Human Reproduction vol.11 no.5 pp. 1049-1054, 1996
Fractal analysis of capacitating human spermatozoa*
S.T.Mortimer1'3, M-A.Swan1 and D-Mortimer2
'Department of Anatomy and Histology and Sydney Institute for
Biomedical Research, University of Sydney, Sydney, NSW 2006
and 2Sydney IVF, Sydney, Australia
^ o whom correspondence should be addressed
While head centroid-derived kinematic values have been
determined for the trajectories of hyperactivated human
spermatozoa, the definitions are not robust with respect to
image sampling frequency and track analysis methods. The
determination of the fractal dimension of a trajectory has
been suggested as an alternative descriptive parameter for
hyperactivated motility. Here, we have investigated two
methods for the determination of the fractal dimension of
a trajectory. A simple but useful equation was found to
be: D = log (n)/[log (n) + log (d/L)], where n is the number
of intervals in the trajectory, d is the planar extent of the
curve and L is the length of the trajectory. This equation
was not influenced by scaling of the trajectory. A fractal
dimension (D) 5*1.30 was found to define hyperactivated
trajectories, and D «£ 1.20 defined non-hyperactivated
trajectories, reconstructed at both 30 and 60 Hz. However,
when circling tracks were studied, all had D > 130,
even though they were classified as non-hyperactivated
by curvilinear velocity and/or amplitude of lateral head
displacement values. An analysis of a series of non-ideal
track segments suggested a relationship between a track's
linearity and its fractal dimension. It was determined by
a linear regression analysis that the fractal dimension of a
trajectory was inversely proportional to its linearity (r =
-0.77, P < 0.001). Although the fractal dimension of a
trajectory is a good indicator of its regularity (describing
its space-filling properties), it should not be used as the
sole criterion for the classification of a trajectory as hyperactivated.
Key words: fractals/human spermatozoa/hyperactivation/kinematics/motility
Introduction
Capacitating human spermatozoa exhibit various patterns of
motility which can be classified generally as forward progressive and hyperactivated, depending on the flagellar movement
at the time of observation (Mortimer and Swan, 1995). It is
believed that the expression of hyperactivated motility is a
•This work forms part of a PhD thesis to be submitted to the
University of Sydney, Sydney, Australia.
© European Society for Human Reproduction and Embryology
prerequisite for fertilization in Eutheria, and the proportion of
spermatozoa with hyperactivated motility under capacitating
conditions in vitro has been correlated with successful in-vitro
fertilization (IVF; Wang et al, 1993; Sukcharoen et al,
1995). Therefore, an accurate estimation of the proportion of
hyperactivated spermatozoa in a preparation could represent a
useful sperm function test
The kinematic parameters used to classify human sperm
movement as hyperactivated have been derived for the various
frame rates and video systems currently used for movement
analysis (Mortimer and Mortimer, 1990; Burkman, 1991).
However, all of the kinematic measurements used rely on an
analysis of the path traced by the sperm head, not changes in
the flagellar movement, so there is an inherent degree of
inaccuracy. The parameters used were originally derived for
studying the movement of spermatozoa in seminal plasma
(David et al, 1981), where the curvilinear velocity (VCL) and
amplitude of lateral head displacement (ALH) are lower than
for spermatozoa incubated under capacitating conditions, hi
addition, the trajectories of spermatozoa in seminal plasma are
more regular, with fewer sharp changes in direction. As a
result, the values derived for ALH, and also perhaps beat/
cross frequency (BCF), for hyperactivated spermatozoa cannot
describe the trajectory accurately, even though the ALH values
of forward progressive and hyperactivated trajectories are
known to be significantly different (Mortimer and Mortimer,
1990; Burkman, 1991). When star-spin hyperactivated motility
is assessed, the ALH value cannot be considered to be a
robust indicator; this is because smoothing cannot provide a
meaningful average path and therefore one cannot derive
legitimate ALH values.
