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1. Title page
The Hip Joint as an Egg Shape: A Comprehensive Study of Femoral and Acetabular
Morphologies
Daniel Simões Lopes ab*, daniel.lopes@inesc-id.pt
Sara M. Pires ab, sara.pires@tecnico.ulisboa.pt
Carolina D. Barata ab, carolina.barata@tecnico.ulisboa.pt
Vasco V. Mascarenhas c, vmascarenhas@hospitaldaluz.pt
Joaquim A. Jorge ab, jorgej@acm.org
a INESC-ID Lisboa, Rua Alves Redol, 9, 1000-029 Lisboa, Portugal
bInstituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa,
Portugal
c Hospital da Luz, UIME, Av. Lusíada 100, 1500-650 Lisboa, Portugal
* - corresponding author, tel.: (351) 214233508, Edifício IST, INESC-ID Taguspark, Room
2N9.19, Avenida Professor Cavaco Silva, 2744-016 Porto Salvo, Portugal
2. Abstract
Understanding the morphological features characterizing a normal femoral head and acetabular
cavity is critical for a more comprehensive and updated definition of hip anatomy. Based on
anatomical observations, MacConaill introduced the notion that spheroidal articular surfaces are
better represented by ovoidal shapes, in comparison with the still very well-established spherical
shape. This work tests MacConaill’s classification by using a surface fitting framework to assess
the goodness-of-fit regarding the largest assortment of sphere-like shapes presented in a single
study (i.e., a total of 10 different shapes: sphere, rotational conchoids, rotational ellipsoid,
ellipsoid, superellipsoid, Barr’s superellipsoid, tapered ellipsoid, Barr’s tapered superellipsoid,
ovoid, superovoid). Anatomical data of the femoral head and acetabular cavity were obtained
from computed tomography scans of a gender balanced, asymptomatic population of 30 adult
subjects. The framework involved image segmentation with active contour methods, mesh
smoothing and decimation, and surface fitting to point clouds was performed with genetic
algorithms. The statistical analysis of the surface fitting errors revealed the superior
approximation of non-spherical shapes: superovoids provided the best fit for each femoral head
and acetabular cavity, whereas spheres presented the worst fitting values. We also addressed
gender variability in bony hip geometry as sphericity, ellipticity, conicity and squareness were
measured. Finally, we consider that this work lays the ground for further biomechanical and
clinical research as the reported morphological findings may serve to inspire new hip prosthetic
shapes and new radiographic measurements that unambiguously characterize femoral head and
acetabular cavity geometries.
1
Key terms
femoral head, acetabular cavity, implicit surfaces, surface fitting, sphere, ovoid.
3. Introduction
The morphology of the hip joint has been a topic of interest for anatomists and physicians during
many centuries [1]. Given its influence on the effective and efficient performance of daily
physical activities, the orthopaedic community has also put a great emphasis on hip joint
morphology to better understand bone geometry in asymptomatic conditions, as normal hip
characterization helps clarify the distinctive morphological features when compared to
pathological deformities, such as femoroacetabular impingement and hip dysplasia [2].
Moreover, understanding the morphology of the femoral head and acetabular cavity may even
inspire new prosthetic devices as form is intimately related to biomechanical joint function.
According to the standard classification of synovial joints [3], the hip articulation falls under the
classification of spheroidal joints given its visual similarities to a sphere or a hemisphere. In fact,
the articular surfaces of this joint (i.e., femoral head and acetabular cavity) were regarded as
being best represented by the spherical shape for a considerable amount of time, and this view
has not yet fallen completely in disuse [3-4]. Nevertheless, there have been studies contradicting
this belief [2,5-10]. It has long been suggested that the articular surfaces of asymptomatic hip
joints are only symmetric in a limited number of axes, presenting an egg-like or ovoidal shape
instead of a spherical one. Attributing an egg shape to a synovial joint is known as MacConaill’s
ovoidal joint classification [5-7].
To assess the veracity of this classification, several authors have conducted studies on the
morphological features of articular surfaces by approximating spheres (S), rotational conchoids
(RC), and ellipsoids (E) to the femoral head and/or the acetabular cavity [8-15]. Studies on
prosthetic designs for the femoral head were also carried out [16-18], in which the artificial
articular surface was approximated by an ellipsoidal shape. Besides these shapes, rotational
ellipsoids (RE) have also been considered as an approximated surface [19-23]. However, these
shapes do not satisfy MacConaill’s classification as they do not account for essential ovoid
characteristics, namely, axial asymmetry and non-homogeneous curvature [24-26]. To this end,
Lopes et al. [27] computationally tested the ovoid conjecture for the femoral head by performing
a shape analysis that compared spheres and ellipsoids against superellipsoids (SE), ovoids (O)
and superovoids (SO), while a more recent work studied the shape of asymptomatic, dysplastic
and impinged hip joints by comparing spheres, ellipsoids and tapered ellipsoids (TE) [2]. (Table
1) summarizes surface fitting results of several papers on shape analysis of femoral head and
acetabular cavity surfaces.
Table 1 - Summary of studies regarding surface fitting of geometric primitives onto spheroidal
articular surfaces of the hip joint. (FH – Femoral Head; AC – Acetabular Cavity; S – Sphere; RC
2
– Rotational Conchoid; RE – Rotational Ellipsoid; E – Ellipsoid; SE – Superellipsoid; O – Ovoid;
SO – Superovoid)
Despite the amount of work reported in the literature, to the authors knowledge, there still lacks a
thorough and exhaustive comparison between a wider variety of shapes that resemble and better
represent hip joint reconstructions taken from CT data-sets. In this paper, we aim to better
evaluate the limit of hip joint sphericity by comparing 10 different shapes through a robust
surface fitting framework, and by measuring egg shape features such as ellipticity and conicity.
