MUSCLE FIBRES MODELLING

Josef Kohout

, Gordon J. Clapworthy

, Saulo Martelli

and Marco Viceconti

Department of Computer Science and Engineering, University of West Bohemia, Plzeˇn, Czech Republic

Department of Computer Science and Technology, University of Bedfordshire, Luton, U.K.

Laboratorio di Tecnologia Medica, Istituto Ortopedico Rizzoli, Bologna, Italy

Keywords:

Muscle Modelling, Muscle Fibres, VTK.

Abstract:

This paper describes a method that represents a muscle by a realistic chaff of muscle ﬁbres that are automat-

ically generated in the volume deﬁned by the surface mesh of the muscle which itself automatically wraps

around bones as they move. Our C++ implementation can decompose the volume into muscle ﬁbres, which

is done by a slice-by-slice morphing of predeﬁned ﬁbres template into the muscle volume, and visualise the

result in only about 1000 ms on commodity hardware. Hence, the method is fast enough to be suitable for

interactive educational medical software. Although a biomechanical assessment has yet to be done, we believe

that the method could be used also in clinical biomechanical applications to extract information on the current

muscle lever arm and ﬁbre path and, thanks to its rapid processing speed, it might be an attractive alternative

to current methods.

1 INTRODUCTION

Knowledge of muscle ﬁbres is essential for physio-

therapists, surgeons and orthopedists, especially for

effective rehabilitation programs that aim at improv-

ing the quality of life of patients suffering from neuro-

muscular disorders (more than 0.1% of general popu-

lation in UK (Pohlschmidt and Meadowcroft, 2010)),

for planning optimal muscle surgery (e.g., muscle

auto-transplantation), and prediction of forces having

impact on joints. Studying traditional anatomical at-

lases (e.g., Gray’s atlas (Gray, 1918)) is a common

option to gain this knowledge. However, the full un-

derstanding requires an excellent imagination skill of

the student since they need to reconstruct 3D mod-

els in their minds. Furthermore, these atlases show

muscles in their rest-poseposition only and, therefore,

they provide a limited insight.

Digital 3D anatomical atlases seem to be a better

alternative. However,as far as we know, most of them

display muscle ﬁbres on the muscle surface only, thus

not providing information about muscle interior, or

do not display them at all. Similarly to traditional

atlases, muscles are displayed in their rest-pose po-

sition, though being able to model muscles in various

positions (e.g., walking, stepping stairs, falling) is an

essential step in the process of looking for an optimal

strategy to provide patients suffering from various

musculoskeletal disorders with better healthcare.

Ng Thow Hing in his research (Ng-Thow-Hing,

2001) represents a muscle with a B-spline solid ﬁtted

to raw muscle surface data extracted from a set of par-

allel images and to internal ﬁbre points obtained from

an autopsy. Muscle ﬁbres can be then generated sim-

ply by constructing iso-curves within the solid. The

simulation of movement is possible through manipu-

lation of control points of B-spline solid. Although

accurate, this technique is rather impractical because

of its the complexity of B-spline ﬁtting procedure.

A different approach is described by Blemker &

Delp in (Blemker and Delp, 2005). In their model,

a muscle is represented by 3D ﬁnite-element hexahe-

dral mesh whose vertices move in reaction to the ex-

ternal force induced by the movement of the bones.

Each cell of the mesh contains information about the

direction of the muscle ﬁbres present in its volume.

When the mesh changes, so do the paths of the ﬁbres.

The process of constructing the mesh is as follows. A

surface model of the muscle to be represented is ob-

tained from the input medical images. The user then

has to create, manually, a hexahedral cubical template

mesh that contains the whole muscle in its interior.

Unlike the approach by Ng Thow Hing, muscle ﬁbres

arrangement is not derived from a real muscle but it

is given in a predeﬁned cubical template that consists

of a set of interpolated rational B´ezier spline curves

Kohout J., J. Clapworthy G., Martelli S. and Viceconti M..

