Calculating curvature through gradient descent and nonlinear regression: A novel mathematical approach to digital anatomical morphometry

Abstract

Background

Angular projection measurements have long been an established approach in anatomical morphometry. However, many described projection angles are in reference to inherently curved structures, often oversimplifying their topologies.

Aims

The objective of this study is to develop a quick, quantitative method for determining structural curvature from digital images. We aim to utilize readily-available software and statistical methods to extrapolate curvature from images and compare this new method to established angular measurements.

Methods

Projection angulation and curvature was modeled on and assessed by the acromia of 50 dry scapulae. Digital images were taken at a known scale, perpendicular to the acromion, and then processed with ImageJ software. Angles were measured by the angle tool and for curvature, seven markers were placed along the external and internal margins of the acromion. Utilizing Excel's Solver function, the coordinate points were passed through a rotation matrix and optimized for second order regression. Solver was instructed to minimize the sum of squared estimated error between our measured and calculated coordinate values by manipulating the angle of point rotation and regression equation coefficients.

Results

Significant differences were found between external, internal, and midline acromion measurements in both angles and curvatures. External angle = 80.8 (14.2)°; internal angle = 130.3 (13.6)°; midline angle = 105.6 (10.5)°; [reported as mean (SD)]. External curvature = 0.055 (0.015) mm−1; internal curvature = 0.035 (0.025) mm−1; midline curvature = 0.046 (0.017) mm−1; [reported as median (IQR)].

Conclusions

Solver allows for researchers and clinicians to quickly characterize morphometric courses and properties of a given structure. Paired with other scalar measurements, curvature can complete the picture of an anatomical structure's pattern.

Full text

Informatics in Medicine Unlocked 43 (2023) 101383
Available online 25 October 2023
2352-9148/© 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-
nc-nd/4.0/).
Calculating curvature through gradient descent and nonlinear regression: A
novel mathematical approach to digital anatomical morphometry
Carl V.L. Olson
*
, David Kachlík, Azzat Al-Redouan
Department of Anatomy, Second Faculty of Medicine, Charles University, V Úvalu 84, Prague 150 06, Czech Republic
ARTICLE INFO
Keywords:
Curvature
Angle
Morphometry
Anatomical structures
Scapula
Acromion
ABSTRACT
Background: Angular projection measurements have long been an established approach in anatomical
morphometry. However, many described projection angles are in reference to inherently curved structures, often
oversimplifying their topologies.
Aims: The objective of this study is to develop a quick, quantitative method for determining structural curvature
from digital images. We aim to utilize readily-available software and statistical methods to extrapolate curvature
from images and compare this new method to established angular measurements.
Methods: Projection angulation and curvature was modeled on and assessed by the acromia of 50 dry scapulae.
Digital images were taken at a known scale, perpendicular to the acromion, and then processed with ImageJ
software. Angles were measured by the angle tool and for curvature, seven markers were placed along the
external and internal margins of the acromion. Utilizing Excel’s Solver function, the coordinate points were
passed through a rotation matrix and optimized for second order regression. Solver was instructed to minimize
the sum of squared estimated error between our measured and calculated coordinate values by manipulating the
angle of point rotation and regression equation coefcients.
Results: Signicant differences were found between external, internal, and midline acromion measurements in
both angles and curvatures. External angle =80.8 (14.2)◦; internal angle =130.3 (13.6)◦; midline angle =105.6
(10.5)◦; [reported as mean (SD)]. External curvature =0.055 (0.015) mm
−1
; internal curvature =0.035 (0.025)
mm
−1
; midline curvature =0.046 (0.017) mm
−1
; [reported as median (IQR)].
Conclusions: Solver allows for researchers and clinicians to quickly characterize morphometric courses and
properties of a given structure. Paired with other scalar measurements, curvature can complete the picture of an
anatomical structure’s pattern.
1. Introduction
Anatomical structures are morphologically complex in terms of their
shapes, patterns, and geometric behaviors. Solid tissues, such as bones,
are composed of topological congurations that do not follow any
classical geometric shapes, while soft tissues are subjected to dynamic
shape changes due to biomechanical force exertions. These biological
dynamics result in osteological reconguration inuenced by tendinous
mechanical stress, which has shaped bones to adapt overly irregular
margins and projections [1]. Characterizing these complex morphol-
ogies of bones in correlation to their soft tissue surroundings, as well as
post-injury remodeling, is of importance in medical practice [2–6].
1.1. Background
Angular projection measurements of bone processes are a common
approach in anatomical morphometry [2,7,8]. However, many
described projection angles are in reference to inherently curved struc-
tures, often oversimplifying their topologies. The scapula is one of the
most commonly analyzed bones exhibiting such inherently curved
processes. This study focused in particular on the acromion as the
candidate structure for morphometric experimentation.
1.1.1. Clinical anatomy of acromion
The acromion projects from the spine of the scapula dorsally,
inclining upward, taking a curved direction as it proceeds laterally.
Despite the variations in the shape of the acromion, it takes a sharp
* Corresponding author.
E-mail address: carl.olson@lfmotol.cuni.cz (C.V.L. Olson).
Contents lists available at ScienceDirect
Informatics in Medicine Unlocked
journal homepage: www.elsevier.com/locate/imu
https://doi.org/10.1016/j.imu.2023.101383
Received 6 September 2023; Received in revised form 16 October 2023; Accepted 17 October 2023

