THE ANATOMICAL RECORD 227:380-386 (1990)
Comparison Between Two Finite-Element Modeliing Methods for Measuring Change in Craniof acial Form SCOTT LOZANOFF Department of Anatomy, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N OW0
ABSTRACT Finite-element modelling of form change is a useful morphometric technique for measuring differences between anatomical patterns. Two different finite-element algorithms currently are used. One method requires normalized coordinates a s input data, while the second method uses globalized coordinates as input data. This study determines whether the two finite-element methods provide equivalent measures of three-dimensional form change when applied to the nasal septa of embryonic mice. Computer models of the nasal septa from mice of 15 and 17 days gestation were generated. Homologous landmarks were identified so that each nasal septum was represented by a tetrahedral finite-element. These elements were subjected to both finite-element modelling methods. Results show that the two algorithms use different interpolation functions and yield dissimilar intermediate results, but generate identical strain matrices as well a s equivalent principal extensions, directions of form change, variables of form change, and graphical displays. Therefore, results are directly comparable from studies using either finite-element modelling method. Finite-element modelling (FEM) is a morphometric technique used to measure form change between anatomical structures. Following this approach, homologous landmarks are identified on a n initial anatomical structure and the coordinates associated with these homologous nodes are used to define a finite anatomical element. Then, homologous landmarks and corresponding coordinates are identified on a second anatomical structure. Form change is viewed as a continuous deformation of the initial finite anatomical element into the final anatomical element. The magnitude and direction of strain experienced by the initial element as it deforms into the final element provides a measure of difference between the two anatomical patterns. The goal of the FEM procedure is to identify the anatomical nodes which experience the greatest rate of change as the initial anatomical element deforms into the final element. Two separate FEM schemes have been applied for the analysis of craniofacial form change. The first approach compares the rates of nodal change relative to a point inside the anatomical element a s it deforms into the final configuration (Bathe and Wilson, 1976). This approach can be referred to as normalized FEM, since i t calculates measures of form change based on normalized points within the two anatomical structures. Computer routines for normalized FEM have been developed specifically for the analysis of two- and three-dimensional craniofacial form change (Hanmer and Bachrach, 1986; Lozanoff and Diewert, 1989). The second FEM scheme determines measures of form change by calculating the rates of change in line segments connecting adjacent nodes relative to the displacement of homologous landmarks as the ini0 1990 WILEY-LISS, INC
tial element deforms into the final configuration (Skalak et al., 1982). The scheme relates the two finite elements based on the displacements of the anatomical nodes during deformation. This approach can be referred to as globalized FEM since the displacement functions are determined within a global coordinate system. An algorithm has been described and implemented for two-dimensional analyses of craniofacial growth and morphology (Skalak et al., 1982; Patel, 1983). Presently, i t is not known whether the normalized and globalized FEM approaches produce equivalent measures of form change. The purpose of this study is to determine whether the normalized FEM scheme provides measures of form change corresponding to those rendered with the globalized technique. Data sets comprising coordinate values of homologous landmarks from the nasal septa of embryonic mice are subjected to both FEM schemes in order to determine whether equivalent numerical results are produced. MATERIALS AND METHODS
Embryonic HeJIC3H mice were used in this study. Specimens were drawn from a larger sample which is being used to study patterns of allometric growth in the developing nasal septum (Fig. 1). Adult female HeJ/C3H mice were subjected to timed matings. Five embryos were collected at 15 gestational days (E-15) while another five embryos were taken a t 17 gesta-
Received J u l y 27, 1989; accepted October 9, 1989.
FORM CHANGE MEASURED BY TWO FEM METHODS
381
implemented for the normalized FEM procedure following Lozanoff and Diewert (1989). A separate algorithm was developed, coded, and implemented for a three-dimensional globalized FEM routine (Appendix A). Data sets were subjected to FEM analysis in a pairwise fashion with the five E-15 specimens serving as the initial elements, while the five E-17 specimens provided the final geometries. Therefore, a total of 25 combinations were performed. Each data set combination was subjected to the normalized FEM routine and subsequently to the globalized FEM scheme. Numerical data from both FEM routines, including the Jacobian matrices, Lagrangian matrices, principal extension ratios, directions of principal extensions, and size and shape variables were generated for all the data set combinations. These values were compared in order to determine whether the two FEM routines produced equivalent results. In addition, the tetrahedral elements and principal extensions were rendered within the histological models. Thus, FEM results were depicted along with the models they described in order to determine whether the visual description of anatomical change coincided between FEM schemes. RESULTS
