^{1}Southwestern University School of Medicine, Cebu City, Philippines
^{2}Gullas College of Medicine, Cebu City, Philippines
*Corresponding author: Rene C Catan, Southwestern University School of Medicine, Cebu City, Philippines
Received Date: 03 May, 2020, 2020; Accepted Date: 10 May, 2020; Published Date: 16 May, 2020
Fibonacci spiral is widely identified in nonbiological fields such as mathematics, architecture and art, but not in human biology; an understanding of its functional role in human anatomy may be essential in making further advances in biomedical engineering specifically knee implant design. This study aims to explore whether the Golden Ratio and Fibonacci Spiral approximate the shape of the distal femoral articular surface of the knee. Twentythree 3D CT scan images of the distal knee were digitalized in 2D sagittal views and best fitted over with Fibonacci spirals computed from the knee anteroposterior dimensions. The differences between the means of the radii of the Fibonacci Spiral and the digitalized images on 5/6 subsectors where not significant at p<0.05 using KrusWallis Test covering 84% of the distal articulating surface. With the Fibonacci presenting three independent axes described a “L” shaped on the sagittal plane, these findings affirm the fixed, flexionextension axes of knee kinematics.
Introduction
The golden ratio, also known as the golden section or golden proportion, is obtained when two segment lengths have the same proportion as the proportion of their sum to the larger of the two lengths. The value of the golden ratio, which is the limit of the ratio of consecutive Fibonacci numbers, has a value of approximately 1.618 which is most commonly represented by the Greek Letter Phi (φ). The Fibonacci sequence is the sum of the two numbers that precede it. So, the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The mathematical equation describing it is
Furthering
this observation, shapes can be created based upon length measurements of the
Fibonacci numbers in sequence. Rectangles created using consecutive numbers
from the Fibonacci sequence can be divided into equally sided squares of such
numbers as well. The length of a single side can be divided into smaller values
of the Fibonacci sequence. Connecting the diagonals of each of the squares
creates a spiral, known as the Golden Spiral (Figure
1). This pattern is seen in several different forms in
nature [2].
Classically,
the Golden Spiral approximates the spiral of the human ear (Figure 2), the shell of a nautilus (Figure 3).
Human
Anatomy
In
1973, Dr. William Littler proposed that by making a clenched fist, Fibonacci’s
spiral can be approximated (Figure
4). However, due a lack of statistical and well documented empirical data,
accurate representations of the golden spiral could not be readily ascertained.
Hamilton and Dunsmuir (2010) confirmed that the phalangeal length ratio data
obtained from their subjects compared to those that were almost arbitrarily
listed by Littler in 1973 [3], were in fact comparable and
approximated a Lucas series, which essentially underscores a Fibonacci sequence
in that the sum of the first two lengths equaled the third length the Fibonacci
value of (φ) of 1.618 [4].
Fibonacci
Series and the underlying Lucas series are observed in several aspects of life
on planet earth and within the cosmos. Although widely identified
in nonbiological fields such as architecture and art, it has not been well
explored in the human biology. More research needs to be performed to explore
its physiological role in biology. Understanding its functional role may be a
keystone to making quantum advances in several fields such as artificial
intelligence, biomedical engineering designs, and human regeneration, amongst
others.
This study aims to explore
whether the Fibonacci Sequence and Fibonacci Spiral approximate the shape of
the distal femoral articular surface of the knee.
Methodology
Twentythree
3D CT Scan images were downloaded from the internet and copied into the
PowerPoint Microsoft Office program of a computer. The following criteria were
used in choosing the images:
1.
Knee images must have at least
2 views: anteroposterior and lateral.
2.
Images must show the distal
anterior and posterior femoral cortices.
3.
Distal articulating surface
including the distal femoral condyle must be traceable on the on the lateral
view
Each
image was assigned a random number from a Google random number generator
(range: 50 to 65) representing the AP dimensions (cm) of the knee. The images
were then digitalized in bundled 3D Paint software application. The distal
femur was cropped out. The outline of the 3D lateral image of the distal knee
was traced using stylus (SPENHP01) and the rendered image was then copied
into a separate PowerPoint file. Images were locked at 1:1 aspect ratio. Each digitalized image was copied into one ppt slide each
(slide 123).