Informatics in Medicine Unlocked 43 (2023) 101383
2
curve at a semi-angular point, followed by a smoother curving pattern
continuation [9] (Fig. 1). The acromion is commonly assessed by its
angles and this has been approached at differing sides [2,8]. One of the
methods of measuring the acromion angle is illustrated in Fig. 1a.
However, one single-angle measurement does not reect the acromion’s
curvature and multi-angle measurements do not sum to a single curva-
ture [10,11]. The internal curvature of the acromion governs the sub-
acromial space. The subacromial space broadly has two important
clinical applications in orthopedics. First, it determines the space ca-
pacity to accommodate the passing soft tissues and this accommodation
correlates to impingement syndrome, according to Asal and S
¸ahan
(2018) [7] and others [12–16]. Second, it is the site where surgeons
navigate their arthroscopic instruments in certain surgical operations
[12,17]. On the other hand, the external curvature of the acromion is
rather supercial and adjacent to the skin surface, making it relatively
prone to trauma and is consequently a site for surgical screws and x-
ation materials placement [18–21]. Measuring the curvature of such a
structure allows for a more accurate description of the structure’s course
and behavior in a given plane that would facilitate the aforementioned
clinical applications.
1.1.2. Principle of measuring anatomical curvatures
Generally speaking, two routinely used methods to measure
anatomical angles in basic anatomical research and some basic clinical
practices are the angular unit and the best-t radius.
1.1.2.1. Angular unit. It is a magnitude, mapping the rotation array
between two lines protracted around a vertex. This approach ts well
with geometric angles between two linear margins and measurements
via this method serve well in assessing the range of joint motions [22,
23]. However, when attempting to measure an anatomical curvature, an
observer may align the rays by placing the two end points on selected
landmarks with the vertex not in actual contact with a landmark site.
Conversely, if the observer chooses to line up the vertex on an
anatomical landmark, the two rays may not actually align with the
anatomical margins (Fig. 1a).
1.1.2.2. Best-t radius. It is determining the radius of the best circle to
t at selected landmark points along the surface of a given curvature
[24]. This approach ts perfectly spherical and evenly concave or
convex surfaces. It serves well in ophthalmology, for example, in
monitoring keratoconus progression [25] and in optimizing a comfort
lens t by matching the concave curvature of a lens with the convex
curvature of the cornea [26,27]. However, that is not the case in the
majority of anatomical structures in which the curves are irregular
(Fig. 1) where a circle tting on one landmark will most likely not t on
another landmark along the same curve. Such an obstacle can be
encountered in snapping scapula, which is a mismatch between the
curved surfaces of the scapula sliding on the convexity of the posterior
thoracic wall [28].
1.2. Objectives
The objective of this study is to develop a quick, quantitative method
for determining structural curvature from digital images with more
precision. Hereby, we provide a novel method experimented on the
acromion. This will aid in the morphological assessments in diagnostics,
as well as in surgical instrumentation and approaches.
2. Materials and methods
2.1. Subject selection
In order to assess acromion angulation and curvature, a digital li-
brary of 50 dry scapulae (25 pairs) was constructed for image analysis
[xage =55.8 (18.2) years; xheight =157 (9) cm; 11 males, 14 females;
reported as mean (SD)]. The scapulae were selected at random without
bias towards age, sex, or pathology from the collection of Charles Uni-
versity and the Czech National Museum. Each scapula was individually
photographed with a focus on the acromion and photos were taken
perpendicular to the plane of anterior projection with a tape measure for
scaling. In our case, the plane of anterior projection was dened as the
horizontal surface formed by the surface of the acromion (Fig. 1a). Using
a level and a carpenter’s square placed along this plane, we were able to
position our camera lens perpendicular to the acromion. Subsequently,
this method was tested on retrospective radiographs composed of 15
lateral projection X-rays for in vivo assessment. All photos were analyzed
using FIJI (ImageJ) software (version 2.1.0/1.53c) [29].
Access to specimens was approved for research and education pur-
poses by the Institutional Review Board (IRB)—The Ethics Committee of
the University Hospital Motol and the Second Faculty of Medicine,
Charles University, Prague, Czech Republic [reference ID number: EK-
353/19].
2.2. Data acquisition with Fiji
Within the FIJI interface, each image of the acromion was scaled by a
10 mm straight line measurement of the in-photo tape measure for
distance-dependent data to be reported as pixels/mm. First, angle
measurements were taken with the angle tool, setting the end of ray 1 to
the proximal acromion inection along the base of the acromion to
where it meets the isthmus of the spine of the scapula [9] and setting the
end of ray 2 to the distal end of the acromion (Fig. 1a). The angle vertex
was then moved to nd the best-t angle, which was determined as
where our angle rays aligned with the most marginal overlap with the
proximal and distal projections. We allowed the best-t angle to be both
within or outside the margins of the acromion itself. Angle data was
measured and reported as degrees (◦).
Next, using the multi-point tool, seven distinct points were marked
along both the external and internal curvatures of the acromion from
proximal to distal ends (Fig. 1b). Once measured, each marked point
corresponded to a pixel coordinate in the Cartesian plane in millimeters,
(x,y), with an origin at the bottom left-hand corner of the taken image,
Fig. 1. Morphological measurements of the acromion. (a) External angle. (b) 7-point sampling for external curvature. (c) 7-point sampling on X-ray image. Legend:
① surface of acromion; ② base of acromion; ③ distal end of acromion; ④ spine of scapula.
C.V.L. Olson et al.

