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Transcript
This article was published in an Elsevier journal. The attached copyis furnished to the author for non-commercial research and
education use, including for instruction at the author’s institution,sharing with colleagues and providing to institution administration.
Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further information
regarding Elsevier’s archiving and manuscript policies areencouraged to visit:
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 8 ( 2 0 0 7 ) 112–122
journa l homepage: www. int l .e lsev ierhea l th .com/ journa ls /cmpb
Computational representation of the aponeuroses as NURBSsurfaces in 3D musculoskeletal models
Florence T.H. Wua, Victor Ng-Thow-Hingb, Karan Singhc,Anne M. Agurd, Nancy H. McKeee,∗
a Division of Engineering Science, University of Toronto, Toronto, Ont., Canadab Honda Research Institute USA, Mountain View, CA, USAc Department of Computer Science, University of Toronto, Toronto, Ont., Canadad Division of Anatomy, Department of Surgery, University of Toronto, Toronto, Ont., Canadae Division of Plastic Surgery, Department of Surgery, University of Toronto, Toronto, Ont., Canada
a r t i c l e i n f o
Article history:
Received 7 September 2006
Received in revised form
11 May 2007
Accepted 31 July 2007
Keywords:
Musculoskeletal modeling
Computational simulation
Aponeuroses
Connective tissue
Anatomy
Architecture
a b s t r a c t
Computational musculoskeletal (MSK) models – 3D graphics-based models that accurately
simulate the anatomical architecture and/or the biomechanical behaviour of organ sys-
tems consisting of skeletal muscles, tendons, ligaments, cartilage and bones – are valued
biomedical tools, with applications ranging from pathological diagnosis to surgical planning.
However, current MSK models are often limited by their oversimplifications in anatomical
geometries, sometimes lacking discrete representations of connective tissue components
entirely, which ultimately affect their accuracy in biomechanical simulation. In particular,
the aponeuroses – the flattened fibrous connective sheets connecting muscle fibres to ten-
dons – have never been geometrically modeled. The initiative was thus to extend Anatomy3D
– a previously developed software bundle for reconstructing muscle fibre architecture –
to incorporate aponeurosis-modeling capacity. Two different algorithms for aponeurosis
reconstruction were written in the MEL scripting language of the animation software Maya
6.0, using its NURBS (non-uniform rational B-splines) modeling functionality for aponeuro-
sis surface representation. Both algorithms were validated qualitatively against anatomical
Computational graphics-based models of human muscu-loskeletal (MSK) systems, with accurate representations oftheir anatomical components – from skeletal muscles, ten-dons, ligaments, cartilage, to bones and joints – can beutilized in diverse biomedical applications. As an educationaltool, interactive MSK models can serve as an inexpensivealternative to cadaveric dissections in teaching anatomical
architecture, as well as a platform for biomechanical sim-ulations in kinesiology courses [1,2]. As a diagnostic tool,comparison of patient data to computerized MSK models rep-resenting the typical healthy subject can lead to quantifiablecharacterization of anatomical or movement abnormalities[3,4], and initiate patient-customizable treatments such asergonomic prosthesis design [5]. As a surgical-planning tool,graphics-based MSK models that allow clinicians to pre-operatively simulate and optimize treatment strategies can
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benefit procedures such as tendon lengthenings, aponeuro-tomy, tendon transfers, hipbone reconstructions and jointreplacements [6–8].
However, current MSK models are usually limited in theirgeometric complexity of anatomical representation, espe-cially in modeling their connective tissue components, whichalso affects the accuracy of the models’ predictive capacity inbiomechanical simulation. In particular, the aponeuroses – thethin fibrous sheets of flattened tendinous extensions on whichmuscle fibres are attached at their origins or insertions – havenever been modeled as distinct connective tissue componentsin MSK models. It is thus imperative that current MSK modelsbe extended to incorporate anatomically and biomechanicallyrealistic aponeuroses before they can be reliably used in sur-gical procedures such as an aponeurotomy—which involves atransection of the aponeurosis in the direction perpendicularto the muscle length to effectively lengthen spastic or abnor-mally short muscles in patients with cerebral palsy or spinalcord injury.
The objective of this paper was thus to develop analgorithm and implementation for the 3D graphics-basedcomputational modeling of the aponeuroses, as an improve-ment to the anatomical representation of connective tissuecomponents in MSK models, and also as a necessary first steptowards accurate simulation of biomechanical behavior.
