Biomech Model Mechanobiol (2012) 11:35–47 DOI 10.1007/s10237-011-0291-5
ORIGINAL PAPER
The application of muscle wrapping to voxel-based finite element models of skeletal structures Jia Liu · Junfen Shi · Laura C. Fitton · Roger Phillips · Paul O’Higgins · Michael J. Fagan
Received: 10 September 2010 / Accepted: 20 January 2011 / Published online: 10 February 2011 © Springer-Verlag 2011
Abstract Finite elements analysis (FEA) is now used routinely to interpret skeletal form in terms of function in both medical and biological applications. To produce accurate predictions from FEA models, it is essential that the loading due to muscle action is applied in a physiologically reasonable manner. However, it is common for muscle forces to be represented as simple force vectors applied at a few nodes on the model’s surface. It is certainly rare for any wrapping of the muscles to be considered, and yet wrapping not only alters the directions of muscle forces but also applies an additional compressive load from the muscle belly directly to the underlying bone surface. This paper presents a method of applying muscle wrapping to high-resolution voxel-based finite element (FE) models. Such voxel-based models have a number of advantages over standard (geometry-based) FE models, but the increased resolution with which the load can be distributed over a model’s surface is particularly advantageous, reflecting more closely how muscle fibre attachments J. Liu · R. Phillips Department of Computer Science, University of Hull, Hull, UK e-mail:
[email protected] R. Phillips e-mail:
[email protected] J. Shi · M. J. Fagan (B) Department of Engineering, University of Hull, Cottingham Road, Hull HU6 7RX, UK e-mail:
[email protected] J. Shi e-mail:
[email protected] L. C. Fitton · P. O’Higgins Hull-York Medical School, The University of York, York, UK e-mail:
[email protected] P. O’Higgins e-mail:
[email protected]
are distributed. In this paper, the development, application and validation of a muscle wrapping method is illustrated using a simple cylinder. The algorithm: (1) calculates the shortest path over the surface of a bone given the points of origin and ultimate attachment of the muscle fibres; (2) fits a Non-Uniform Rational B-Spline (NURBS) curve from the shortest path and calculates its tangent, normal vectors and curvatures so that normal and tangential components of the muscle force can be calculated and applied along the fibre; and (3) automatically distributes the loads between adjacent fibres to cover the bone surface with a fully distributed muscle force, as is observed in vivo. Finally, we present a practical application of this approach to the wrapping of the temporalis muscle around the cranium of a macaque skull. Keywords Muscle wrapping · Finite element analysis · Voxel-based · Stress analysis
1 Introduction Engineering techniques such as finite element analysis (FEA) and more recently multibody dynamics analysis (MDA) are now commonly used to investigate the form and function of the skeletal system (Curtis et al. 2008, 2009; Dumont et al. 2005; Kupczik et al. 2007, 2009; McHenry et al. 2007; Moazen et al. 2008, 2009a,b; Rayfield 2007; Moreno et al. 2008; Richmond et al. 2005; Strait et al. 2009; Tsubota et al. 2009; Wroe et al. 2007). With the ready availability of microCT scanners, it is now easy to produce geometrically accurate 3D models of whole bones, including small surface features and internal geometries and cavities. At present however, the majority of FE models use smoothed, and usually simplified, geometry-based meshing with tetrahedral elements, rather than voxel-based meshing—which can be
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created directly from CT (voxel) data with minimum loss of detail. Apart from this simplicity of model creation, working directly with voxel data offers a number of other advantages when modelling bony structures: • • •
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the CT voxels are converted into identical cuboid elements, with no meshing, distortion or scaling issues; resampling of the model is straightforward, allowing the resolution of the geometry and the model to be readily increased or decreased; the material properties (especially Young’s modulus) of each element can be straightforwardly inferred from the voxel CT grey scale (Hounsfield) value; (e.g. Austman et al. 2008; Harrison et al. 2008; Helgason et al. 2008b; Keyak et al. 1994; Linde et al. 1992; Rho et al. 1995); parallelized FE solvers are relatively easy to write for the voxel-based approach, allowing very large-scale models to be analysed with modest high-performance computers; with very large-scale models, very fine details of the bone structure, such as small internal cavities and trabeculae, can be included automatically. This micro-structural variation then leads in large part to the macro-structural inhomogeneity observed in bones; adaptation of the model (i.e. through simulated resorption or formation of the bone where elements are stripped away or added, respectively) is facilitated with a voxelbased approach.