Informatics in Medicine Unlocked 43 (2023) 101383
3
(0,0). These points were then transferred into a spreadsheet for regres-
sion and curve tting.
2.3. Model equations
The transferred (x,y) points were inputted as a 2 ✕ 7 matrix, A
p
:
Ap=[x1
y1
…
…
x7
y7](1)
where each pair [xn
yn]corresponds to the (x,y) pairs measured in FIJI.
Since there is variability between each sampled scapula and each im-
age’s rotational orientation could potentially limit and interfere with the
functional calculations for curve tting, matrix A
p
was multiplied by a
rotation matrix, R:
RAp =[cos θ−sin θ
sin θcos θ]•[x1
y1
…
…
x7
y7](2)
whose output was a rotated 2 ✕ 7 matrix.
With θ initially set to 0, the graph output of our data modeled the
curvature of the sampled acromion in the same rotational orientation as
the original photo. In order to asses curvature, nonlinear quadratic
regression was performed using the equation:
f(x) = Ax2+Bx +C(3)
where x is the measured x value of each pixel coordinate and f(x)is a
calculated y value for each pixel coordinate. We selected a parabola as
our regression model since it is an even-order polynomial and therefore
would yield one true vertex, from which we could calculate curvature
without the ambiguity of dening separate local and global maxima. The
constants A, B, and C were initially set to 0. The x outputs from the
rotated matrix R
Ap
, [x1(rot.)
y1(rot.)
…
…
x7(rot.)
y7(rot.)], were set as inputs for the
regression Equation (3) to initially yield seven points at (0,0).
For assessing our function’s t between our measured y points (Eq.
(1)), yi, and calculated y points (Eq. (3)), 
y, R2 was measured by the
equation:
R2=1−SSE
SST (4)
which can be expanded as:
R2=1−∑7
i=1(yi−yi)2
∑7
i=1(yi−y)2(5)
Here, SSE is the sum of squared estimated error, SST is the sum of
squared total error, yi is the y value of our rotated i-th sampled point, 
yi
is the corresponding predicted y value from our model equation for the
rotated i-th sampled point, and y is the recorded y average from our
rotated samples; y was calculated by taking the mean of our rotated y
values, 1
7∑7
i=1yi(rot.).
2.4. Rotation and nonlinear regression with excel
Utilizing Microsoft Excel’s Solver Add-In [2016 MSO, version
16.0.14026.20270, 64-bit], Generalized Reduced Gradient (GRG)
Nonlinear Regression was selected and Solver was instructed to mini-
mize the SSE between the rotated measured y values and the expected
model y values by manipulating the values of A, B, C, and θ. Solver was
set to use central derivatives and Multistart (pop. size =10
4
, random
seed =0) to maximize the chances of nding the global minimum of the
function’s SSE. Constraint precision was set to 10
−6
and Multistart
convergence was set to 10
−9
. The resulting rotations in radians and
equations of the quadratic functions of each acromion measurement, as
well as their ts, R2 s (Eqs. (4) and (5)), were recorded for further
analysis. From these calculations, external acromion curvature R2 =
0.99 (0.01) and internal acromion curvature R2 =0.98 (0.04) [reported
as mean (SD)].
2.5. Determining curvature
In the special case where the graph of y=f(x)is a twice differen-
tiable planar parametrized curve, the formula for signed curvature can
be written as:
K=y
″
(1+y
′
2)3
2
(6)
where y
′
and y
″
are the rst and second derivatives of y, respectively
[30], and f(x)is the output of Eq. (3). Therefore, the signed curvature of
our outputted parabola becomes:
K(x) = 2A
(1+ (2Ax +B)2)3
2
(7)
To specify a single point of internal and external curvatures of each
acromion, as well as for inter-scapula comparisons, we chose to report
absolute maximal curvature, that is, curvature at the vertex of our
parabola, Kmax =K(x)vertex. Substituting the vertex formula xvertex = − B
2A
into Equation (7), the nal form the curvature equation becomes:
Kmax = |2A|(8)
where A is the constant from the now-Solver-optimized Eq. (3) which
yielded the lowest SSE. This value, Kmax, corresponds to 1
radius of the best-
t circle at the vertex of the parabola for each given acromion mea-
surement, reported as mm
−1
, and a nal output graph was generated for
each scapula (Fig. 2).
2.6. Statistical analysis
All statistical analyses were performed using GraphPad Prism sta-
tistical software package [version 9.1.2.226 for Windows, GraphPad
Software, San Diego, California USA, www.graphpad.com]. Pvalues <
0.05 were considered signicant in all tests. Post hoc power analyses
determined all groups were of adequate sample size; power >90%.
3. Results and discussion
3.1. Structure measurements are margin specic
Distributions and measured values for both acromion angulation and
curvature are presented in Fig. 3, with results listed in Table 1. With
Fig. 2. Solver output graph from GRG nonlinear curve tting.
C.V.L. Olson et al.

