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ERGONOMICS, 1995, VOL. 38, No.9, 1911-1920
Analysis of foot shape variation based on
the medial axis of foot outline
MAKIKO KOUCHI
Human-Environment System Department,
National Institute of Bioscience and Human-Technology,
I-I Higahi, Tsukuba, Ibaraki 305, Japan
Keywords: Foot; Pronation; Overhang of navicular bone;
Foot outflare; Shoe.
The variations in foot outline forms are analyzed by using flexion angles of the
medial axis of foot outline. Foot outline and 12 conventional measurements taken
on the right foot of 443 male and 297 female subjects with no visible pathological
deformation of the foot were used for analyses. The results indicate that the foot is
outflared in most of the subjects. Medial bulge and lateral concavity of foot outline
are responsible for the foot outflare, and they are not correlated with each other.
Medial bulge is due to the overhang of navicular bone that is caused by the pronation
of the foot. Its intensity is negatively correlated with dorsal arch height. Lateral
concavity is partly due to the abduction of talus and calcaneus relative to the
tarsometatarsal bones anterior to them. These three-dimensional morphological
characteristics of outflared feet intimately relate to the fit and comfort of the shoe.
The flexion angles of medial axis of foot outline provide a useful tool in
morphological analysis of the foot for the following reasons; (I) they carry the
information on the three-dimensional foot shape that cannot be represented by
conventional measurements; and (2) the data is easily obtained and calculations are
easily made with minimum expense.
1. Introduction
The variation of human foot shape plays an important role in the fitting between foot
and shoe, for, even in the same shoe, feet of different shape receive localized pressure
at different parts. A shoe last has been designed and modified manually to fit to a
particular foot based on the experience of craftsmen. Systematic analysis of foot shape
variation is necessary to identify the causes of misfit and to establish an objective
method to modify a shoe last to fit to feet of different shapes. If the morphological
characteristics of a human foot are important in shoe fitting and their range of variation
are clarified, then engineering may be introduced into the shoe last design and
modification. The lack of an appropriate method to extract the shape characteristics of
a human foot has hindered this type of approach.
The outline of a foot contour projected on the standing plane, which has been used
by shoe manufacturers, contains information on the morphological characteristics of the
foot that cannot be represented by conventional variables such as dimensions, angular
measurements and proportions. Therefore, the analysis of foot outline forms may
provide useful information to construct a morphology-based strategy for improving the
fit between the foot and shoe.
An outline form can be numerically expressed by using Fourier descriptors (Lestrel
and Brown 1976) or by the vector angle method (Kouchi and Yamazaki 1990). In the
0014-0139195 $10·00 © 1995 Taylor & Francis Ltd.
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M. Kouchi
latter, an outline is approximated by a polygon with equidistant vertices, and a sequence
of angles between the edges and the x-coordinate axis is used to represent the outline
form. In both cases multivariate analysis methods can be applied for identifying the
important morphological features and for type classification. This type of approach,
however, does not provide more information than that carried by the variables used for
the analysis.
Medial axis, or skeleton, is useful for the analysis of irregular form on which
anatomical landmarks are not determined (Bookstein 1991). Foot outline has an
elongated irregular form, and the landmarks on it are defined as the maximum
protrusion, and are difficult to define precisely. Also, the intuitive midline of a foot
outline is not straight, and the results of previous studies suggest the importance of
midline bends as a morphological feature of the foot (Freedman et al. 1946, Kouchi and
Yamazaki 1990).
The aims are: (I) to show the usefulness of medial axis of foot outline in extracting
and summarizing the morphological features of the foot; and (2) to analyze the variation
in foot shape by using the flexion angles of medial axis.
Subjects and method
2.
2.1. Subjects and data
Subjects are 443 males and 297 females without visible pathological deformations of
the fool. Mean age, height and weight of the subjects are shown in table I. Foot outline
Table 1.
Age, height and weight of subjects.
Male (n
= 443)
Female (n
= 297)
Item
Mean
SD
Min
Max
Mean
SD
Min
Max
Age (years)
Height (em)
Weight (kg)
41·0
166·6
62·7
11·29
5·76
9·12
18·0
145·0
40·5
75·0
181·0
92·9
26·0
156·5
52·8
10·24
5·82
8·65
18·0
140·0
35·0
64·0
170·0
91·2
Figure I. Landmarks and measurements of the foot. D, dorsal arch point; MF, metatarsale
fibulare; MT, metatarsale tibiale; P, ptemion; SF, sphyrion fibulare, A I, ball flex angle;
A2, toe I angle; A3, toe Vangie; B I, foot breadth; B2, heel breadth; C I, foot
circumference; C2, instep circumference; HI, dorsal arch height; H2, sphyrion fibulare
height; Ll, foot length; L2, fibular instep length; and L3, instep length.
