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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.

10. References

[1] - Vesalius, A. (1998). De humani corporis fabrica (No. 4). Norman Publishing.

[2] - Daniel Simões Lopes, Sara M. Pires, Vasco V. Mascarenhas, Miguel T. Silva, Joaquim A.

Jorge, On a “Columbus’ Egg”: Modeling the shape of asymptomatic, dysplastic and impinged hip

joints, Medical Engineering and Physics , September 2018, Volume 59 , 50 - 55, DOI:

doi.org/10.1016/j.medengphy.2018.07.001

[3] - Rouvière, H., Delmas, A., Götzens García, V., Testut, J. L., Jacob, R., Velayos, J. L. S., &

Testut, O. L. (2005). Anatomía humana: descriptiva, topográfica y funcional (No. 611). Masson.

[4] - Williams, G. M., Chan, E. F., Temple-Wong, M. M., Bae, W. C., Masuda, K., Bugbee, W.

D., & Sah, R. L. (2010). Shape, loading, and motion in the bioengineering design, fabrication,

and testing of personalized synovial joints. Journal of biomechanics, 43(1), 156-165.

17

[5] - MacConaill, M. A. (1966). The geometry and algebra of articular kinematics. Biomed. Eng,

1, 205-212.

[6] - MacConaill, M. A. (1973). A structuro-functional classification of synovial articular units.

Irish journal of medical science, 142(1), 19-26.

[7] - Standring, S. (Ed.). (2015). Gray's anatomy: the anatomical basis of clinical practice.

Elsevier Health Sciences.

[8] - Menschik, F. (1997). The hip joint as a conchoid shape. Journal of biomechanics, 30(9),

971-973.

[9] - Kang, M. (2004). Hip joint center location by fitting conchoid shape to the acetabular rim

region of MR images. In Engineering in Medicine and Biology Society, 2004. IEMBS'04. 26th

Annual International Conference of the IEEE (Vol. 2, pp. 4477-4480). IEEE.

[10] - Kang, M. J., Sadri, H., Stern, R., Magnenat-Thalmann, N., Hoffmeyer, P., & Ji, H. S.

(2010). Determining the location of hip joint centre: application of a conchoid's shape to the

acetabular cartilage surface of magnetic resonance images. Computer methods in biomechanics

and biomedical engineering,14(1), 65-71.

[11] - Anderson, A. E., Ellis, B. J., Maas, S. A., & Weiss, J. A. (2010). Effects of idealized joint

geometry on finite element predictions of cartilage contact stresses in the hip. Journal of

biomechanics, 43(7), 1351-1357.

[12] - Berryman F., Pynsent P., and McBryde C. (2014), A semi-automated method for

measuring femoral shape to derive version and its comparison with existing methods, Int. J.

Numer. Meth. Biomed. Engng., 30, pages 1314–1325, doi: 10.1002/cnm.2659

[13] - Xi, J., Hu, X., & Jin, Y. (2003). Shape analysis and parameterized modeling of hip joint.

Journal of Computing and Information Science in Engineering, 3(3), 260-265.

[14] - Cerveri, P., Manzotti, A., & Baroni, G. (2014). Patient-specific acetabular shape

modelling: comparison among sphere, ellipsoid and conchoid parameterisations. Computer

methods in biomechanics and biomedical engineering, 17(5), 560-567.

[15] - K.Subburaj, B. Ravi, Manish Agarwal, Computer-aided methods for assessing lower limb

deformities in orthopaedic surgery planning, Computerized Medical Imaging and Graphics,

Volume 34, Issue 4, June 2010, Pages 277-288, DOI:

doi.org/10.1016/j.compmedimag.2009.11.003

[16] - Jiang, H. B., Liu, H. T., Han, S. Y., & Fen, L. I. U. (2010). Biomechanics Characteristics

of New Type Artificial Hip Joint. Advances in Natural Science,3(2), 258-262.

18

[17] - Liu, B., Hua, S., Zhang, H., Liu, Z., Zhao, X., Zhang, B., & Yue, Z. (2014). A personalized

ellipsoid modeling method and matching error analysis for the artificial femoral head design.

Computer-Aided Design, 56, 88-103.

[18] - Liu, B., Zhang, H., Hua, S., Jiang, Q., Huang, R., Liu, W., ... & Yue, Z. (2015). An

automatic segmentation system of acetabulum in sequential CT images for the personalized

artificial femoral head design. Computer methods and programs in biomedicine, 127, 318-335.

[19] - Gu, D., Chen, Y., Dai, K., Zhang, S., & Yuan, J. (2008). The shape of the acetabular

cartilage surface: A geometric morphometric study using three-dimensional scanning. Medical

engineering & physics, 30(8), 1024-1031.

[20] - Gu, D. Y., Dai, K. R., Hu, F., & Chen, Y. Z. (2010). The shape of the acetabular cartilage

surface and its role in hip joint contact stress. In 2010 Annual International Conference of the

IEEE Engineering in Medicine and Biology (pp. 3934-3937). IEEE.

[21] - Gu, D. Y., Hu, F., Wei, J. H., Dai, K. R., & Chen, Y. Z. (2011). Contributions of

non-spherical hip joint cartilage surface to hip joint contact stress. In 2011 Annual International

Conference of the IEEE Engineering in Medicine and Biology Society (pp. 8166-8169). IEEE.

[22] - Cerveri, P., Marchente, M., Chemello, C., Confalonieri, N., Manzotti, A., & Baroni, G.

(2011). Advanced computational framework for the automatic analysis of the acetabular

morphology from the pelvic bone surface for hip arthroplasty applications. Annals of biomedical

engineering, 39(11), 2791-2806.