MUSCLE FIBRES MODELLING.

DOI: 10.5220/0003847400580066

In Proceedings of the International Conference on Computer Graphics Theory and Applications (GRAPP-2012), pages 58-66

ISBN: 978-989-8565-02-0

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

connecting two muscle attachment areas highlighted

on this template. This template of the ﬁbre geome-

try is mapped into this template hexahedral mesh. Fi-

nally, the template mesh is projected into the volume

of muscle by a proprietary mapping method available

in the commercial TrueGRID (XYZ Scientiﬁc Appli-

cations) software. Although much easier deﬁnition of

muscle ﬁbres is presented, the dependence on an ex-

pensive commercial tool together with the large mem-

ory consumption (because of 3D mesh) renders this

approach also quite impractical.

Representing a muscle with a triangular surface

mesh is very popular, especially, due to its simplic-

ity and low memory requirements. Although vari-

ous technique for deforming this mesh as bones move

were described, such as mesh-skinning based ap-

proaches (Kohout et al., 2011), (Aubel and Thalmann,

2000), mass-spring systems (Thalmann, 1998) or gra-

dient domain (Kohout et al., 2012), as far as we know,

no method is available at present for a decomposition

of the volume of the deformed muscle (deﬁned by its

mesh surface) into muscle ﬁbres.

Hence, in this paper, we propose a simple tech-

nique that can decompose the muscle volume into

muscle ﬁbres by a slice-by-slice morphing of prede-

ﬁned ﬁbres template (proposed by Blemker & Delp)

into the interior of the muscle, employing mapping

technique described in (Hormann and Floater, 2006).

The remainder of this paper is structured as fol-

lows. In the next section, we give a brief brings an

overview of our method; details are described in sec-

tions 3. Section 4 presents the experiments that were

performed. Section 5 concludes the paper and pro-

vides an overview of possible future work.

2 METHOD OVERVIEW

Our method is designed to process any muscle rep-

resented by its surface mesh for which information

about its attachment areas, i.e., about sites at which

the muscle is attached to the bone by a tendon, is

available. This information is typically provided as

two sets of landmarks, one for the origin and the other

for the insertion area, speciﬁed by an expert. Every

landmark is ﬁxed to an underlying bone, so that when

bones move (during the simulation of various activi-

ties), so do landmarks. The number of landmarks in

the set deﬁne the accuracy of the ﬁbres generated. We

note that for many muscles, it is sufﬁcient to spec-

ify just one landmark. Figure 1 shows an example of

muscles and their landmarks.

For each muscle to be decomposed, it is also nec-

essary to specify (again by an expert) what type of

Figure 1: Gluteus Medius (left) and Semimembranosus

(right) with their attachment areas. The origin area is blue,

the insertion area is red.

muscle ﬁbres it contains, if parallel, pennate, curved,

fanned or something else. According to this informa-

tion, the method selects a predeﬁned template con-

taining the description of ﬁbres geometry. As we

seek to decompose the muscle into an arbitrarily large

number of ﬁbres, the requested number of ﬁbres to be

constructed and their resolution must be also deﬁned.

We note that the resolution of a ﬁbre is the number

of segments along its length, e.g., if the resolution is

19 (the default value in our case), each ﬁbre is repre-

sented by a poly-line of 20 points. We would expect

that the higher the resolution, the smoother the curve

of the ﬁbre will be and, therefore, the better will be

the correspondence with anatomical ﬁbres. It is im-

portant to point out that these settings (i.e, the type of

ﬁbres, their number and resolution) can be speciﬁed

by an expert and then stored with the atlas data, so

that no input is required from an ordinary user (e.g., a

student of medicine).

The decomposition method starts with the pro-

duction of muscle ﬁbres of the requested resolution

within the unit template cube. It involves exploitation

of Sobol points (Joe and Kuo, 2008) and muscle ﬁ-

bre geometry templates (Blemker and Delp, 2005). In

the next step, this template cube is subject to an afﬁne

transformation to form the best ﬁt to the muscle to be

decomposed. The transformed template is an oriented

bounding box (OBB) of the muscle whose attachment

areas are aligned with those speciﬁed as input.