We also introduce as a third shape feature, squareness, as supra-quadratic shapes have indicated
better results when compared to spheres and ellipsoids [27]. In addition, it is well known that the
bony geometry of the pelvis is complex and that gender differences exist. While multiple studies
have catalogued these differences, most of the measurements are not directly related to ellipticity
or conicity, much less to squareness. In line with MacConaill’s classification, we hypothesized
that gender differences existed in terms of ellipticity and conicity of the articular surfaces of the
hip joint.
Therefore, the major goals of this work are to address the following morphological questions: (i)
how aspherical are the femoral heads and acetabular cavities of healthy hip joints when compared
to ellipsoidal and ovoidal shapes? (ii) which shape primitive can be considered the most
representative of a healthy hip joint? (iii) how does sphericity, ellipticity, conicity and squareness
vary among sexes?; and (iv) can we identify morphological similarities between genders?
Therefore, this work intends to carry on previous research on hip joint morphology [2,27] by
convening, in a single paper, the shape analyses of femoral heads and acetabular cavities of
asymptomatic hips. Such a comprehensive study compares a broad spectrum of shape primitives
either reported in the literature (e.g., sphere, rotational conchoid, rotational ellipsoid, ellipsoid,
superellipsoid, ovoid, superovoid) or newly introduced shapes (e.g., Barr’s superellipsoids (SEB),
tapered ellipsoids (TE) and Barr’s tapered superellipsoids (TSEB)). In order to address which
shape primitive best portrays the morphological features of a normal hip joint, an optimization
scheme was developed to compute the signed Euclidean distances between each point in the
reconstructed 3D scan and the optimally fitted shapes. Error-of-fit statistical analyses were then
performed to sort out the best and worst shapes. Finally, the shape features to describe sphericity,
ellipticity, conicity and squareness are compared between both sexes.
4. Material and methods
The image-based anatomical workflow followed the pipeline presented in work previously
carried out by Lopes et al. [27] (Figure 1). CT data sets of asymptomatic hip joints were used to
extract the geometric information necessary for 3-D reconstruction. Semi-automatic segmentation
techniques were used to extract the anatomical information. In order to guarantee homogeneous
nodal distribution and attenuate artifacts resulting from model creation, mesh adjustment
operations such as smoothing and decimation filters were applied to the reconstructed models.
From the 3-D models, only the regions corresponding to the articulating surfaces are of interest
and were manually selected and stored as point clouds. A surface fitting tool was then used to
adjust 10 different smooth convex shapes, using a genetic algorithm to solve a non-linear
least-squares minimization problem, to the point clouds of femoral heads and acetabular cavities
3
of 30 subjects. Here, surface fitting is formulated as a minimization problem in which the
objective function is highly non-linear presenting a large number of local minima. Standard
optimization algorithms are not well suited for such minimization problems [Bazaraa et al.,
1993]. A genetic algorithm was considered as it consists of a powerful tool to find optimal
solutions for highly non-linear surface fitting problems [Ahn, 2004]. Finally, statistical analyses
were performed to compare the goodness-of-fit between different shape models that best
characterize the articular surfaces of the hip joint in normal conditions.
Figure 1 - Sequence of computational applications used for anatomical and geometric
information extraction and modeling of spheroidal articular surfaces of the hip joint. White boxes
represent the file formats used as input in the software tools referenced in the blue boxes.
4.1 Hierarchy of shape models
The macroscopic features of the articular surfaces of the hip joint can be considered as
spheroidal, convex, limited, closed, topologically similar to a sphere, and present second-degree
continuity for most of the surface range. The chosen geometric primitives do, in fact, display
these properties (with exception of the rotational conchoids which are slightly concave at one
end). The considered mathematical models are drawn from previous studies [8,26,30] and consist
of the following 10 shapes: sphere (S), rotational conchoid (RC), rotational ellipsoid (RE),
ellipsoid (E), superellipsoid (SE), Barr’s superellipsoid (SEB), tapered ellipsoid (TE), Barr’s
tapered superellipsoid (TSEB), ovoid (O), superovoid (SO).
However, it is important to stress the hierarchical connection between all the shape primitives, in
order to fully understand how the surface fitting process is built. Considering the initial and
simplest surface represented in (Figure 2), the sphere, it is possible to obtain the remaining
surfaces through non-linear morphing operations, such as rescaling, exponentiation, and
asymmetrization. The changes generated by these actions are easily identified on the resulting
surface, being the higher level of squareness exhibited by surfaces such as superellipsoids,
superovoids and tapered superellipsoids an example of the modifications introduced by variation
in exponentiation. The orientation of the arrows composing the hierarchical graph (Figure 2)
indicate which shape models constitute generalizations and which are particular cases within a
given geometric primitive. For example, superovoids are a generalization of superellipsoids and
ovoids, whereas ellipsoids are a particular case of both superellipsoids and tapered ellipsoids.
Figure 2 - Hierarchical organization of the different geometric primitives represented in the form
of a graph revealing the morphological relationships between the various shapes.
