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Patient-Specific Modeling of Facial Soft Tissue based on Radial Basis
Functions Transformations of a Standard Three-Dimensional Finite
Element Model
1
Abstract
This article describes a novel way to build patient-specific models of facial soft
tissues by transforming a standard finite element (FE) model into one that has two
stages: a first transformation and a second transformation. The radial basis functions
(RBFs) interpolation method is used to transform the standard FE model into a
patient-specific one based on landmark points. A combined strategy for selecting
landmark points was developed in this study: manually picking them for the first
transformation and automatically picking them using a program designed for the
second transformation. Four typical patients were chosen to validate the effectiveness
of this transformation method. The results show good similarity between the
transformed FE models and the computed tomography (CT) models. The absolute
values of average deviations were in the range of 0.375–0.700 mm at the lip–mouth
region after the first transformation, and they decreased to a range of 0.116–0.286 mm
after the second transformation, which is even less than the system error (the CT
image’s pixel size was 0.300 mm). With some further research, this patient-specific
FE model can be applied to predict orthodontic treatment results.
Keywords: Finite element model, Facial soft tissue, Model transformation, Radial
basis functions, Landmark points
2
Introduction
An important goal of orthodontic treatment is to gain a harmonic soft tissue
profile. Sometimes this goal is difficult to achieve, with the difficulty lying partly in
the great variations in the thickness of facial soft tissue.1 The profile of facial soft
tissue changes after orthodontic treatment, and the relation between soft tissue and
hard tissue is complicated. The movement of soft tissue accompanying teeth
movement cannot yet be accurately predicted, which is a problem encountered by
orthodontists.
Various approaches have been applied to investigate the relation of lips and teeth,
but in orthodontics applicable procedures for profile prediction are mainly
two-dimensional. The proportional relations of lip changes and teeth movement were
calculated by Rudee2 and Waldman3 among others. These authors used statistical
analysis to study the correlations of upper and lower lip changes with teeth movement.
Others, including Talass et al.4 and Kasai,1 analyzed the statistical data of subjects by
multiple regression analysis and found that changes in the lower lip were more
predictable than those in the upper lip. The ratios between lip changes and teeth
movement vary significantly owing to sample differences (e.g., age, race, sex). Some
researchers tried multiple regression analysis to evaluate the relation. The researchers
found that it is difficult to reveal the real relations of complicated three-dimensional
facial structures using two-dimensional methods. In addition, the full spatial prognosis
of a patient’s posttreatment appearance is a highly desired goal that cannot be
accomplished with two-dimensional methods.
With the development of computer technology, the finite element method (FEM)
has been become more and more prevalent in orthodontic studies, especially since
Bookstein5 introduced two-dimensional FEM into the cephalometrics of skeletal
3
change in 1982. In contrast to two-dimensional FEM, a three-dimensional FE model
could provide a more realistic modeling of cephalofacial tissue. FE models were then
first used to study the stress distribution caused by orthodontic tooth movement.
Recently, many others have reported using FE models to predict soft tissue
deformation after orthognathic surgery.6-9 Some FE models of craniofacial tissues that
were built in previous studies were quite complicated, consisting of bone, muscle, and
facial skin.9-12 The development of fine FE models for clinical use would be
extremely costly over time, and it is an unreal expectation to build an individual
model for each patient. In addition, there are few reports in the literature on modeling
of facial soft tissue and its application in orthodontic treatment. A possible reason is
that orthodontic treatment is done much more often than orthognathic surgery, and
orthodontists do have not enough time to build individual models for each patient.
To address this problem, we have developed a method to build patient-specific
FE models via transformation from a standard finite element (FE) model based on
landmark points.13 Figure 1 shows the standard FE model, which includes facial skin,
subcutaneous fat, facial muscles, skull bone, and teeth. Although the transformation
method described in the literature13 functions quite well with a regular face type, its
error increases when applied to patients with other face types. Our model, which is a
two-stage transformation technique, can produce better transformation outcomes with
a patient-specific model.
The ethics committee at Peking University approved this study, and relevant
consent was given by the patients.