However, there are two main disadvantages with voxel-based models. First, the number of elements required to achieve adequate resolution is one or two orders of magnitude greater than with the geometry-based approach; hence, significantly more computing resource is required to solve the FE equations. But this is becoming less of an issue as processing power continues to increase while hardware costs continue to decrease. Second, cuboid elements result in rough surfaces that can lead to discontinuous and inaccurate surface stresses. However, a number of studies have examined this issue, showing that voxel-based models produce accurate answers if a sufficient number of elements are used (Camacho et al. 1997; Charras and Guldberg 2000; Guldberg et al. 1998; Ulrich et al. 1998; Verhulp et al. 2006; Yeni et al. 2005). Thus, there have been an increasing number of voxel-based studies (and pixel-based studies in 2D) reported in the literature (e.g. Camacho et al. 1997; Gröning et al. 2009; Kupczik et al. 2009; McDonnell et al. 2009; Tsubota et al. 2009). The effects of different skeletal model geometries and material properties on the accuracy of the FE predictions have been the subject of investigation for a number of groups (e.g. Helgason et al. 2008a; Strait et al. 2005). However, to date, the same attention has not been paid to the loadings and
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constraints applied to the models. While there have previously been some efforts to model the behaviour of individual muscles (i.e. Blemker and Delp 2005; Blemker et al. 2005; Lemos et al. 2005), detailed modelling of muscle geometry and of the consequent loading applied to bones that underlie muscles in FE models is very rare; yet the muscles’ interactions with the bone will be important in correctly estimating bone deformations under load. Normally, muscles are approximated as simple force vectors attached piecemeal to the surface nodes of finite element models (e.g. Curtis et al. 2008; Dumont et al. 2005). The selection of loaded nodes is usually guided by anatomical knowledge of muscle attachment sites and sometimes by distributing the muscle forces over the attachment areas as fields of point forces (Ross 2005; Strait et al. 2009; Wroe et al. 2007). However, muscles lie over and around many bones and are intimately attached to bone surfaces with fibres passing over bony surfaces and loading the attachment areas. As such, representations of muscles as having only point attachments emerging from bone are gross approximations in many cases. Recently, Grosse et al. (2007) proposed a more sophisticated approach to modelling muscle forces on curvilinear bone structures and compared various loading methods, from simple point loads to fully distributed tangential and normal forces, on the stress distribution in a bat skull. They found reasonable qualitative agreement between models with and without wrapped muscle but appreciable differences in some high-stress regions of skull. Further, significantly less total applied muscle force was required in wrapped relative to unwrapped models to generate equivalent bite-point reaction forces. In this study, we report on the development of novel muscle wrapping algorithms for voxel-based finite element models, applied first to a simple validation model and then used to wrap the temporalis muscle around a macaque skull. This required the development of procedures to calculate the shortest path of a muscle fibre over complex irregular voxelbased surfaces, and the calculation of the resulting normal and tangential loadings applied to the surface of the bone by the fibre, assuming the fibre is uniformly anchored to the bone over its length as well as an algorithm for evenly distributing the load between adjacent ‘fibres’.
2 Methods The muscle wrapping algorithm is implemented in our own voxel-based FE software, VOX-FE (Fagan et al. 2007), which is based on earlier BMU-SIM software developed by some of the present authors to simulate bone modelling activity in cancellous bone samples (Fagan et al. 1999; Langton et al. 1998, 2000). VOX-FE has been developed to allow very large models to be visualized interactively in 3D and sectioned orthogonally to allow the internal geometry and results to be
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examined. Model geometry is generated directly from CT (usually microCT) data, where voxels are converted directly into hexahedral finite elements. VOX-FE allows complex loading and constraint conditions to be readily applied to the models, with the model and load data from VOX-FE then passed to a bespoke FE solver which is parallelised and implemented on a high-performance cluster. Output from the FE solver can then be visualized in VOX-FE. Model 3D geometry is approximated with a fine mesh of voxels (elements) so that when a fibre is wrapped around the model, it must lie over this digitized surface forming a smooth, continuous curve. To determine the path of a fibre, we need to find the shortest path over the voxels between user-defined start and end points. Finding the shortest (geodesic) path between points on a curved surface is a wellknown problem for smooth polyhedral surfaces, with several methods proposed (e.g. Mitchell et al. 1987; Chen and Han 1990). However, a different approach is used here for our digitized data, to avoid the complexities of converting the voxelbased model into a triangular surface and then mapping the muscle attachment points back from tetrahedral to voxel form for the FE solver. It is based on a method first proposed by Kiryati and Szekely (1993).
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Fig. 1 The chain code scheme—red, green and blue lines show the direct, minor and major link types
2.1 Calculation of the muscle path The first step of the process is to build a weighted graph of the surface over which the muscle acts. The nodes of the surface voxels are regarded as vertices of the graph, and two neighbouring nodes are connected by edges in the graph. A 26 chain code scheme (Freeman 1974) is then applied to define the neighbours of each node and its connection types (see Fig. 1). Since only the exterior surface nodes are considered, the real number of edges incident on a vertex is much less than 26. There are three types of connections: direct, minor diagonal and major diagonal, shown as red, green and blue lines respectively in Fig. 1. Therefore, the shortest path on the surface can be approximated by the set of surface voxels that the path transverses, i.e. the length L of the shortest path can be approximated by: L ≈ w 1 N 1 + w2 N 2 + w3 N 3
(1)
where N1 , N2 and N3 are the number of direct, minor and major links respectively the path transverses in the graph; and w1 , w2 and w3 are different weights assigned to these different connection types that take account of their different lengths. Here, weights of w1 = 0.9016, w2 = 1.289 and w3 = 1.615, are used as suggested by Kiryati and Szekely (1993). Choosing the true connection lengths of the links (i.e. √ √ 1, 2, 3) as the weights would lead to consistent overestimation of length L (Kiryati and Kübler 1995). A resulting weighted graph for a simple curved section is shown on the
Fig. 2 a Shows the surface graph for a cross-section of a cylinder; and b is an enlarged view, where red, green and blue lines show the different weights (w1 = 0.9016, w2 = 1.289 and w3 = 1.615 respectively) of the edges of the graph as those in Fig. 1, and black squares are the nodes of voxels. For example, sample node A has eight edges incident on it, shown as four direct (red) lines and four minor diagonal (green) lines