Numerical results from all pairwise comparisons embryo of 17 gestational days viewed from a lateral perspective. showed that both globalized and normalized FEM routines produced different Jacobian matrices, but generated identical Lagrangian matrices, principal extentional days (E-17). None of the specimens were litter- sion ratios, values of size change, values of shape mates. Specimens were weighed, placed in 1%phos- change, and graphical displays. Rather than tabulate phate buffer, and photographed. The heads were lists of identical numerical data derived from the two removed and fixed in 10% neutral buffered formalin for FEM analyses, one illustrative case will be presented a t least 1 week. Specimens were dehydrated in a in detail. Three-dimensional surface models of the nasal septa graded series of ethanol (50%, 70%, 95%, 100%) and embedded in plastic (JB-4 embedding kit). Three holes from a n E-15 individual (HeJ/C3H/5a) and one E-17 were drilled (.013 inch diameter) in each block, provid- specimen (HeJ/C3H/lOa) are provided in Figure 3. The ing alignment guides for subsequent computerized re- models, viewed from a lateral perspective, correconstruction. Heads were sectioned in a coronal plane sponded well to the nasal septal models of the other at 6 pm on a Sorvall (JB-4) microtome. Five sections embryonic HeJIC3H mice with equivalent gestational were mounted on each slide and alternate slides were ages used in this study. The models were rotated to various perspectives and viewed. Based on these modstained with toludine blue or hematoxylin and eosin. Histologic sections from each specimen were viewed els, the major form change appeared to be oriented with a Leitz compound microscope and subjected to vid- along a n anteroposterior axis rather than superoinfeeomicroscopy. Images were viewed on a computer mon- rior or mediolateral axes. Coordinate values of the homologous nodes for both itor (Apollo DN-4000), and the nasal septum was identified. Outlines of the nasal septum and the alignment the initial and final elements are given in Table 1. holes were traced directly from the monitor. Approxi- Numerical results indicate the two procedures yield mately 30 nasal septal contours were used for each different Jacobian matrices (Table 2). However, the Laspecimen. These tissue contours were assembled pro- grangian strain matrices are identical (Table 3). The viding three-dimensional models for both the E-15 and principal extension ratios are also the same since these E-17 specimens. Four homologous landmarks were values are derived from the Lagrangian strain matrix identified on each model which included one superior, in both procedures (Table 4). The models and their corresponding finite elements one anterior and two posterior nodes. The landmarks selected for analysis included the midpoint of the crista are graphically displayed in Figure 4. Principal extengalli (superior node), the anterior junction between the sion axes, generated using the normalized FEM procebilateral limbs of the anterior transverse lamina (an- dure, are rendered a t the centroid of the initial finite terior node), and the right and left junctions between element. The major stretch (1.84) is oriented along a n the pars posterior and the nasal septum (two posterior anteroposterior axis directed near to the anterior nodes). These landmarks were verified histologically transverse lamina. This value indicates that the E-15 (Fig. 2). These homologous landmarks were digitized specimen experienced a n 84% extension along this anfrom the models providing ten sets of four (x,y,z) coor- teroposterior axis when compared with the E-17 model. dinates. Hence, ten sets of coordinate values estab- The second axis is oriented along a superoinferior line, but the model of the E-15 specimen increased only 25% lished ten tetrahedral finite elements. Computer routines were developed, coded in “C,” and when compared to the older individual. Finally, the Fig. 1. Schematic representation of the nasal septum in an HeJIC3H
382
S. LOZANOFF
Fig. 2. Anatomical landmarks from the nasal septum used to construct the finite elements for the FEM analysis include A, the crista galli (node 1);B, the junction between the limbs of the anterior transverse lamina (node 2); C, the left (node 3) and right (node 4) points of
posterior attachment between the lateral capsular wall and the nasal septum. ATL, anterior transverse lamina; CG, crista galli; NS, nasal septum; PP, pars posterior. Bar = 0.2 mm.
FORM CHANGE MEASURED BY TWO FEM METHODS
383
Fig. 3. Three-dimensional reconstructions of the E-15 specimen (A) and the E-17 specimen (B) viewed from the lateral aspect. S, superior; i, inferior; a, anterior; p, posterior. Bar = 0.5 mm
TABLE 1. Coordinate values of the homologous nodes used for the FEM analysis E-15 specimen Node 1 2 3 4
X
-9.97 94.24 - 129.81 -145.00
Y -150.15 -29.51 6.88 6.00
E-17 specimen Z
X
60.03 - 124.43 131.82 114.93
-50.38 209.71 -278.21 -268.00
Y -181.08 142.95 -90.06 -89.00
Z
34.65 128.11 -110.66 -133.54
TABLE 2. Jacobian matrix loadings from the normalized and globalized FEM analysis Normalized FEM routine Initial element Final element 27.8105 -58.4378 -31.8032 4.3290 -28.8605 -35.5667 29.1260 29.8455 28.0057 -4.5476 -32.0323 60.9898 -74.9000 45.5291 0.3588 50.3718 35.7134 -73.7466
Globalized FEM routine -0.2848 0.5420 1.2875
-0.3428 0.2751 -0.0336
-1.2301 -0.6164 -0.8013
TABLE 3. Lagrangian strain matrix loadings from the normalized and globalized FEM analvsis ~~~
Normalized FEM routine -.478987 ,731515 .201393 -.185602 .201393 ,372220 -.478987 -.185602 ,466267
TABLE 4. Variables of form change derived from the Langrangian strain matrix Major principal extension First minor principal extension Second minor principal extension Angles' Size change Shape change
-55.7
-45.7
1.84 1.25 1.09 -252.10 2.52 1.02
'Angles through which the principal extension cross hairs must be rotated such that they coincide with the plane normal formed by nodes 2-4.