A
line (X) was draw parallel to the anterior cortex of the digitalized outlined
image of the knee. And another line (Y) was drawn at the posterior tip of the
distal femoral condyle and parallel to the line X. The three images were arranged
in group image (W) formatted with aspect ratio 1:1 (Figure 5). Each downloaded,
digitalized and grouped image was copied into one ppt slide each (slide 123).
Drawing
the Fibonacci Spiral (Figure 6)
Note:
The pattern for creating the Fibonacci Spiral follows the Fibonacci sequence of
numbers are generated by setting F0 =
0, F1 = 1, and then using the recursive formula Fn = Fn1 + Fn2 to get the
rest. Thus, the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, … This sequence
of Fibonacci numbers arises all over mathematics and also in nature.
On a separate file a square (b1) with
sides (1x1) was inserted with no fill with dashed outline. Another square (b2)
was drawn alongside b1 thus creating a 1x2 rectangle (r1). A 2x2 box (b3) is
inserted alongside the 1x2 rectangle (r1) following the sequence of 1,1,2
forming a 2x3 rectangle (r2). The sum of the previous two numbers in the
sequence being 1 and 2 has a sum of 3, therefore, a fourth square (b4) with a
3x3 configuration is drawn and is attached alongside rectangle 2 (r2) creating
third rectangle (r3) with a 3x5 configuration. Then a fifth 5x5 square (b5) is
inserted alongside the preceding rectangle (r3) forming a 5x8 rectangle (r4).
And the final square (b6) is an 8x8 square alongside the (r4) creating a an
8x13 rectangle (r5).
The
Fibonacci Spiral is then drawn to approximate the golden spiral using
quartercircle arcs on each square derived from the Fibonacci sequence.
Fibonacci
Spiral Template
The
distance between the parallel lines x and y is the Anteroposterior dimensions
(AP) in cm of the distal femoral articulating is referred to as A1 corresponding to the measurement of square b6 in Figures 6, 7.
The
Fibonacci sequence describes the relationship with the Golden Rectangle as
follows:
b6/b5≈1.618.
Therefore, where b6=A1, b5= A2.
Subsequently,
b/c≈ 1.618
b5/b4≈1.618
Therefore, b4≈b5/1.618 (2), b4= A3.
The rest of the squares can be calculated by
A1’,
A2’, A3’, A4” are the Axes
of Rotation (AoR) of the quarter
circle in each square. Black
arrows represent distance covered by the arc from the AoR in each square.
The
center of rotation for each arc drawn in each square and hereto referred as
A1’, A2’, A3’, A4’and A5’.
The arcs are formed one quarter of circle shaped by the radius that equal to
the dimension of the side
of the square. These are referred hence ArcA1, ArcA2, ArcA3, ArcA4 and ArcA5
respectively (Figure 8).
Each
arc is further subdivided into two subgroups thus we have the following: A2.1,
A2.1, A3.1, A3.2, A4.1, and A4.2, respectively. There are six (6) subsectors (Figure
9).
Posterior tilt of the template if necessary is measured to allow best fit of outlines of the template over the digitalized image. Increments of 5 degrees up to 20 degrees is merged into the template with its vertex at A2’ (Figure 9).
All
lines in figure 9
are formatted, grouped and its aspect ratio locked at 1:1. This template (“T”)
hence is used to ascertain the shape of the distal femoral articulating
surface.
Procedure
The
slide is set at 116% magnification to mark actual centimeter values on the
computer screen. The digitalized image of the distal femur is enhanced or
reduced with aspect ratio locked at 1:1 based on its actual AP dimension in
centimeters.
Template
“T” is copied into each slide of the rendered image (slide 123) and its A1
dimension enhanced or reduced to equalize or closely approximate the AP
dimension of the digitalized image.
Template
T is set on the digitalized image of the distal femur to approximate fit of
ArcA2, ArcA3 and ArcA4 over the articulating surface of the femur. Posterior or
anterior rotation of the template is permitted to allow as much as possible
100% fit of the two images. Once it is acceptable the two images are grouped.