The implicit surface expressions for all 10 shapes, in the canonical form, are written as
sphere
FS(x, , )y z = (a x)
−1 2+ (a y)
−1 2+ (a z)
−1 2
(1)
rotational conchoid
FRC (x, , )y z = (x x)
2+y2+z2−a2−b2(x)
2+y2+z2
(2)
4
rotational ellipsoid
FRE (x, , )y z = (a x)
−1 2+ (b y)
−1 2+ (b z)
−1 2
(3)
ellipsoid
FE(x, , )y z = (a x)
−1 2+ (b y)
−1 2+ (c z)
−1 2
(4)
superellipsoid
FSE (x, , )y z = (a x)
−1 2
ε1+ (b y)
−1 2
ε2+ (c z)
−1 2
ε3
(5)
Barr’s
superellipsoid
FSEB (x, , )y z = (a x)
[−1 2
ε1+ (b y)
−1 2
ε1]ε2
ε1
+ (c z)
−1 2
ε2
(6)
tapered ellipsoid
(c z) FT E (x, , )y z =(a x
−1
T z+1
x)2
+(b y
−1
T z+1
y)2
+ −1 2
(7)
Barr’s tapered
superellipsoid
FT SE B (x, , )y z =[(a x
−1
T z+1
x)2
ε1+(b y
−1
T z+1
y)
2
ε1]ε2
ε1
+ (c z)
−1 2
ε2
(8)
ovoid
FO(x, , )y z =
(a x
−1
c+c z+c z +c z
0x1x2x23x3)2
+(b y
−1
c+c z+c z +c z
0y1y2y23y3)2
+ (c z)
−1 2
(9)
superovoid
FSO (x, , )y z =
(a x
−1
c+c z+c z +c z
0x1x2x23x3)2
ε1+(b y
−1
c+c z+c z +c z
0y1y2y23y3)
2
ε2+ (c z)
−1 2
ε3
(10)
where are the local coordinates of the point in space that belongs to the surface; , , ∈R x y z
represent shape dimensions or semi-axis radii; are the
,b,c a ∈R+ ,, ∈]0, 1] ε1ε2ε3
squareness parameters; , , , , , , and are ovoidal shape coefficients, c0x c1x c2x c3x c0y c1y c2y c3y
where the zero and first degree coefficients , , , are restricted to the range , c0x c1x c0y c1y 0, 1
[ ]
while the second and third degree coefficients , , , are limited to the interval c2x c3x c2y c3y
; and are the tapering values in the and directions, restricted between
− .1, 0.1
[0 ] Tx Ty x y
-1.0 and 1.0.
The surfaces defined by (Equations 1-10) are represented in their respective local systems, where
the referential origin corresponds to the surfaces’ centre. For modeling purposes it is important to
guarantee the possibility of granting the surface any spatial configuration. This geometrical
modification consists in applying transformations to the surface’s coordinate system, such as
translation, rotation, and scaling.
Affine transformations are applied to the unit shape model, described by (Equations 1-10), by
converting local coordinates, x, to global coordinates, x*, by an affine matrix transformation that
incorporates a scaling matrix, D, that contains shape coefficients and dimension parameters (e.g.,
in millimeters) a
,b
, and calong the x
,y
, and zdirections, a rotation matrix, R, and a translation
column vector, t, which is expressed as a set of linear algebraic equations, which can be
described in vector form by (Equation 11):
x x*=x y z 1 [ * * * ]T=RD t 0 1
[1x3]
(11)
where xand x* are written in homogeneous coordinates. Note that, the rotation matrix Rcontains
the information about the orientation of each local coordinate with respect to the global frame.
5
4.2 Image-based anatomical modeling
The articular surface geometry was extracted from CT data sets of a gender balanced,
asymptomatic population composed of 30 adult hip joints (14 male and 16 female subjects; 13
right sided and 17 left sided). All subjects are Caucasian and ages ranged from 18 to 44 years old
(male: 32.7 ± 5.7; female: 28.6 ± 8.5). Young and relatively young healthy subjects were selected
for this purpose, as older individuals have a higher risk of hip joint pathology. Data sets were
gathered from two other studies [2,31]: (i) , 20 CT scans of asymptomatic hip joints (512x512
acquisition matrix, in-plane and resolutions = 0.602–0.869 mm, slice thickness = 1.5–2 mm,
262–929 slices) acquired from the Hospital da Luz (Lisboa, Portugal) with a Siemens Emotion 16
(Siemens Healthineers, Germany) [2]; and (ii) 10 multi-detector CT scans of the pelvic region
(512x512 acquisition matrix, in-plane xand yresolutions = 0.2155-0.2637 mm, slice thickness =
0.70-1.0mm, and 241-357 slices) acquired from the University of Utah Hospital [29], which are
available from the Musculoskeletal Research Laboratories at the University of Utah1[31] All data
sets were anonymized. All subjects had been informed of the intention to use their respective
image sets and provided their written informed consent. The data sets used in our study resulted
from the approval by the Ethics Research Committee of the Nova Medical School
(nr.61/2014/CEFCM) [2], and also by the University of Utah Institutional Review Board #10983
[31].
The modeling pipeline begins with image segmentation of the bone-cartilage interface composing
the femoral head and acetabular cavity. This process was performed with ITK-SNAP2(version
3.4) by using a combination of a semi-automatic method, that relied on 3-D active contour
evolution [32], and manual segmentation to correct errors. The segmented images were then
imported into ParaView3(version 4.3.1), in order to create a triangle mesh with the marching
cubes algorithm [33]. Mesh decimation and Laplacian filtering were then applied to remove the
excessive vertice number and smooth mesh artifacts, such as voxelized features. Furthermore, the
regions corresponding to the articular surfaces of the hip joint were manually selected from the
surface model in Blender4(version 2.75), so that the underlying point cloud resulting from the
remaining vertices of the surface model could be stored.
4.3 Surface fitting analysis
Implicit surface fitting of the 10 geometric primitives is performed taking the extracted point
clouds as input. Goodness-of-fit or surface error is analyzed and compared based on the
Euclidean distance between the input points and the optimally fitted surfaces. To obtain these
errors, an orthogonal distance optimization framework is taken into account, which needs to
satisfy a non-linear equality constraint given by the implicit surface equation. Both surface fitting
and the surface error calculations are accomplished in Matlab® (version R2014a) using the
Genetic Algorithm and Direct Search ToolboxTM and the code ran on an Intel® CoreTM i5
processor 2.4 GHz and 5 GB of RAM. To better understand the goodness of fit of each of the
geometric primitives, a qualitative and quantitative analysis is performed on the fitting results
using the surface errors.