Materials and methods
4
Standard FE model and landmark points
The standard FE model described in the literature13 consists of facial skin,
subcutaneous fat, skull bone, and facial muscles. Because of the importance of
muscles in the profile of the face, seven pairs of muscles were reconstructed in the
standard FE model (Table 1). The soft tissue part of the standard FE model has
122,496 elements for skin, 178,529 for fat, and 8,699 for muscles. The standard FE
model is then transformed into four individual models based on selected landmark
points from the computed tomography (CT) data of four typical patients: three Angle
class I and II patients and one with bimaxillary protrusion. The resolution of four
patients’ CT data is 0.300 mm, slice thickness 0.300 mm, and image size 512 × 512.
Three types of landmark point are used in this study: anatomical landmark point,
mathematical landmark point, pseudo landmark point. Anatomical landmark points
are chosen on the surface of skin and skull based on the facial measurement method
developed by Farkas.14 Mathematical landmark points are mainly of equated or
extremal points of curvature diversification. A pseudo landmark point usually has no
anatomical or mathematical meaning, and it is selected for a particular region to
improve the local accuracy of transformation. In all, there are about 120 landmark
points for soft tissue and 16 for hard tissue. Figure 2 shows the main landmark points,
which are little different from those noted in the literature.13 Element checking shows
that the quality of new elements remains stable after model transformation.
Transformation method
The standard FE model is meshed into tetrahedral elements because of the
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geometric complexity of the facial soft tissue structure, and each element is connected
by four vertices (i.e., nodes). When the FE model undergoes geometrical deformation,
all nodes move to new positions in the three-dimensional Euclidean space. This can
be explained as a transformation of the FE model from its initial configuration to a
new configuration. The mathematical method for this transformation is, by nature,
scattered data interpolation of the nodes’ positions in a new configuration because the
nodes distribute irregularly in space. There are several interpolation methods for this
kind of scattered data,15 including the Shepard method,16,17 the thin plate spline
method,18 and the radial basis functions (RBFs)-based method.19-21 Compared with
other methods, the RBFs interpolation has excellent approximation properties15,21 and
can interpolate the displacement of a node caused by large deformations in the
two-dimensional or three-dimensional (3D) FE model with high quality.22
Therefore, the RBFs interpolation is chosen to transform the standard FE model
into a patient-specific FE model. The node displacement between the two models
can be calculated by the RBFs interpolation through selecting some nodes with
known displacements as landmark points. The interpolation function u, which
describes the displacement of nodes p = ( x, y, z )∈R3 in the standard FE model, is
approximated by a sum of radial basis functions21
n
u( p)  i ( p  pi )  s( p)
(1)
i 1
where nodes pi = ( xi, yi ,zi ) ∈R3 (1 ≤ i ≤ n) are landmark points; ai represents the
coefficients; φ is a given radial basis function with respect to the Euclidean distance
||p-pi||; s is a polynomial; and n is the number of landmark points. For landmark
points pi, the coordinates (xi, yi , zi ) are measured separately on the standard FE model
and on the patient CT data, so
6
u( pi )  d pi
(2)
could be easily calculated as the coordinate differences of pi. There is an additional
requirement20
n
 a q( p )  0
i 1
i
i
(3)
for all polynomials q with a degree less or equal to that of polynomial s. The
minimum degree of the polynomial s depends on the choice of the radial basis
function φ. Here, the radial basis function is chosen to be Hardy’s multiquadric
function19 {φ = (c2+r2)1/2} with conditionally positive definite order m ≤ 2; then the
polynomial s can be linear polynomials23; that is, s now has the form
s( p)  0  1 x  2 x  3 z .
(4)
When linear polynomials are used, it means that the rigid body translations in
this research are recovered. The values of the coefficients αi and the linear polynomial
s can be obtained by solving the following square matrix system:
 d n   M n, n Sn   
    T
 
 0   Sn 0    
(5)
Here, α contains the coefficients αi; β contains the coefficients of the linear
polynomial s; Mn,n is an n × n matrix containing the evaluation of the radial basis
function φninj = φ(||pni - pnj||); Sn is an n × 4 matrix with row j given by (1, xn, yn, zn);
and dn is an n ×1 matrix with row j equal to u(pj). Equations (2) and (5) are solved
separately for coordinates x, y, and z to obtain the corresponding coefficients α and β.