After the template cube has been ﬁtted to the mus-

cle, the poly-lines representing the ﬁbres undergo the

MUSCLE FIBRES MODELLING

same afﬁne transformation and are then morphed into

the muscle so that the template box becomes the sur-

face of the muscle and its ﬁbres are mapped into the

muscle volume. The morphing is done successively

by slicing both the transformed cube with its ﬁbres

and the muscle surface and mapping the contour of

the cube onto the contour of the muscle using generic

barycentric coordinates (Hormann and Floater, 2006).

An example of the morphing is given in Figure 2.

Figure 2: Decomposition of the muscle volume into muscle

ﬁbres – above the template with parallel ﬁbres, below the

result of mapping of template into the interior of muscle.

For muscles with wide attachment areas, the paths

of muscle ﬁbres generated by the process described so

far are unrealistic in a proximity of such an area since

the ﬁbres tend to meet in a common point instead of

spreading over the whole area. To correct this, we cut

out the part of the ﬁbre that is close to the attachment

area and replace it by a line segment whose end-point

lies on the surface of muscle in the region deﬁned by

the attachment area and that has the direction derived

from the trimmed ﬁbre. Finally, the muscle ﬁbre is

smoothed to eliminate noise that might be present in

the produced muscle ﬁbres.

A simpliﬁed and unoptimised version of the

method written in a pseudo-code is given in Figure 3.

Please note that the pseudo-code does not perfectly

match the description that was given above because

of the performed simpliﬁcation.

3 DECOMPOSITION OF

WRAPPED MUSCLE

In this section, we describe all steps of the method,

we have just outlined, in detail.

3.1 Template Generation

The template for the ﬁbres follows the ideas presented

by Blemker & Delp (Blemker and Delp, 2005). This

template is a unit cube with deﬁned the origin and in-

sertion areas on its bottom and top faces, respectively.

The areas are connected by B´ezier curves (of degree

varying from 2 to 4 depending on the muscle type)

which will represent the muscle ﬁbres – see Figure 4.

Figure 3: A simpliﬁed and unoptimised version of our

method written in a pseudo-code.

Generally, the cube can contain an inﬁnite number of

curves, so there is no limit on the number of ﬁbres

that can be represented; each ﬁbre is a B´ezier curve

C(t) of real parameter t whose control polygon can

be identiﬁed by a pair of real parameters r, s.

To create the requested number of ﬁbres, the para-

metric space r, s must be sampled. Ng-Thow-Hing

(Ng-Thow-Hing, 2001) suggests the use of Sobol

sampling (Joe and Kuo, 2008) which produces a bet-

ter distribution of ﬁbres within the muscle volume

than random or uniform sampling, especially when

GRAPP 2012 - International Conference on Computer Graphics Theory and Applications

Figure 4: The template of parallel (left) and pennate (right)

ﬁbres. The origin area is blue, the insertion area is red.

the number of ﬁbres is relatively low (up to hundreds),

which is typical in this context. The ﬁbre curves re-

trieved are sampled in the parameter space, t, to pro-

duce poly-lines of as many segments as the value of

the required resolution.

3.2 Template Fitting

We assume that the principal axis of the muscle coin-

cides with one of the axes of the cube. The origin of

the principal axis is calculated as the centroid of the

muscle, i.e., the mean of coordinates of surface ver-

tices. The direction of the principal axis can be deter-

mined easily as the difference between the centroid of

the insertion area and the centroid of the origin area.