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1913
Medial axis offoot outline
and 12 measurements of right foot were used for the analysis. All the data were taken
with the subjects standing barefoot, with their weight distributed equally on both feet.
All the data were taken in 1987 with a foot-measuring system developed in a project
by The Japan Leather and Leather-Goods Industries Association. Data were obtained
to update the Japanese Industrial Standard for shoes (Japan Leather and Leather-Goods
Industries Association 1987, 1988). This system takes foot outline, foot print, lateral
foot outline, foot cross-sections at instep and ball, and 24 measurements optically
without touching the subject (Kouchi and Yamazaki 1990).
The twelve measurements used are shown in figure J. Foot axis is the line connecting
pternion (P) and the tip of the second toe. Length measurements are projected length
on foot axis. Dorsal arch point is the highest point on the dorsum at 54% of the foot
length from P. The following two indices are calculated: foot index = foot
breadth X 100lfoot length; and foot axis index = MT-faX JOOIMF-fa, where MT-fa
and MF-fa are distances from MT (metatarsale tibiale) or MF (metatarsale fibuJare) to
the foot axis (figure I).
2.2. Medial axis (skeleton)
Figure 2 shows an example of medial axis and its flexion angles. A foot outline is
described by data points at intervals of about 2 mm. The coordinate system is defined
as follows: the x-axis corresponds to the foot axis, and the y-axis is perpendicular to
the x-axis crossing P, the origin. Indentations between the toes are enveloped before
the calculation of medial axis.
Medial axis corresponds to the intuitive midline of an outline form. To avoid
confusion with bones, the term medial axis is used instead of skeleton. The medial axis
is calculated according to the algorithm of subroutine DIST of SPIDER, a subroutine
package for image processing (Electrotechnical Laboratory 1980). Using an image
processing method, an internal shape of the outline is divided into pixels at intervals
of I mm for each direction, x and y. Then a distance from the outline is calculated for
each pixel based on four-neighbour distance. On the four-neighbour distance, the pixel
adjoins only four directions, upper, lower, left, and right direction. The unit of the
distance is defined as a distance between the adjoining pixels. The medial axis is
obtained as the points that have local maximum of the distance.
The medial axis of foot outline has two triple points, heel triple point (HTP) and
toe triple point (TIP), and between them it curves like an'S' with two inflexion points
in most of the cases. To represent the flexions of the medial axis between the two triple
points, two angular measurements are calculated according to the following procedure.
A regression line is calculated using data points of the medial axis whose x coordinate
belong to each of the following three ranges (figure 2): (I) regression line I(RL I), from
x
Figure2. Flexionanglesof medialaxisof foot outline.A, one-thirdof foot lengthfrom P; AFA,
anterior flexion angle; AFP, anterior flexion point; B, inflexion point determined by eye
inspection; C, medial bulge; D, lateral concavity; HTP, heel triple point; PFA, posterior
flexion angle; PFP, posterior flexion point; RLI - RL3, regression lines; and TIP, toe
triple point.
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1914
M. Kouchi
HTP to one-third of foot length from origin, P (figure 2, A); (2) regression line 2 (RL2).
from one-third of foot length from origin to the anterior intlexion point (AlP)
determined by eye inspection (figure 2. B); and (3) regression line 3 (RL3), from AlP
to TIP. Acute angles made by RL I and RL2, and by RL2 and RL3 are named posterior
tlexion angle (PFA) and anterior flexion angle (AFA) respectively and are used for
further analysis. The intersection of RL I and RL2 and that of RL2 and RL3 are named
posterior flexion point (PFP) and anterior tlexion point (AFP) respectively.
One-third of foot length from P is used instead of the posterior intlexion point (PIP)
because there are cases in which the posterior tlexion angle is too small to determine
PIP by eye inspection. AlP is determined as the data point with the smallesty and whose
x is largerthan one-third of foot length and smaller than two-thirds offoot length. When
more than two data points satisfy this condition, the one whose x is close to the median
of them is chosen. Interobserver error in determining AlP is very small. Mean absolute
differences of PFA and AFA calculated for 40 randomly selected subjects by two
different observers are 0·14 and 0·29° respectively. The maximum absolute differences
of PFA and AFA are < 1°.