As a consequence, different definitions of hyperactivated
motility are being sought; in particular, methods that do not
employ smoodiing algorithms in their derivation (because the
curvilinear path must be 'smoothed' to allow an approximation
of die average path; Boyers et al, 1989). The average path is
used in the derivation of several other 'traditional' movement
characteristics, including ALH and BCF (David et al, 1981).
One method for defining the trajectory independently of
traditional kinematic values is to calculate its fractal dimension
(D). The concept of fractals was developed by Mandelbrot
(1983). A curve or trajectory is a fractal if more details are
observable the more closely it is inspected. For a sperm
trajectory, it has been shown that as the image sampling
frequency increases, more track details are revealed (Mortimer
et al., 1988), so the basic requirement for it to be a fractal is
met Another requirement for a fractal is that it must have a
fractal dimension (Mandelbrot 1983).
1049
S.T.Mortiiner, MjV.Swan and D.Mortimer
The fractal dimension is an expression of the degree to
which a line fills a plane. If a line is straight, with no deviations,
its fractal dimension will be 1.0 because it is only in the first
dimension (length). However, if the line is meandering, then
it starts to cover more of the plane, and introduces the concept
of breadth. Since the line itself still has a single dimension, it
cannot be described as a two-dimensional plane, so the fractal
dimension of the line is between one and two because it takes
up more space than a single dimension, but it cannot have a
second dimension because it is only a line. Similarly, if the
line begins to fold back on itself, e.g. when a trajectory is
recursive, the concept of depth can be introduced, even though
the line is one-dimensional — allowing the line to have a
fractal dimension above two (Katz and George, 1985).
Therefore, it may be considered that the fractal dimension
of a curve indicates its regularity. A curve with a low fractal
dimension would be regular and predictable. Similarly, a curve
with a high fractal dimension would have irregularly spaced
changes in direction, apparently at random. The concept of a
curve which contains random directional changes has been
defined in mathematics as a 'random walk'. Curves with D
values approaching two have been defined as random walks.
The 99% confidence limits for D values of random walks are
1.26-2.61 for a 30-point trajectory and 1.33-2.37 for a 60point trajectory (Katz, 1988).
If one looks at the 'random walk' path, it is possible
to make comparisons between it and the trajectory of a
hyperactivated spermatozoon (Katz, 1988). In the same way,
a plot of Brownian motion, which has been used as a classic
example in the derivation of the fractal dimension (Mandelbrot,
1983), is similar to the path of a hyperactivated human
spermatozoon. These similarities have suggested the possibility
of the fractal dimension being a new kinematic definition for
human sperm motility (Schoevaert-Brossault and David, 1984;
Davis and Siemers, 1995).
The aims of this study were to determine whether D values
were consistent with the results obtained by a traditional
kinematic analysis, and to determine whether there was any
effect of frame rate on the fractal dimension.
Materials and methods
Sperm preparation and video recording
The trajectories used in our study were those used in previous
studies of human sperm hyperactivation. Hence, the methods used
for sperm preparation, video recording and track reconstruction have
been presented in detail elsewhere (Mortimer and Mortimer, 1990;
Mortimer and Swan, 1995). Briefly, semen was collected by masturbation from five healthy volunteers. After the semen had liquefied (37°C
for 30 min), 0.25 ml aliquots were gently layered under 1 ml volumes
of IVF culture medium [either T6 medium supplemented with 10%
(v/v) serum or human tubal fluid (HTF) medium supplemented with
30 mg/ml human serum albumin]. The motile spermatozoa were
allowed to swim-up into the culture medium during a 30 min
incubation period (37°C, 5% CO2 humidified atmosphere). The upper
0.5 ml portions of the preparations were harvested, placed in fresh
tubes and re-incubated until video taping.
For video taping sperm movement, 5 ul aliquots of the prepared
sperm suspensions were placed in ~30 urn deep chambers which had
1050
been prewarmed to 37°C. For the 30 Hz hyperactivated track study,
and for the 30 versus 60 Hz comparison, a Leitz Orthoplan microscope
with a X40 phase-contrast objective was used For the circling and
segmented track studies, either the Lcitz Orthoplan microscope or
Olympus optics with a X10 phase-contrast objective was used. The
final magnifications when the video tapes were replayed were X354O
for those made using the Leitz microscope and X1270 for those
made using the Olympus optics. Sperm movement was recorded
using an NTSC video system, and the video tapes were replayed on
either a Sony SL-HF75O VCR (which gave 30 images/s) or a Panasonic
F66 VCR (which gave 60 images/s) on freeze-frame playback.