In a case that the attachment area is so complex that

its centroid does not ﬁt the data well, the direction

is determined differently as the eigenvector with the

largest eigenvalue obtained for the ﬁrst-order covari-

ance matrix:

N − 1





0,0

0,1

0,2

0,0

1,1

1,2

2,1

2,2





(1)

i, j

N−1

∑

k=0

k,i

− O

) · (V

k, j

− O

) (2)

where O = (O

) is the origin of the principal

axis andV

i,0

i,1

i,2

are the Euclidean coordinates of

the surface vertex V

(i = 0...N − 1; N is the number

of surface vertices).

Having aligned u

with the principal axis, any two

vectors v

and w

so that (u

) forms an orthog-

onal set are chosen. These vectors are successively

rotated around the principal axis by a small angle (we

use 5

◦

), which results in a set of frames (u

, v

, w

as depicted in Figure 5. For each frame, the minimal

axis-aligned bounding box of the muscle surface ver-

tices is constructed based on vectors u

, v

, w

From these bounding boxes, we must choose the

one whose origin and insertion areas (see Figure 4)

best match the muscle origin and insertion areas that

Figure 5: The principal axis (dashed red line) oriented

bounding boxes (grey) of the object. Red axis + green and

blue arrows denote local coordinate frames. Parallel pro-

jection on to the plane perpendicular to the principal axis is

shown below.

are speciﬁed either by the action lines of the muscle or

manually by an expert. To do so, for each point of the

template origin area (deﬁned as a rectangle for most

templates) we ﬁnd the closest point on the muscle ori-

gin area and, likewise, for each point of the template

insertion area, the closest point on the muscle inser-

tion area. The Euclidean distances between the pairs

of points found are summed – the best conﬁguration

is the one with the minimal sum. The result of ﬁtting

the sartorius muscle is shown in Figure 6.

Figure 6: The best ﬁtting of the parallel ﬁbres template for

the sartorius muscle. The origin area deﬁned in the template

(a rectangle with four points) and speciﬁed by the user (a

cloud of points) on the bone is blue, while the insertion area

is red.

3.3 Template Morphing

After the template cube has been ﬁtted to the mus-

cle, the poly-lines representing the ﬁbres undergo the

same afﬁne transformation and are then morphed into

the muscle so that the template box becomes the sur-

face of the muscle and its ﬁbres are mapped into the

muscle volume.

MUSCLE FIBRES MODELLING

The morphing uses a sweeping paradigm: a plane

perpendicular to the principal axis of the muscle

moves from one face of the template box to the oppo-

site face, stopping at each point F

of ﬁbre poly-lines

that has not yet been processed. This plane cuts both

the template box and the surface of the muscle pro-

ducing a rectangle from the template and a polygon

contour from the surface.

Let us assume that vertices of both polygons are

oriented clockwise and that the contour polygon is

formed of m segments, where m ≥ 1. Our task is to

subdivide the sides of the template rectangle in such a

manner that the resulting polygon is also formed of m

segments, and to establish a correspondence between

the vertices of both polygons (rectangle template and

contour polygon).

The algorithm starts with the detection of the ver-

tex of the muscle contour that is closest to the ﬁrst

vertex Q

of the template rectangle. The chain of ver-

tices P

...P

must be split into four parts, where each

part corresponds to one side of the input rectangle.

The split must be such that the overall error, given as

the sum of errors for every part, is minimal.

Let k be the ratio of the rectangle perimeter to the

contour perimeter. The error for a given part of the

chain is computed as the square of the difference be-

tween the size of the rectangle side associated with

the part and the sum of the lengths of the segments

formed by the vertices in the part, scaled by the con-

stant k. After that, a side of the rectangle can be easily

subdivided into as many segments as there are seg-

ments in the corresponding part of the chain. The ra-

tios between the segment lengths are, of course, pre-

served. Establishing the correspondence between the

polygons is straightforward: the vertex P

corresponds

to the vertex Q

– see Figure 7.

Figure 7: Establishing correspondence between the rectan-

gle of the template and the muscle contour.