The
values of A2, A3, A4 and A5 of template “T” are determined using equations (1),
(2), (3), and (4) which are equal to the dimensions of the squares and the
radius of the arc, respectively.
On
of each subsector: A2.1, A2.2, A3.1, A3.2, A4.1 and A4.2 the digitalized image
of the femur is measured from the center of the arc of rotation on template “T”
to the maximum distance protracted by the image (radius) (Figure 10).
The
preceding procedure is performed on each slide (Slide1 to 23) and the values
are documented on excel file.
Statistical
Analysis
The
mean of the radius of template “T” (A2, A3, A4) are compared to the mean of the
radius on each subsector using MannWhitney/Wilcoxon TwoSample Test
(KruskalWallis test for two groups), values p < 0.05 is significant.
1)
A2 cf A2.1
2)
A2 cf A2.2
3)
A3 cf A3.1
4)
A3 cf A3.2
5)
A4 cf A4.1
6)
A4 cf A4.2
Results
Data
Comparing the Shapes (Tables 1 to 4)
Discussion
Geometry of the Femoral Condyles
The tibiofemoral joint is formed by the distal end of the
femur and the proximal surfaces of the tibia. The distal aspect of the femur
has two surfaces. Both are convex, asymmetrical, saddleshaped condylar
surfaces are coated with cartilage. They are separated by a Ushaped notch. The
femur viewed laterally, is flattened in its anterior surface and curved on its
posterior aspect. The medial femoral condyle has a smaller transverse diameter
but longer longitudinal length than the lateral condyles.
As early as the late 19th century, the axis of knee ﬂexion
and extension was derived from the geometry of the femoral condyles, by
analysis of true sagittal plane sections through the femoral condyles.
The femoral condyles were described as spirals and the changing
curvature of the condyles seen on sagittal sections results in an axis that
moves as the knee ﬂexes and extends. This was described as “the instant center
of motion” moving along a predictable curved pathway during knee ﬂexion.
Fick in 1911 reanalyzed the condyle shapes by using 3
dimensions and concluded that the ﬂexionextension axis of the knee was offset
that resulted in a single, ﬁxed axis, rather than an instant center.
Contemporary movement toward the concept of a ﬁxed,
ﬂexionextension axis began in the ﬁeld of total knee arthroplasty. Use of the
epicondylar axis to align the prosthesis during surgery indicated that a
conceptual shift was taking place in the view of knee kinematics at this time.
In vitro research has led to the development of a model of
the tibiofemoral joint with 3 independent axes of motion. One, the posterior
condylar axis is effective from approximately 15° to
150° of knee ﬂexion. It closely approximates the epicondylar line and is offset
from the sagittal plane by 7° . Two, as the knee reaches extension, the axis of
motion shifts from the posterior condylar axis to the larger distal condylar
axis coupled with an anterior shear motion of the the tibia on the distal femur
[5]. The reverse occurs from full extension to the first 20° of
flexion where there is posterior gliding of the tibia until it reaches
posterior condylar axis. And three is the longitudinal axis of rotation of the
knee in coronal plane [4].
Fibonacci Spiral
The investigator recreated a Fibonacci Spiral based on the
anteroposterior dimesion of the distal femur on the sagittal plane (A1).
Subsequent measurements of of the Golden Rectagle were derived by dividing by
the value with Ø = 1.618 to generate template “T”. The latter represented the “idealized” shape of the
distal femoral articulating surface.
Thru bestfit technique and by statistical analysis
(KrusWallis H, p<0.05) comparing template “T” with the digitalized femoral
image, the differences of the means of all subsectors (A2.1, A2.2, A3.1, A4.1
A4.2) are p>0.05. Therefore, the null hypothesis is accepted in all
subsectors except A3.2 where p value= 0.0375. This investigator has shown that
the distal femoral shape is consistent with a Fibonacci Spiral (5/6 subsectors
or 84%). Furthermore, the Fibonacci Rectangle is rotated posteriorly from the
horizontal axis of the distal anterior femoral shaft by an average of 21.3°.