Track reconstruction
The video tapes were replayed and sperm tracks classified as either
forward progressive or hyperactivated, depending on the flagellar
motility, or as circling. After a spermatozoon was judged to be
suitable for inclusion in the study, on the basis of movement pattern
and length of time in the field of view, the tape was rewound and
successive positions of the centroid (centre of the spermatozoon's
head) were plotted onto acetate sheets attached to the monitor's
screen. The acetates were placed over mm graph paper and the (x, y)
coordinates of each track point were recorded. The movement
characteristics of each track were derived from these coordinates as
described previously (Mortimer and Mortimer, 1990; Mortimer and
Swan, 1995). A trajectory was classified as hyperactivated if it
met all of the kinematic criteria, i.e. for 60 Hz trajectories: VCL 5=
180 um/s, linearity (LIN) =£45%, wobble (WOB) <50% and either
ALHmetn > 6 urn or ALHm,^ > 10 |im (Mortimer and Swan, 1995);
or for 30 Hz trajectories: VCL 2= 100 um/s, LIN < 60% and ALHmc,
> 5 um (Mortimer and Mortimer, 1990). These kinematic definitions
included both 'star-spin' and 'transitional' patterns of hyperactivated motility, and were chosen for use in this study because
they were free of machine bias, being derived using manually
reconstructed trajectories. The average path was calculated by
determining the seven-point running average for each point
('smoothing') for the 60 images/s trajectories, and the five-point
running average for the 30 images/s trajectories. For the phaseswitching
tracks,
DANCEMEAN
was also
calculated
[DANCEMEAN = ALH^XCVCL/VSL); Robertson et a/.,
1988], where VSL = straight line velocity.
Calculation of the fractal dimension (D)
It was possible to calculate a fractal dimension value for each
trajectory using its (x, y) coordinates. Initially, two equations were
compared for the calculation of D (Mandelbrot, 1983; Katz, 1988).
These were:
D = log (Lyiog (d),
(1)
where D —fractaldimension, L = length of the curvilinear path (Jim):
and d = planar extent of the curve [maximum distance between the
origin and any plotted point (Jim)]:
d = <Kxx - xf
+ (yi
-y^\.
And
D = log («ynog (n) + log (dJL)],
(2)
where D = fractal dimension, n = number of track intervals
('steps') = [(number of track points) — 1], d — planar extent of the
curve (um) and L — length of the curvilinear path (um).
Worked examples for the calculation of D using Equation 2 are
given in Figure 1.
Fractal analysis of sperm movement
Non-hyperactfvated track
Start
Rnish
n » number of track intervals = 30
d - maximum distance of any pointfromstart = 104.0 \wn
L • sum of distances ab, be, etc = 105.6 um
D - tog (30) / (tog (30) +tog(104.0/105.6))» 1.00
Hyperactivated track
Start
d » 26.1 \m\
L-180.2 um
D =tog(30) / (log (30) + log (26.1/1802))
1.59
Figure 1. Worked examples for the calculation of D using Equation 2 (see text). The trajectories were reconstructed from 30 Hz images.
This equation incorporates n, meaning that the average step length is considered. The step length is the distance between consecutive points
along the trajectory (which can also be used to calculate instantaneous velocity values), while the average step length is the mean of the
distances between all pairs of consecutive points along the trajectory.
Table L Comparison of fractal dimension (D) values for 30 Hz
hyperactivated tracks for both the correct magnification and for X2
magnification (mcf2)
Parameter
Range
Median
Mean
SD
D, mcf2
1.44-1.89*
1.66'
1.651
0.121
1.74-3.45
2.47"
2.43b
0.43"
b
1.56-2.22°
1.84C
1.84C
0.18°
mcf2
1.74-3.45b
2.47b
2.43b
0.43b
D\ and £>j were derived using Equations 1 and 2 respectively (see text for
details). Within each parameter, values with different superscripts were
statistically different by paired /-tests (all P < 0.001); values with the same
superscripts were identical.