left in Fig. 2. Due to the fine resolution of the surface voxels, an enlarged view of part of the weighted graph is shown on the right. Here, red, blue, green lines show the different weights attributed to edges of the graph. For example, for sample node A in Fig. 2b, there are eight edges incident on it, i.e. four direct (red) lines and four minor diagonal (green) lines. The second step is to find the shortest path between a source and a destination vertex on the weighted graph for a particular fibre, where the source vertex can be considered to be the origin of the muscle fibre, and the destination vertex to
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Fig. 3 Construction of the shortest fibre path over a surface, where red and green spheres represent the start and end points of the fibre, respectively. a The original path follows the actual voxel-based surface profile (red jagged curve); and b the approximated continuous curve using NURBS (black smooth curve) so that fibre stretches over depressions on the surface
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third-degree NURBS curve (cubic curve) fitting the 140 data points on the shortest path with an error bound of twice the size of a voxel. The mapping relationship between points on the NURBS curve (Fig. 3b) and those on the shortest path (Fig. 3a) is stored after this process, thus prior to the FEA calculation, the force obtained from the smoothed curve can be applied to all points on the shortest path i.e. the nodes of the voxel-based model. Once the path of each fibre has been obtained, the next stage is to calculate the forces to be distributed along muscle ‘fibre’ paths. We use the term ‘fibre’ to refer to the way in which we model muscles, simplifying them for MDA as a series of single fibres that span the regions of origin and insertion. 2.2 Calculation of the forces along the fibre
be where the fibre leaves the surface of the bone to attach to an adjacent bone. For such a weighted graph, the Dijkstra algorithm (Dijkstra 1959) can be applied to find the shortest path. However, due to the voxel-based representation of the surface, and the connection restrictions of voxel nodes (direct, minor, major links), the shortest path is a jagged poly-line as shown in Fig. 3a. Therefore, post-processing is required to find a smooth curve that is representative of the surface path. This is important because the tangent and normal vectors of the curve are used to determine the muscle force directions with the curvature of the fibre playing a very important role in the calculation of normal forces. There are many techniques for curve approximation, but here we use a Non-Uniform Rational B-Spline (NURBS; Piegl and Tiller 1997) because it is a piecewise rational polynomial curve and can fit data with a complex shape. In approximating the curve, we specify a maximum bound on its deviation from the shortest path (see Fig. 3a). Starting with a first-degree interpolated curve of the shortest path, the end points are constrained and the remaining points approximated using a least squares approach. Figure 3b shows a Fig. 4 a A fibre bundle wraps around the bone, starting at point A and leaving at point B; and b the muscle forces acting on an isolated infinitesimal section of muscle spans an angle dθ of the bone at point B. (Refer to the text for all symbols; adapted from Grosse et al. 2007)
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The calculation of the tangential and normal forces that each fibre exerts on the skull follows the same approach as that used by Grosse et al. (2007). First, consider the fibre wrapping around the bone from point A to point B (see Fig. 4a). There is more muscle bulk on the right-hand side of the muscle section, thus there will be a greater tensile force (T + dT) on that side than on the left-hand side (T). Assuming that the fibre is uniformly anchored onto the bone, τt is the tangential traction of an infinitesimal section of muscle fibre. If Ttotal is the total tensile force exerted by the muscle bundle, then we will have: τt =
Ttotal s
(2)
where s is the arc length of the fibre from A to B. Second, consider an isolated infinitesimal section of muscle that spans an angle dθ (see Fig. 4b) and comprises fibres that arise from (and act on) the underlying bone. The fibre exerts both normal (τn ) and tangential tractions (τt ) on the underlying surface. For equilibrium of the infinitely small
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section at point B, (Grosse et al. 2007): s(θ ) τt τn (θ ) = R(θ )
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(3)
where θ is the angle the muscle fibre spans from its attachment point, R(θ ) is the radius of curvature of the bone, and s(θ ) is the arc length. For the more general 3-D case, the above can be written as: τn (r) = s(r)κ(r)τt
(4)
where r is the position vector of the current point on the fibre, s(r) is the arc length from the muscle attachment point to the current point on the curve, and κ(r) is the curvature of the fibre curve at the current point. Finally, for a small section of muscle with length of ds, the tangential and normal components of muscle forces can be calculated from d Fn = τn ds d Ft = τt ds
(5)
The net muscle force exerted by the fibre at any point can be obtained from: F = Ft + Fn
(6)
For a parametric NURBS curve C(u), its first derivative C (u) gives the tangent vector direction, and the curvature κ can be calculated from its first and second derivatives: κ=
||C (u) × C (u)|| ||C (u)||3
(7)
One can also obtain normal vectors: N(u) =
C (u) × (C (u) × C (u)) C (u) C (u) × C (u)
Fig. 5 Five muscle fibres wrapped around a cylindrical model. The red and green spheres show, respectively, each fibre’s point of attachment and the point at which it leaves the surface of bone to connect to an adjacent bone, with the blue shaded area showing the elements over which the forces are distributed. a The full cylindrical model (red, green, and blue lines show the x, y and z coordinate axes respectively); b shows the normal (red) and tangential (green) components of muscle forces along the NURBS curves; and c also shows the net muscle forces (black) for just one fibre
(8)
Therefore, given the magnitude of muscle loading and by calculating a NURBS curve, the tangential and normal components of the muscle forces can be derived from Eq. (5). 2.3 Distribution of the forces between the fibres After obtaining the loading along each fibre, the final stage of muscle wrapping is to automatically distribute the loads between adjacent fibres so that the muscle activity is fully distributed over the bone surface, as observed in vivo. This is achieved by repeating the following algorithm until the whole surface between fibres is loaded. The bone surface between adjacent fibres is divided by a poly-line into equal halves. A vector v is calculated perpendicular to the plane on which a fibre lies. A walk on the bone surface is then started from the current point on the fibre in the vector direction (see the blue arrows in Fig. 5b and in the opposite direction −v) until it reaches the closest node on the dividing line in that
direction. The direction of every step is selected by comparing neighbour directions (the number of comparisons is less than 26, all possible directions, since only surface nodes are considered) to the vector direction v and choosing the closest one. The nodes between fibres are thus found, and the forces at the relevant points along each fibre are averaged among them (see blue nodes in Figs. 5a, 6). Note that for the boundary cases (the first and last fibres, see Fig. 5), we can optionally count the nodes in one direction only, or walk the same number of nodes in the opposite direction as above.