2. Background
2.1. Limitations of current MSK models in theliterature
The MSK models presented in the literature commonlyoversimplify geometrical representations of anatomical archi-tecture for the sake of minimizing computational costs. Thecomplex 3D architecture of muscles is often reduced to seriesof line segments that represent muscle–tendon paths in“lumped-parameter” models, carrying the false assumption ofuniform moment arms and muscle fibre lengths within eachmuscle segment [9]. An attempt to incorporate higher musclefibre complexity was made by Blemker et al. in mapping theboundaries of uniform and pre-defined fibre geometry tem-plates onto that of imaged target meshes to reproduce the3D fibre architecture in the target muscles, but even then, the
internal diversity in fibre arrangements was not fully capturedas architecturally distinct regions within the same muscle[9,10].
There has not been much evidence of correspondingprogress made in modeling the connective tissue components.Connective tissue components often do not have anatomicallyrealistic representations in MSK models, replaced instead bysimple geometric constructs, such as spherical “wrapping sur-faces” and zero-dimensional “via points”, to reproduce thephysiological functions of fascia, tendons and aponeurosesin guiding muscle contractions and constraining surroundingtissue from penetrating each other [7,9]. Fascia, which keepsthe various muscles in tight contact as a group during jointmotion, are not being modelled explicitly as distinct boundingsurfaces, but instead have been represented as “sticky linkers”or line segments embedded between adjacent muscle bellies.Neither are aponeuroses or tendons – the fibrous extensionsat the end of muscle bellies – modelled as separate entities.Instead, they have only been functionally modelled by assign-ing inhomogeneous material properties, such as increasedstiffness and additional resistance to elongation, on the mus-cle locations where the muscle–tendon boundaries should be[3]. Recent work by Epstein et al. has shown that the commonsimplistic models of muscle fibres, aponeuroses and tendonsacting mechanically in series leads to erroneous force trans-mission calculations, and that proper considerations of thecomplex 3D spatial and structural relations between musclefibres and connective tissue components are necessary [11].
2.2. Prior development of Anatomy3D software
Our lab has previously developed a protocol for the digitizationof muscle data and a software bundle called Anatomy3D forthe reconstruction of computerized 3D models of muscle fibrearchitecture based on the digitized templates [10].
For muscle data digitization, formalin embalmed cadavericmuscle specimens, such as the extensor carpi radialis bre-vis (ECRB) specimen shown in Fig. 1a, were serially dissectedin multiple layers, from superficial to deep. At each layer,fibre bundles were demarcated at regular intervals along theirlengths, the 3D spatial coordinates of which were then digi-tized using a Microscribe 3DX digitizer (Immersion, San Jose,CA, USA) and collected into separate fibres files [10,12]. This
Fig. 1 – Digitization and reconstruction of muscle fibres in Anatomy3D. (a) Digitization of anatomical specimens, such as thesagittally dissected ECRB shown here, gives a set of templates. (b) Streamlines (blue) are interpolated from templates overmuscle volume to represent muscle fibre bundles. The most superificial layer of streamlines is highlighted (light green) toillustrate the layers hierarchy. Insertion ends (red dots) of reconstructed muscle fibres are mathematically represented byu, v coordinates. (For interpretation of the references to color in this figure legend, the reader is referred to the web versionof the article.)
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process of digitizing by layers captured the internal diversityof fibre arrangements, and thus did not suffer from the geo-metric oversimplification limitation common to other imagesegmentation methods in the literature.
To reconstruct computerized 3D models of muscle fibrearchitecture, a software bundle called Anatomy3D was previ-ously developed to run as a module within the Maya animationplatform. The fibres files containing digitized 3D coordinatesof muscle data were imported into Anatomy3D, and lay-ers of template curves called fibre template objects were thencreated from the original sampled representative fibre datapoints. A routine within Anatomy3D then fitted a B-splinesolid model over the volume enclosing the fibre templateobjects, and interpolated additional fibre bundles called fibrestreamlines evenly distributed throughout the solid muscle vol-ume. The resulting collection of streamlines thus representedthe reconstructed muscle model (Fig. 1b). It should be notedthat streamlines could be generated at any user-specifieddensity, defined through two parameters: (a) the numberof streamline layers and (b) the number of streamlines perlayer.
