=Paper=
{{Paper
|id=None
|storemode=property
|title=An Articulated Statistical Shape Model of the Human Knee
|pdfUrl=https://ceur-ws.org/Vol-715/bvm2011_14.pdf
|volume=Vol-715
}}
==An Articulated Statistical Shape Model of the Human Knee==
https://ceur-ws.org/Vol-715/bvm2011_14.pdf
An Articulated Statistical Shape Model of the
Human Knee
Matthias Bindernagel, Dagmar Kainmueller, Heiko Seim, Hans Lamecker,
Stefan Zachow, Hans-Christian Hege
Zuse Institute Berlin
bindernagel@zib.de
Abstract. In this work we present an articulated statistical shape mo-
del (ASSM) of the human knee. The model incorporates statistical shape
variation plus explicit degrees of freedom that model physiological joint
motion. We also present a strategy for segmentation of the knee joint
from medical image data. We show the potential of the model via an
evaluation on a set of 40 clinical MRI datasets with manual expert seg-
mentations available.
1 Introduction
For biomechanical analysis or surgery planning it can be beneficial to reconstruct
an estimated healthy joint anatomy from medical image data in the presence
of strong pathological changes or implants. For total knee arthroplasty, for
instance, knowledge about the morphology of the joint before osteoarthritis may
impact the choice of implant. Since the respective image data is typically not
available, the need for an estimation from the pathological case arises. Secondly,
an accurate reconstruction of the actual patient-specific joint anatomy is often
needed.
Statistical shape models (SSMs) [1] are a powerful tool for reconstructing
both estimated healthy as well as actual patient-specific anatomies. Multiple
SSMs of different bones may be applied successfully for the reconstruction of
joints [2]. However, such an approach does not model knowledge about joint
posture or correlated morphology of the involved bones, and consequently lacks
robustness. Robustness is crucial for the reconstruction of the estimated healthy
anatomy, as the pathological or implanted region in the image data, which usu-
ally provides significant information about the joint morphology, is deliberately
ignored here. In case of poor contrast in the joint region, robustness also plays
an important role for the reconstruction of actual patient-specific anatomies.
One may model joint flexibility implicitly by capturing joint motion statisti-
cally [3, 4]. This approach is beneficial only if relative transformations between
individual objects are a statistical property of anatomy, which is, e.g., not the
case for knee bending. Instead, we follow the approach presented for the hip
joint in [5] and propose an articulated SSM (ASSM) of the knee, where we
60 Bindernagel et al.
model knee joint posture explicitly as a combination of characteristic transfor-
mations [6]. With an evaluation on 40 MRI datasets, we show that our knee
ASSM outperforms reconstruction based on separate SSMs.
2 Materials and Methods
2.1 Model of the Knee Joint
Our knee joint model consists of the femur (thighbone) and the tibia (shinbone),
represented by triangular surface meshes. As proposed in [6] we model the joint
motion by (i) a rotation around the epicondylar axis, which is defined by the
lateral and medial epicondyles of the femur that are represented as vertices of
the femur mesh, and (ii) a translation in direction of the intersection of the tibial
plateau and the epicondylar axis shape model of the human knee normal plane
(Fig. 1a). Accordingly, the transformation K of the tibia relative to the femur
can be written as
K(α, t) = rot(r epi , depi , α) ◦ trans(dplateau , t) (1)
where rot denotes the rotation around the epicondylar axis with origin r epi and
direction depi by an angle α, and trans represents a translation in direction of
dplateau by a distance t.
In addition to the global transformation Tg of the knee bone compound the
model incorporates a local transformation K for the tibia to keep track of varying
joint postures. That is, the overall transformation of the femur is Tg and the
overall transformation of the tibia equals Tg ◦ K. K = I represents a reference
bending. To adjust the knee posture to a particular α and t under motion of the
tibia, one has to set K ← K(α, t). In order to adjust the joint under motion of
the femur, one may apply Tg ← Tg ◦ K −1 (α, t) and K ← K(α, t).
Fig. 1. Knee joint motion is modeled by rotation Repi around the epicondylar axis and
translation Ttrans (a). Instances of the knee ASSM: Shape and joint state I (b), shape
II and joint state I (c), shape I and joint state II (d), and shape and joint state II (e).
Shape Model of the Human Knee 61
2.2 Adjusting Transformations by Alignment of Bones
We employ the root mean square distance of all vertices v i of a given model
instance to corresponding vertices v ref,i of a reference mesh to measure alignment
∑ 2
∑ 2
D(Tg , K) = ∥Tg ◦ v i − v ref,i ∥ + ∥Tg ◦ K ◦ v i − v ref,i ∥ (2)
i∈Ifemur i∈Itibia
where Ifemur and Itibia are the sets of vertex indices of the femur and tibia,
respectively. Hence, to align a model instance to a reference mesh in terms of
transformation, one must optimize D(Tg , K) with respect to Tg and K.
We propose an iterative scheme to optimize Tg and K by repeated alignment
of the object compound, the femur, and the tibia until ∆D falls below a user-
defined stopping criterion. Here, one iteration is composed of a sequence of
alignment steps: One may, e.g., align the whole compound, then align the tibia
by joint adjustment while keeping the femur fixed, and then align the femur by
joint adjustment while keeping the tibia fixed (sequence CTF), align the whole
compound with regard to femur vertices only, and then align the tibia by joint
adjustment (sequence CF T), or vice versa (sequence CT F).