Informatics in Medicine Unlocked 43 (2023) 101383
4
respect to the acromion, there are three possible measurements which
can be taken from 2D projections: external margin, internal margin, and
midline. The current state of literature is not specic with regards to
which margins are being reported, but for most analog and digital
measurements, some midline approach is used [31]. However, esti-
mating a true midline of a skewed structure can be difcult and incon-
sistent [19,32]. Therefore, both the external and internal margins were
measured by both angulation and curvature methods and a simple
average between the two for each scapula was taken to determine the
midline “margin” (mean).
The goal in this study was to develop a quick, quantitative method
for determining structural curvature from digital images. Then, this new
method of curve tting was compared to the existing standard of simply
taking an angular measurement. The rst obstacle in this comparison
was determining which margin of the acromion to measure, as no set
standard across all studies currently exists.
A one-way ANOVA was performed to compare the effect of measured
acromion margin on angulation and revealed that there was a statisti-
cally signicant difference in angulation between at least two groups [F
(2, 147) =185.3, p <0.0001]. Tukey’s multiple comparisons test found
that the mean values of angulation were signicantly different between
all three groups (external vs. mean [adjusted p <0.0001, 95% CI of diff.
= − 30.84, −18.66]; external vs. internal [adjusted p <0.0001, 95% CI
of diff. = − 55.58, −43.41]; mean vs. internal [adjusted p <0.0001, 95%
CI of diff. = − 30.84, −18.66]). Normality of residuals was determined
by Shapiro–Wilk method [W =0.99, p =0.41].
Subsequently, a Kruskal–Wallis H-test was performed to compare the
effect of measured acromion margin on curvature and revealed that
there also was a statistically signicant difference in curvature between
at least two groups [H (2) =37.28, p <0.0001]. Dunn’s multiple
comparisons test found that the median values of curvature were
signicantly different between all three groups (external vs. mean
[adjusted p =0.0016, mean rank diff. =30.13]; external vs. internal
[adjusted p < 0.0001, mean rank diff. =52.88]; mean vs. internal
[adjusted p =0.027, mean rank diff. =22.75]).
The signicant differences between external, internal, and mean
measurements across both angulation and curvature (Fig. 3) indicate
that this lack of specicity may be masking the true reported values of
these measurements in the published literature [2,6,12,32,33]. And
while the external and internal margins may vary, the midline mea-
surement between these two will also be different and can therefore not
be substituted in place of the other margins.
3.2. Curvature is correlated with angulation
Based on the results of the distribution tests (Table 1), curvature
measurements were log transformed for simple linear regression anal-
ysis against angulations (Table 2). The data were then transformed back
into the normal space and plotted with the nonlinear trendlines calcu-
lated in semi-log space to preserve the original units (Fig. 4). All three
margins showed signicant non-zero slopes and low adjusted co-
efcients of determination (adjusted R2s), indicating that while specic
curvatures are not accurate predictors of angulation, curvature does
have a signicant effect on angulation.
From our regressions in Fig. 4, we can infer that a relationship exists
between angular projection and curvature in a given plane; as the cur-
vature of a structure increases, its angle decreases. And while a struc-
ture’s measured curvature cannot accurately predict its angular
projection, by demonstrating the relationship between these two
morphometric parameters, we hope to begin a discussion about
continuously changing shapes, rather than singular, xed projections for
other structures.
3.3. Reproducibility and replicability
The selection of the optimal number of sample points for each cur-
vature measurement was determined by taking the calculated curvature
and R2 from 3 to 20 points, three corresponding to the number of points
for angle measurements. Calculated curvature and model R2 were
plotted against the number of samples (Fig. 5). The 7-sample measure-
ment was determined optimal as this was the lowest number of points
which was within the SEM of curvature from all sample variants at the
highest R2. Seven also has the advantage of being an odd number,
allowing the observer to gauge approximately the apex of the arc being
Fig. 3. Violin plots of morphometric measurements of the acromion. (a)
External, mean, and internal acromion angulations. (b) External, mean, and
internal acromion curvatures. Dashed lines denote the median; dotted lines
denote the rst and third quartiles; signicance stars report post hoc multiple
comparisons test results. Legend: *p <0.05, **p <0.01, ****p <0.0001.
Table 1
Morphometric measurements of the acromion and their distributions.
Structure Measurement
a
Normality test
b
Lognormality test
b
Distribution
External angle 80.8 (14.2) W =0.98, p =0.40 W =0.97, p =0.49 normal
External curvature 0.055 (0.015) W =0.98, p =0.43 W =0.97, p =0.96 lognormal
Internal angle 130.3 (13.6) W =0.98, p =0.67 W =0.98, p =0.70 normal
Internal curvature 0.035 (0.025) W =0.73, p <0.0001 (****) W =0.97, p =0.25 lognormal
Mean angle 105.6 (10.5) W =0.98, p =0.65 W =0.98, p =0.43 normal
Mean curvature 0.046 (0.017) W =0.83, p <0.0001 (****) W =0.96, p =0.073 lognormal
a
Values reported as mean (SD)◦for normally distributed data; median (IQR) mm
−1
for lognormally distributed data; n =50.
b
Shapiro–Wilk method.
Table 2
Summary of log transformed regression results.
Structure Slope [95% CI]
(◦/Log [mm
−1
])
Normality of
residuals
a
Non-zero slope Adjusted
R
2
External
acromion
−85.01 [-134.9,
−35.10]
W =0.98, p =
0.67 (ns)
F (1, 48) =12,
p =0.0013 (*)
0.20
Mean
acromion
−52.14 [-71.85,
−32.43]
W =0.97, p =
0.19 (ns)
F (1, 48) =28,
p <0.0001 (*)
0.37
Internal
acromion
−44.23 [-56.82,
−31.64]
W =0.98, p =
0.50 (ns)
F (1, 48) =50,
p <0.0001 (*)
0.51
a
Shapiro–Wilk method.
C.V.L. Olson et al.

Informatics in Medicine Unlocked 43 (2023) 101383
5
outlined. Similar studies which have calculated curvature based on line
intersections have opted for similar odd-number values, such as ve
[34].
To test the intra-rater reliability, one observer took two separate
measurements for all six parameters on 10 random scapulae from the
overall set. For assessing inter-rater reliability, two observers repeated
the intra-rater measurements. The results of both groups of tests were
visualized as Bland–Altman plots in Figs. 6 and 7 with the results listed
in Table 3. For log-distributed data, the limits of agreement were
calculated as functions of the bias [35].
The replicability and reproducibility of both angular and curvature
measurements are similar. Table 3 shows that a single, experienced
observer can accurately replicate their measurements, both with the
established angle method and with the new curvature one. This dem-
onstrates that the potential level of variability and sensitivity to input
starting conditions of our model can be equated to the ambiguity of
taking repeated straight-line measurements. However, once introducing
a second observer, the variability in the data increased. This variability
was assessed by taking a one-sample t-test between the calculated biases
of each assessment and an ideal bias of 0 (Table 3). Both the internal
angle and internal curvature had signicant deviations between the two
observers, with a greater difference detected in angle measurements.
Thus, these results have two interpretations. First, the internal
margin of the acromion has a greater chance for ambiguity and uncer-
tainty between two observers attempting to replicate the same mea-
surements. Second, due to the perceived curved nature of the internal
acromial margin, this level of uncertainty is greater when attempting to
t a straight-line measurement to the structure. Combined with the
complete data set results (Fig. 3), not only are these measurements site-
specic, but the internal marginal measurements themselves may vary
each time the data is replicated, adding to the overall uncertainty in the
published literature.
3.4. Limitations
One encountered limitation was the problem of scaling and depth of
eld distortion in taking accurate images to analyze. We set out the scale
using a measuring tape on the same plane as the acromion (Fig. 1), but as
Fig. 4. Semi-log regression of acromion angulations versus curvatures on the
original scale. Legend: Dotted lines indicate the 95% CI for the lines.
Fig. 5. Acromion curvature and model R
2 versus number of sampled points.
Legend: Dashed line corresponds to the mean curvature across the range of
3–20 samples taken from the same image; gray band indicates standard error of
the mean [Kmax =0.059 (0.0013) mm
−1
; reported as mean (SEM)].
Fig. 6. Bland–Altman plots for repeated acromion measurements by one observer. (a) External acromion angle. (b) Internal acromion angle. (c) Mean acromion
angle. (d–f) Log transformed data plotted on the original scale. (d) External acromion curvature. (e) Internal acromion curvature. (f) Mean acromion curvature.
Legend: Solid line corresponds to the bias, with shading indicating the 95% CI. Dashed line at 0 is the line of identity. Lower and upper dotted lines indicate the 95%
limits of agreement.
C.V.L. Olson et al.