2.3. Extending Anatomy3D: aponeurosis-modeling
It was hypothesized that collectively, the insertion ends ofthe computationally reconstructed muscle fibre bundles orstreamlines, as generated by Anatomy3D, should theoreticallymap out the spatial landscape of aponeurotic surfaces. Fur-thermore, the mathematical model of Non-Uniform RationalB-Splines (NURBS) was proposed to be ideal for the geometricsurface representation of aponeuroses.
3. Design considerations
3.1. NURBS theory
NURBS geometries, based on B-spline basis functions, are idealfor modeling anatomical shapes that cannot easily be reducedto analytical forms such as straight lines, ellipses, or spheres,and have long been the industry standard for computer-aidedmodeling of free-form geometries [13–16].
The curvature of a NURBS curve (Fig. 2a) is defined by itsdegree, a set of control vertices (CVs), a weight associated witheach CV and a knot vector. The CVs are indexed by the order
they appear along the curve, so that there are two CVs whichcorrespond to the end points and the rest being internal CVs.The degree controls the smoothness and continuity propertiesof a curve: degree-1 curves are linear interpolations betweenthe set of CVs, while degree-3 curves are cubic interpolations,whose shapes are influenced, but not necessarily interpolatingthe set of internal CVs. In contrast to CVs, edit points (EPs) areactual points located exactly on the curves and are parameter-ized in the u-direction, such that the first EP is defined whereu = 0, etc. The algorithms presented in this paper specificallyimplemented a subset of NURBS for aponeuroses representa-tion called uniform non-rational B-splines, where knots wereequally spaced along the curves (i.e. uniform) and the weightsof all CVs were set to 1 (i.e. non-rational) [14].
Fig. 2b shows a typical NURBS surface, with a two-dimensional parameterization, in U (yellow arrow) and V(orange arrow), where the concepts of degree, knot vectors,weights and CVs are extended in two dimensions. Isopa-rameteric curves (shortened as isoparms), such as the onehighlighted in pink are lines created by proceeding alongeither the U- or V-direction while holding the other parameterat a constant value. NURBS surfaces also have directionality(inside versus outside) that can be visualized by the directionof surface normals such as the one highlighted in purple.
3.2. Principal design: surface modeling strategies
Two surface design strategies [13] for manipulating NURBSare particularly relevant to the aponeurosis-modeling devel-opments to be presented: (1) Surface fitting: Given an orderedset of CVs and two knot vectors in the U- and V-directions, aNURBS spline surface (rational or non-rational) can be fitted orapproximated to go near the specified CVs. (2) Surface lofting:Given an ordered set of NURBS curves, a surface is skinned orinterpolated exactly through these sectional or profile curves.
3.3. Aponeurosis design criteria and performancegoals
To evaluate the validity of the aponeurosis models developedin this study, three main design criteria have been identi-fied as follows: (1) Physical representation: The reconstructedaponeuroses have to exist as physical entities in the formof surfaces, as materials or objects distinct from the mus-
Fig. 2 – Components of: (a) a NURBS curve and (b) a NURBS surface. Abbreviations: EP, edit point; CV, control vertex.
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cle fibres, for realistic visual representation and to facilitateassignment of biomechanical properties to them that are dis-tinct from that of muscle fibres. (2) Anatomical smoothness:While local deformations, such as bumps and folds, due toirregularity of muscle fibre arrangements are acceptable whenconsidering the anatomical validity of aponeurosis surfaces,sharp-edged surfaces as computational artefacts should beminimized wherever possible. (3) Attachment to muscle fibres:To be anatomically correct, it is necessary that the fibre bun-dle insertion points, represented computationally as the firstendpoints of the streamline NURBS curves, are positionedspatially exactly on the modelled aponeurosis surface. Inaddition, while biomechanical simulation of the musculo-tendinous models is not an immediate objective of this study,it would be a natural future direction in the development ofAnatomy3D once all the anatomical architecture is correctlymodelled. It is imperative then, that the aponeuroses mod-elled in this paper maintain its positional continuity with themuscle fibres throughout biomechanical movements—suchthat when the streamlines are simulated to contract in thefuture, the aponeuroses should deform accordingly and fol-low the muscle contractions, rather than just staying in itsoriginal position.