stretch of least magnitude is directed along a mediolateral axis and it indicates that the finite element representing the E-15 specimen increased by only 9% along this axis when compared to the E-17 specimen. Graphic results generated with the globalized FEM are
~
Globalized FEM routine - ,478987 .731515 .201393 ,372220 -.185602 .201393 - ,478987 .185602 .466267 ~
given in Figure 5 and this display is identical to that rendered with the normalized FEM routine (Fig. 4). DISCUSSION
Finite-element modelling methods were developed within a mechanical engineering context and became attractive to morphologists since they provide a means to quantify form change corresponding to a Thompsonian grid transformation (Thompson, 1917).The application of FEM to craniofacial growth analysis stems from the biorthogonal grid method developed by Bookstein (1978). This earlier method provided numerous advantages over more traditional morphometric techniques using linear measurements. However, FEM confers certain advantages over the biorthogonal grid method since it can be extended to three-dimensional analysis and the mathematical functions are more easily programmed (Lewis et al., 1980). In addition, the
384
S. LOZANOFF
Fig. 4. Graphical displays of the nasal septa with corresponding finite elements and principal extensions for the E-15 specimen (A) and the E-17 specimen (B) viewed from a lateral perspective. These results were generated using the normalized FEM algorithm. Anterolateral (C,D) and inferolateral (E,F) perspectives of the models with finite elements and principal extensions are also displayed. The major
stretch is represented by the solid line, the first minor stretch is displayed as a dashed line while the second minor stretch is seen as the dotted line. Gray boxes connected to the ends of the principal extensions signify the legs of the cross hairs which project toward the viewer. S, superior; i, inferior; a , anterior; p, posterior. Bar = 0.5 mm.
numerical solution is interfaced easily with computer graphics routines so that form change can be quantified and visualized simultaneously (Lozanoff and Diewert, 1989). Both normalized and globalized FEM schemes have been applied to the study of craniofacial growth. The normalized FEM scheme has been applied to the analysis of two-dimensional form change in the craniofacial region of humans (Richtsmeier and Cheverud, 1986; Richtsmeier, 1987, 1988), rats (Lozanoff and Diewert, 1986), and mice (Diewert and Lozanoff, 1988) a s well as
three-dimensional applications using samples of nonhuman primates (Cheverud et al., 1983; Cheverud and Richtsmeier, 1986) and rats (Lozanoff and Diewert, 1989). The globalized FEM approach has been used to study two-dimensional form change in the craniofacial region of rats (Moss et al., 1985, 1986, 1987; Moss, 1988) and humans (Book and Lavelle, 1988). The method used in the present study is a n extension of the two-dimensional case described by Moss et al. (1985). Further, results of the present study indicate that the two different FEM schemes provide equivalent results
FORM CHANGE MEASURED BY TWO FEM METHODS
Fig. 5. Graphic display ofthe E-15 model when compared to t h e E-17 specimen along with the corresponding finite element and principal extensions using t h e globalized FEM algorithm. This display is identical to that generated with the normalized FEM routine.
so data derived from these two approaches can be compared directly. The angle of the form change and a meaningful shape variable are two aspects of FEM analysis which remain problematic. Studies using normalized FEM generally reference the angles of the principal extensions (strains) to the initial element. However, the direction of strain generally is referenced to the final object when globalized FEM is used. Therefore, a major strain may be directed through a particular node when the principal extensions are plotted within the initial object. The major strain may not be directed through this particular node if the principal extension is plotted in the final object where the nodes have moved differentially. Therefore, the direction of form change may appear different depending on whether the principal extensions are plotted in the initial or final finite element. Similarly, a measure of shape change is difficult to determine for the three-dimensional case. The ratio of the major and minor stretches provides a measure of distortion or shape change, while a value of size change is represented by the product of the principal extensions for the two-dimensional case (Bookstein, 1983). A size variable is determined a s the product of the three principal extensions in the three-dimensional case. However, a n appropriate measure of three-dimensional shape change has not been established. Cheverud and Richtsmeier (1986) use the invariants of the strain tensor to provide a measure of shape change. However, these values are not on the same scale a s the size invariant which obviates the direct comparison of size and shape change. Lozanoff and Diewert (1989) calculate shape by computing the total sum of differences between In-transformed extension values. However, this approach is applicable only when the direction of form change is similar between specimens. In the present study, a measure of shape is determined by calculating the surface area of the volume formed by the three principal extensions. The surface area of the minimal volume, i.e., cube, capable of being produced by the same principal extensions is then computed. The ratio of the actual surface area and the minimal sur-
385
face area is determined providing a measure of distortion. This measure of shape change is easier to visualize than that derived from the strain invariants. However, measures of size and shape change are scaled differently, and they cannot be compared directly, corresponding to the problem arising when strain invariants are used to calculate measures of shape change. Both normalized and globalized FEM schemes provide equivalent loadings for the Lagrangian strain matrix. This strain matrix is used to calculate the magnitudes of the principal extensions, variables of size and shape change, and the angles of the principal extensions. Therefore, the strain matrix provides the common factor relating normalized and globalized FEM analyses. It is suggested that Langrangian strain matrices should be provided with FEM analyses so that results can be compared directly regardless of the scheme employed. Future research needs to determine whether FEM analysis of global form change successfully predicts cell growth activity in the developing craniofacial cartilages. ACKNOWLEDGMENTS