Informatics in Medicine Unlocked 43 (2023) 101383
6
this structure projects through 3D space, a perfectly at representation is
not possible across the entire photograph. However, this distortion was
able to be combated by simply aiming the focus on the few centimeters
of interest, ignoring any depth of eld effects on the periphery. Also, the
simple 2D analysis was recognized to be not optimal. Anatomical
structures are inherently three-dimensional objects and the acromion
does not project into just one plane. Our aim was therefore to rst
develop a robust method for calculating curvature just in the 2D space
for easier translation to other digital anatomical photographs and X-
rays.
3.5. Future directions
In the future, all of these concerns will be addressed by expanding
both the methodology and the model. The plan is to reevaluate 2D
curvature analysis using a telecentric lens, which will almost entirely
remove image distortion. In addition, the rotation matrix used in the
Solver algorithm can be expanded into three dimensions, allowing us to
nd optimal curve paths through this method on CT reconstructions.
Lastly, the series of equations used for the regression function in Solver
can be expanded upon to nd other nonlinear ts. One avenue of
exploration is the idea of modeling spirals as a series of piecewise
functions [36], which would have the advantage of allowing curve
overlap, and could be used to measure the projection curvature of co-
chlea in both two and three dimensions.
3.6. Other potential clinical application
In addition to the aforementioned clinical applications in (1.1.2.),
calculating curvature as an index value would optimize a wide range of
clinical applications.
The Latarjet procedure, by which the coracoid process of the scapula
is relocated in cases of shoulder instability, received attention lately in
the literature [4,37–39] and morphometric values of the coracoid pro-
cess would optimize the bone transfer surgical manipulation [37]. The
glenoid, as well, is spherical in shape, yet irregular with complex cur-
vatures. Bankart lesion, which affects the glenoid labrum composed of
the soft tissue and extends to a fracture of the underlying bone is
Fig. 7. Bland–Altman plots for acromion measurements by two observers. (a) External acromion angle. (b) Internal acromion angle. (c) Mean acromion angle. (d–f)
Log transformed data plotted on the original scale. (d) External acromion curvature. (e) Internal acromion curvature. (f) Mean acromion curvature. Legend: Solid line
corresponds to the bias, with shading indicating the 95% CI. Dashed line at 0 is the line of identity. Lower and upper dotted lines indicate the 95% limits
of agreement.
Table 3
Inter-rater and intra-rater analyses.
Structure Comparison Bias [LLA,
ULA]
a,b
One sample t-
test
Bias
signicance
External
angle
A vs. B 7.9 [-19.1,
35.0]
t (9) =1.8, p =
0.10
ns
A vs. A 0.28 [-3.8, 4.3] t (9) =0.43, p
=0.68
ns
External
curvature
A vs. B 0.0048 [-0.68
X, 0.68 X]
t (9) =1.3, p =
0.22
ns
A vs. A −0.0013 [-0.14
X, 0.14 X]
t (9) =1.1, p =
0.30
ns
Internal
angle
A vs. B 12.8 [-8.7,
41.4]
t (9) =4.0, p =
0.0029
**
A vs. A 0.0020 [-16.4,
16.4]
t (9) =2.1, p =
0.062
ns
Internal
curvature
A vs. B −0.014 [-0.97
X, 0.97 X]
t (9) =2.4, p =
0.039
*
A vs. A 0.0036 [-0.85
X, 0.85 X]
t (9) =0.0008,
p =0.9994
ns
Mean angle A vs. B 3.4 [-6.9, 13.7] t (9) =20, p =
0.072
ns
A vs. A 0.14 [-9.38,
9.66]
t (9) =2.3, p =
0.052
ns
Mean
curvature
A vs. B −0.0005 [-0.55
X, 0.55 X]
t (9) =1.6, p =
0.28
ns
A vs. A 0.0012 [-0.43
X, 0.43 X]
t (9) =0.09, p
=0.93
ns
Curvature measurement one sample t-tests were performed on the log trans-
formed data.
All comparisons n =10. Legend: *p <0.05, **p <0.01.
a
Values reported as ◦for angle measurement comparisons; mm
−1
for curva-
ture measurement comparisons.
b
For angle measurements, limits of agreement (LAs) are calculated as ±1.96
SD of the bias; for curvature measurements, the antilogs of the LAs computed in
the transformed space are expressed on the original scale as functions of the bias
X (calculated according to Euser et al., 2008 [35]).
C.V.L. Olson et al.