4. System description
4.1. Software specifications
The software extensions to incorporate aponeurosis-modelingfunctionality into the existing Anatomy3D bundle as pre-sented in this paper were written in the Maya EmbeddedLanguage (MEL) to be run within the animation software,Maya 6.0 (Autodesk, previously Alias, San Rafael). OtherNURBS-modeling platforms or programming languages couldalso have been used to implement and test our proposedaponeurosis-reconstructing algorithms presented below.
4.2. Algorithm 1: surface fitting
Our first algorithm, as outlined in flowchart form in Fig. 3,implemented the surface fitting strategy, calling Maya’ssurface command to surface-fit an aponeursis over theinsertion-ends of the muscle streamlines. The surface com-mand requires the input of two knot vectors in the U- andV-directions, as well as the ordered matrix of CVs over whichthe surface is to be fitted.
The beginning portion of the algorithm is thus devoted tocalculating: (1) the knot vector sizes, which are dependent onthe degree, the number of streamline-layers for the U-vectorand the number of streamline per layer for the V-vector and(2) the actual knot values that constitute uniform knot vectorsin U and V.
The latter portion of the algorithm codes for the orderedspecification of the set of CVs to be surface fitted. The inser-tion ends of each muscle streamline curve, represented as itsfirst CV or EP by the parameter u = 0, are taken as the inputCVs for the aponeurosis surface. The resulting U-direction ofthe aponeurosis surface follows the direction of increasingstreamline-layer number, while the V-direction of the aponeu-
rosis surface follows the direction of increasing streamlinenumber in each streamline-layer. Starting with streamlinelayer #1 (light green in Fig. 1b), the first CV of each stream-line curve is sequentially fed into the surface command. Thisprocess repeats for each layer until the last layer is reached.
4.3. Algorithm 2: surface lofting with insertion sorting
Our second script, as illustrated in flowchart form in Fig. 4,implemented the surface lofting strategy, calling Maya’s loftcommand to loft an aponeursis over the insertion-ends of themuscle streamlines. The loft command required the orderedinput of a set of NURBS curves through which the surfaceis created by interpolation of these profile curves. Theseinput curves were taken as the insertion-edge isoparms ofeach streamline-layer-surface (white lines in Fig. 5c), whichwere themselves created by surface lofting over each stream-line in that particular layer (Fig. 5a and b). Duplicates ofthese insertion-edge isoparms, defined as u = 0 of each result-ing streamline-layer-surface (the highlighted green edgesin Fig. 5a and b), were sorted in an subroutine describedbelow before they were fed into another loft command tocollectively form the sectional curves of the final loftedaponeurosis.
The objective of the insertion sorting subroutine (Fig. 6)was to tackle a computational artifact of the muscle fibrestreamlines-generating plug-in in Anatomy3D. The manifes-tation of this artifact is illustrated in Fig. 7, where the fiveinterpolated streamline layers (color-coded in the order theywere labelled by the plug-in: 1, red; 2, orange; 3, yellow; 4,green; 5, blue) were not volumetrically concentric and in factintersected each other significantly in 3D. The apparent orderby which the streamline layers were generated and labelleddid not correspond to a consistent spatial order, e.g. in Fig. 7b,layer #2 was superficial to layer #3 but layer #3 was deeperthan layer #4. Without sorting, the input isoparms fed intothe surface-lofting algorithm would have been taken as theorder by which streamline layers were generated and loft-ing could occur from a more spatially superficial layer to adeeper layer and back to a superficial layer at times, result-ing in unnatural-looking and excessively convoluted loftedaponeuroses.
The sorting procedure “SurfaceOrderCheck” (Fig. 6) wasthus created, as summarized below, to determine by vectoralgebra whether the order of streamline-layers as given fromdigitization and computational reconstruction correspondedto a correct spatial order from superior to deep. To verifywhether a surface computationally labelled as “top” is in factspatially superficial to a “bottom” surface (Fig. 7b):
1. Pick a pair of corresponding points on the insertion edge ofthe two muscle-layer surfaces.
2. Let nbottom be the normal vector at the point on the “bot-tom” surface.
3. Let pbottom-top be the vector from the point on the bottomsurface to the point on the top surface.
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Fig. 3 – Flowchart for algorithm 1. Aponeurosis-modeling by surface fitting.