Results
Comparison of methods for the calculation of D
To compare Equations 1 and 2 for the estimation of D, the
fractal dimensions of 30 previously analysed 30 Hz hyperactivated tracks were derived (Table I; Mortimer and Mortimer,
1990). Equations 1 and 2 were used to derive D\ and Dj
respectively. The D values obtained by each equation were
significantly different, with Equation 2 yielding higher D
values. However, when the magnification correction factor was
doubled, the D values from Equation 1 were changed, while
those from Equation 2, incorporating n, were not This result
was predicted because Equation 1 did not make use of the
average length of track intervals (average step length), while
Equation 2 did. Thus, as the magnification of the track
increased, the curve appeared to take up more of the plane,
thus increasing the fractal dimension, whereas when the
average step length was considered, there was a concept of
scale already inherent in the calculation. For this reason,
Equation 2 was used in the remainder of this study because it
was not affected by the magnification.
Effect of image sampling frequency on hyperactivated and
forward progressive tracks
The reconstructed trajectories used in the comparison of
30 and 60 Hz values for both forward progressive and
hyperactivated tracks have been shown previously to meet
published criteria for 30 and 60 Hz forward progressive and
hyperactivated tracks (Mortimer and Swan, 1995). Therefore,
it may be assumed that these tracks were sufficiently representative for the determination of discriminatory D values. A
total of 40 forward progressive and 40 hyperactivated tracks
were analysed at both 30 and 60 Hz.
There was a significant difference between the D values
obtained at 30 and 60 Hz for both forward progressive and
hyperactivated tracks (P < 0.001; Figure 2). For forward
progressive tracks, the median D value was higher for the
60 Hz tracks (1.05 versus 1.09), whereas the 30 Hz median
was higher for the hyperactivated tracks (1.61 versus 1.54).
Therefore, the frame rate used for the analysis affected the
D value.
However, the range of D values for forward progressive
tracks was below, that of hyperactivated tracks, regardless
of the frame rate used (Figure 2; P < 0.001). For forward progressive tracks, D =£ 1.20 (1.00-1.15, 30 Hz; 1.031.20, 60 Hz); for hyperactivated tracks, D 5= 1.30 (1.31-2.31,
30 Hz; 1.30-2.05, 60 Hz). Therefore, for selected or 'ideal'
forward progressive and hyperactivated tracks, it was possible
to discriminate between them on the basis of D value alone.
Figure 3 shows ten 60 Hz tracks representing D values of
between 1.03 and 2.05. From Figure 3 it can be seen that
relatively small changes in D reflect substantial changes in
movement pattern.
Circling tracks
Because hyperactivated motility is denned by a number of
different kinematic criteria, including VCL, LIN and ALH, it
1051
S.TJVlortiiner, MA.Swan and D.Mortimer
2.5-1
Table IL Kinematic and fractal values for six circling tracks plotted at 60
imagcs/s
I 2.0 1
E
•o
I 1.51
Track
VCL (UJTI/S)
LIN (%)
WOB (%)
A L H , ^ (Jim)
1
2
3
4
5
6
168.4
199.0
101.9
187.1
133.7
107.7
7
13
13
39
17
4
54
54
71
58
58
59
2.6
2.4
1.3
2.3
1.6
1.3
D
.69
.50
.42
.32
.45
.53
ALH = amplitude of lateral head displacement; D = fractal dimension;
U N = linearity; VCL = curvilinear velocity; WOB = wobble.
1.0 30 Hz
60 Hz
Non-hyptractivated
30 Hz
60 Hz
Hypcractivated
Figure 2. Comparison of fractal dimensions (D) of forward
progressive and hyperactivated tracks analysed at both 30 and
60 Hz (n = 40 in each motility group). The bars show the ranges
of D values, the spots represent the medians. The two broken lines,
at D = 1.20 and 1.30, emphasize the lack of overlap between
hyperactivated and non-hyperactivated D values. There was a
highly significant difference between the 30 and 60 Hz values.