There are two causes for the medial axis of foot outline to flex at PIP, which is
located at about one-third of the foot length from P. One is medial bulge (figure 2. C)
and the other is lateral concavity (figure 2. D) of foot outline. To determine which is
responsible in an individual case. medial axis is calculated for the corrected foot outline
with medial bulge removed when PFA is equal to or larger than 4°, and the corrected
posterior tlexion angle (CPFA) is calculated as shown in figure 3. The difference
between PFA and CPFA represents the intensity of medial bulge, and will be referred
to as 1MB in the following text. CPFA is due to the lateral concavity of foot outline,
or eversion of the heel (forefoot).
2.3. Statistics
A r-test is used for testing the significance of sex difference. In order to clarify the
relation between foot outline form and conventional variables. correlation coefficients
between angular variables of medial axis and conventional variables are calculated.
Also, small-PFA group and large-1MB group are compared by using a r-test. The former
consists of foot outlines with minimal intlare or outtlare having PFA smaller than the
15th centi Ie. The latter consists of foot outlines showing conspicuous outtlare due to
the medial bulge, whose 1MB are larger than the 85th centile. Figure 4 shows an example
of a foot outline belonging to a small-PFA group (figure 4A) and examples belonging
to a large-1MB group (figure 4B, C). In large-1MB group, foot outlines
with large 1MB and small CPFA (figure 4B) and those with large 1MB and large CPFA
(figure 4C) are included.
3. Results
Figure 5. which shows the distribution of PFA, indicates that most of the feet are more
or less outtlared. There is no significant sex difference in the position of PFP and AFP
of the medial axis. The mean position ofPFP is 31·4% of foot length from P (calculated
when PFA;;" 4°), and that of AFP is 57-4% of foot length from P.
Table 2 presents the basic statistics of the angular measurements of medial axis. PFA
and 1MB are significantly larger, and AFA is significantly smaller in females. No sex
difference is observed in CPFA.
PFA is highly correlated with CPFA and 1MB, but CPFA and 1MB are not correlated
with each other. AFA is not correlated with other angular measurements of medial axis.
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Medial axis offoot outline
1915
Figure 3. Original (A) and corrected (B) foot outlines. a. Medial bulge: CPFA. corrected
posterior flexion angle: and PFA. posterior flexion angle.
c
Figure4. Examples of medialaxis of foot outline. A, straightat the heel, both 1MB and CPFA
are small; B, with large1MB and very small CPFA; and C, markedly outflared, both 1MB
and CPFA are large.
The angular measurements of medial axis have very low correlations with age, body
size (height and weight), Rohrer index, or foot size (foot length, foot breadth and foot
circumference) and most of the correlation coefficients are statistically insignificant.
Table 3 presents the correlation coefficients between the angular measurements of
medial axis and conventional foot variables with absolute value > 0·3. Most of the
variables that show higher correlation with angular measurements of medial axis are
not dimensions, but angles or indices. PFA is significantly correlated with foot axis
index in both sexes. Correlation coefficient between PFA and fibular instep length
observed in females becomes much smaller when foot length is partialled out.
Abduction of heel (CPFA) is not correlated with foot measurements or indices. AFA
is correlated with toe Vangie, foot axis index and foot index in both sexes. Negative
correlation between 1MB and dorsal arch height indicates a tendency that the stronger
the medial bulge, the lower the dorsal arch height.
The results of the comparison between a small-PFA group and a large-1MB group
are presented in table 4. In males, foot size is not significantly different between the
two groups, and the only dimension showing significant difference between the two
groups is dorsal arch height. In females, the small-PFA group has a significantly larger
foot, and thus several dimensions show significant differences between the two groups.
Among these dimensions. dorsal arch height has highly significant differences between
the two groups (p < 0·001). When the foot size difference between the two female
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M. Kouchi
1916
N
60
Male
N=443
50
40
30
20
In-
10
o
o
-5
10
PFA
~
15
20
25
(dogl
N
40
Female
N=297
30
20
10
0
n-5
-nf
0
Figure 5_
Table 2.
10
PFA
15
PFA
AFA
CPFA
1MB
25
(dogl
Histogram of PFA.
Flexion angles (0) of medial axis of foot outlilne.
Male
Item
20
Female
n
Mean
SD
Min
Max
n
Mean
SD
Min
Max
Difference
r-test
443
443
392
392
8·01
9·26
2·98
5.80
3·60
2·81
1-98
2·22
-1-4
0·5
-1·5
0·6
20·7
18·7
8·4
14·2
297
297
270
270
8·84
8-65
3·22
6-38
3·80
2·87
2·11
2-21
-4·8
1·4
-1·8
0-5
18·8
16·4
10·4
13-3
- 2-98t
2-82t
- 1·44 ns
- 3·43t
tSignificant at the I % level.
ns, Not significant.