Then, calculating Eq. (1), the displacement vectors for all nodes (p) can be calculated,
and the new position of the nodes after transformation can be determined by adding
the displacement to the original coordinates. Thus, each node is moved individually
based on its position in space according to the interpolation function. New tetrahedral
7
elements in the current configuration have their original element–node relation and
can be used to build an individual FE model for a particular patient.
Selection of landmark points
The RBFs interpolation employs the Euclidean distance as the relation of
landmark points between the standard FE model and the patient’s CT data; it does not
take the topological and geometrical information of the face into consideration. Thus,
the selection of landmark points significantly influences the interpolation result. The
landmark points of the first transformation are selected manually. It is a necessary and
effective method, but it has also several shortcomings. First, the number of landmark
points is limited because of the large amount time needed and the high cost when the
landmarks are picked manually. Second, the picking error is difficult to control
because it is difficult to pick the corresponding position exactly on the patient’s CT
model; this is especially true for mathematical and pseudo landmark points.
Furthermore, sometimes the picking error is much greater at some facial regions
because of the face type difference between the patient and the standard model.
Finally, selecting landmark points at some facial regions is impossible owing to the
lack of picking criterion; that is, the spot cannot be categorized into one of the three
types of landmark points described above. As shown in Fig. 2, the landmark points
distribute quite sparsely on the face.
In this study, the landmark points for the second transformation are automatically
chosen on the facial surfaces of the first transformation FE model and the patient CT
model according to certain selection criterion. The facial surface of the first
transformed FE model is divided into many areas of cells. The cells in the lip–mouth
region are much smaller than in other regions of the facial surface, so there can be
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more landmark points to describe the area’s appearance. For each area cell, a
landmark point is selected by two steps: (1) selecting all nodes whose deviations from
the facial surface of the CT model are between –0.1 mm and +0.1 mm; (2) if any
nodes are found by step 1, the node with the lowest absolute deviation value is
selected as the landmark point in this area. The corresponding position of this
landmark point on the CT model is its projection on the CT model’s facial surface.
Figure 4 shows the landmark points selected for the second transformation on a
patient’s facial surface.
Results
First Transformation
The purpose of this study was to provide a fast approach to building individual
models for preoperative orthodontic predictions. The lip–mouth region of the face is a
major target for orthodontic treatment. Figure 3 compares the results of the
transformed models and their CT models on the lower half of the face (lip–mouth
region). The average absolute deviation values vary from 0.375 to 0.700 mm, and the
standard deviations range from 0.662 to 0.946 mm (Table 2).
Second Transformation
Figure 3 shows the deviation outcomes of the second transformation FE model
for four patients compared with their corresponding CT models. The absolute average
deviation values varied from 0.116 to 0.286 mm and the standard deviations from
9
0.199 to 0.431 mm (Table 2). Figure 5 shows the lateral views of CT models and the
second transformation FE models together; it indicates similar findings for all four
patients.
Discussion
Posttreatment facial appearances of orthodontic patients are of great interest to
orthodontists. In previous studies, some approaches based on two-dimensional
analysis have been used to predict the outcome of orthodontic treatment, but the real
head structure is three-dimensional, and the two-dimensional approaches have
inherent flaws. Therefore, a three-dimensional FE model with realistic anatomical
structure is necessary to achieve a precise prediction of tissue changes according to
the realignment of the teeth and bone after orthodontic treatment.
Building a three-dimensional model for individual patients is time-consuming and
requires skilled technicians. It is not realistic for orthodontists to work at model
building, even when further mechanical analysis of tissues is required. For these
reasons, a standard three-dimensional FE model with realistic anatomical structures,
including facial skin, muscles, fat, bone, and teeth, was described in the literature.13
Two patient-specific FE models, after the first transformation, were applied to predict
the outcomes of orthodontic treatment. The average prediction deviations were > 1
mm compared with the corresponding postoperative CT models at the lip–mouth
region, and the transformation error was about half of the prediction error. Obviously,
prediction accuracy is influenced by the transformation quality. Therefore, to improve
the accuracy of individual models, a second transformation was introduced into the
FE model.