6
For a point cloud with points in Cartesian space belonging to the outer cortical bone n∈N
surface of the hip joint, the vector of geometric parameters , where is the λ ∈Rm m∈N
number of parameters characterizing a given implicit surface, which minimizes the EOF
objective function, , was determined. This objective function is defined as the square OF (λ) E
sum of residual function for each point , where is the difference between the f 1, , } i= { … n f
shape model function and the -th point datum, as formulated by the following expressioni
OF E(λ) = ∑
n
i=1
fi
2x, , ;
(gygzgλ) = ∑
n
i=1
1
(−Fix, , ;
(gygzgλ))2
(12)
under the restriction
l≤ λ ≤ u
(13)
where is the implicit surface representation of a given shape model and are the F , l u ∈Rm
lower and upper bound column vectors, respectively, setting the limits for the solution presented
in . In addition to the parameters needed to define each shape model, such as curvature and, in λ
the case of ovoids, conicity, vector also includes the rotation and translation factors used in the λ
affine transformations. It is, therefore, a vector of global anatomical information. As for the EOF
objective function domain,, Ω ⊆ℝM
, or surface parameter space, it can be expressed as the
following:
(14)
where I
kis a real-valued interval of the k
th surface parameter, {0,1} subscript indices designate
the start and end value of the k
th interval, and M= 4,8,8,9,12,11,11,13,17,20 is the total number of
surface parameters for the S, RC, RE, E, SE, SEB, TE, TSEB, O, and SO models, respectively
(Table 2). As for spheroidal surfaces, the shape parameters must be constrained, where the values
for a
, b
, and c
are all positive.
(Table 2) summarizes the different shape models used in the studies and the vector of geometric
parameters associated to each of them. For the shape models that include squareness parameters,
and since are confined to be greater than 0 and lesser than 1, the gamma exponents are , , ε1ε2ε3
represented as . Note that for surfaces exhibiting exponents larger than 2, the change in γ = ε
2
exponent representation means that and are restricted to the range .
, γ1γ2γ32,+ [ [ ∞
Table 2 - Vector of geometric parameters for all shape models considered and respective number
of degrees of freedom, given by the total number of surface parameters, m
.
7
The surface fitting process of the remaining geometric primitives considered used as an initial
approximation the optimal solutions obtained from the fitting process of the shape models
hierarchically linked to them, as determined in (Figure 2). For instance, superellipsoid, tapered
ellipsoids, and ovoid fittings were initiated with resource to the optimal ellipsoid parameters. In
turn, the optimal parameters found for ovoids and superellipsoids were used as the initial
approximation for superovoids.
The surface fitting error is expressed as the minimum distance between each point of the point
cloud and the optimally fitted surface, also called signed Euclidean distance, SED
, was computed
as:
SED sign || ( x; )
OS xOP = min
xOS
(F)(x)
OP || xOP − xOS ||2= min
xOS
dP S ||2
(14)
and must respect the non-linear equality constraint
1 F( x; λ )
OS *=
(15)
where is a point belonging to the fitted surface and is a point from the point xOS ∈R3 xOP
cloud which can lie inside, outside or on top of the surface; is the sign function; ign(.) s
represents the distance vector between point P of the point cloud and the iterateddP S ∈R3
surface point S, is the implicit surface representation for each of the geometric primitives F
given by (Equations 1-10); and is the vector of geometric parameters characterizing the λ*
optimally fitted surface. Note that (Equation 14) expresses the physical distance between each
point of the reconstructed hip surface to the optimally fitted shape, hence, can be used to measure
the surface fitting error.
4.4 Shape metrics
Several metrics that quantify deviations from the sphere shape were used to characterize hip
joints. Given the close resemblance between synovial ball-and-socket joint morphology and
egg-like shapes [5-7], we adopted metrics found in ornithology [24-26,34]. Interestingly,
zoologists and ornithologists consider that there are three types of avian egg shapes: spherical,
elliptical and conical [35]. Correspondingly, each avian shape presents a main feature: sphericity,
ellipticity (or flattening), and conicity (or asymmetry).
Sphericity quantifies how closely the shape of an object approaches that of a sphere. It can be
measured by comparing the difference between the surface fitting errors of each shape to those of
the sphere (Equation 14). As for ellipticity, it refers to how much a shape deviates from being
spherical as if it resulted from compressing a sphere along a given diameter to form an ellipsoid
of revolution. Regarding conicity, it measures how pointy a shape is, i.e., how much a shape is
axial asymmetric. Although ellipticity and conicity are standard metrics to quantify egg-like
shapes, other parameters can be introduced to attempt better descriptions of hip joint
8
morphology. Following previous work on femoral head morphology [27], we also considered
squareness that measures how close a shape is to a box form. In short, the performed shape
analysis took into account four shape metrics: sphericity, ellipticity, conicity and squareness.
(Table 3) lists the formulas for each shape and associated metric.
Table 3 – Shape metric formulas of the non-spherical shapes. All metrics are strictly less than 1.
Near zero values for ellipticity and conicity correspond to shapes similar to the perfect sphere,
while squareness values to 2.0 are more similar to a sphere. With the exception of the rotational
conchoid, shape dimensions satisfy the inequality expression of .