Excellent technical assistance was provided by Jayne Johnston and John Deptuch. This research was supported by grant MA-10269 from the Medical Research Council of Canada and by the Saskatchewan Health Research Board. LITERATURE CITED Bathe, K.-J., and E.L. Wilson 1976 Numerical Methods in Finite Element Analysis. Prentice Hall, Englewood Cliffs, NJ. Book, D., and C. Lavelle 1988 Changes in craniofacial size and shape with two modes of orthodontic treatment. J . Craniofac. Genet. Dev. Biol., 8t207-223. Bookstein, F.L. 1978 The Measurement of Biological Shape and Shape Change. Lecture Notes in Biomathematics, No. 24 Springer-Verlag, New York. Bookstein, F.L. 1983 The geometry of craniofacial growth invariants. Am. J . Orthod., 83221-234. Cheverud, J.M., J.L. Lewis, W. Bachrach, and W.D. Lew 1983 The measurement of form and variation in form: An application of three-dimensional quantitative morphology by finite-element methods. Am. J. Phys. Anthropol., 62t151-165. Cheverud, J.M., and J.T. Richtsmeier 1986 Finite-element scaling applied to sexual dimorphism in Rhesus macaque lMacaca mulatta) facial growth. Syst. Zool., 35:381-399. Diewert, V.M., and S. Lozanoff 1988 Finite element methods applied to analysis of facial growth during primary palate formation. In: Craniofacial Morphogenesis and Dysmorphogenesis. K.W. Vig and A. Burdi, eds. Monograph 21, Craniofacial Growth Series, Center for Human Growth and Development, University of Michigan, Ann Arbor, MI, pp. 53-71. Hanmer. R.S., and W.E. Bachrach 1986 A Three-Dimensional Nonhomogeneous Scaling and Deformation Analysis System Using Interactive Computer Graphics. Northwestern University, Evanston, IL. Lewis, J.L., W.D. Lew, and J.R. Zimmerman 1980 A nonhomogeneous anthropometric scaling method based on finite element principles. J: Biomech., I3:815-824. Lozanoff, S., and V.M. Diewert 1986 Measuring histological form change with finite element methods: An application using diazo0x0-norleucine (DONbtreated rats. Am. J. Anat., I77:187-201. Lozanoff, S., and V.M. Diewert 1989 A computer graphics program for measuring two- and three-dimensional form change in developing craniofacial cartilages using finite element methods. Comput. Biomed. Res., 22:63-82. Moss, M.L. 1988 Finite element method comparison of murine mandibular form differences. J. Craniofac. Genet. Dev. Biol., 8t3-20. Moss, M.L., L. Moss-Salentijn, and R. Skalak 1986 Finite-element modelling of craniofacial growth and development. In: Orthodontics State of the Art, Essence o f Science. L.W. Graber, ed. C.V. Mosby, Toronto, pp. 143-168.
S.LOZANOFF
386
Moss, M.L., R. Skalak, H. Patel, K. Sen, L. Moss-Salentijn, M. Shinozuka, and H. Vilmann 1985 Finite element method modeling of craniofacial growth. Am. J. Orthod., 87t453-472. Moss, M.L., H. Vilmann, L. Moss-Salentijn, K. Sen, H. Pucciarelli, and R. Skalak 1987 Studies on orthocephalization: Growth behavior of the rat skull in the period 13-49 days a s described by the finite element method. Am. J . Phys. Anthropol., 72t323-342. Patel, H. 1983 Growth Analysis by Non-Linear Continuum Theory. Ph.D. dissertation. Columbia University, New York. Richtsmeier, J.T. 1987 Comparative study of normal Crouzon, and Apert craniofacial morphology using finite element scaling analysis. Am. J. Phys. Anthropol., 74:473-493. Richtsmeier, J.T. 1988 Craniofacial growth in Apert Syndrome as measured by finite-element scaling analysis. Acta Anat. (Basel), 133t50-56. Richtsmeier, J.T., and J.M. Cheverud 1986 Finite element scaling analysis of human craniofacial growth. J . Craniofac. Genet. Dev. Biol., 6t289-323. Skalak R. G. Das upta M.L. Moss E.. Otten P. Dullemei.er and H. Vil'mahn 198gAnafytical descr'iption of krowth. J. Tiedr. Blol., 94:555-577. ThomDson. D'A. 1917 On Growth and Form. Cambridge University ' P k s , Cambridge. I
coefficients of the plane. The resulting interpolation function returns a value of 1 for node i and 0 a t the other nodes. A Jacobian matrix (J)is formulated relatinn the displacement functions to the interpolation functions -
-
du du du _-dx dy dz dv _ -dv- dv dx dy dz
J=
-
dw dw dw _ -dx dy dz
The Lagrangian strain matrix (E) is expressed in terms of the Jacobian matrix,
APPENDIX A
The following procedure is used to formulate the globalized FEM routine used in this study. Two anatomical structures are defined based on four homologous landmarks, or nodes. One structure is identified as the initial element while the second anatomical feature is termed the final element. A displacement vector is defined as
where subscripts i j , k are cyclic rotations of Jacobian matrix loadings. The Lagrangian strain tensor is expressed a s a n eigenmatrix and principal extensions are determined by
U=X'-x
v, = (1 + 2A,)1'2
where X' is the vector composed of x,y,z coordinates from each of the four nodes in the final element; X is the vector of u,v,w coordinates from the corresponding nodes of the initial element; and U is the vector of linear displacements. An interpolation function is derived in order to relate all points in each element. Using the initial element, one node (i) is arbitrarily selected and the three nodes remaining provide a bounding surface opposite to the first node. Cross products of the vectors provide the
where vi is the extension ratio of the major or minor axes and hi is the corresponding eigenvalue derived from E. Associated eigenvectors are used to determine the angles of the principal extensions with nodes 2-4. Finally, a size variable is calculated as the product of the principal extensions. The shape variable is determined by comparing the surface area of the parallelpiped formed by the principal extensions with the surface area of the minimal volume corresponding to the same principal extension values.