5. In a correctly ordered pair of surfaces (e.g. “top” is superfi-cial to “bottom”):
90◦ < �bottom-top < 180◦ → cos(�bottom-top) < 1
→ nbottom · pbottom-top < 1
6. In an incorrectly ordered pair of surfaces (e.g. “top” isdeeper than “bottom”):
0◦ < �bottom-top < 90◦ → cos(�bottom-top) > 1
→ nbottom · pbottom-top > 1
The above calculation is performed at regular intervalsalong the aponeurotic edge for n number of times, wheren = number of streamlines per layer. If the majority of the
calculations registered a violation (i.e. positive dot products),the order of isoparm specification would be swapped priorto aponeurosis-lofting. Note that the restriction of samplingto the aponeurotic edge, rather than the entire depth of themuscle layers, allows for physiological inter-penetration ofmuscle fibres in the muscle belly while minimizing unnec-essary convolution of the lofted aponeuroses at the fibreends.
5. Status report
The following discussion serves as a validation of the sim-ulated aponeuroses, using the ECRB – a forearm musclefunctionally involved in the extension and abduction of thehand at the wrist – as the model muscle [17]. Precisely, it is the
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Fig. 4 – Flowchart for algorithm 2. Aponeurosis-modeling by surface lofting.
aponeurotic interface between the muscle belly and distal ten-don of the ECRB (Fig. 1a) that was modelled and qualitativelyvalidated. During performance testing of the aponeurosis-modeling scripts, 5 streamline layers of 12 streamlines per
layer were consistently used in muscle reconstruction of theECRB, for the sake of simplicity, despite being a much lowerdensity than that seen in normal physiology. Execution timesof each script on the ECRB dataset were averaged over 5 runs,
Fig. 5 – Illustration of surface lofting algorithm. NURBS surfaces are sequentially lofted over each muscle fibre streamline instreamline-layer #1 (a) and so on until the deepest layer (b) is reached. The duplicated first ispoarms of the lofted layers(white curves in (c)) are themselves lofted into an aponeurotic surface.
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Fig. 6 – Flowchart for insertion sorting subroutine of algorithm 2. Connects to Fig. 4 at the asterisks.
measured on a computer equipped with a Intel Pentium 43 GHz processor with 1 GB of RAM. Fig. 8 compares the resultsobserved for each algorithm.
5.1. Performance of algorithm 1: surface fitting
The execution time of algorithm 1 averaged at 0.14 ± 0.00 s.Fig. 8d shows an example of a surface-fitted aponeurosis onthe ECRB muscle. Qualitatively, these results were reasonableas a first attempt, and were accurate in that the aponeurosisroughly spans over the insertion ends of streamlines. But itcan be seen that some muscle fibres penetrated the aponeu-rosis, as grey lines crossing the purple surface in Fig. 8d, whichshould not be physiologically possible.
The modeling results were further evaluated according tothe design criteria. Surface-fitted aponeuroses satisfied the
first requirement in that they have their own physical repre-sentation as distinct NURBS surfaces. However, the other tworequirements – anatomical smoothness and attachment tomuscle fibres – could not be satisfied simultaneously throughthis surface fitting strategy.
To satisfy the anatomical smoothness requirement, thesurfaces had to be fitted in a degree greater than 1. But asalluded to in Section 3.1 and illustrated in Fig. 8a using degree-3, when surfaces were generated by CV-specification, thesurface created through cubic smoothing only approximatedthe positional location of the CVs and found its smoothestpossible path around the CVs between the end points. Thismeant that smooth aponeuroses (e.g. degree-3) naturally vio-lated the final design criterion, in that they did not exactlyspatially attach to muscle fibre streamlines. Conversely, asshown in Fig. 8b, when the code was modified to surface
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Fig. 7 – Rationale for insertion sorting procedure. (a)Spatially intersecting streamline-layers. Numbers asassigned by streamlines-generating plug-in: 1, red; 2,orange; 3, yellow; 4, green; 5, blue. (b) Discrepancy betweenspatial order and computationally assigned orderdetermined using vector algebra: three layers have beenassigned computationally as layers #2, #3 and #4. Layers #2and #3 are in correct spatial order, giving a negative dotproduct when the “SurfaceOrderCheck” procedure isperformed on them. Layers #3 and #4 are in incorrectspatial order, giving a positive dot product when the“SurfaceOrderCheck” procedure is performed on them. Theresulting order after insertion-sorting would then be 2, 4, 3rather than the unsorted 2, 3, 4. (For interpretation of thereferences to color in this figure legend, the reader isreferred to the web version of the article.)
fit aponeuroses in degree-1 and effectively sacrificing sur-face smoothness, linear interpolation between the input CVsachieved the positional attachment requirement, in that thestreamlines were exactly spatially positioned on the generatedaponeurosis.