D-1.03
D-1.10
Figure 3. Tracks of 10 spermatozoa (reconstructed at 60 Hz)
displaying a range of movement patterns to illustrate how
apparently small changes in the fractal dimension, D, reflect
substantial alterations in trajectory complexity.
1052
is necessary to test whether a new kinematic criterion for
hyperactivation is actually an indicator for hyperactivated
motility or merely low track LIN. An example of a nonhyperactivated pattern of movement with low LIN is a circling
track. Circling tracks would be expected to have most kinematic
values in the forward progressive range, except for LJN. Thus,
if the D value was low for these tracks, it would suggest that
LIN, rather than hyperactivated motility per se, was the primary
influence on D.
Six non-hyperactivated circling tracks were identified on
the same video tapes used for the above analyses and the
trajectories were reconstructed at 60 Hz. They were analysed
as above, using (x, y) coordinates. Employing the traditional
kinematic values, all of the tracks were classified as forward
progressive because although the LIN values were very low,
the ALH and WOB values were outside the ranges for
hyperactivated motility (Table II). Only four out of six of the
tracks had VCL values below the threshold for hyperactivated
motility. The range of D values was 1.32-1.69, with the track
with the lowest D value having the highest LIN. These results
suggested that the fractal dimension of a trajectory was related
to its LIN. This hypothesis was tested by studying a series of
'non-ideal' trajectories.
Long tracks
Because it appeared that D was only able to discriminate
between the 'ideal' hyperactivated and forward progressive
tracks based upon LJN rather than hyperactivation, 29 long
tracks were analysed in which both hyperactivated and
forward progressive motility occurred. The trajectories were
reconstructed at 60 Hz and (x, y) coordinates assigned. They
were then split into 0.5 s segments (31 points/segment), with
the last point of one segment being the first point of the next
A total of 255 track segments were analysed.
The segments were classified as hyperactivated when a
group of kinematic definitions were met, namely VCL >
180 nm/s, LIN « 45%, WOB < 50% and either ALH™*,, 3=
6.0 |im or ALHma, s» 10.0 |im. If these criteria were not met,
the track segments were described as forward progressive,
even if they showed some but not all of the criteria for
hyperactivation.
There were a total of 152 track segments classified as
hyperactivated. Of these, 149 (98%) had D 5= 1.30, with all
hyperactivated tracks having D s= 1.20.
Of the remaining 103 segments, 43 had LIN =s 45% but
Fractal analysis of sperm movement
Table ID. Correlation coefficients between kinematic values and fractal dimension (D) for long tracks, 0.5 s segments plotted at 60 images/s
Kinematic values
VCL versus D
VAP versus D
VSL versus D
UN versus D
STR versus D
WOB versus D
A L H ^ versus D
ALHna, versus D
BCF versus D
DANCEMEAN versus D
Hyperactivated only
All track segments
Non-hyperactivated only
r
P-
r
r2
r
P-
0.50*
-0.32
-0.68 b
-0.77 b
-0.67 b
-O.67b
O.^
0.55*
-0.24
0.72b
0.25
0.10
0.46
0.59
0.45
0.45
0.36
0.31
0.06
0-52
0.21
-0.20
-0.6 l b
-0.65 b
-0.56 b
-0.34
0.22
0.14
-0.17
0.63b
0.05
0.04
0.37
0.42
0.32
0.12
0.05
0.02
0.03
0.39
0.06
-0.55 1
-0.6711
-O.78b
-0.59 b
-0.74 b
0.57b
0.45*
-0.18
0.75b
0.003
0.30
0.45
0.61
0.35
0.55
0.32
0.20
0.03
0_57
ALH = amplitude of lateral head displacement; BCF = beat/cross frequency; DANCEMEAN = ALH racM X(VCLA'SL); STR = straightness, i.e.