Hormann & Floater (Hormann and Floater, 2006)

proposed an approach that allows to express the co-

ordinates of the ﬁbre poly-line point F

(where the

cutting plane has currently stopped) with respect to

the coordinates of the segmented rectangle as the sum

of λ

· Q

, where the λ

are real non-negative weights

such that their sum equals 1. Once the weights λ

are

computed, the new coordinates of the poly-line point

within the muscle contour P

...P

are simply given as

the sum of λ

· P

As the cutting of the muscle surface is clearly the

bottleneck of the decomposition, we process not only

all ﬁbre poly-line points lying on the cutting plane but

also those in its close proximity. Hence, the minimal

number of slices used is equalled to the speciﬁed res-

olution of ﬁbres. Naturally, the worst-case number of

slices used is given as the requested number of ﬁbres

times their resolution.

3.4 Fibres Filtering

The process described so far may produce ﬁbres with

unrealistic paths close to their attachment areas. This

is a problem especially for muscles with large attach-

ment areas, as it is illustrated in Figure 8. Hence, we

need to change the path in the proximity of the at-

tachment area to better correspond with the reality.

To do so, we construct two cutting planes perpendic-

ular to the principal axis passing through the extremal

(in the direction of the principal axis) landmarks of

attachment areas. These planes are used to cut out

the unwanted parts of ﬁbres, i.e., parts close used to

attachment areas – see Figure 9.

Figure 8: Muscle ﬁbres of Gluteus Medius that were pro-

duced by the basic method without ﬁltering.

In the next step, it is necessary to reconstruct the

missing parts of ﬁbres. Having a ﬁbre P

...P

whose

part in proximity of the insertion area must be recon-

structed, we add a new segment (P

m+1

), where

m+1

is a new point such that it lies on the surface

of muscle, in the insertion area, and is closest the ray

deﬁned by the segment (P

m−1

). We note that the

extraction of triangles belonging to the insertion area

can be done by cutting out the larger part of the sur-

face at the place of surface contour deﬁned by the

points obtained from projecting the landmarks onto

GRAPP 2012 - International Conference on Computer Graphics Theory and Applications

Figure 9: Muscle ﬁbres of Gluteus Medius after their parts

close to the origin or insertion area were removed (left) and

reconstructed (right).

the surface of muscle. The reconstruction of the part

in the proximity of the origin area is similar. Figure 10

demonstrates the overall process. The result of recon-

struction could be seen in Figure 9.

Figure 10: Illustration of the trimming and reconstruction

of a ﬁbre – ﬁbres are yellow, the attachment area red, the

cutting plane is dotted black.

Clearly, the smoothness of the produced ﬁbres de-

pends upon both the resolution and the shape of the

muscle. If the muscle model is coarse, perhaps due

to imprecise image data segmentation, as in the sar-

torius muscle shown in Figure 6, the ﬁbres may be

too noisy, in which case the poly-lines may have to

be smoothed. We use an iterative process – the more

smoothing steps it takes, the smoother the poly-lines

become. At each step, the coordinates of the inner

points P

of the poly-line are modiﬁed according to

the equation: P

′

= (P

i−1

+k·P

i+1

)/(k+2), where

k is a smoothing constant – we use 4.

The resulting smoothed muscle ﬁbres can be then

visualise using any rendering technique for poly-lines

visualisation – we use a VTK (Schroeder et al., 2004)

ﬁlter that generates a tube (represented by a triangu-

lar surface mesh) of the given radius around each in-

put line segment. We also believe that these ﬁbres can

be passed to any solver predicting lever arms char-

acteristics from the paths of the action lines, which,

in most cases, should bring an increased accuracy to

these predictions because the accuracy generally in-

creases with the number of input poly-lines passed to

the solver and whilst a muscle is typically represented

by a couple of action lines only (it is because action

lines cannot be constructed automatically), one could

easily generate an arbitrary number of ﬁbres using the

proposed technique.