2.4 Validation: cylindrical model A simple cylindrical model was used to validate our approach to muscle wrapping. The model was analysed in VOX-FE, and the results using the muscle wrapping algorithm described above were compared with those from loading,
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Fig. 6 Wrapping of the temporalis muscle around a Macaque cranium. The red spheres show the points of attachment of the fibres, and green spheres show the points at which fibres leave the surface of bone to connect to the blue spheres on the coronoid process (not shown). The blue shaded element faces indicate the area over which the muscle forces are distributed
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vectors Ft at the points on the NURBS curves in red and green arrows respectively. The right image shows the net muscle forces acting along the fibre with black arrows. From Eq. (4), if the curvature is the same along the fibre, normal forces increase with increasing distance along the fibre. These forces are then averaged between fibres, being applied among the blue nodes (see Fig. 5a) as indicated in the previous section. The ANSYS model geometry is likewise defined to represent a quarter of a cylinder of the same dimensions as the VOX-FE model, but with a smooth geometry. The model is meshed using 23,229 hexahedral elements. For the ANSYS cylinder model, we are unable to directly apply the forces computed from the above in the same way. Instead of calculating the shortest paths, the NURBS curves and averaging the forces in-between the fibres, we use 60 fibres over the middle third as the muscle wrapping region. Along each fibre, 80 evenly distributed data points are evaluated (to match the surface mesh of the model). Again, using Eqs. (2), (3) and (4) accurate normal and tangential forces were obtained and applied. The quarter-circular ends of this ANSYS model were rigidly constrained as in the VOX-FE model. In both models, isotropic material properties were assumed with a Young’s modulus of 17 GPa and Poisson’s ratio of 0.3. Both models were solved, and the results are compared by computing and plotting displacements and principal strains around the cylindrical profile. 2.5 Sample application: macaque skull model
based on exact calculation, using commercial FE software ANSYS (ANSYS Inc., Canonsburg, PA, USA). For this test model, in VOX-FE only one quarter of a cylinder is modelled, and each of the quarter-circular ends is rigidly constrained in all directions. The diameter of the cylinder was 100 mm and length 180 mm with each voxel (element) being of size 0.5 × 0.5 × 0.6 mm; thus, the VOX-FE model consisted of 2,420,550 elements. The resulting stepped surface of the VOX-FE model is seen in Fig. 5a. For this model, the point of origin (start) and the point at which it leaves the surface of the bone to insert into another bone (end) specified by the user are shown as red and green spheres respectively in Fig. 5a. The first fibre (leftmost blue curve in the figure) lies on the surface z = 66 mm, and the last fibre at z = 114 mm (since the muscle wrapping region extends 6 mm beyond the first and last fibres) with the three intermediate fibres evenly spaced in between. Thus, a total of one-third (from z = 60 to 120 mm) of the surface is wrapped as shown by the blue nodes in Fig. 5a. For each fibre, there are 140 nodes on the shortest path. A total load of 100 N is evenly applied by these five equal length fibres, thus each fibre carries 20 N. The left image in Fig. 5b shows the normal force vectors Fn and tangent force
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In order to evaluate the effect of wrapping muscle fibres in a realistic simulation, two FEA models of a macaque skull were compared. In the first, the muscle wrapping algorithms were applied in modelling the forces directly exerted by the temporalis muscle on the temporal region of the cranium of an adult male Macaca fascicularis, which is the same specimen (Mac-14) as that used in the studies by Kupczik et al. (2007) and Curtis et al. (2008). In the second, the temporalis was modelled using point insertions of 6 fibres exerting the same force as in the wrapping simulation but directly into the temporal fossa. The cranium was constructed in VOX–FE from 2,985,507 cuboidal elements with the x, y, z dimensions of each element being 0.227 × 0.227 × 0.3 mm respectively. In this demonstration, model constraints were simply applied in the vertical direction at the temporomandibular joints and in all directions at the second molar. Isotropic properties were assumed with a Young’s modulus of 17 GPa and Poisson’s ratio of 0.3. In the wrapping simulation, the origins of the fibres in the temporal fossa (red spheres) and insertions in the coronoid (blue spheres) were exported from a multibody dynamics analysis (MDA) model (see Fig. 6). Since the insertions are not present in the cranial FEA model, straight line vectors
The application of muscle wrapping to voxel-based finite element models of skeletal structures
are constructed from the fibre end points (green spheres) towards the location of the coronoid. The algorithm described above was then applied to compute the path between the start and the intersection point of each fibre around the temporal region. The point at which each fibre leaves the cranium (green spheres in Fig. 6) was calculated as being the point where the tangent to the surface intersects the insertion point without passing through any other bone. The normal and tangential forces were then calculated for the fibre in contact with the bone. Where a group of fibres is defined to represent a muscle or section of a muscle, the force carried by each fibre is weighted according to its proportion of the total fibre length of the group. Alternatively, since area determines force, we might reasonably have equally divided the force among fibres; for this methodological presentation, the key issue is the effect of distribution of loads over surfaces between rather than among fibres and as such, sensitivity analyses of different approaches to force distribution among fibres are left to future studies. Here, a total load of 100 N was applied by 6 fibres representing the temporalis muscle, as shown in Fig. 6. In this simulation, a uniform distribution is required and so the fibres are distributed uniformly, again future sensitivity analyses might profitably assess the impact of alternative distributions. In the non-wrapping simulation, 30 forces providing a total load of 100 N were directed towards the coronoid process, without any muscle wrapping and applied at points