(VSL/VAP)X1OO; VAP = average path velocity; VCL = curvilinear velocity; VSL = straight line velocity. See the legend to Table II for other abbreviations.
Superscripts denote P values CP < 0.01; bP < 0.001); correlation coefficients without superscripts were not significant.
did not meet the VCL and/or ALH criteria for hyperactivated
motility. Of these 43 track segments, 36 (84%) had D s* 1.30,
indicating that LIN rather than hyperactivation per se was
influencing the magnitude of D.
The correlation between LIN and D was tested for all
255 track segments, and a correlation coefficient of r = -0.77
was found (r2 = 0.59, P < 0.001). This indicated that LIN and
D were significantly but inversely correlated. When only the
hyperactivated segments were considered, the correlation coefficient for D and LIN was r = -0.65 (r2 = 0.42, P < 0.001); for
non-hyperactivated tracks, r = -0.78 (r2 = 0.61, P < 0.001).
Other kinematic criteria found to be significantly correlated with
D were DANCEMEAN, VSL and WOB (Table III).
Discussion
To compare results between laboratories, all investigators must
understand what is meant by the descriptive terms used to
define the motility patterns of capacitating human spermatozoa.
In the past, this has been attempted by assigning ranges for
the kinematic values for each track type (Robertson et al., 1988;
Mortimer and Mortimer, 1990; Burkman, 1991; Mortimer and
Swan, 1995). However, there is still potential for confusion,
given that different computer-aided sperm analysis (CASA)
instruments may assign different kinematic values to the same
track, and that many of the kinematic parameters used in the
discrimination between forward progressive and hyperactivated tracks are frame rate dependent, i.e. they will vary
according to the image sampling frequency used for the
analysis (Mortimer et al, 1988). One of the main areas of
difference between instruments is in the method used to
calculate the average path. The average path may be determined
by calculating its coordinates from fixed-point running averages
of the coordinates of the track points, or by applying adaptive
smoothing algorithms to the curvilinear trajectory, depending
on the degree of deviation from the general direction of the
curvilinear path. Because ALH and BCF are calculated with
respect to the average path, it is obvious that changes in the
method of derivation of the average path will change its length
and trajectory, thus altering the ALH and BCF values calculated
from it As ALH is a commonly used indicator for hyper-
activated motility, differences in its calculation by different
instruments may lead to differences in the perceived proportion
of hyperactivated cells estimated in a population.
The derivation of ALH for trajectories of hyperactivated spermatozoa is further complicated by the star-spin type of hyperactivated motility. When the spermatozoa exhibit this nonprogressive form of hyperactivated motility, it is not possible to
discern a meaningful average path when the two-dimensional
reconstructed track is considered. Because the sperm head is
moving significantly in the third dimension, it is not always
possible to recognize separate waves in the centroid trajectory,
which makes any ALH values derived subject to a great deal of
error. It would be preferable to define hyperactivated motility,
or perhaps the motility of all capacitating spermatozoa, without
using ALH as a major characteristic.
In response to these difficulties, a fractal analysis of the trajectory has been proposed as an alternative kinematic value (Schoevaert-Brossault and David, 1984; Davis and Siemers, 1995).
The fractal dimension (£>) of a track was thought both to be
independent of the image sampling frequency and to allow discrimination between forward progressive and hyperactivated
paths by relating the length of the spermatozoon's trajectory in
a given period to the maximum distance it travels from the origin
in that time (Davis and Siemers, 1995). Straight paths with low
amplitudes would be expected to have D values approaching
1.0, because the distance travelled and the maximum distance
from the origin would be almost identical. Conversely, in high
amplitude paths, the D value would approach 2.0 because the
trajectory could still be quite long, while having a very low net
space gain (Figure 3).
Many different methods have been published for the
derivation of fractal dimensions. In sperm track analysis,
the emphasis is on the speed and simplicity of calculation, as
well as the ability to employ a post-hoc analysis for the
derivation of kinematic values using centroid coordinates,
allowing incorporation into present generation CASA instruments. Our decision to use Equation 2 for the derivation of D
was based on our observation that it was independent of the
magnification of the trajectory (Table I).