4 EXPERIMENTS AND RESULTS

Our approach was implemented in C++ (MS Visual

Studio 2010) under the Multimod Application Frame-

work – MAF (Viceconti et al., 2004), which is a vi-

sualisation system based mainly on VTK (Schroeder

et al., 2004) and other specialised libraries. This

framework is designed to support the rapid develop-

ment of biomedical software. It is particularly useful

in multimodal visualisation applications, which sup-

port the fusion of data from multiple sources and in

which different views of the same data are synchro-

nised, so that when the position of an object changes

in one view, it is updated in all the other views. Our

implementation was integrated into the MuscleWrap-

ping software

, which is a part of the larger LHP-

Builder software being developed within the VPHOP

project (VPHOP, 2010). We tested our implementa-

tion on various real data sets of muscles with typical

sizes about 15K vertices on Intel Core i7 2.67 GHz,

12 GB DDR3 1.3GHz RAM with Windows 7 Pro x64.

Figure 11 and Figure 12 show the results of de-

composition of muscles into a chaff of ﬁbres and com-

pares them with ﬁbres illustrated in anatomical at-

lases. Parallel ﬁbres were generated for all muscles

apart from semimembranosus, for which the pennate

ﬁbre template was used (Blemker and Delp, 2005).

Figure 13 shows the wrapping of a small se-

lected set of muscles (sartorius, rectus femoris, bi-

ceps femoris and semimembranosus) at frames t =

0.00, 0.25, 0.50 and 0.75 of the walk sequence of 1.56

s produced by our wrapping method (Kohout et al.,

2011). Fifty parallel ﬁbres represented by poly-lines

http://graphics.zcu.cz/Projects/Muskuloskeletal-

Modeling

MUSCLE FIBRES MODELLING

Figure 11: Comparison of generated ﬁbres of Gluteus

Medius (top) and of Obturator Externus (bottom) with those

in Richardson’s anatomical atlas (Richardson, 2011).

Figure 12: Comparison of generated ﬁbres of biceps

femoris (red), semitendinosus (orange), semimembranosus

(pink), gracilis (blue) and sartorius (light green) with those

in Gray’s anatomical atlas (Gray, 1918).

of 14 line segments were generated for all four mus-

cles considered – see Figure 14 – in about 300 ms per

frame, which makes the proposed method suitable for

interactivevisualisation, especially, if we take into ac-

count that our implementation could be easily paral-

lelized to run faster.

Figure 13: Four selected muscles (sartorius – green, rectus

femoris – yellow, biceps femoris – red and semimembra-

nosus – fuchsia) during the movement at frames t = 0.00,

0.25, 0.50 and 0.75.

Figure 14: Fibres of four selected muscles (sartorius –

green, rectus femoris – yellow, biceps femoris – red and

semimembranosus – fuchsia) during the movement at two

different frames.

A comparison of visualisation of muscles using

our approach and using an online digital anatomical

atlas is brought in Figure 15. As this online atlas rep-

resents a muscle by a surface of a quite low level of

detail and its muscle ﬁbres only as a texture mapped

onto the surface of the muscle, there cannot be doubt

that our method provides users with a more realistic

visualisation. We note that similar conclusions could

be drawn also for other anatomical atlases.

When we compare results produced by the method

by Blemker et al. (Blemkerand Delp, 2005) – see Fig-

ure 16 with our results – see Figure 17, it is clear that

GRAPP 2012 - International Conference on Computer Graphics Theory and Applications

Figure 15: Our visualisation of gluteus minimus mus-

cle in comparison with the visualisation obtained from

http://www.biodigitalhuman.com/ online anatomical atlas

(bottom right corner).

although our method may produce unrealistic path for

a couple of ﬁbres, the majority of produced ﬁbres re-

semble those produced by its much slower, and, there-

fore, impractical, counterpart. This is also conﬁrmed

by orthopedists with whom we cooperate.

Figure 16: Muscle ﬁbres of psoas, gluteus maximus, illiacus

and glutes medius produced by the method by Blemker et

al. This ﬁgure was taken from (Blemker and Delp, 2005).