Comparison Between Two Finite-Element Modeliing Methods for Measuring Change in Craniof acial Form SCOTT LOZANOFF Department of Anatomy, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N OW0
ABSTRACT Finite-element modelling of form change is a useful morphometric technique for measuring differences between anatomical patterns. Two different finite-element algorithms currently are used. One method requires normalized coordinates a s input data, while the second method uses globalized coordinates as input data. This study determines whether the two finite-element methods provide equivalent measures of three-dimensional form change when applied to the nasal septa of embryonic mice. Computer models of the nasal septa from mice of 15 and 17 days gestation were generated. Homologous landmarks were identified so that each nasal septum was represented by a tetrahedral finite-element. These elements were subjected to both finite-element modelling methods. Results show that the two algorithms use different interpolation functions and yield dissimilar intermediate results, but generate identical strain matrices as well a s equivalent principal extensions, directions of form change, variables of form change, and graphical displays. Therefore, results are directly comparable from studies using either finite-element modelling method. Finite-element modelling (FEM) is a morphometric technique used to measure form change between anatomical structures. Following this approach, homologous landmarks are identified on a n initial anatomical structure and the coordinates associated with these homologous nodes are used to define a finite anatomical element. Then, homologous landmarks and corresponding coordinates are identified on a second anatomical structure. Form change is viewed as a continuous deformation of the initial finite anatomical element into the final anatomical element. The magnitude and direction of strain experienced by the initial element as it deforms into the final element provides a measure of difference between the two anatomical patterns. The goal of the FEM procedure is to identify the anatomical nodes which experience the greatest rate of change as the initial anatomical element deforms into the final element. Two separate FEM schemes have been applied for the analysis of craniofacial form change. The first approach compares the rates of nodal change relative to a point inside the anatomical element a s it deforms into the final configuration (Bathe and Wilson, 1976). This approach can be referred to as normalized FEM, since i t calculates measures of form change based on normalized points within the two anatomical structures. Computer routines for normalized FEM have been developed specifically for the analysis of two- and three-dimensional craniofacial form change (Hanmer and Bachrach, 1986; Lozanoff and Diewert, 1989). The second FEM scheme determines measures of form change by calculating the rates of change in line segments connecting adjacent nodes relative to the displacement of homologous landmarks as the ini0 1990 WILEY-LISS, INC
tial element deforms into the final configuration (Skalak et al., 1982). The scheme relates the two finite elements based on the displacements of the anatomical nodes during deformation. This approach can be referred to as globalized FEM since the displacement functions are determined within a global coordinate system. An algorithm has been described and implemented for two-dimensional analyses of craniofacial growth and morphology (Skalak et al., 1982; Patel, 1983). Presently, i t is not known whether the normalized and globalized FEM approaches produce equivalent measures of form change. The purpose of this study is to determine whether the normalized FEM scheme provides measures of form change corresponding to those rendered with the globalized technique. Data sets comprising coordinate values of homologous landmarks from the nasal septa of embryonic mice are subjected to both FEM schemes in order to determine whether equivalent numerical results are produced. MATERIALS AND METHODS
Embryonic HeJIC3H mice were used in this study. Specimens were drawn from a larger sample which is being used to study patterns of allometric growth in the developing nasal septum (Fig. 1). Adult female HeJ/C3H mice were subjected to timed matings. Five embryos were collected at 15 gestational days (E-15) while another five embryos were taken a t 17 gesta-
Received J u l y 27, 1989; accepted October 9, 1989.
FORM CHANGE MEASURED BY TWO FEM METHODS
381
implemented for the normalized FEM procedure following Lozanoff and Diewert (1989). A separate algorithm was developed, coded, and implemented for a three-dimensional globalized FEM routine (Appendix A). Data sets were subjected to FEM analysis in a pairwise fashion with the five E-15 specimens serving as the initial elements, while the five E-17 specimens provided the final geometries. Therefore, a total of 25 combinations were performed. Each data set combination was subjected to the normalized FEM routine and subsequently to the globalized FEM scheme. Numerical data from both FEM routines, including the Jacobian matrices, Lagrangian matrices, principal extension ratios, directions of principal extensions, and size and shape variables were generated for all the data set combinations. These values were compared in order to determine whether the two FEM routines produced equivalent results. In addition, the tetrahedral elements and principal extensions were rendered within the histological models. Thus, FEM results were depicted along with the models they described in order to determine whether the visual description of anatomical change coincided between FEM schemes. RESULTS