≥ c a≥b
Note that shape metrics are expressed, by definition, as a ratio between parameters of
two-dimensional curves (e.g., eccentricity of an ellipse). Since we are dealing with
three-dimensional objects, each formula in (Table 3) accounts for shape measurements in the
x
Oy
, x
Oz
, and y
Oz
planes expressed in local coordinates.
4.5 Statistical analyses
The quantitative analyses of the limit of hip sphericity relied on the surface fitting errors that are
quantified by the signed Euclidean distances, i.e., distance of each point in a point cloud to the
optimally fitted surface of each shape model (points laying on the surface have zero valued
distance, points inside the surface have ‘negative distances’ while points outside have positive
valued distance). Surface fitting errors were estimated across the 30 pelvic bones and compared
between the 10 shapes. To compare the surface fitting results of the different shape models, we
conducted two statistical analyses [36,37]. First, we measured descriptive statistics to describe
the main features of data in quantitative terms, e.g., first-order statistics such as mean, standard
deviation, minimum, maximum, and root mean square (RMS) error values. Second, we aimed to
verify if different shapes have an effect on shape morphology through statistical hypothesis
testing to verify which shape fits best. Gender variability was also quantified by comparing
ellipticity, conicity and squareness among several different shapes.
Regarding statistical hypothesis testing, we took surface fitting error as a continuous
measurement variable and shape as a nominal variable whilst we assume, as a null hypothesis,
that different shapes do not affect the surface fitting error, equivalently, as an alternate hypothesis
that different shapes have different averages of surface fitting error. To verify if the surface
fitting errors represent observable differences between the means of the surface fitting errors, it
was necessary to check for normality to decide which type of statistical test is more appropriate
for statistical reasoning. Normality tests indicate that the sample data does not follow a
well-modeled normal distribution. All surface fitting error datasets were evaluated for normality
using the Shapiro–Wilk test that provides sufficient statistical confidence that the population is
far from normally distributed. By computing population kurtosis, k
, we verified that data was
substantially skewed data or flat (k> 0.5). Since the surface fitting error does not follow a normal
distribution, we applied the Kruskal-Wallis test [36]. Pairwise comparisons among the shape
groups was accomplished by selecting two groups at a time and by running a separate
Kruskal-Wallis test for each pair. A statistically significant result was given a p-value
<0.05.
9
5. Results
The initial assessment of the overall goodness-of-fit of the approximated surfaces, performed by
visual examination, suggests that the chosen shape models adjust well to the global anatomy of
the articular surfaces for all data sets of the femoral head (Figure 3) and acetabular cavity (Figure
4). Along with visual inspection, the statistical metrics provided insight on whether the surface
parameters have anatomical meaning and how well adjusted shapes performed in terms of
dispersion and central tendency. In order to improve the general understanding of the discussion,
statistical results for femoral and acetabular cases are presented separately. The full list of
estimated femoral head and acetabular cavity shape parameters are presented in (Supporting
Information: Table 1) and (Supporting Information: Table 2), respectively.
Figure 3 - 3-D view of the optimally fitted surfaces for the femoral head of subject 11. Point
clouds are coloured according to the Euclidean distance between the point and approximated
surface. Surface error color map: inner points are represented in blue; outer points are colored in
red; points close to the surface are represented in a gray scale.
Figure 4 - 3-D view of the optimally fitted surfaces for the acetabular cavity of subject 11. Point
clouds are coloured according to the Euclidean distance between the point and approximated
surface. Surface error color map: inner points are represented in blue; outer points are colored in
red; points close to the surface are represented in a gray scale.
The point clouds in (Figure 3) and (Figure 4) are colored as a function of each point’s Euclidean
distance to the optimally fitted surface. The color code uses three gradients: points located inside
the surface below -1.0 mm are given the color red whose intensity increases with the distance to
the surface; exterior points located above 1.0 mm of the surface are given the color blue
following the same intensity criterion as the interior points; and all points within -1.0 mm and 1.0
mm of distance to the approximated surface are colored in a grayscale, where brighter shades
correspond to smaller distances with white being the zero distance.
5.1 Limit of sphericity of the femoral head
(Table 4) promptly represents how the fitting errors distribute among the different shapes and it is
possible to observe the overall similarity and variability in range values, mean and standard
deviation between all shape models. To better illustrate the distribution of fitting errors between
genders and ages, a heat map representing the individual values of RMS for each subject and
shape is presented in (Figure 5). Sorted by age, the heat map is divided in top and bottom sections
to list male and female subjects, respectively.
Table 4 - Surface fitting errors statistical analysis of the femoral head for each shape model, for
each gender and the whole population present in the study (30 subjects). All metrics are
represented in millimeters (mm). The mean and standard deviation are calculated for the absolute
value of the surface error. Min and Max values are represented based on the minimal signed
Euclidean distances calculated between each point and the optimal fitted shape.
10
Figure 5 – Heat map of the surface fitting errors of the femoral heads for the study population. A
single row contains the RMS fitting errors for a subject (in mm), while each column corresponds
to a different shape.
From (Table 4) and (Figure 5) it is possible to easily discern that spheres provided the worst fit,
whereas egg-like shapes present the lowest fitting errors. Interestingly, there was no statistical
significance between sphere and all the remaining shapes (p≥ 0.05), hence the overall medians of
each shape group are not different for both male and female subjects (Table 5). In its turn,
rotational conchoids, rotational ellipsoids and ellipsoids are statistically different from
superellipsoidal and egg-liked shapes, although the differences were not significant between
superellipsoidal shapes (SE, SEB) or between egg-like shapes (TE, TSEB, O, SO).
Table 5 - Statistical significance of the differences between fitting errors for all shape models for
the femoral head, using paired Kruskal-Wallis tests with statistical significance set at p
< 0.05.
The lower triangular half and upper triangular half correspond to the male and female groups,
respectively.