PFA, posterior flexion angle; AFA, anterior flexion angle; CPFA, corrected posterior flexion
angle; and 1MB, intensity of medial bulge.
Table 3. Correlation coefficients between angular measurements of medial axis offoot outline,
foot measurements and indices. Coefficients with absolute values> 0·3 are shown.
Male
r(PFA,
r(AFA,
r(AFA,
r(AFA,
r(AFA,
r(IMB,
Female
foot axis index)
toe I angle)
toe VangIe)
foot axis index)
foot index)
dorsal arch h.)
0-37
-0·33
0·71
- 0·55
0·32
-0·38
r(PFA, fibular instep I.)
r(PFA, ball flex angle)
r(PFA, foot axis index)
r(AFA, toe Vangie)
r(AFA, foot axis index)
r(AFA, foot index)
r(lMB, dorsal arch h.)
-0·36
- 0·36
0-33
0·60
-0·42
0-43
-0·37
PFA, posterior flexion angle; AFA, anterior flexion angle; and 1MB, intensity of medial
bulge.
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Medial axis offoot outline
Table 4.
1917
Variables with significant difference between the smaJl-PFA and large-1MB groups.
Small-PFA
Large-IMB
Difference
Item
Mean
SD
Mean
SD
r-test
Male (n = 66)
Height (em)
H I Dorsal arch height (mm)
A I Ball flex angle (0)
A2 Toe I angle
A3 Toe Vangie
Foot axis index
PFA (0)
1MB
167·7
60·6
77·6
6·4
16·1
79·8
2·6
6·55
4·48
2·10
5·41
4·68
8-48
1·51
165·5
55·8
76·2
10·4
12·7
89·6
12·6
9·2
5·63
4·46
2·41
4·35
5·40
10·18
2·61
1·22
:j:
:j:
:j:
:j:
:j:
:j:
Female (n = 45)
Height (em)
Weight (kg)
L1 Foot length (mm)
L2 Fibular instep length
L3 Instep length
B2 Heel breadth
C2 Instep circumference
H I Dorsal arch height
AI Ball flex angle (0)
Foot axis index
PFA (0)
AFA
1MB
159·2
56·8
230·3
147·5
166·4
62·6
225·9
54·7
78·7
81·6
2·7
7·8
6·07
11·36
9·82
7·67
7·90
3·45
13-16
3·92
2·97
10·73
2·25
3·05
154·7
51·2
225·4
141·1
163·3
60·1
219·6
49·7
76·5
90·3
12·8
9·8
9·7
5·38
8·70
10·22
6·88
8·36
2·75
10·05
3·85
2·99
11·28
2·31
2·80
1·18
t
:j:
:j:
t
:j:
t
:j:
t
:j:
:j:
:j:
:j:
:j:
Significant at the t 5% and :j: I % levels.
PFA, posterior flexion angle; AFA, anterior flexion angle; and 1MB, intensity of medial
bulge.
groups is taken into consideration, it is suggested that the large-1MB group has lower
plantar arch height and flatter cross-section shape at instep than the small-PFA group,
since there is minimal difference in instep circumference in spite of significantly lower
dorsal arch height in the large-1MB group.
4.
Discussion
4.1. Foot outline form and skeletal structure of the foot
The present findings agree with those by Freedman et al. (1946) in that the great majority
of the subjects are characterized by a greater or lesser degree of foot outflare, and the
degree of foot flare correlated poorly with foot length and breadth. This fact indicates
the importance of the flexion of the medial axis of the foot outline as a morphological
characteristic of the foot.
For the angular measurements of medial axis of foot outline to be useful in
summarizing morphological features of the foot, foot outline form must reflect the
three-dimensional skeletal structure of the foot. The main cause of medial bulge of foot
outline is the overhang of navicular bone as shown in figure 6. The overhang of navicular
bone is one of the characteristics of a pronated foot (Kelly 1947), and it becomes more
prominent when the foot is intentionally pronated. The negative correlation between
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1918
M. Kouchi
Figure 6. Dorsoplantar roentgenogram of a foot with marked medial bulge. PFA = 9·3;
CPFA = 2·2; and 1MB = 7·1.
1MB and dorsal arch height also implies the relation between medial bulge of foot
outlline and the pronation of the fool.
Nishino (1959) divided the foot into four groups: normal, spread, excavated (high
arched), and flat foot groups, based on 13 angular measurements and one index taken
from dorsoplantar roentgenograms. The characteristic feature of his flat foot group is
the medial shift of the head of talus, navicular bone and medial cuneiform in relation
to the line connecting the central point of the first metatarsal head and the midpoint of
the posterior margin of calcanean eminence. This situation causes the overhang of the
navicular bone, and thus a medial bulge of the foot outline. Also, Nishino's flat foot
group includes a much higher percentage of low-arch subjects judged from lateral
roentgenograms than other groups. The relationship between overhang of navicular
bone and low dorsal arch height observed by Nishino corresponds to the present finding.