Numerical results from all pairwise comparisons embryo of 17 gestational days viewed from a lateral perspective. showed that both globalized and normalized FEM routines produced different Jacobian matrices, but generated identical Lagrangian matrices, principal extentional days (E-17). None of the specimens were litter- sion ratios, values of size change, values of shape mates. Specimens were weighed, placed in 1%phos- change, and graphical displays. Rather than tabulate phate buffer, and photographed. The heads were lists of identical numerical data derived from the two removed and fixed in 10% neutral buffered formalin for FEM analyses, one illustrative case will be presented a t least 1 week. Specimens were dehydrated in a in detail. Three-dimensional surface models of the nasal septa graded series of ethanol (50%, 70%, 95%, 100%) and embedded in plastic (JB-4 embedding kit). Three holes from a n E-15 individual (HeJ/C3H/5a) and one E-17 were drilled (.013 inch diameter) in each block, provid- specimen (HeJ/C3H/lOa) are provided in Figure 3. The ing alignment guides for subsequent computerized re- models, viewed from a lateral perspective, correconstruction. Heads were sectioned in a coronal plane sponded well to the nasal septal models of the other at 6 pm on a Sorvall (JB-4) microtome. Five sections embryonic HeJIC3H mice with equivalent gestational were mounted on each slide and alternate slides were ages used in this study. The models were rotated to various perspectives and viewed. Based on these modstained with toludine blue or hematoxylin and eosin. Histologic sections from each specimen were viewed els, the major form change appeared to be oriented with a Leitz compound microscope and subjected to vid- along a n anteroposterior axis rather than superoinfeeomicroscopy. Images were viewed on a computer mon- rior or mediolateral axes. Coordinate values of the homologous nodes for both itor (Apollo DN-4000), and the nasal septum was identified. Outlines of the nasal septum and the alignment the initial and final elements are given in Table 1. holes were traced directly from the monitor. Approxi- Numerical results indicate the two procedures yield mately 30 nasal septal contours were used for each different Jacobian matrices (Table 2). However, the Laspecimen. These tissue contours were assembled pro- grangian strain matrices are identical (Table 3). The viding three-dimensional models for both the E-15 and principal extension ratios are also the same since these E-17 specimens. Four homologous landmarks were values are derived from the Lagrangian strain matrix identified on each model which included one superior, in both procedures (Table 4). The models and their corresponding finite elements one anterior and two posterior nodes. The landmarks selected for analysis included the midpoint of the crista are graphically displayed in Figure 4. Principal extengalli (superior node), the anterior junction between the sion axes, generated using the normalized FEM procebilateral limbs of the anterior transverse lamina (an- dure, are rendered a t the centroid of the initial finite terior node), and the right and left junctions between element. The major stretch (1.84) is oriented along a n the pars posterior and the nasal septum (two posterior anteroposterior axis directed near to the anterior nodes). These landmarks were verified histologically transverse lamina. This value indicates that the E-15 (Fig. 2). These homologous landmarks were digitized specimen experienced a n 84% extension along this anfrom the models providing ten sets of four (x,y,z) coor- teroposterior axis when compared with the E-17 model. dinates. Hence, ten sets of coordinate values estab- The second axis is oriented along a superoinferior line, but the model of the E-15 specimen increased only 25% lished ten tetrahedral finite elements. Computer routines were developed, coded in “C,” and when compared to the older individual. Finally, the Fig. 1. Schematic representation of the nasal septum in an HeJIC3H
382
S. LOZANOFF
Fig. 2. Anatomical landmarks from the nasal septum used to construct the finite elements for the FEM analysis include A, the crista galli (node 1);B, the junction between the limbs of the anterior transverse lamina (node 2); C, the left (node 3) and right (node 4) points of
posterior attachment between the lateral capsular wall and the nasal septum. ATL, anterior transverse lamina; CG, crista galli; NS, nasal septum; PP, pars posterior. Bar = 0.2 mm.
FORM CHANGE MEASURED BY TWO FEM METHODS
383
Fig. 3. Three-dimensional reconstructions of the E-15 specimen (A) and the E-17 specimen (B) viewed from the lateral aspect. S, superior; i, inferior; a, anterior; p, posterior. Bar = 0.5 mm
TABLE 1. Coordinate values of the homologous nodes used for the FEM analysis E-15 specimen Node 1 2 3 4
X
-9.97 94.24 - 129.81 -145.00
Y -150.15 -29.51 6.88 6.00
E-17 specimen Z
X
60.03 - 124.43 131.82 114.93
-50.38 209.71 -278.21 -268.00
Y -181.08 142.95 -90.06 -89.00
Z
34.65 128.11 -110.66 -133.54
TABLE 2. Jacobian matrix loadings from the normalized and globalized FEM analysis Normalized FEM routine Initial element Final element 27.8105 -58.4378 -31.8032 4.3290 -28.8605 -35.5667 29.1260 29.8455 28.0057 -4.5476 -32.0323 60.9898 -74.9000 45.5291 0.3588 50.3718 35.7134 -73.7466
Globalized FEM routine -0.2848 0.5420 1.2875
-0.3428 0.2751 -0.0336
-1.2301 -0.6164 -0.8013
TABLE 3. Lagrangian strain matrix loadings from the normalized and globalized FEM analvsis ~~~
Normalized FEM routine -.478987 ,731515 .201393 -.185602 .201393 ,372220 -.478987 -.185602 ,466267
TABLE 4. Variables of form change derived from the Langrangian strain matrix Major principal extension First minor principal extension Second minor principal extension Angles' Size change Shape change
-55.7
-45.7
1.84 1.25 1.09 -252.10 2.52 1.02
'Angles through which the principal extension cross hairs must be rotated such that they coincide with the plane normal formed by nodes 2-4.