10
As described above, patient-specific FE models are gained by transforming the
standard FE model twice, each time based on different landmark points. This article
introduces a combined strategy of selecting landmark points: (1) manually selecting
landmark points for the first transformation; and (2) selecting them automatically
using a CT program for the second transformation. This strategy acquires more shape
information at the facial region with a complicated appearance, such as the lip–mouth
region.
The average deviations of the second transformation FE model are less than half
that of the first transformation deviations, as shown in Table 2 and Fig. 3. They are
also less than the CT image’s pixel size (0.300 mm). The second transformation
described in this article can be characterized as an iterative transformation technique.
Therefore, if necessary, a third transformation can be carried out based on the results
data of the second transformation using the same selection process for identifying
landmark points. Figure 6 shows the deviation results of the first, second, and third
transformations for one of the four patients using the method introduced in the above
paragraphs. It was found that the second transformation achieved much greater
accuracy than the first transformation, and the third transformation achieved even
better results. The reason might be that the average deviation of the second
transformation was in the range of the CT image’s pixel size, a system error. A third
transformation cannot avoid the influence of the CT resolution ratio. Figure 7 presents
frontal views of the standard FE model of one patient, after the second transformation
with the FE model, and the corresponding CT model. It shows great similarity
between the transformed models and the CT models.
The RBFs interpolation is a well-established tool for interpolating scattered data,
but it employs only the Euclidean distance as the relation between a face vertex and a
11
landmark point. It does not take into consideration the topological and geometrical
information of the facial model. Thus, when the standard FE model is transformed
into a patient’s facial soft tissue with an open mouth, the points between the upper and
lower lips in the standard FE model are pulled in opposite directions by the lips. This
could result in stretched lips and increase the deviation of the transformation outcome.
This is true especially at the corner of mouth; there are not enough points in that area
to fill in the gap of the separated lips, leading to a large deviation. This phenomenon
can be observed in Fig. 3 and Fig. 6. It is a problem that needs further attention in a
future study.
With the commercial development of software for CT image processing, the
modeling of craniofacial tissues is becoming easier and thus more prevalent. It is
noteworthy that the software generated by commercial efforts have millions of
elements and unstructured tissues.7-9,12 Building an FE model that addresses
anatomical structures and an appropriate number of elements is still a goal. The
patient-specific model introduced in this article maintains the same anatomical
structures and number of elements as the standard FE model. Compared with the
methods described in literature,6-12 the patient-specific technique developed in this
study is fast and is highly accurate.
In this study, hard tissue only provided the boundaries for soft tissue. The
landmark points of hard tissue ensure that the inner surface of the soft tissue is
transformed accurately, meeting the protruding part of the skull bone in the sagittal
plane. This is critical for further application of the transformed FE model to
preoperative prediction of orthodontic treatment outcomes, which would be the next
step in this study.
12
Conclusions
A novel way to build a patient-specific FE model for facial soft tissue modeling
based on a two-stage transformation method is presented. The standard FE model is
transformed twice by the RBFs interpolation. The key point of this novel method is
the manner in which landmark points are selected for each transformation. The
modeling results show that the second transformation resulted in enhanced accuracy
compared with the first transformation. Because of these results, a third
transformation is usually not necessary. The patient-specific FE model after two
transformations can be applied to predict orthodontic treatment, which will be
addressed in a future study.
ACKNOWLEDGMENT
This work was supported by the National Natural Science Foundation of China
(NSFC) under grants 30672350 and 10872007.
13
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Orthod. 52(2)(1982) 129-134.
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in maxillofacial surgery including soft tissue prediction, J. Craniofacial Surg.
16(1)(2005) 100-104.
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face soft tissues for computer-assisted maxillofacial surgery, Medical Image
Analysis 7(2)(2003) 131-151.
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biomechanical model of the face: a study of muscle coordination for speech lip
gestures. In:8th International Seminar on Speech Production ISSP. 2008,321-324.
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NewYork, 1994.
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computation. 1982, 38(157), 181-200.
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Proc.23rd Nat. Conf. ACM (1968) 517-523.
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Numer Meth Eng. 1980, 15, 1691-1704.