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distributed over the area of attachment of the temporalis muscle. Thus, a large part of the muscle force passed directly through the bone. Both models were solved and the results compared.
3 Results 3.1 Cylindrical model Figure 7 compares the displacement contour plots predicted by the ANSYS and VOX-FE models respectively (using the same contour levels), while the y displacements around the circumference of the models at three different locations are compared graphically in Fig. 8. Further inspection of the results confirms that there is good agreement between the ANSYS and VOX-FE models, with a maximum error of 8%. However, there are some differences in the raw principal strain results between the ANSYS and VOX-FE models, as shown in Fig. 9a and b, with the voxel-based model predicting more fluctuations rather than the smooth distributions observed with the ANSYS solution. To reduce the effect of the jagged surface, a smoothing step is undertaken, which simply takes the average of the immediately neighbouring nodal values, the results of which is shown in Fig. 9c. Figure 10 shows in more detail the variations in maximum principal strains around the cylindrical profile for ANSYS,
Fig. 7 Component displacements of the cylindrical model as predicted by ANSYS and VOXFE. a x displacement [range −0.974e-3 to 0.731e-3 mm]; b y displacement [range −0.361e-4 to 0.1891e-2 mm]; and c z displacement [range −0.2391e-3 to 0.240e-3 mm]
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Fig. 8 Plot of the y displacement around the cylindrical profile at the three locations shown (at z = 30, 70, 90 mm) as predicted by ANSYS and VOX-FE
and smoothed and unsmoothed VOX-FE results at the three positions considered in Fig. 8. The effect of the irregular surface of the voxel-based model is more obvious here, with significant oscillations in the results, but these are largely removed by the smoothing process. 3.2 Macaque model Figure 11 shows the variations of the tangential, normal and resultant forces acting along 6 muscle fibres laid over the macaque cranium when the same muscle wrapping algorithms are applied—shown as red, green and black arrows respectively. Note that the normal compressive forces (red arrows) increase from zero at the attachment of the fibre (red sphere) to a maximum when the fibre leaves the side of the skull. The exception is the most posterior fibre which runs over a surface whose curvature tends to zero in one region; as such, the normal force decreases and then increases again before the fibre leaves the skull (green sphere). The resultant forces (black arrows in Fig. 11b) also vary in complex ways, but generally increasing as fibres wrap around the bone. Thus, unlike the simple cylindrical model where the curvature is constant and the forces vary in a consistent manner, the forces on the skull clearly vary in more complex ways, especially as the curvature of the surface varies. Contour plots of the minimum (compressive) and maximum (tensile) principal strains produced by Fn , Ft and the total force F are shown in Fig. 12. Note the smooth, continuous regular contours following application of the load distribution and smoothing routines described previously. For the loading considered here, the compressive normal muscle loads produce a high compressive strain under the muscle fibres as expected, with highest values under the
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Fig. 9 Maximum principal strain plots (nodal values). a From VOXFE with no smoothing; b from ANSYS; and c from VOX-FE with smoothing [Contour range 0.08 to 21 μS]
posterior fibres (Fig. 12a). The tangential forces also produce a region of high tensile strain under the most posterior
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Fig. 10 Plot of the maximum principal strain around the cylindrical profile at the three locations shown (at z = 30, 70, 90 mm) as predicted by ANSYS and VOX-FE
muscle fibres (Fig. 12d), but in addition, a region of both high tensile and high compressive strain behind the orbit; (resulting in an area of high shear strain—not plotted here). An area of high compressive strain is also observed above the loaded tooth as expected. When the normal and tangential muscle force components are combined, the principal strains shown in Fig. 12e and f are produced. Note that the high tensile and compressive strains under the posterior fibres are reduced, but the area behind the orbit is still highly strained. In comparison, Fig. 13 shows the minimum and maximum principal strains when the non-wrapping, non-distributed loading condition is considered. In general aspects, these show considerable similarities to the plots of Fig. 12e and f, with the obvious exception of the muscle attachment areas. Figure 14a and b plots the differences (i.e. subtracting the simple, non-wrapped from the wrapped values) in the minimum and maximum principal strains, i.e. the differences between Figs. 12e, f and 13a, b, for the face in particular. These differences are presented as absolute percentages in Fig. 14c and d. This shows that the simple loading slightly overestimates the minimum principal strains above the loaded tooth and overestimates the minimum and underestimates the maximum strains on the browridge.