5.2 Limit of sphericity of the acetabular cavity
Similarly to the femoral case, a quantitative analysis of the differences between shapes and
respective goodness-of-fit to acetabular point clouds was performed. (Table 6) lists first-order
statistics of the surface fitting errors and the RMS of total fitting errors for each shape model
which contribute to understanding of how error values are distributed along each shape. A
heatmap of the RMS for each subject and shape was also generated to visually represent the
distribution of fitting errors between genders and ages (Figure 6).
Table 6 - Surface fitting errors statistical analysis of the acetabular cavity for each shape model,
for each gender and the whole population present in the study (30 subjects). All metrics are
represented in millimeters (mm). The mean and standard deviation are calculated for the absolute
value of the surface error. Min and Max values are represented based on the minimal signed
Euclidean distances calculated between each point and the optimal fitted shape.
Figure 6 – Heat map of the surface fitting errors of the acetabular cavities for the study
population. A single row contains the RMS fitting errors for a subject (in mm), while each
column corresponds to a shape.
A closer look at the results of (Table 6) and (Figure 6) bespeaks the same tendency previously
found in the femoral cases for both male and female groups. Goodness-of-fit improves
progressively for shape primitives which present increasing asphericity, culminating in egg-like
shapes with the best fitting values. By comparing the differences between the surface fitting
errors (Table 4; Table 6), an expected result emerges as the acetabular cavity is indeed better
established as a non-spherical articular surface than the femoral head. A paired Kruskal-Wallis
was also used to classify the significance of the differences between fitting errors of all shape
models. Statistical significance was once again set at p< 0.05. The pairs which demonstrated
significant results are highlighted in (Table 7). Curiously, the acetabular cavity presents a
11
slightly greater amount of shape pairs where the differences between fitting errors are
significantly different. Even so, the difference between fitting errors was still significantly
different between either TE or SO and ellipsoidal shapes (RC, RE, E) and superellpsoidal shapes
(SE, SEB).
Table 7 - Statistical significance of the differences between fitting errors for all shape models for
the acetabular cavity, using paired Kruskal-Wallis tests with statistical significance set at p<
0.05. The lower triangular half and upper triangular half correspond to the male and female
groups, respectively.
5.3 Other shape metrics
By evaluating the different shape metric formulas described in (Table 3), it is possible to address
how hip joint ellipticity, conicity and squareness vary among both genders when comparing each
gender metric for a given shape (Table 8; Table 9). These tabulated results indicate that male and
female hips have very similar shape metrics. Only minute differences indicate that the female
femoral head is slightly more asymmetric and squared than male hips. The same occurs for the
acetabular cavity, with the addition of female hips being slightly more flattened although
notoriously more asymmetric than male acetabular cavities.
Table 8 - Shape metrics measured for the femoral head of male and female subjects. All metrics
are normalized with exception of squareness. Shape position within x
,y
, and zcoordinate space is
represented by t
1,t
2,t
3, respectively, whereas
ϕ
,θ
,ψare the angles of rotation along the same
coordinate system.
Table 9 - Shape metrics measured for the acetabular cavity of male and female subjects. All
metrics are normalized with exception of squareness. Shape position within x
,y
, and zcoordinate
space is represented by t
1,t
2,t
3, respectively, whereas
ϕ
,θ
,ψare the angles of rotation along the
same coordinate system.
In addition, shape parameters (Supporting Information: Table 1; Supporting Information: Table
2) reveal that the calculated ovoidal coefficients are within the boundaries established by Todd
and Smart [26] to describe avian eggs. Moreover, the calculated exponent values of
superellipsoids, Barr’s superellipsoids, Barr’s tapered superellipsoids and superovoids are
extremely close to the quadratic values, despite the the maximum of = 2.25. As for the γ
rotational conchoids, all optimally shapes have a ratio between the largest and smallest shape
parameters lesser or equal to 2, hence, each computed shape is a convex limaçon.
6. Discussion
The human femoral head and acetabular shape are commonly represented as a sphere or
hemisphere, but there have been no extensive quantitative assessments of this assumption in the
literature. The work by MacConaill introduced the idea that the hip joint, along with other
12
spheroidal joints, did not present geometrical features most consistent with a sphere, but with
ovoidal shapes, instead [4-5]. In this work, we evaluated shape variation and tested the limit of
the hip joint sphericity assumption by comparing the largest set of shapes adjusted to 3D
reconstructions of femoral heads and acetabular cavities. Our aim was to contribute to the
ongoing debate and to test the limits and validity of this hypothesis by comparing 10 different
parameterisations. The considered shape primitives present a compact number of geometric
modeling parameters which are intuitive, easily controllable, are able to describe macroscopic
features of the femoral head and acetabular cavity, namely, ellipticity, conicity and squareness.
We also addressed how such shape features vary among both sexes.
In conclusion, we can synthesise that the osseous morphology of the femoral head and acetabular
cavity, of both genders, can be parameterised by superovoids with superior quality than the
simple sphere shape. There exists a clear distinction between spheres and egg-like shapes:
spheres have the worst fitting metrics while superovoids have the lowest surface fitting errors
throughout the entire study population. As for shapes previously considered in hip morphology
literature (e.g., rotational conchoids, rotational ellipsoids and ellipsoids), they lie within the
spherical and oval extremities of the surface fitting error spectrum, as such shapes are more
limited in terms of representing morphological inter- and intra-subject-specific variations. In its
turn, ovoidal shapes present a greater level of generalization, brought in some extent by the
conicity parameters which account for greater shape complexity and individual morphology
variation. Therefore, ovoidal shapes exhibited better fitting results for the study population.