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15
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16
Table 1. Basic characteristics of muscles in the standard model
Muscles
Thickness
Origin
(upper lip
Wildth
From basis nasi to
Orbicularis
oris
Insertion
2.5mm
One side of
angulus oris
muscle)
The other
oris rimae (about
side of
22mm at the middle
angulus oris
part and 3mm at
angulus oris)
Length
Adequate to
the length of
oris rimae
(about 60mm)
From oris rimae to
Orbicularis
oris
(lower lip
2.5mm
One side of
angulus oris
muscle)
The other
side of
angulus oris
the middle part of
Adequate to
mandible (about
the length of
22mm at the middle
oris rimae
part and 3mm at
(about 60mm)
angulus oris)
Prefrontal
bone of von
Levator labii
superioris
1.5mm
Bardeleben,
suborbital ,
About 30mm at the
Ranging from
top and 10mm at the
40mm to
bottom
60mm
Upper lip
About 8 mm
About 60 mm
Upper lip
About 5 mm
About 50mm
Anguli oris
About 8 mm
About 58 mm
Upper lip
interior of
zygoma
Zygomaticus
major
1.5mm
Zygomatic
arch
The lateral
Zygomaticus
minor
1.5mm
face of the
zygomatic
bone
Fascia
Risorius
1.5mm
parotidea
masseterica
About 30mm
Depressor
anguli oris
The oblique
1.5mm
line of the
Anguli oris
mandible
About 25mm at the
bottom part
from anguli
oris to the
oblique line of
the mandible
About 22mm
Depressor
labii oris
from the
The oblique
1.5mm
line of the
mandible
Lower lip
About 25mm
oblique line of
the mandible
to the lower
lip
17
Table 2. Deviations of transformed models versus corresponding computed
tomography data
The first
Patient A
transformation
The second
transformation
The first
Patient B
transformation
The second
transformation
The first
Patient C
transformation
The second
transformation
The first
Patient D
transformation
The second
transformation
Average
Average
deviations
deviations
(positive)
(minus)
(mm)
(mm)
0.437
-0.539
0.724
0.137
-0.116
0.199
0.700
-0.525
0.946
0.286
-0.211
0.431
0.593
-0.549
0.867
0.254
-0.191
0.364
0.554
-0.375
0.662
0.219
-0.184
0.374
Standard
deviations
(mm)
18
FIGURE LEGENDS
Fig. 1. Standard finite element (FE) model with skull bone, teeth, facial muscles, fat,
and skin
Fig. 2. Landmark points for the first transformation. Stars: anatomical landmark point;
triangles: mathematical landmark point; diamonds: pseudo landmark point.
Fig. 3. Deviations (+3.0 to –3.0 mm) of transformed models compared with the
corresponding computed tomography (CT) data. Top row: Outcome of the first
transformation: patient A, 0.488 ± 0.724 mm; patient B, 0.615 ± 0.946 mm; patient C,
0.571 ± 0.867 mm; patient D, 0.511 ± 0.73 9 mm. Bottom row: Outcome of the second
transformation: patient A, 0.157 ± 0.334 mm; patient B, 0.254 ± 0.499 mm; patient C
0.165 ± 0.333 mm; patient D, 0.251 ± 0.488 mm
Fig. 4. Landmark points (n = ~340) with the second transformation. Note that some
points at the border are not shown
Fig. 5. Right lateral views of the CT models (dark gray) and the second
transformation models (light gray). A. Patient with bimaxillary protrusion and normal
skeletal relations. B. Patient with an Angle class I condition with protrusive upper
incisors. C. Patient has an Angle class II1 condition with a retrusive mandible. D.
Patient has an Angle class II1s condition with protrusive upper incisors
Fig. 6. Deviation outcomes (+1.0 to –1.0 mm) of the first (A), second (B), and third
19
(C) transformations in a patient with bimaxillary protrusion and normal skeletal
relations
Fig. 7. A. Standard FE model. B. FE model for C after the second transformation. C.
Patient’s CT model (bimaxillary protrusion with normal skeletal relations)
20
Fig. 1
levator labii superioris
zygomaticus minor
zygomaticus major
orbicularis oris
risorius
tendon
orbicularis oris
depressor anguli oris
depressor labii inferioris
21
Fig. 2
22
Fig. 3
23
Fig. 4
24
Fig. 5
25
Fig. 6
26
Fig. 7
27