4 Discussion
Fig. 11 a Variation of the normal Fn forces (red arrows) and tangential Ft forces (green arrows); and b the resultant F forces, acting along the fibres as they contact the skull surface
In this study, we have developed and applied novel muscle wrapping algorithms for voxel-based finite element models. In brief, it computes and distributes the normal and tangential forces generated by muscles represented as fibres over the local surface of the FE model. The current method assumes that the each fibre is anchored uniformly to the surface over
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44 Fig. 12 Principal strain plots in the skull from the wrapped muscle loads (units μS). a Minimum and b maximum principal strains from the normal Fn components; c minimum and d maximum principal strains from the tangential Ft components; and e minimum and f maximum strains from the combined F loading. (Note the different contour ranges, which are selected to produce the most revealing plots. Regions in which strains are below or above the minimum and maximum limits are also shown blue and red respectively in these and all subsequent plots. Figures (c) and (d) have the same scales as (a) and (b))
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Fig. 13 Principal strain plots in the skull from muscle loads that are not wrapped (units μS): a minimum principal strain from the combined loading; and b maximum principal strains from the combined loading
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which it wraps, but this need not be the case. The algorithms could be developed further to allow wrapping with only partial attachment. In this situation, the normal forces would be maintained, but the tangential forces would be redistributed. In order to evaluate the effects of this wrapping on our FE models, we first validated our FE software and the wrapping algorithms by comparing results with those obtained from commercial software, ANSYS using a simple quarter cylindrical model. This study confirms accuracy of our calculation of strains using VOX-FE and achieves highly comparable results for wrapping of a simple muscle. The main reasons for the differences between analyses are, first, that the forces calculated on the approximated curve (Fig. 3b) have to be mapped back to the nodes of the surface voxels prior to FEA calculation; whereas for ANSYS model, the forces are applied at locations separated by equal arc lengths (80 data points along the circle); second, and more significantly, the jagged surface of voxelized model results in discontinuities. After smoothing using averages, the significant oscillations
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in the strain values due to voxellation are largely removed, as demonstrated in Fig. 10. More sophisticated post-processing smoothing algorithms are currently being investigated which should improve these results further, but they are not discussed further here. The impact of muscle wrapping relative to point distribution of muscle loading of FE results is explored in a more biorealistic context by applying the methodology to an adult macaque skull. While overall distributions of strains are similar, there are clear differences that arise as described above (Figs. 12, 13, and 14). The extent to which these are biologically important requires extensive sensitivity analyses in studies that compare FEA results between specimens; does wrapping lead to different interpretations of how different crania respond to loads? These studies are ongoing. However, our findings parallel those of Grosse et al. (2007) who compared various loading methods, from simple point loads through to fully distributed tangential and normal forces, on the stress distribution in a bat skull. As in this study, they
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noted reasonable agreement between models with and without wrapped muscle but with appreciable differences in some high-stress regions of skull. Additionally, their results indicated that significantly less total applied muscle force was required in wrapped relative to unwrapped models to generate equivalent bite-point reaction forces. Note, for simplicity in the skull analyses considered here, the total temporalis muscle force was distributed to each fibre according to its proportion of the total fibre length of the group. However, we have shown in recent MDA modelling studies (Curtis et al. 2008; Shi et al. 2009) that there will be a non-uniform distribution of forces through the muscle, with more load applied by the anterior fibres. While this effect will not change the findings of the study carried out here, future applications of this and other muscle wrapping algorithms should incorporate this variation if accurate stress and strain values are to be predicted through the skull. As discussed previously, to be confident in these predictions, a sufficient number of elements must be used in the model. In Camacho’s study of the meshing of a human skull (Camacho et al. 1997), he showed that at least five voxels were needed through the bone’s thickness to produce reasonable results in very thin structures. For the macaque skull considered here, the voxel size was 0.227 × 0.227 × 0.3 mm and thus defined with three to four voxels per mm (i.e. 27–64 voxels per mm3 ). This suggests that features as small as 1.0– 1.5 mm will be modelled accurately, although more detailed mesh convergence tests are required to prove this. Thus, to model something as complex as a skull, to accurately replicate its geometry and reliably predict the strains through the bone, very many voxel-based elements are required; the sample skull model considered here used nearly 3 million elements. Geometry-based meshing of such intricate and complex geometries is very demanding and often the fine details and smaller structures are lost as the geometry is resampled and simplified to allow meshing. As a result, geometry-based FE meshes of skulls invariably use very many low-order tetrahedral elements, with linear displacement functions and therefore constant strain predictions throughout each element, and obvious compromises in accuracy. As computing power increases, it is likely that voxel-based modelling will become increasingly popular in the analyses of bones, because of the ease and accuracy of meshing, and its ability to easily represent the fine level of detail which is such an important aspect of bony structures. Incorporation of accurate musculoskeletal loading in such models is essential, because of the complex, intimate nature of muscle attachments and their wrapping over and around the surfaces of the bones. Therefore, the development of methods to simulate muscle wrapping over complex geometries is a key step in the realisation of this next generation of musculoskeletal models. The methods and examples presented in this paper
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demonstrate how this can be achieved and the type of results that can be expected. Acknowledgments We are grateful to colleagues who have contributed in various ways through discussion and critique to the development of the methods and ideas we present; Neil Curtis, Flora Groening, Kornelius Kupczik, Charles Oxnard, Lee Page and Ulrich Witzel. The work is supported by research grants from BBSRC (BB/E013805 BB/E014259; BB/E007813, BB/E009204) and initially from the Leverhulme Trust (F/00224).