stretch of least magnitude is directed along a mediolateral axis and it indicates that the finite element representing the E-15 specimen increased by only 9% along this axis when compared to the E-17 specimen. Graphic results generated with the globalized FEM are
~
Globalized FEM routine - ,478987 .731515 .201393 ,372220 -.185602 .201393 - ,478987 .185602 .466267 ~
given in Figure 5 and this display is identical to that rendered with the normalized FEM routine (Fig. 4). DISCUSSION
Finite-element modelling methods were developed within a mechanical engineering context and became attractive to morphologists since they provide a means to quantify form change corresponding to a Thompsonian grid transformation (Thompson, 1917).The application of FEM to craniofacial growth analysis stems from the biorthogonal grid method developed by Bookstein (1978). This earlier method provided numerous advantages over more traditional morphometric techniques using linear measurements. However, FEM confers certain advantages over the biorthogonal grid method since it can be extended to three-dimensional analysis and the mathematical functions are more easily programmed (Lewis et al., 1980). In addition, the
384
S. LOZANOFF
Fig. 4. Graphical displays of the nasal septa with corresponding finite elements and principal extensions for the E-15 specimen (A) and the E-17 specimen (B) viewed from a lateral perspective. These results were generated using the normalized FEM algorithm. Anterolateral (C,D) and inferolateral (E,F) perspectives of the models with finite elements and principal extensions are also displayed. The major
stretch is represented by the solid line, the first minor stretch is displayed as a dashed line while the second minor stretch is seen as the dotted line. Gray boxes connected to the ends of the principal extensions signify the legs of the cross hairs which project toward the viewer. S, superior; i, inferior; a , anterior; p, posterior. Bar = 0.5 mm.
numerical solution is interfaced easily with computer graphics routines so that form change can be quantified and visualized simultaneously (Lozanoff and Diewert, 1989). Both normalized and globalized FEM schemes have been applied to the study of craniofacial growth. The normalized FEM scheme has been applied to the analysis of two-dimensional form change in the craniofacial region of humans (Richtsmeier and Cheverud, 1986; Richtsmeier, 1987, 1988), rats (Lozanoff and Diewert, 1986), and mice (Diewert and Lozanoff, 1988) a s well as
three-dimensional applications using samples of nonhuman primates (Cheverud et al., 1983; Cheverud and Richtsmeier, 1986) and rats (Lozanoff and Diewert, 1989). The globalized FEM approach has been used to study two-dimensional form change in the craniofacial region of rats (Moss et al., 1985, 1986, 1987; Moss, 1988) and humans (Book and Lavelle, 1988). The method used in the present study is a n extension of the two-dimensional case described by Moss et al. (1985). Further, results of the present study indicate that the two different FEM schemes provide equivalent results
FORM CHANGE MEASURED BY TWO FEM METHODS
Fig. 5. Graphic display ofthe E-15 model when compared to t h e E-17 specimen along with the corresponding finite element and principal extensions using t h e globalized FEM algorithm. This display is identical to that generated with the normalized FEM routine.
so data derived from these two approaches can be compared directly. The angle of the form change and a meaningful shape variable are two aspects of FEM analysis which remain problematic. Studies using normalized FEM generally reference the angles of the principal extensions (strains) to the initial element. However, the direction of strain generally is referenced to the final object when globalized FEM is used. Therefore, a major strain may be directed through a particular node when the principal extensions are plotted within the initial object. The major strain may not be directed through this particular node if the principal extension is plotted in the final object where the nodes have moved differentially. Therefore, the direction of form change may appear different depending on whether the principal extensions are plotted in the initial or final finite element. Similarly, a measure of shape change is difficult to determine for the three-dimensional case. The ratio of the major and minor stretches provides a measure of distortion or shape change, while a value of size change is represented by the product of the principal extensions for the two-dimensional case (Bookstein, 1983). A size variable is determined a s the product of the three principal extensions in the three-dimensional case. However, a n appropriate measure of three-dimensional shape change has not been established. Cheverud and Richtsmeier (1986) use the invariants of the strain tensor to provide a measure of shape change. However, these values are not on the same scale a s the size invariant which obviates the direct comparison of size and shape change. Lozanoff and Diewert (1989) calculate shape by computing the total sum of differences between In-transformed extension values. However, this approach is applicable only when the direction of form change is similar between specimens. In the present study, a measure of shape is determined by calculating the surface area of the volume formed by the three principal extensions. The surface area of the minimal volume, i.e., cube, capable of being produced by the same principal extensions is then computed. The ratio of the actual surface area and the minimal sur-
385
face area is determined providing a measure of distortion. This measure of shape change is easier to visualize than that derived from the strain invariants. However, measures of size and shape change are scaled differently, and they cannot be compared directly, corresponding to the problem arising when strain invariants are used to calculate measures of shape change. Both normalized and globalized FEM schemes provide equivalent loadings for the Lagrangian strain matrix. This strain matrix is used to calculate the magnitudes of the principal extensions, variables of size and shape change, and the angles of the principal extensions. Therefore, the strain matrix provides the common factor relating normalized and globalized FEM analyses. It is suggested that Langrangian strain matrices should be provided with FEM analyses so that results can be compared directly regardless of the scheme employed. Future research needs to determine whether FEM analysis of global form change successfully predicts cell growth activity in the developing craniofacial cartilages. ACKNOWLEDGMENTS
