structure from 3-D images. IEEE Transact. Pattern Anal. Machine Intell. 12:833– 854; 1990. Simon, J. H.; Schiffer, R. B.; Rudick, R. A.; Hernonn, R. M. Quantitative determination of MS induced corpus callosum atrophy in vivo using MRI imaging. AJNR 8:599 – 604; 1987. Steinmetz, H.; Ja¨ncke, L.; Kleinschmidt, A.; Schlaug, G.; Volkman, J.; Huang, Y. Sex but no hand difference in the isthmus of the corpus callosum. Neurology 42:749 –752; 1992. Tomasch, J. S. Size, distribution and number of fibres in the human corpus callosum. Anat. Rec. 119:119 –135; 1954. Van Essen, D. C. A tension-based theory of morphogenesis and compact wiring in the central nervous system. Nature 385:313–318; 1997. Weiss, S.; Kimbacher, M.; Wenger, E.; Neuhold, A. Morphometric analysis of the corpus callosum using MR: Correlation of measurements with aging in healthy individuals. AJNR 14:637– 645; 1993. Witelson, S. F. The brain connection: The corpus callosum is larger in left-handers. Science 229:665– 668; 1985. Witelson, S. F. Cognitive neuroanatomy: A new era. Neurology 42: 709 –713; 1992. Yamauchi, H.; Fukuyama, H.; Nabatame, H.; Kiyoshi, H.; Jun, K. Callosal atrophy with reduced cortical oxygen metabolism in carotid artery disease. Stroke 24:88 –93; 1993.