References Austman RL, Milner JS, Holdsworth DW, Dunning CE (2008) The effect of the density-modulus relationship selected to apply material properties in a finite element model of long bone. J Biomech 41(15):3171–3176 Blemker S, Delp S (2005) Three-dimensional representation of complex muscle architectures and geometries. Ann Biomed Eng 33(5):661–673 Blemker S, Pinksy P, Delp S (2005) A 3D model of muscle reveals the cause of non-uniform strains in the biceps brachii. J Biomech 38:657–665 Camacho DLA, Hopper RH, Lin GM, Myers BS (1997) An improved method for finite element mesh generation of geometrically complex structures with application to the skull base. J Biomech 30:1067–1070 Charras GT, Guldberg RE (2000) Improving the local solution accuracy of large-scale digital image-based finite element analyses. J Biomech 33:255–259 Chen J, Han Y (1990) Shortest paths on a polyhedron. In: SCG ’90: Proceedings of the sixth annual symposium on Computational geometry. ACM, New York, NY, USA, pp 360–369 Curtis N, Kupczik K, O’Higgins P, Moazen M, Fagan MJ (2008) Predicting skull loading: applying multibody dynamics analysis to a macaque skull. Anat Rec 291:491–501 Curtis N, Jones M, Evans SE, Shi JF, O’Higgins P, Fagan MJ (2009) Predicting muscle activation patterns from motion and anatomy: modelling the skull of Sphenodon (Diapsida: rhynchocephalia). J Roy Soc Interface. doi:10.1098/rsif.2009.0139 Dijkstra E (1959) A note on two problems in connection with graphs. Numer Math 1:269–271 Dumont ER, Piccirillo J, Grosse IR (2005) Finite-element analysis of biting behavior and bone stress in the facial skeletons of bats. Anat Rec A 283(2):319–330 Fagan MJ, Dobson CA, Ganney PS, Sisias G, Phillips R, Langton CM (1999) Finite element analysis of cancellous bone resorption. Comput Methods Biomech Biomed Eng 2(4):257–270 Fagan MJ, Curtis N, Dobson CA, Kupczik K, Moazen M, Page L, Phillips R, O’Higgins P (2007) Voxel-based finite element analysis— working directly with microCT scan data. J Morph 268(12):1071 Freeman H (1974) Computer processing of line drawing images. ACM Comput Surv 6(1):57–94 Gröning F, Liu J, Fagan MJ, O’Higgins P (2009) Validating a voxelbased finite element model of a human mandible using digital speckle pattern interferometry. J Biomech 42(9):1224–1229 Grosse IR, Dumont ER, Coletta C, Tolleson A (2007) Techniques for modelling muscle-induced forces in finite element models of skeletal structures. Anat Rec 290:1069–1088 Guldberg RE, Hollister SJ, Travers GT (1998) The accuracy of digital image-based finite element models. J Biomech Eng 120:289–295 Harrison NM, McDonnell PF, O’Mahoney DC, Kennedy OD, O’Brien FJ, McHugh PE (2008) Heterogeneous linear elastic
The application of muscle wrapping to voxel-based finite element models of skeletal structures trabecular bone modelling using micro-CT attenuation data and experimentally measured heterogeneous tissue properties. J Biomech 41(11):2589–2596 Helgason B, Perilli E, Schileo E, Taddei F, Brynjólfsson S, Viceconti M (2008) Mathematical relationships between bone density and mechanical properties: a literature review. Clin Biomech 23(2):135–146 Helgason B, Taddei F, Pálsson H, Schileo E, Cristofolini L, Viceconti M, Brynjólfsson S (2008) A modified method for assigning material properties to FE models of bones. Med Eng Phys 30(4):444–453 Keyak JH, Lee IY, Skinner HB (1994) Correlations between orthogonal mechanical properties and density of trabecular bone: use of different densitometric measures. J Biomed Mater Res 28:1329– 1336 Kiryati N, Szekely G (1993) Estimation shortest paths and minimal distances on digitized three-dimensional surfaces. Pattern Recogn 26(11):1623–1637 Kiryati N, Kübler O (1995) Chain code probabilities and optimal length estimators for digitized three-dimensional curves. Pattern Recogn 28(3):361–372 Kupczik K, Dobson CA, Crompton RH, Phillips R, Oxnard CE, Fagan MJ, O’Higgins P (2009) Masticatory loading and bone adaptation in the supraorbital torus of developing macaques. Am J Phys Anthr 139(2):193–203 Kupczik K, Dobson CA, Fagan MJ, Crompton R, Oxnard CE, O’Higgins P (2007) Assessing mechanical function of the zygomatic region in macaques: validation and sensitivity testing of finite element models. J Anat 210:41–53 Langton