Excellent technical assistance was provided by Jayne Johnston and John Deptuch. This research was supported by grant MA-10269 from the Medical Research Council of Canada and by the Saskatchewan Health Research Board. LITERATURE CITED Bathe, K.-J., and E.L. Wilson 1976 Numerical Methods in Finite Element Analysis. Prentice Hall, Englewood Cliffs, NJ. Book, D., and C. Lavelle 1988 Changes in craniofacial size and shape with two modes of orthodontic treatment. J . Craniofac. Genet. Dev. Biol., 8t207-223. Bookstein, F.L. 1978 The Measurement of Biological Shape and Shape Change. Lecture Notes in Biomathematics, No. 24 Springer-Verlag, New York. Bookstein, F.L. 1983 The geometry of craniofacial growth invariants. Am. J . Orthod., 83221-234. Cheverud, J.M., J.L. Lewis, W. Bachrach, and W.D. Lew 1983 The measurement of form and variation in form: An application of three-dimensional quantitative morphology by finite-element methods. Am. J. Phys. Anthropol., 62t151-165. Cheverud, J.M., and J.T. Richtsmeier 1986 Finite-element scaling applied to sexual dimorphism in Rhesus macaque lMacaca mulatta) facial growth. Syst. Zool., 35:381-399. Diewert, V.M., and S. Lozanoff 1988 Finite element methods applied to analysis of facial growth during primary palate formation. In: Craniofacial Morphogenesis and Dysmorphogenesis. K.W. Vig and A. Burdi, eds. Monograph 21, Craniofacial Growth Series, Center for Human Growth and Development, University of Michigan, Ann Arbor, MI, pp. 53-71. Hanmer. R.S., and W.E. Bachrach 1986 A Three-Dimensional Nonhomogeneous Scaling and Deformation Analysis System Using Interactive Computer Graphics. Northwestern University, Evanston, IL. Lewis, J.L., W.D. Lew, and J.R. Zimmerman 1980 A nonhomogeneous anthropometric scaling method based on finite element principles. J: Biomech., I3:815-824. Lozanoff, S., and V.M. Diewert 1986 Measuring histological form change with finite element methods: An application using diazo0x0-norleucine (DONbtreated rats. Am. J. Anat., I77:187-201. Lozanoff, S., and V.M. Diewert 1989 A computer graphics program for measuring two- and three-dimensional form change in developing craniofacial cartilages using finite element methods. Comput. Biomed. Res., 22:63-82. Moss, M.L. 1988 Finite element method comparison of murine mandibular form differences. J. Craniofac. Genet. Dev. Biol., 8t3-20. Moss, M.L., L. Moss-Salentijn, and R. Skalak 1986 Finite-element modelling of craniofacial growth and development. In: Orthodontics State of the Art, Essence o f Science. L.W. Graber, ed. C.V. Mosby, Toronto, pp. 143-168.
S.LOZANOFF
386
Moss, M.L., R. Skalak, H. Patel, K. Sen, L. Moss-Salentijn, M. Shinozuka, and H. Vilmann 1985 Finite element method modeling of craniofacial growth. Am. J. Orthod., 87t453-472. Moss, M.L., H. Vilmann, L. Moss-Salentijn, K. Sen, H. Pucciarelli, and R. Skalak 1987 Studies on orthocephalization: Growth behavior of the rat skull in the period 13-49 days a s described by the finite element method. Am. J . Phys. Anthropol., 72t323-342. Patel, H. 1983 Growth Analysis by Non-Linear Continuum Theory. Ph.D. dissertation. Columbia University, New York. Richtsmeier, J.T. 1987 Comparative study of normal Crouzon, and Apert craniofacial morphology using finite element scaling analysis. Am. J. Phys. Anthropol., 74:473-493. Richtsmeier, J.T. 1988 Craniofacial growth in Apert Syndrome as measured by finite-element scaling analysis. Acta Anat. (Basel), 133t50-56. Richtsmeier, J.T., and J.M. Cheverud 1986 Finite element scaling analysis of human craniofacial growth. J . Craniofac. Genet. Dev. Biol., 6t289-323. Skalak R. G. Das upta M.L. Moss E.. Otten P. Dullemei.er and H. Vil'mahn 198gAnafytical descr'iption of krowth. J. Tiedr. Blol., 94:555-577. ThomDson. D'A. 1917 On Growth and Form. Cambridge University ' P k s , Cambridge. I
coefficients of the plane. The resulting interpolation function returns a value of 1 for node i and 0 a t the other nodes. A Jacobian matrix (J)is formulated relatinn the displacement functions to the interpolation functions -
-
du du du _-dx dy dz dv _ -dv- dv dx dy dz
J=
-
dw dw dw _ -dx dy dz
The Lagrangian strain matrix (E) is expressed in terms of the Jacobian matrix,
APPENDIX A
The following procedure is used to formulate the globalized FEM routine used in this study. Two anatomical structures are defined based on four homologous landmarks, or nodes. One structure is identified as the initial element while the second anatomical feature is termed the final element. A displacement vector is defined as
where subscripts i j , k are cyclic rotations of Jacobian matrix loadings. The Lagrangian strain tensor is expressed a s a n eigenmatrix and principal extensions are determined by
U=X'-x
v, = (1 + 2A,)1'2
where X' is the vector composed of x,y,z coordinates from each of the four nodes in the final element; X is the vector of u,v,w coordinates from the corresponding nodes of the initial element; and U is the vector of linear displacements. An interpolation function is derived in order to relate all points in each element. Using the initial element, one node (i) is arbitrarily selected and the three nodes remaining provide a bounding surface opposite to the first node. Cross products of the vectors provide the
where vi is the extension ratio of the major or minor axes and hi is the corresponding eigenvalue derived from E. Associated eigenvectors are used to determine the angles of the principal extensions with nodes 2-4. Finally, a size variable is calculated as the product of the principal extensions. The shape variable is determined by comparing the surface area of the parallelpiped formed by the principal extensions with the surface area of the minimal volume corresponding to the same principal extension values.