[23] - Wu, H. H., Wang, D., Ma, A. B., & Gu, D. Y. (2016). Hip joint geometry effects on

cartilage contact stresses during a gait cycle. In IEEE 38th Annual International Conference of

the Engineering in Medicine and Biology Society (EMBC), 2016 (pp. 6038-6041).

[24] - Carter, T. C. (1968). The hen's egg: a mathematical model with three parameters. British

Poultry Science, 9(2), 165-171.

[25] - Paganelli, C. V., Olszowka, A., & Ar, A. (1974). The avian egg: surface area, volume, and

density. The condor, 76(3), 319-325.

[26] - Todd, P. H., & Smart, I. H. (1984). The shape of birds' eggs. Journal of theoretical biology,

106(2), 239-243.

[27] - Lopes, D. S., Neptune, R. R., Gonçalves, A. A., Ambrósio, J. A., & Silva, M. T. (2015).

Shape Analysis of the Femoral Head: A Comparative Study Between Spherical,(Super)

Ellipsoidal, and (Super) Ovoidal Shapes. Journal of Biomechanical Engineering, 137(11),

114504.

[28] - Bazaraa, M.S., Sherali, H.D., Shetty. C.M. (1993). Nonlinear Programming: Theory and

Algorithms, Wiley, Hoboken, New Jersey.

19

[29] - Ahn, S.J. (2004), Least Squares Orthogonal Distance Fitting of Curves and Surfaces in

Space, Series: Lecture Notes in Computer Science, Vol. 3151, Springer, USA.

[30] - Barr, A. H. (1981). Superquadrics and angle-preserving transformations. IEEE Computer

graphics and Applications, 1(1), 11-23.

[31] - Harris, M. D., Anderson, A. E., Henak, C. R., Ellis, B. J., Peters, C. L., and Weiss, J. A.,

2012, “Finite Element Prediction of Cartilage Contact Stresses in Normal Human Hips,” J.

Orthop. Res., 30(7), pp. 1133–1139.

[32] - Yushkevich, P. A., Piven, J., Hazlett, H. C., Smith, R. G., Ho, S., Gee, J. C., & Gerig, G.

(2006). User-guided 3D active contour segmentation of anatomical structures: significantly

improved efficiency and reliability. Neuroimage, 31(3), 1116-1128.

[33] - Lorensen, W. E., & Cline, H. E. (1987). Marching cubes: A high resolution 3D surface

construction algorithm. In ACM siggraph computer graphics (Vol. 21, No. 4, pp. 163-169).

ACM.

[34] - Afoke, N. Y., Byers, P. D., & Hutton, W. C. (1980). The incongruous hip joint. A casting

study. Bone & Joint Journal, 62(4), 511-514.

[35] – Mary Caswell Stoddard, Ee Hou Yong, Derya Akkaynak, Catherine Sheard, Joseph A.

Tobias, L. Mahadevan, Avian egg shape: Form, function, and evolution, Science, 23 Jun 2017,

pp: 1249-1254

[36] - Ghasemi, Asghar and Saleh Zahediasl. “Normality tests for statistical analysis: a guide for

non-statisticians” International journal of endocrinology and metabolism vol. 10,2 (2012): 486-9.

[37] - Marusteri, Marius and Vladimir Bacarea. "Comparing groups for statistical differences:

how to choose the right statistical test?." Biochemia Medica, vol. 20, no. 1, 2010, pp. 15-32.

https://hrcak.srce.hr/47847. Accessed 4 Mar. 2019.

[38] - Azmeri Khan and Glen D. Rayner, “Robustness to non-normality of common tests for the

many-sample location problem,” Journal of Applied Mathematics and Decision Sciences, vol. 7,

no. 4, pp. 187-206, 2003. https://doi.org/10.1155/S1173912603000178.

[39] - Cooper R. J., Mengoni M., Groves D., Williams S., Bankes M. J. K., Robinson P., and

Jones A. C. (2017) Three-dimensional assessment of impingement risk in geometrically

parameterised hips compared with clinical measures, Int. J. Numer. Meth. Biomed. Engng., doi:

10.1002/cnm.2867.

[40] - Anderson AE, Ellis BJ, Peters CL, Weiss JA. Cartilage thickness: factors influencing

multidetector CT measurements in a phantom study. Radiology. 2008; 246:133–141. [PubMed:

18096534]

20

[41] - Allen BC, Peters CL, Brown NA, Anderson AE. Acetabular cartilage thickness: accuracy

of three-dimensional reconstructions from multidetector CT arthrograms in a cadaver study.

Radiology. 2010; 255:544–552. [PubMed: 20413764]

[42] - Harris MD, Anderson AE, Henak CR, Ellis BJ, Peters CL, Weiss JA. Finite Element

Prediction of Cartilage Contact Stresses in Normal Human Hips. Journal of Orthopaedic

Research. 2012;30(7):1133-1139. doi:10.1002/jor.22040.

[43] - Clohisy JC, Carlisle JC, Trousdale R, et al (2008) Radiographic Evaluation of the Hip has

Limited Reliability. Clin Orthop Relat Res 467:666–675. doi: 10.1007/s11999-008-0626-4

[44] - Jamali AA, Mak W, Wang P, et al (2013) What Is Normal Femoral Head/Neck Anatomy?

An Analysis of Radial CT Reconstructions in Adolescents. Clin Orthop Relat Res

471:3581–3587. doi: 10.1007/s11999-013-3166-5

[45] - Chloe E. Haldane, Seper Ekhtiari, Darren de SA, Nicole Simunovic. Olufemi R. Ayeni,

(2017) Preoperative physical examination and imaging of femoroacetabular impingement prior to

hip arthroscopy—a systematic review, Journal of Hip Preservation Surgery Vol. 0, No. 0, pp.

1–13, doi: 10.1093/jhps/hnx020

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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