Knee Kinematics and Fibonacci Spiral Model
Based on Fibonacci Spiral Model, A2’ and A3’ represent the
larger condylar and smaller posterior condylar axes, respectively. With
posteior tilt of Template “T” to accommodate best fit of Fibonacci Spiral over
the digitalized image, the axes A2 and A3 are not aligned horizontally but are
off set by about 21°. This offset translates to
movement of the tibial perpendicular to the axes.
Beginning at full extension (0°) of the tibio femoral joint
in alignment with A2’ the tibia glides posteriorly and distally to about 21°
flexion until the tibial axis reaches the posterior condylar axis (A3). The axis
of rotation shifts distally and posteriorly from A2’ to A3’ to reach 110°
(20°+90°). In order to achieve flexion beyond 110° the axis of rotation
following the Fibonacci Spiral Model then shifts to more posterior and more
superior condylar axis at A4’ but ith a shorter arc. The axes hence follow an
“L” shaped.
Although it is beyond the scope of this investigation, it is
worth mentioning that the medial and collateral ligaments are taut at complete
extension, permitting no varus or valgus motion and no rotation on the axial
plane. The anterior fibers of the medial collateral ligaments incline forward
as they descenf on the tibia, blocking its rotation. After 20° of flexion the
ligaments become relaxed permitting both gliding and axial rotation. The degree
of rotation increases as flexion increases and is maximum at full extension [6].
Conclusion
The Fibonacci Spiral approximate the shape of the distal
femoral curvature; and shifting axes explains the knee flexion extension
movement through the sagittal plane.
Figure 1: Golden
Spiral fashioned out from Fibonacci sequence of numbers.
Figure 2: Fibonacci Spiral
approximates the spiral of a human ear.
Figure 3:
Shell of a Nautilus approximates a Golden Spiral.
Figure 4:
Fibonacci Spiral approximated by a clenched fist.
Figure
5:
Anteroposterior ((AP) Dimensions of digitalized image distal femur.
Figure 6:
Fibonacci Spiral.
Figure
7: Drawing the Fibonacci
Spiral. A1 corresponds to b6; A2=b5; A3=b4; A4=b3 and A5=b2.
Figure
8:
Six subsectors: A2.1, A2.2, A3.1, A3.2, A4.1 and A4.2.
Figure 9: Template
“T” with angulation showing degrees of posterior tilt with vertex at A2’.
Figure
10:
Approximation of Fit of Template “T” on Digitalized Image.
Note: Arrow indicates distance from center of rotation to the maximum distance protracted by the digitalized image on subsector.
Table
1:
Data Comparing the Shape of the Distal Femoral Articulating Surface with
Fibonacci Spiral (N=23).
Descriptive Statistics for Each Value of Crosstab
Variable 


Observed 
Total 
Mean 
Variance 
Std Dev 

Female 
11 
510 
46.3636 
150.2545 
12.2578 

Male 
12 
528 
44 
123.4545 
11.111 


Minimum 
25% 
Median 
75% 
Maximum 
Mode 
Female 
33 
35 
45 
60 
62 
33 
Male 
32 
35 
41.5 
52.5 
64 
32 
Obs 
Total 
Mean 
Variance 
Std Dev 

23 
1038 
45.1304 
131.482 
11.4666 

Minimum 
25% 
Median 
75% 
Maximum 
Mode 
32 
35 
43 
59 
64 
33 
Comparison of Means 
KrusWallis H (Equivalent to Chi Square) 
Degrees of Freeedom 
p Value 
p < 0.05 
A2 cf A2.1 
21.7272 
13 
0.0597 
not significant 
A2 cf A2.2 
20.5623 
15 
0.1515 
not significant 
A3 cf A3.1 
20.9044 
14 
0.1014 
not significant 
A3 cf A3.2 
22.0000 
12 
0.0375 
significant 
A4 cf A4.1 
15.0578 
12 
0.2383 
not significant 
A4 cf A4.2 
16.4247 
10 
0.0881 
not siginififant 
MannWhitney/Wilcoxon TwoSample Test (KruskalWallis test for two
groups) 
Citation: Catan RC, Pastor NI (2020) The Fibonacci Spiral and the Shape of the Distal Femoral Articulating Surface of the Knee. Human Anat Physiol Open Acc 3: 102. DOI: 10.29011/HPAOA.100002