Informatics in Medicine Unlocked 43 (2023) 101383
7
evaluated and reconstructed based on assessing the curvature of the site
of the lesion [40].
Other than the scapula, lower limb deformities are rather common,
either through congenital or elderly complications, affecting the
straightness of long bones, causing them to bow. For instance, this
method can potentially apply to tibial recurvature constructive pro-
cedures [41], hallux valgus, which is a lateral deviation of the great toe
brought on by metatarsophalangeal joint pathologies [23], and radio-
graphic assessments of calcaneal angles [42], as well as of talocrural
joint surfaces [43]. Proximally, the tibia and femur condyles at the knee
joint are curved structures that are commonly replaced by prosthetic
implants in the elderly due to osteoporosis and other bone and joint
degenerative diseases [44,45]. The same concept can be applied to the
humerus condyle and epicondyles and adjacent forearm structures [46].
Imaging assessment of curvature can optimize osteochondral ap pro-
cedures, such as reconstructing the articular surfaces of the wrist at the
scaphoid fossa of the radius and the articular circumference of the
scaphoid by grafting a femoral trochlea ap [47]. The hip joint also faces
common pathological deformities affecting the convex surface of the
head of the femur, as well as the concave acetabulum of the hip bone.
The morphological typing of these structures and implant replacements
relies not just on the curved surfaces of these structures, but also on
angular projections of the neck of the femur bearing the weight of the
hip joint [48]. In addition, the head of the femur contains a fovea that is
concave in shape and itself requires attention and morphological clas-
sication [49].
Other than the extremities, bones of the head, such as the angle of the
mandible, are indeed inherently curved structures. The curved surface of
the mandible is subjected to reconguration during aging and patho-
logical formations [50]. A very distinguishably curved bone is the
petrous part of the temporal bones and especially their cavities. The
spiral anatomy of the cochlea and its internal canals are evaluated with
challenges by sophisticated CT and MRI techniques [34,51]. Finally, the
vertebral column is composed of curved concave-convex intervals that
are subjected to kyphosis, lordosis, and scoliosis. Cobb angle determi-
nation is a long-standing clinical method of measurement, performed by
measuring in degree angles the array between the rst and last vertebrae
of the misaligned segment of the vertebral column [52]. This method is
not reliable and has been supplemented by additional methods of
radius-t and arc plotting analysis [33]. Collectively, employment of
new methods would optimize the early detections of vertebral column
deformity [53].
This curvature calculating method can be extended into more ap-
plications, e.g., it can optimize certain assessments in cardiovascular
disorders. Vessels take curved courses at certain anatomical localiza-
tions. For instance, the coronary arteries of the heart take a curved
course around the heart, which plays a key role in blood ow dynamics.
The dynamicity of the curved coronary arteries can be characterized by
such morphological quantications [54,55]. Aneurysm severity can be
evaluated by assessing the degree of its curvature growth [56]. Likewise,
with new breakthroughs in fractional derivative modeling, fractal
growth functions of aneurisms, dened by rates of change in curvature,
could follow similar 2D procedures as has been shown for tumor growth
[57]. This application is not limited to morphological assessments of
aneurysms as it can potentially also serve in the physiological evaluation
of blood ow through the ascending aorta followed by the curved aortic
arch in correlation to the distally formed aneurysm in the descending
aorta [58]. Another potential application in functional anatomy was
illustrated in assessing the diaphragm movement in terms of its curva-
ture changes over time during breathing cycles in restrictive pulmonary
disorders [59].
4. Conclusions
Measuring curved structures is an inevitable step in morphological
assessments in biomedicine. What have been regarded and measured as
angles are rather inherent curvatures and require more precision in
describing their trajectories. Here, we have shown how angulation and
curvature conceptually may refer to different surface structures (Fig. 1a
and b) and, in fact, these parameters show different distributions, even
within the same measured structures (Fig. 3a and b; Table 2). Solver
allows for both researchers and clinicians to quickly characterize
morphometric courses and properties of a given structure, and these
characterizations and measurements share similar reliabilities as other
methods (Figs. 5 and 6; Table 3). Paired with other scalar measurements,
curvature can complete the picture of an anatomical structure’s pattern,
which could lead to future disease evaluation modalities that may be
better monitored and evaluated based of pathological shape change,
such as skeletal deformations, vascular conditions, and soft tissue
restrictions.
Ethical approval
The study and access to specimens was approved for research and
education purposes by the Institutional Review Board (IRB)—The Ethics
Committee of the University Hospital Motol and the Second Faculty of
Medicine, Charles University, Prague, Czech Republic [reference num-
ber: EK-1175.1.18/22].
Funding
This work was supported by the Grant Agency of Charles University
(GAUK) [grant numbers: 1720119 and 2120319].
CRediT authorship contribution statement
Carl V.L. Olson: Methodology, Software, Formal analysis, Data
curation, Visualization, Writing – original draft. David Kachlík: Su-
pervision, Writing – review & editing. Azzat Al-Redouan: Conceptu-
alization, Methodology, Investigation, Validation, Visualization, Project
administration, Writing – original draft.
Declaration of competing interest
The authors declare the following nancial interests/personal re-
lationships which may be considered as potential competing interests:
Azzat Al-Redouan reports a relationship with Charles University
Second Faculty of Medicine that includes: employment, funding grants,
and speaking and lecture fees. Carl VL Olson reports a relationship with
Charles University Second Faculty of Medicine that includes: employ-
ment and funding grants.David Kachlik reports a relationship with
Charles University Second Faculty of Medicine that includes: employ-
ment, funding grants, and speaking and lecture fees.
Acknowledgments
The authors want to thank Mohamad Dib Itani for assistance with the
data reliability tests, Radovan Hud´
ak for help with X-ray images, and
Ehsan Abbaspour and ˇ
S´
arka Salavov´
a for help with specimen
observations.
References
[1] Klingenberg CP. Evolution and development of shape: integrating quantitative
approaches. Nat Rev Genet 2010;11:623–35.
[2] Alfaro-Gomez U, Fuentes-Ramirez LD, Chavez-Blanco KI, Vilchez-Cavazos JF,
Zdilla MJ, Elizondo-Omana RE, et al. Anatomical variations of the acromial and
coracoid process: clinical relevance. Surg Radiol Anat 2020;42:877–85.
[3] Alraddadi A, Alashkham A, Lamb C, Soames R. Examining changes in acromial
morphology in relation to spurs at the anterior edge of acromion. Surg Radiol Anat
2018;41:409–14.
[4] Jacxsens M, Elhabian SY, Brady SE, Chalmers PN, Tashjian RZ, Henninger HB.
Coracoacromial morphology: a contributor to recurrent traumatic anterior
glenohumeral instability? J Shoulder Elbow Surg 2019;28:1316–1325.e1.
C.V.L. Olson et al.