Another violation of the aponeurosis-muscle attachmentcriteria existed in that when streamlines were translated,the surface-fitted aponeurosis failed to follow the stream-line movements (Fig. 8d and e). This means spatial continuitybetween aponeuroses and muscle fibres could not be main-tained during biomechanical simulation even if they werespatially exactly attached (i.e. degree-1) to begin with.
5.2. Performance of algorithm 2: surface lofting withinsertion sorting
Surfacing lofting by itself averages 0.05 ± 0.00 s in executiontime, and together with insertion sorting gives an aver-age runtime of 0.18 ± 0.01 s. The lofted aponeuroses werefound to simultaneously satisfy all three design requirements.Firstly, the lofted surfaces possessed physical representa-tion and anatomical smoothness as they were generatedas distinct degree-3 NURBS surfaces. In addition, the spa-tial attachments between the aponeurosis and muscle fibreswere exact (Fig. 8c). This was accomplished by the fact thatNURBS surfaces lofted exactly through the streamlines in eachstreamline layers to begin with, resulting in edge isoparmsthat string together the insertion ends of streamlines exactly,and allowing the final aponeurosis-lofting procedure to skin aNURBS surface exactly through these input isoparm curves, asdictated by NURBS theory. Thirdly, lofted aponeuroses are ableto deform according to spatial movements of the streamlinesfrom which they were created (Fig. 8f and g). While naturalmuscle contraction involves more than a simple translationof a single muscle fibre, this served as a proof-of-concept inthat spatial continuity between lofted aponeuroses and mus-cle fibres could be maintained during future biomechanicalsimulation.
While fibre streamline penetration through the aponeuro-sis had also been observed with surface lofting, the problemhas been significantly reduced with the implementation of theinsertion sorting algorithm (Fig. 8f). This was expected sinceinsertion sorting rearranged the lofted isoparms in at leastone direction – superficial to deep – so that the aponeurosisno longer folded back on itself in this direction. However, asthe density of the streamlines was increased, the improve-ment in this regard became less evident (Fig. 8h). This was alsoexpected as the lofted isoparms could still be disorganizedin the distal to proximal direction, the probability of whichincreased with increased number of streamline-layers.
6. Discussion
6.1. Merits and limitations of the aponeurosis models
Table 1 summarizes the performance characteristics of thetwo aponeurosis-modeling algorithms based on the qual-itative, anatomical and functional assessment presentedpreviously. The first model, implementing the NURBS surfacefitting algorithm, was flawed in both the visual and functionalrepresentation of the aponeuroses because of its inability tosimultaneously be smooth and position exactly on the mus-cle fibres, and its inability to deform according to muscle fibremovements, respectively. The second model implemented aNURBS surface lofting algorithm that addressed the limita-tions of the first model, and utilized an insertional sortingprocedure to minimize the convolution of the lofted aponeuro-sis and its associated streamline-penetration problem, whichwas successful in simple cases when the streamline den-sity was kept sufficiently low (below physiological density). Atthis point, it is concluded that algorithm #2 has more meritsthan #1. However, if strategies can be found to alleviate the
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Fig. 8 – Comparison of aponeurosis reconstruction results on the ECRB from surface fitting and surface lofting strategies. (a)Surface fitting with degree-3 produces smooth aponeuroses (purple) that does not necessarily attach to the insertion ends(pink dots) of muscle fibres (hidden). (b) Surface fitting with degree-1 produces sharp aponeuroses that attach to musclefibres directly. (c) Surface lofting produces smooth aponeuroses that attach directly to muscle fibre streamlines (grey). (d ande) Surface-fitted aponeuroses fail to deform along with streamline translations, with red arrows indicating the insertion endof the translated streamlines (green). (f and g) Surface-lofted aponeuroses deform according to streamline movements andalso had significantly less streamline penetration due to insertion sorting prior lofting. (h) However, the improvement instreamline penetration due to insertion sorting is less significant with increasing number of streamline-layers. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
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Table 1 – Performance characteristics of the two aponeurosis models
Characteristics Model #1: surface fitting Model #2: surface loftingand insertion sorting
Average run time (s) 0.14 ± 0.00 0.18 ± 0.01Distinct physical representation Yes YesAnatomical smoothness Yes (degree-3); no (degree-1) Yes (degree-3)Positional attachment to muscle fibre bundles Inexact (degree-3); exact (degree-1) ExactPenetration of aponeurosis by muscle streamlines Extensive Less extensiveSustained positional continuity with muscle fibre
bundles during muscle contractionNo Yes
shortcomings of algorithm #1 (e.g. muscle detachment, fibrepenetrations) directly without needing the sorting proceduresin algorithm #2, algorithm #1 could in the future become thebetter choice given its faster execution time.