From the comparison between the femoral head surface fitting errors listed in (Table 4) and
(Table 6), it is possible to conclude that the goodness-of-fit of the distinct shape models follows
the same pattern for individual subjects as for the population as a whole. Relatively to the femoral
head, (Table 4) allows the establishment of the asphericity relationships between the different
shape models, ordered according to decreasing RMS surface fitting error:
Male
S > RC > RE > E > SEB > SE > TSEB > TE > O > SO
Female
S > RE > E > TE > SEB > RC > SE > TSEB > O > SO
(16)
Study population
S > RE > E > RC > SEB > TE > SE > TSEB > O > SO
The inequality condition (Equation 16) express a clear division between two sets of surfaces,
namely, spheres and ovoids. In addition, (Equation 16) reveals that ellipticity alone does not lead
to cost-efficient analyses, given that RE, E, SE and SEB shapes presented worse fitting results
than ovoids and superovoids. Although the differences in the statistical measures presented are
not of a high magnitude, the geometric features of the two shapes with highest and lowest fitting
errors are undoubtedly distinct, despite the morphometric changes which transform one into the
other.
Regarding the acetabular cavity, (Table 6) reveals the comparison between the RMS of the
surface fitting errors for the 10 different shapes, which results in the following inequality relation
established after decreasing RMS surface fitting error:
13
Male
S > RC > RE > SE > E > SEB > TSEB > O > TE > SO
Female
S > RC > RE > E > SE > SEB > TE > O > TSEB > SO
(17)
Study population
S > RC > RE > SE > E > SEB > O > TSEB > TE > SO
When cross-checked with the comparison drawn for the femoral case, a shape polarization
becomes very evident (Equation 16 and equation 17): spheres on one end and ovoids on the other
of the surface fitting spectrum, with the remaining shapes occupying in-between positions. It is
also worthwhile to highlight that the RMS values of the surface fitting errors of the femoral head
were lower than the ones observed for the acetabular cavity, which emphasizes the notion that the
femoral head is a more spherical structure than the acetabulum.
In short, the results from surface fitting analyses demonstrate that the sphere is not the most
representative hip joint shape. In fact, the best fit is an egg shape which contains a well balanced
combination of ellipticity, conicity and squareness. The performed shape metric analyses also
reveal how hip anatomy differs between males and females regarding sphericity, ellipticity,
conicity and squareness vary among sexes. An overall comparison of the shape metrics
performed on the study population revealed just minute gender differences (Table 8; Table 9). On
average, the female femoral head is more asymmetric and squared comparatively to the male
counterpart, which in turn is slightly more flat. Whereas, the female acetabular cavity is more
flat, asymmetric and squared when compared to male hips. Therefore, the distribution of
observed shape metrics indicates morphological similarity between genders.
The results indicate that the femoral head and acetabular cavity, in asymptomatic conditions,
approximate better to ovoidal geometries, in detriment to spherical ones, hence, corroborates the
idea introduced by MacConaill [5-6] and reinforces the need to change the global understanding
of the hip joint established within the radiologic orthopaedic community, considering that the
computer-aided tools used currently for orthopaedic pre-surgical planning rely on spherical
geometries for the articular surfaces of the hip joint [2].
Moreover, the shape primitives with the lowest RMS of surface fitting errors for the femoral head
and the acetabular cavity are not especially distant from each in both orders of goodness-of-fit.
This lack of shape model match between the articular surfaces is frequently described in the
orthopaedic community as “incongruity” and it implies a difference in the contact area between
the two surfaces dependent on the applied stress/load on the joint. Lighter or lower loads lead to
limited contact, while heavier or higher loads conduce to an increase in contact area. The
existence of this incongruity generates space between the two articular surfaces, which is thought
to be a way of distributing load and protecting the cartilage from undue stress while giving
synovial fluid access for lubrification and nutrition of the joint. Also, incongruity is commonly
determined by an arched acetabulum and a rounded femoral head [34,39].
An important aspect that rarely appears in the literature, but deserves to be mentioned are the
issues related to systematic error due to 3D reconstruction. Although implicit surface
14
representations allow for infinite resolution and accuracy-controlled point-surface distance
computations (Equations 1-10), the surface fitting errors depend on the reconstructed mesh
obtained by image segmentation and mesh processing. Moreover, CT image data were segmented
semi-automatically, hence, observer-dependence may taint the resulting segmentations. To this
end, [30-42] have performed research on reconstruction reproducibility and quantified geometric
errors associated with 3D reconstruction. In particular, [42] provided the community with reliable
estimates of the systematic error induced by 3D reconstruction from volume data (<0.5 mm).
Such errors are lower than the computed RMS errors. This study also reported that the
distribution of error throughout the articulating surface is locally consistent and varies smoothly.
Towards the contribution of exponentiation to joint morphology, the exponent values of SE, SEB,
TSEB, and SO did not differ greatly from the quadratic surfaces from which they originate.
Given that the upper boundary set for these parameters was 4, to accommodate the approximation
of the articular surfaces to quadrics, the fact that the maximum value observed for the
supraquadric exponents remained this close to its lower boundary leads to the conclusion that the
interval was well set and that a quadratic-to-quartic exponent interval is enough to achieve good
fitting results.
Concerning conicity, because of the more sphere-like appearance of the femoral head, the
asphericity of the optimally adjusted surfaces in these cases is more difficult to identify by naked
eye than for acetabular articular surfaces. The geometric properties which endow asphericity and
higher geometric modeling freedom to the shape models originating from morphing
transformations applied to the sphere are not clearly pronounced, for instance, in the set of
surfaces represented in (Figure 2), even though the non-spherical primitives allow for a better fit
to the femoral head point cloud, as demonstrated by the higher number of grayscale points in
these surfaces’ adjustments (Figure 3).