Informatics in Medicine Unlocked 43 (2023) 101383
8
[5] Li Z, Ji C, Wang L. Development of a child head analytical dynamic model
considering cranial nonuniform thickness and curvature – applying to children
aged 0–1 years old. Comput Methods Progr Biomed 2018;161:181–9.
[6] Lu Y-C, Untaroiu CD. Statistical shape analysis of clavicular cortical bone with
applications to the development of mean and boundary shape models. Comput
Methods Progr Biomed 2013;111:613–28.
[7] Asal N, S¸ahan MH. Radiological variabilities in subcoracoid impingement: coracoid
morphology, coracohumeral distance, coracoglenoid angle, and coracohumeral
angle. Med Sci Mon Int Med J Exp Clin Res 2018;24:8678–84.
[8] Guo X, Ou M, Yi G, Qin B, Wang G, Fu S, et al. Correction between the morphology
of acromion and acromial angle in Chinese population: a study on 292 scapulas.
BioMed Res Int 2018;2018:1–6.
[9] Al-Redouan A, Kachlik D. Scapula revisited: new features identied and denoted by
terms using consensus method of Delphi and taxonomy panel to be implemented in
radiologic and surgical practice. J Shoulder Elbow Surg 2021.
[10] Fu G, Bo W, Pang X, Wang Z, Chen L, Song Y, et al. Mapping shape quantitative
trait loci using a radius-centroid-contour model. Heredity 2013;110:511–9.
[11] Fu G, Huang M, Bo W, Hao H, Wu R. Mapping morphological shape as a high-
dimensional functional curve. Briengs Bioinf 2017;19:461–71.
[12] Arag˜
ao JA, Silva LP, Reis FP, dos Santos Menezes CS. Analysis on the acromial
curvature and its relationships with the subacromial space and types of acromion.
Revista Brasileira de Ortopedia (English Edition) 2014;49:636–41.
[13] Dugarte AJ, Davis RJ, Lynch TS, Schickendantz MS, Farrow LD. Anatomic study of
subcoracoid morphology in 418 shoulders: potential implications for subcoracoid
impingement. Orthop J Sports Med 2017;5:232596711773199.
[14] Leschinger T, Wallraff C, Müller D, Hackenbroch M, Bovenschulte H, Siewe J. In
vivo analysis of coracoid and subacromial shoulder impingement mechanism
during clinical examination. Eur J Orthop Surg Traumatol 2017;27:367–72.
[15] Morelli KM, Martin BR, Charakla FH, Durmisevic A, Warren GL. Acromion
morphology and prevalence of rotator cuff tear: a systematic review and meta-
analysis. Clin Anat 2018;32:122–30.
[16] Torrens C, Alentorn-Geli E, Sanchez JF, Isart A, Santana F. Decreased axial
coracoid inclination angle is associated with rotator cuff tears. J Orthop Surg 2017;
25:230949901769032.
[17] Altintas B, K¨
a¨
ab M, Greiner S. Arthroscopic lateral acromion resection (ALAR)
optimizes rotator cuff tear relevant scapula parameters. Arch Orthop Trauma Surg
2016;136:799–804.
[18] Hess F, Zettl R, Smolen D, Knoth C. Anatomical reconstruction to treat acromion
fractures following reverse shoulder arthroplasty. Int Orthop 2017;42:875–81.
[19] Sun Z, Li H, Wang B, Yan J, Han L, Han S, et al. A guideline for screw xation of
coracoid process base fracture by 3D simulation. J Orthop Surg Res 2021;16:58.
[20] Voss A, Dyrna F, Achtnich A, Hoberman A, Obopilwe E, Imhoff AB, et al. Acromion
morphology and bone mineral density distribution suggest favorable xation
points for anatomic acromioclavicular reconstruction. Knee Surg Sports Traumatol
Arthrosc 2017;25:2004. –12.
[21] Yoon JP, Lee YS, Song GS, Oh JH. Morphological analysis of acromion and hook
plate for the xation of acromioclavicular joint dislocation. Knee Surg Sports
Traumatol Arthrosc 2016;25:980–6.
[22] Lea RD, Gerhardt JJ. Range-of-motion measurements. J Bone Joint Surg 1995;77:
784–98.
[23] Moulodi N, Kamyab M, Farzadi M. A comparison of the hallux valgus angle, range
of motion, and patient satisfaction after use of dynamic and static orthoses. Foot
2019;41:6–11.
[24] Tan K-L, Ooi BC, Thiang LF. Indexing shapes in image databases using the
centroid–radii model. Data Knowl Eng 2000;32:271–89.
[25] Gupta N, Trindade BL, Hooshmand J, Chan E. Variation in the best t sphere radius
of curvature as a test to detect keratoconus progression on a scheimpug-based
corneal tomographer. J Refract Surg 2018;34:260–3.
[26] Young G, Hall L, Sulley A, Osborn-Lorenz K, Wolffsohn JS. Inter-relationship of soft
contact lens diameter, base curve radius, and t. Optom Vis Sci 2017;94:458–65.
[27] Tang C, Wu Q, Liu B, Wu G, Fan J, Hu Y, et al. A multicenter study of the
distribution pattern of posterior-to-anterior corneal curvature radii ratio in Chinese
myopic patients. Front Med 2021;8:724674.
[28] Totlis T, Konstantinidis GA, Karanassos MT, Sodis G, Anastasopoulos N, Natsis K.
Bony structures related to snapping scapula: correlation to gender, side and age.
Surg Radiol Anat 2013;36:3–9.
[29] Rueden CT, Schindelin J, Hiner MC, DeZonia BE, Walter AE, Arena ET, et al.
ImageJ2: ImageJ for the next generation of scientic image data. BMC Bioinf 2017;
18:529.
[30] Weisstein EW. Curvature. From MathWorld - A Wolfram Web Resource n.d. https
://mathworld.wolfram.com/Curvature.html (accessed July 10, 2021).
[31] Leite MJ, Pinho AR, S´
a MC, Silva MR, Sousa AN, Torres JM. Coracoid morphology
and humeral version as risk factors for subscapularis tears. J Shoulder Elbow Surg
2020;29:1804–10.
[32] Jamali AA, Deuel C, Perreira A, Salgado CJ, Hunter JC, Strong EB. Linear and
angular measurements of computer-generated models: are they accurate, valid, and
reliable? Comput Aided Surg 2007;12:278–85.
[33] Goh S, Price RI, Leedman PJ, Singer KP. A comparison of three methods for
measuring thoracic kyphosis: implications for clinical studies. Rheumatology 2000;
39:310–5.