The object compound is aligned via rigid and scale transformations. For a
description of well-established methods [1]. To align femur and tibia by means
of knee joint adjustment, one has to find
[ ∑ 2
]
argmin Dfemur (α, t) = Tg ◦ K −1 (α, t) ◦ v i − v ref,i (3)
α,t
i∈Ifemur
[ ∑ ]
2
argmin Dtibia (α, t) = ∥Tg ◦ K(α, t) ◦ v i − v ref,i ∥ (4)
α,t
i∈Itibia
for femur and tibia, respectively. A simple and efficient way to solve these
problems is by iterative optimization with respect to α and t separately, since
the respective partial derivatives of Dfemur (α, t) and Dtibia (α, t) as well as their
roots can be computed analytically. Optimization is stopped when ∆D falls
below a user-defined convergence criterion.
2.3 ASSM Generation
We generated the articulated knee model from 40 training shapes made available
by [7]. In contrast to the generation of SSMs of single objects, the preparation of
training shapes for an ASSM requires alignment via transformations according
to the joint model. To this end, we use the method described in Sec. 2.2, after we
have determined vertex correspondences on the set of training shapes as in [2].
We then generate the ASSM by applying principal component analysis on the
aligned and corresponding training meshes. An example of varying joint posture
and shape is shown in Fig. 1, b-e.
62 Bindernagel et al.
Fig. 2. Results. Con-
tour: separate SSMs
(black) and ASSM
(white); rectangle: ex-
trapolated purely from
the model (white).
2.4 Segmentation Framework
Commonly, an iterative segmentation process repeats the following steps to ad-
just an SSM to image data:
1. Analyze the normal intensity profiles of the current SSM instance within the
image data. Here, we employ the strategy presented in [2].
2. Displace vertices of the instance to positions that better fit the image data,
resulting in a triangular mesh that is in general not an instance of the SSM.
3. Adjust the overall transformation of the instance to fit the displaced mesh.
4. Adjust shape parameters of the instance to fit the displaced mesh.
In order to cope with the relative joint transformation we extend step 3 to adjust
not only the overall transformation of the compound shape, but also the relative
transformation within the joint, as proposed in Sec. 2.2. Note that in general the
shape adjustment in step 4 changes the transformation axes of the model, which
results in a slight change of the relative transformation. This effect is currently
only compensated for by the iterative nature of the approach.
3 Results
To evaluate the model with regard to different alignment sequences as suggested
in Sec. 2.2, we initially built three different ASSMs: For each ASSM, the training
shapes were aligned according to one of the three sequences CTF, CF T and CF T.
We conducted a leave-one-out study for each model, using the respective “left-
out” training shape in its unaligned form as target. There are no significant
differences among the three ASSMs in terms of average reconstruction accuracy.
However, employing alignment sequence CTF results in the fastest convergence.
We employed the respective ASSM for all further experiments.
To evaluate the reconstruction capability of the knee ASSM, we conducted a
leave-one-out study on 40 clinical MRI datasets that were used for model gen-
eration. To simulate the task of estimating healthy anatomy in the presence
of pathologies, we manually labeled a region of interest around the joint gap
(Fig. 2): In this region image features are not considered during model adapta-
tion, i.e., the joint anatomy is purely extrapolated from the model. We compare
the results to reconstructions obtained with single-object SSMs as in [2]. The
results are presented in Tab. 1. We measured the accuracy by comparison to
given manual expert segmentations in terms of Dice’s coefficient (DICE), rela-
tive volume difference (RVD), average surface distance (AD), and average root
mean square surface distance (RMS).
Shape Model of the Human Knee 63
Table 1. Average accuracy measures for ASSM and single-object SSM segmentation.
DICE RVD AD RMS
Femur (ASSM) 0.94 (±0.02) 0.05 (±0.03) 1.12 (±0.28) 1.52 (±0.44)
Tibia (ASSM) 0.89 (±0.05) 0.06 (±0.05) 2.01 (±0.91) 2.65 (±1.22)
Femur (Single) 0.94 (±0.02) 0.05 (±0.04) 1.16 (±0.37) 1.59 (±0.60)
Tibia (Single) 0.86 (±0.10) 0.12 (±0.12) 2.61 (±2.08) 3.52 (±2.77)
4 Discussion
The comparison between ASSM and two separate SSMs shows that both methods
perform similar for the femur, whereas the reconstruction quality of the tibia was
significantly improved by use of the ASSM (Fig. 2). The similar results for the
femur might be attributed to its distinguished shape outside the extrapolated
joint region: This seems to determine the extrapolated region sufficiently also
for a femur SSM. In contrast, the shape of the tibia is less distinguished outside
the extrapolated region. While the tibia SSM suffers from this ambiguity, the
ASSM deals with it via the knowledge about the relative positioning of the
tibia encoded in the joint model. Furthermore, part of the anterior tibia is
ignored during adaptation by design of the profile analysis strategy [2], which
adds to the above mentioned effect, but may also explain worse results for the
tibia as compared to the femur for both methods. In summary, ASSMs are
a promising tool for an accurate reconstruction of anatomical structures from
poorly contrasted, incomplete or pathological medical image data.
Acknowledgement. This work was partially supported by the EU-FP7 Project
MXL (ICT-2009.5.2), the DFG Research Center Matheon and the DFG Collab-
orative Research Center SFB760.
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