Figure 17: Muscle ﬁbres of psoas, gluteus maximus, illiacus

and glutes medius produced by our method.

5 CONCLUSIONS

This paper has presented an approach that can auto-

matically generate an arbitrary number of muscle ﬁ-

bres within the volume of muscle represented by its

surface in a convenient time. Although the main goal

of our work was to enhance educational tools used

by both medical experts and physiotherapists, we be-

lieve that, since the produced ﬁbres quite well cor-

relate with those depicted in anatomical atlases (even

for muscles with large attachment sites), the ﬁbres can

be used (instead of action lines) to predict the muscle

lever arm on the articular joints and the distribution of

ﬁbre length (which is an input for the muscle force-

length-velocity relationship to deﬁne the boundaries

within which the force is constrained) with an ex-

pected accuracy somewhere in between predictions

provided by action-line methods and the more accu-

rate, but due to their large time-consumption imprac-

tical, ﬁnite-element methods. A biomechanical valid-

MUSCLE FIBRES MODELLING

ation is, however, still required. This is a part of

our future work. In the future, we would like also to

speed up the decomposition process by parallelization

to make it run in almost real time.

ACKNOWLEDGEMENTS

This work was supported by the Information Soci-

ety Technologies Programme of the European Com-

mission under the project VPHOP (FP7-ICT-223865).

The authors would like to thank the various people

who contributed to the realisation of the MAF and

LHPBuilder software and to various people who pro-

vided condition under which the work could be done.

REFERENCES

Aubel, A. and Thalmann, D. (2000). Efﬁcient muscle shape

deformation. In IFIP, pages 132–142.

Blemker, S. S. and Delp, S. L. (2005). Three-dimensional

representation of complex muscle architectures and

geometries. Annals of Biomedical Engineering,

33(5):661–673.

Gray, H. (1918). Anatomy of the Human Body. Lea &

Febiger.

Hormann, K. and Floater, M. S. (2006). Mean value coor-

dinates for arbitrary planar polygons. ACM Transac-

tions on Graphics, 25(4):1424–1441.

Joe, S. and Kuo, F. Y. (2008). Constructing sobol se-

quences with better two-dimensional projections. So-

ciety, 30(5):2635–2654.

Kohout, J., Clapworthy, G. J., Martelli, S., Wei, H., Vice-

conti, M., and Agrawal, A. (2011). Fast muscle wrap-

ping. Computers & Graphics. Submitted for publica-

tion.

Kohout, J., Kellnhofer, P., and Martelli, S. (2012). Fast de-

formation for modelling of musculoskeletal system. In

Proceedings of Proceedings of the International Con-

ference on Computer Graphics Theory and Applica-

tions: GRAPP 2012.

Ng-Thow-Hing, V. (2001). Anatomically-based models for

physical and geometric reconstruction of humans and

other animals. PhD thesis, University of Toronto,

Canada.

Pohlschmidt, M. and Meadowcroft, R. (2010). Muscle dis-

ease: the impact. http://www.muscular-dystrophy.org,

Muscular Dystrophy Campaign.

Richardson, M. (2011). Muscle Atlas of the Extremities.

Amazon Whispernet.

Schroeder, W., Martin, K., and Lorensen, B. (2004). The

Visualization Toolkit, Third Edition. Kitware Inc.

Thalmann, L. P. N. D. (1998). Real time muscle defor-

mations using mass-spring systems. In Proceedings

Computer Graphics International, volume 98, pages

156–165. IEEE Comput. Soc.

Viceconti, M., Astolﬁ, L., Leardini, A., Imboden, S.,

Petrone, M., Quadrani, P., Taddei, F., Testi, D., and

Zannoni, C. (2004). The multimod application frame-

work. Information Visualisation, International Con-

ference on, 0:15–20.

VPHOP (2010). the osteoporotic virtual physiological hu-

man, http://vphop.eu.

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