[34] Manoussaki D, Chadwick RS, Ketten DR, Arruda J, Dimitriadis EK, O’Malley JT.
The inuence of cochlear shape on low-frequency hearing. Proc Natl Acad Sci USA
2008;105:6162–6.
[35] Euser AM, Dekker FW, le Cessie S. A practical approach to Bland-Altman plots and
variation coefcients for log transformed variables. J Clin Epidemiol 2008;61:
978–82.
[36] Taponecco F, Alexa M. Piecewise circular approximation of spirals and polar
polynomials. In: Wscg ’2003: posters: the 11-th international conference in central
europe on computer graphics, visualization and computer vision 2003. Plzeˇ
n,
Czech Republic: UNION Agency; 2003. p. 137–40.
[37] Buda M, D’Ambrosi R, Bellato E, Blonna D, Cappellari A, Delle Rose G, et al. Failed
Latarjet procedure: a systematic review of surgery revision options. J Orthop
Traumatol 2021:22.
[38] Dolan CM, Hariri S, Hart ND, McAdams TR. An anatomic study of the coracoid
process as it relates to bone transfer procedures. J Shoulder Elbow Surg 2011;20:
497–501.
[39] Nagar M, Tiwari V, Joshi A, Ahmed M, Patel M. A cadaveric morphometric analysis
of coracoid process with reference to the Latarjet procedure using the “congruent
arc technique.”. Arch Orthop Trauma Surg 2020;140:1993–2001.
[40] DeHaan A, Munch J, Durkan M, Yoo J, Crawford D. Reconstruction of a bony
bankart lesion. Am J Sports Med 2013;41:1140–5.
[41] Scheidegger P, Horn Lang T, Schweizer C, Zwicky L, Hintermann B. A exion
osteotomy for correction of a distal tibial recurvatum deformity. The Bone & Joint
Journal 2019;101-B. 682–90.
[42] Guo J, Mu Y, Xue D, Li H, Chen J, Yan H, et al. Automatic analysis system of
calcaneus radiograph: rotation-invariant landmark detection for calcaneal angle
measurement, fracture identication and fracture region segmentation. Comput
Methods Progr Biomed 2021;206:106124.
[43] Nozaki S, Watanabe K, Kato T, Miyakawa T, Kamiya T, Katayose M. Radius of
curvature at the talocrural joint surface: inference of subject-specic kinematics.
Surg Radiol Anat 2018;41:53–64.
[44] Jade S, Tamvada KH, Strait DS, Grosse IR. Finite element analysis of a femur to
deconstruct the paradox of bone curvature. J Theor Biol 2014;341:53–63.
[45] Koh Y-G, Nam J-H, Chung H-S, Kang K-T. Difference in coronal curvature of the
medial and lateral femoral condyle morphology by gender in implant design for
total knee arthroplasty. Surg Radiol Anat 2019;42:649–55.
[46] de Chaves L, Fonseca GV, da Silva FH, Acioly MA. Osseous morphology of the
medial epicondyle: an anatomoradiological study with potential clinical
implications. Surg Radiol Anat 2021;43:713–20.
[47] Daruwalla JH, Skrok J, Pet MA, Giladi AM, Higgins JP. Magnetic resonance
imaging to investigate the clinical applicability of the medial femoral trochlea
osteochondral ap within the wrist. Hand 2022:155894472110643.
[48] Wang P, Liu X, Xu J, Li T, Sun W, Li Z, et al. Deep learning for diagnosing
osteonecrosis of the femoral head based on magnetic resonance imaging. Comput
Methods Progr Biomed 2021;208:106229.
[49] Yarar B, Malas MA, Çizmeci G. The morphometry, localization, and shape types of
the fovea capitis femoris, and their relationship with the femoral head parameters.
Surg Radiol Anat 2020;42:1243–54.
[50] Direk F, Uysal II, Kivrak AS, Unver Dogan N, Fazliogullari Z, Karabulut AK.
Reevaluation of mandibular morphometry according to age, gender, and side.
J Craniofac Surg 2018;29:1054–9.
[51] Heutink F, Koch V, Verbist B, van der Woude WJ, Mylanus E, Huinck W, et al.
Multi-Scale deep learning framework for cochlea localization, segmentation and
analysis on clinical ultra-high-resolution CT images. Comput Methods Progr
Biomed 2020;191:105387.
[52] Singer KP, Edmondston SJ, Day RE, Breidahl WH. Computer-assisted curvature
assessment and cobb angle determination of the thoracic kyphosis. Spine 1994;19:
1381–4.
[53] Kluszczy´
nski M, Pilis A, Czaprowski D. The importance of the size of the trunk
inclination angle in the early detection of scoliosis in children. BMC Muscoskel
Disord 2022;23:5.
[54] Biglarian M, Larimi MM, Afrouzi HH, Moshfegh A, Toghraie D, Javadzadegan A,
et al. Computational investigation of stenosis in curvature of coronary artery
within both dynamic and static models. Comput Methods Progr Biomed 2020;185:
105170.
[55] Govindaraju K, Viswanathan GN, Badruddin IA, Kamangar S, Salman Ahmed NJ,
Al-Rashed AA. The inuence of artery wall curvature on the anatomical assessment
of stenosis severity derived from fractional ow reserve: a computational uid
dynamics study. Comput Methods Biomech Biomed Eng 2016;19:1541–9.
[56] Akkoyun E, Gharahi H, Kwon ST, Zambrano BA, Rao A, Acar AC, et al. Dening a
master curve of abdominal aortic aneurysm growth and its potential utility of
clinical management. Comput Methods Progr Biomed 2021;208:106256.
[57] Singh R, Mishra J, Gupta VK. Dynamical analysis of a tumor growth model under
the effect of fractal fractional Caputo-Fabrizio derivative. International Journal of
Mathematics and Computer in Engineering 2023;1:115–26.
[58] van Hout MJP, Juffermans JF, Lamb HJ, Kr¨
oner ESJ, van den Boogaard PJ,
Schalij MJ, et al. Ascending aorta curvature and ow displacement are associated
with accelerated aortic growth at long-term follow-up: a MRI study in Marfan and
Thoracic Aortic Aneurysm Patients. IJC Heart and Vasculature 2021;38:100926.
[59]] Harlaar L, Ciet P, van Tulder G, Brusse E, Timmermans RGM, Janssen WGM, et al.
Diaphragmatic dysfunction in neuromuscular disease, an MRI study. Neuromuscul
Disord 2021.
C.V.L. Olson et al.