In a pronated foot, the whole foot tilts inward and dorsal arch height and plantar
arch are low. These characteristics correspond with those of the large-1MB group.
The lateral concavity of foot outline is due to the deformation of soft tissue by weight
bearing as well as to the alignment of foot bones, and the latter has greater effects. In
Nishino's classification, subgroup PI of the flat foot group is characterized by smaller
fibular instep length and abduction of talus and calcaneus in relation to the
tarsometatarsal part of the foot anterior to them. In this situation the foot looks flexed
at Chopart's joint, and its outline form shows the lateral concavity. In other words, the
foot is prominently outflared as the example shown in figure 4C.
The foot outline form thus provides information regarding the three-dimensional
foot structure such as pronation and the alignment of foot bones. Further study on the
relation of medial axis of foot outline and three-dimensional foot shape is currently
being conducted by the present author.
4.2. Importance of morphology in the fitting between the foot and shoe
The present results suggest that there is great inter-individual variation in threedimensional skeletal structure, and that the foot is pronated and outflared in most of the
present subjects. On the contrary, in all the shoe lasts so far measured, medial axis has
no posterior flexion; a shoe last is shaped for foot with neither pronation nor outflare.
Therefore, the inner border of the shoe's upper is pressed down by the low plantar arch
of the pronated foot, which has large PFA and large 1MB.
When outflare is conspicuous as the example shown in figure 4C, everted forefoot
causes pressure against the outer forepart of the shoe, and the fifth toe in turn receives
uncomfortable pressure. When both 1MB and CPFA are large, the medial bulge deforms
the topline, and gaping around the top line of the shoe results owing to everted heel.
The tarsal bones are rigidly articulated against each other and are far less movable than
phalangeal bones, even when the tarsal part causes the morphological misfit, thus
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Mediai axis offoot outline
1919
phalangeal parts are under uncomfortable pressure. In other words, the undue
pressure on toes cannot be relieved necessarily by changing the design of the forepart
of the last.
Since the majority of Japanese have pronated and outflared foot whose threedimensional shape is considerably different from that of the 'ideal' foot, development
of a shoe last for such feet will improve the fit between feet and shoes in the general
population. In addition classifying foot shape into a few types based on proper
morphological characteristics, and developing shoe lasts for each type of foot may be
a realistic solution to cope with the wide variation of the human foot shape.
4.3. Advantage of medial axis
The variation of foot morphology has been investigated by using dimensions, angular
measurements, and indices taken on living subjects (Baba 1975, Kouchi 1989, Okada
et ai. 1990), or on roentgenograms (Yokokura 1928, 1929, Nishino 1959, Steel et ai.
1980, Takai 1984). These methods have the shortcomings of being unable to fully utilize
the shape information and/or difficulties in obtaining data.
One of the advantages of medial axis of foot outline is in its simplicity. Foot outline
is taken by a foot measurement system in the present study, but almost the same data
can be obtained by digitizing a foot outline taken with a scriber (Kouchi et al. 1992).
Medial axis can be calculated by a general image processing software.
Another advantage is that it carries information on the three-dimensional skeletal
structure of the foot, such as pronation, which is almost independent of foot size and
has been ignored.
For these advantages, the flexion characteristics of the medial axis of foot outlline
will provide a useful tool in morphological studies of the foot.
5. Conclusions
In the study of the fit between the foot and shoe, morphological variations in the human
foot have not been taken into consideration. Medial axis of foot outline provides a useful
and powerful method to study the variations in human foot shape. Through the analyses
of the flexion angles of medial axis calculated for the right foot outline of 443 male and
297 female subjects, the following conclusions are obtained: (I) the variability of
human foot shape caused by the differences in the skeletal structure is not negligible;
and (2) the most important factors in determining foot shape are pronation of the foot
and abduction of talus and calcaneus in relation to the tarsometatarsal part of the foot.
Since these characteristics intimately relate to the fit and comfort of the shoe, by
focusing attention on these characteristics, it may be easier to evaluate the fit between
the foot and shoe, and to determine the strategy to improve the shoe last.
Acknowledgement
The author expresses her gratitude to Keio University Tsukigase Rehabilitation Center
for the roentgenogram.
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Manuscript received 20 November 1993
Manuscript accepted 5 July 1994