CM, Ganney PS, Dobson CA, Fagan MJ, Sisias G, Phillips R (2000) Stochastically simulated assessment of anabolic treatment following varying degrees of cancellous bone resorption. Bone 27(1):111–118 Langton CM, Haire TJ, Dobson CA, Fagan MJ (1998) A dynamic stochastic simulation of cancellous bone resorption. Bone 22(4):375– 380 Lemos R, Rokne O, Baranoski G, Kawakami Y, Kurihara T (2005) Modeling and simulating the deformation of human skeletal muscle based on anatomy and physiology. Comput Animat Virtual Worlds 16:319–330 Linde F, Hvid I, Madsen F (1992) The effect of specimen geometry on the mechanical behaviour of trabecular bone specimens. J Biomech 25:359–368 McDonnell P, Harrison N, Liebschner MA, Mc Hugh PE (2009) Simulation of vertebral trabecular bone loss using voxel finite element analysis. J Biomech. doi:10.1016/j.jbiomech.2009.07.038 (in press) McHenry CR, Wroe S, Clausen PD, Moreno K, Cunningham E (2007) Super-modeled sabercat, predatory behaviour in Smilodon fatalis revealed by high-resolution 3-D computer simulation. Proc Natl Acad Sci USA 104:16010–16015 Mitchell J, Mount D, Papadimitriou C (1987) The discrete geodesic problem. SIAM J Comput 16(4):647–668 Moazen M, Curtis N, Evans SE, O’Higgins P, Fagan MJ (2009) Biomechanical assessment of evolutionary changes in the lepidosaurian skull. Proc Natl Acad Sci USA 20:8273–8277
47
Moazen M, Curtis N, Evans SE, O’Higgins P, Jones MEH, Fagan MJ (2009) Assessment of the role of sutures in a lizard skull— a computer modelling study. Proc Roy Soc B 276:39–46 Moazen M, Curtis N, Evans SE, O’Higgins P, Fagan MJ (2008) Combined finite element and multibody dynamics analysis of biting in a Uromastyx hardwickii lizard skull. J Anat 213:499–508 Moreno K, Wroe S, Clausen P, McHenry C, D’Amore DC, Rayfield EJ, Cunningham E (2008) Cranial performance in the Komodo dragon (Varanus komodoensis) as revealed by high resolution 3-D finite element analysis. J Anat 212:736–746 Piegl L, Tiller W (1997) The NURBS book, 2nd edn. Springer, Berlin Rayfield E (2007) Finite element analysis and understanding the biomechanics and evolution of living and fossil organisms. Annu Rev Earth Planet Sci 35:541–576 Rho JY, Hobatho MC, Ashman RB (1995) Relations of mechanical properties to density and CT numbers in human bone. Med Eng Phys 17:347–355 Richmond BG, Wright BW, Grosse I, Dechow PC, Ross CF, Spencer MA, Strait DS (2005) Finite element analysis in functional morphology. Anat Rec 283(2):259–274 Ross CF (2005) Finite element analysis in vertebrate biomechanics. Anat Rec 283:253–258 Shi JF, Curtis N, Fitton L, O’Higgins P, Fagan MJ (2009) The effect of variations in muscle positions in a complex biomechanical model of a macaque skull. Am J Phys Anth 138:S48 Strait DS, Wang Q, Dechow PC, Ross CF, Richmond BG, Spencer MA, Patel BA (2005) Modelling elastic properties in finite element analysis: how much precision is needed to produce an accurate model?. Anat Rec 283:275–287 Strait DS, Weber GW, Neubauer S, Chalk J, Richmond BG, Lucas PW, Spencer MA, Schrein C, Dechow PC, Ross CF, Grosse IR, Wright BW, Constantino P, Wood BA, Lawn B, Hylander WL, Wang Q, Byron C, Slice DE, Smith AL (2009) The feeding biomechanics and dietary ecology of Australopithecus africanus. Proc Natl Acad Sci USA 106(7):2124–2129 Tsubota K, Suzuki Y, Yamada T, Hojo M, Makinouchi A, Adachi T (2009) Computer simulation of trabecular remodeling in human proximal femur using large-scale voxel FE models: approach to understanding Wolff’s law. J Biomech 42(8):1088–1094 Ulrich D, van Rietbergen B, Weinans H, Rüegsegger P (1998) Finite element analysis of trabecular bone structure: a comparison of image-based meshing techniques. J Biomech 31:1187–1192 Verhulp E, van Rietbergen B, Huiskes R (2006) Comparison of microlevel and continuum-level voxel models of the proximal femur. J Biomech 39(16):2951–2957 Wroe S, Moreno K, Clausen P, McHenry C, Curnoe D (2007) Highresolution three-dimensional computer simulation of hominid cranial mechanics. Anat Rec 290(10):1248–1255 Yeni YN, Christophersin GT, Dong XN, Kim D-G, Fyhrie DP (2005) Effect of microcomputed tomography voxel size on the finite element model accuracy for human cancellous bone. J Biomech Eng 127:1–8
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