6.2. Significance of contributions
While most MSK models in the literature have focused onmodeling the biomechanics or kinematics aspects of MSKsystems, often at the expense of architectural complexity,Anatomy3D had been developed with anatomical fidelity asfirst priority. With regard to muscle fibre architecture mod-eling, Anatomy3D already had higher anatomical complexitythan most other MSK models to begin with. This cur-rent extension of Anatomy3D towards aponeurosis-modelingcan be in effect a step towards claiming the same withregard to connective tissue architecture modeling. While thereare plans for further improvements to the reconstructedaponeurses, as discussed in the last section, the qualita-tive insights already obtained from this aponeurosis-modelingstudy, on the functional merits and limitations of utiliz-ing various NURBS surface design strategies to model ofanatomical structures, can be useful in guiding future endeav-ours in modeling other connective tissue components. Theextended Anatomy3D thus has potential for eventually pos-sessing both active and passive load-bearing componentsof MSK systems and being compatible for biomechanicalsimulation.
7. Future plans
The performance of the aponeurosis-generating algorithmsdeveloped here is greatly dependent on quality of the inputdigitization dataset representing the muscle fibre bundles ortendons architecture. It would be ideal if cleaner datasetscould be obtained that would not inherit the computationalartefacts of the streamlines-generating algorithms and in turnminimize the non-physiological folding or convolution duringthe creation of aponeuroses surfaces. It has been suggestedthat anatomical architecture data obtained through cadavericdissection and digitization are inherently noisy and somewhatunreliable due to factors including postmortem muscle con-tractions (rigor mortis), embalming procedures and long-termtissue degradation [18]. Methods for improving the physi-ological relevance of cadaveric data, such as comparativeextrapolation by in vivo ultrasonography [18], could perhapsbe explored.
Assuming digitization errors during the layer and stripdissection process are not the cause of spatially mislabelledstreamlines, further investigation is necessary to elucidate theroot of this computational artefact from streamline recon-struction and to determine whether streamlines can bere-labelled based on heuristics. Given the current state ofreconstructed streamlines, some of the issues encounteredsuch as aponeurosis-muscle detachment upon fibre trans-lation or fibres penetrating aponeurotic surfaces can stillpotentially be solved by applying geometric or connectivityconstraints. Another possible approach to circumvent fibrepenetrations would be to trim fibre streamlines at their firstpoints of intersection with the aponeurosis.
To assess the geometrical dimensions of the modelledaponeuroses, their surface areas were validated implicitlythrough the measure of “anatomical smoothness” in Table 1,since in the absence of artificial surface convolution, the mod-elled surface would span the digitized insertions points ofmuscle fibres perfectly with a surface area that converges tothe actual physiological measurement. It should be noted thataponeurosis thickness is not currently represented. This canonly be addressed with the help of explicit tendon-modeling,as aponeuroses are really flat extensions from the tendons.For instance, in the case of Fig. 1a, the aponeurosis thicknessshould theoretically be modelled to have a tapered distribu-tion, thinnest at the distal end from the tendons and thickestat the proximal end as it merges with the tendon. Further workis needed to investigate whether this simplistic tapering thick-ness would be adequate for representing in vivo physiologicalmeasurements.