There was a small, yet significant, effect of image
sampling frequency upon the fractal dimension of a track
1053
S.TMortimer, MA^wan and D-Mortimer
(Figure 2). When the 40 forward progressive trajectories were
compared, the D values were higher for the 60 than for the
30 Hz trajectories. Conversely, for the hyperactivated tracks,
the range of D in the 60 Hz tracks was smaller than for the
30 Hz tracks, and the median was lower. However, despite
these small frame rate-dependent differences, the fractal
dimension was always >1.30 for hyperactivated tracks and
<1.20 for forward progressive tracks. The threshold value of
D 5» 1.30 for hyperactivated trajectories corresponds well with
the 99% confidence limits for a random walk (30 point track
1.26-2.61; 60-point track 1.33-2.37) (Katz, 1988). These
results would indicate that the hyperactivated trajectories
studied were 'random walks', where it is not possible to predict
the angle of direction change from one interval to the next
nor the length of consecutive intervals.
The reason for the good discrimination between forward
progressive and hyperactivated trajectories using D became
obvious when circling tracks were considered. The circling
tracks all had low LJN values because LIN is the ratio of the
straight line path to the curvilinear path, which is low for both
circling and hyperactivated paths. The circling tracks did not
have high ALH values, so even though LIN was low, they
could not be considered hyperactivated because a group of
kinematic criteria (VCL, LIN, WOB and ALH) must all be met
for classification as hyperactivated using an image sampling
frequency of 60 Hz. The D values for all of the circling tracks
were above the threshold value of D = 1.30, i.e. within the
range established for hyperactivated spermatozoa, indicating
that D may have been directly related to LIN. The absolute
value of VCL was not an influence on D, because four of the
circling trajectories had VCL values below the hyperactivation
threshold.
Further evidence for an inverse relationship between LIN
and D was provided by the analysis of a series of 'nonideal' trajectories comprising both hyperactivated and forward
progressive segments. In this study, it was found that some
non-hyperactivated segments had D values above the hyperactivation threshold, but coincidentally had LJN values below
the hyperactivation threshold. Correlation coefficients calculated for all kinematic values and D for each track segment
revealed a significant inverse correlation between LIN and D
(r = -0.77, P < 0.001) (Table ffl), thus supporting the
hypothesis.
The fractal dimension of a trajectory is an important parameter because it gives an indication of the pattern of the
trajectory, even though some patterns (such as the circling
tracks) will give random walk values when they are in fact
not This observation was verified by Katz and George (1985)
in their derivation of the equation for the calculation of the
fractal dimension: 'As we use them, fractal dimensions are
quantitative assessments of the 'space-filling' properties of
curves in a plane. Like any single automated procedure, the
fractal characterization is not a complete pattern recognition
system: it will not distinguish between all classes of curves
that humans naturally distinguish.'
The equation we used for the derivation of D required only
the number of track intervals, die length of the curvilinear
path and the maximum distance of any point on the curve
1054
from the origin (the first point). The effect of frame rate
was also considered in this study because different CASA
instruments use different image sampling frequencies for track
analysis. The kinematic criteria used at present are known to
be influenced by the image sampling frequency, and threshold
values for hyperactivated motility have been derived for each
frequency used. We found that although the D values were
influenced by the image sampling frequency, inclusion of the
number of images (steps) in the calculation of D allowed a
threshold value to be derived which was identical for both 30
and 60 Hz trajectories (D 3= 1.30).
In consideration of these comments, and of our results,
the fractal dimension cannot be the sole criterion for the
classification of a trajectory as hyperactivated. However,
because it does not rely on any calculation of the average path
for its derivation, the value of the fractal dimension is that it
could remove much of the variability between CASA instruments and allow comparison between laboratories.
Acknowledgements
The authors are grateful to Associate Professor Neville de Mestre
(School of Information Technology, Bond University, Gold Coast,
Queensland, Australia) for his advice on the calculation of the fractal
dimension, and for his critical reading of the first draft of the
manuscript.
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Received on October 30, 1995; accepted on February 16, 1996