Unlike the femoral cases, the differences between the geometric features of shape are more easily
distinguishable when acetabular point clouds are adjusted. As the fitting proceeds to more
non-spherical shapes, the approximation of the point clouds improves drastically, especially
when we move closer to the acetabular rim. Although visual inspection of the optimally fitted
surfaces depicted in (Figure 4), particularly in the cases of sphere, rotational ellipsoid, and
rotational conchoid, might suggest that the points located on the outer edge of the acetabulum are,
in fact, exterior to the surface adjusted to them, the colour code used to discriminate points based
on their Euclidean distance to the fitted surface clearly indicates that that is not the case. Given
that points are coloured based on the same criteria described for the femoral case, points closer to
the acetabular rim are, in truth, positioned below the surface, distanced more than 1.0 mm from it.
The illusion that these points are located above the surface arises from the fact that the acetabular
cavity is more planar than the femoral head. Therefore, surfaces with more pronounced
curvatures overlap with the point cloud, as observable in the areas seemingly absent of points
corresponding to the approximation of the sphere, rotational ellipsoid, ellipsoid, superellipsoid,
and rotational conchoid. This higher level of asphericity inherent to the acetabular cavity is
supported by the more straightforwardly identifiable geometric differences between all shape
models. Such differences are particularly notable in ovoidal, superovoidal, and tapered ellipsoidal
shapes.
15
There are a few limitations to the methodology presented in this work whose understanding can
motivate future developments and improvements. Firstly, it is necessary to take into account that
bone-cartilage interface is not clearly delimited in CT images [13], increasing the difficulty in
identifying the true contour of the articular surfaces of the hip joint and extracting the relevant
anatomical and geometrical data which should be used in the surface fitting framework. Even so,
the surface fitting errors indicate that the bone-cartilage boundary of the femoral heads and
acetabular cavities closely resembled the idealized geometric primitives, as the error metrics were
very small (i.e., on the order of 10-1 mm). As shape fitting only considered points from the
bone-cartilage boundary, the shape fitting reflects a pure bony structure, which itself does not
reflect the true articulation since the cartilage thickness is not uniform. By not taking into account
the free surface of the articular surface, we eliminate any confounding effects from the cartilage.
However, to assess the true articular shape, we should rely on MRI since it allows for mapping of
the cartilage geometry, whereas CT does not allow this. We consider this as future work, since it
would be interesting to assess if there is a correlation between cartilaginous surface and bony
surface, namely, if cartilage thickness compensates the lack of bony asphericity or simply follows
the underlying bony shape, and if this hypothetical correlation is verified for both healthy and
unhealthy hips.
Secondly, this work lacks a biomechanical contribution per se
, besides alerting the community
that more representative shapes than the sphere may better describe hip biomechanics as “form
follows function
”. Although more work is required to achieve biomechanical relevance (e..g, hip
joint simulations through Finite Element Analysis, Multibody or Discrete Element Analysis), we
consider that this work does lay ground for further biomechanical research as the reported
morphological findings may serve to inspire new hip prosthetic shapes. New shapes may even
elucidate the effects of hip joint morphology on predictions of cartilage contact mechanics from a
validated, subject-specific Finite Element model of the human hip.
Thirdly, although we did not consider pathological hips, the clinical relevance of our study
consists of introducing new shapes that define morphological parameters from CT scans. Note
that, understanding the subjacent morphological features of a normal, asymptomatic hip joint is
the first step in identifying abnormal and potentially pathological morphologies, such as the ones
characterizing femoroacetabular impingement and hip dysplasia. Our findings are relevant when
compared to radiographic measurements which have been reported do not properly characterize
the fémur and acetabulum, raising concerns on defining hip disorders and anatomy based on
radiographic measurements alone [43-45]. It is possible that clinicians are not only
overdiagnosing and over treating hip conditions but paradoxically missing the diagnosis entirely.
Therefore, we consider that this study opens new research lines for clinical relevance as more
representative shapes propose new metrics that unambiguously characterize femoral head and
acetabular cavity geometries.
Finally, a Statistical Shape Modeling (SSM) approach based on Principal Component Analysis
(PCA) would be necessary to better differentiate anatomical variations in the hip, thus, providing
more useful insights into shape variation across the population. SSM data could be used to
identify more differentiated or even novel shape variations between groups, which could in turn
16
be used to develop more sensitive and specific clinical measurements. Besides comparing
variations between male and female subjects, SSM could accurately describe, reproduce, quantify
variation and compare morphologic differences between asymptomatic and symptomatic bone
shapes. However, additional research of SSM as a clinical tool is required. Although a
PCA-based statistical shape analysis is out of the scope of the presented work, we consider this
topic to be a very interesting future work.
7. Conflicts of interest
The authors have no conflicts of interest to declare.
8. Ethical approval
The data sets used in our study resulted from the approval by the Ethics Research Committee of
the Nova Medical School | Faculdade de Ciências Médicas da Universidade Nova de Lisboa
(CEFCM) under the Project entitled "DEFORMIDADES COXO-FEMURAIS E CONFLITO
FEMUROACETABULAR: contributo epidemiológico, diagnóstico e prognóstico" with reference
nr.61/2014/CEFCM, and also by the University of Utah Institutional Review Board #10983.
9. Acknowledgments
All authors are thankful for the financial support given by Portuguese Foundation for Science and
Technology through national funds with references UID/CEC/50021/2019 and STREACKER
UTAP-EXPL/CA/0065/2017.
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10. Footnotes
1 – https://mrl.sci.utah.edu/software/normal-hip-image-data/
2 – www.itksnap.org/
3 – www.paraview.org/
4 – https://www.blender.org/
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