Quantitative validation of the biomechanical functional-ity of the modelled aponeuroses, in terms of their surfacedeformation characteristics and load-transmission behaviour,is also desirable. But until Anatomy3D is further developedto incorporate tendons and bones, or to simulate the biome-chanical behaviour, it is difficult to perform such quantitativecharacterization. A possible strategy to model the tendonscould begin with digitizing cadaveric tendons and againutilitze NURBS surfaces to bridge the gap between the digi-tized tendon and the reconstructed aponeurosis. Once all theactive and passive load-bearing elements of skeletal musclehave accurate anatomical representations in the Anatomy3Dmodel, biomechanical simulation can be incorporated intoAnatomy3D. Future directions in this regard may involveattributing differential material properties such as elasticityand stiffness to each material: the muscle fibre streamlines,the aponeurosis, the tendons, etc. Scripted or time-framedcontraction and relaxation of muscle fibre streamlines can
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then be analyzed for their effects on connective tissue defor-mation [2].
r e f e r e n c e s
[1] K. Gielo-Perczak, Musculoskeletal modeling in biomedicaleducation, Theor. Issues Ergonom. Sci. 6 (2005)299–304.
[2] W. Tsang, Helping hand: an anatomically accurate inversedynamics solution for unconstrained hand motion, M.A.Sc.dissertation, University of Toronto, 2005, 97 pp.
[3] J. Teran, E. Sifakis, S.S. Blemker, V. Ng-Thow-Hing, C. Lau, R.Fedkiw, Creating and simulating skeletal muscle from thevisible human data set, IEEE Trans. Vis. Comput. Graph. 11(2005) 317–328.
[4] D.G. Thelen, F.C. Anderson, S.L. Delp, Generating dynamicsimulations of movement using computed muscle control, J.Biomech. 36 (2003) 321–328.
[5] K. Gielo-Perczak, State-of-the-art musculoskeletal modelingand prognosis of its influence on the future directions ofergonomics theory, Theor. Issues Ergonom. Sci. 6 (2005)213–216.
[6] R.T. Jaspers, R. Brunner, U.N. Riede, P.A. Huijing, Healing ofthe aponeurosis during recovery from aponeurotomy:morphological and histological adaptation and relatedchanges in mechanical properties, J. Orthop. Res. 23 (2005)266–273.
[7] A.S. Arnold, S. Salinas, D.J. Asakawa, S.L. Delp, Accuracy ofmuscle moment arms estimated from MRI-basedmusculoskeletal models of the lower extremity, Comput.Aided Surg. 5 (2000) 108–119.
[8] A.M. Herrmann, S.L. Delp, Moment arm and force-generating capacity of the extensor carpi ulnaris after
transfer to the extensor carpi radialis brevis, J. Hand Surg. 24(1999) 1083–1090.
[10] A.M. Agur, V. Ng-Thow-Hing, K.A. Ball, E. Fiume, N.H. McKee,Documentation and three-dimensional modelling of humansoleus muscle architecture, Clin. Anat. 16 (2003)285–293.
[11] M. Epstein, M. Wong, W. Herzog, Should tendon andaponeurosis be considered in series? J. Biomech. 39 (2006)2020–2025.
[12] A. Shapiro, V. Ng-Thow-Hing, P. Faloutsos, DynamicAnimation and Control Environment, ACM InternationalConference Proceedings Series, vol. 112, Proceedings of theGraphics Interface 2005, pp. 61–70.
[13] L. Piegl, On NURBS: a survey, IEEE Comput. Graph. Appl. 11(1991) 55–71.
[14] McNeel North America, Rhinoceros: NURBS Modelling forWindows: “What is NURBS?”,http://www.rhino3d.com/nurbs.htm, 2005.
[15] E. Dimas, D. Briassoulis, 3D geometric modelling based onNURBS: a review, Adv. Eng. Software 30 (1999) 741–751.
[16] G. Farin, Curves and Surfaces for CAGD A, fifth ed.,Morgan-Kaufmann, Tempe AZ, 2002.
[17] N. McKee, T. Hess, Y. Shkula, L. Hess, K. Ravichandiran, Z.Harley, A. Agur, Extensor carpi radialis longus and brevis: a3D study of musculotendinous architecture, J. Reconstr.Microsurg. 20 (2004) 340 (Abstracts from American Societyfor Peripheral Nerve Annual Meeting, Rancho Mirage, CA,USA).
[18] D.C. Martin, M.K. Medri, R.S. Chow, V. Oxorn, R.N. Leekam,A.M. Agur, N.H. McKee, Comparing human skeletal musclearchitectural parameters of cadavers, J. Anat. 199 (2001)429–434.
