Applied Mathematics, 2013, 4, 1673-1681
Published Online December 2013 (
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Applying a Mathematical Model to the Performance of a
Female Monofin Swimmer
Elisée Gouba1, Balira Ousmane Konfe2, Ousseynou Nakoulima3, Blaise Some1, Olivier Hue4
1Laboratoire LANIBIO, U.F.R. S.E.A., Université de Ouagadougou,
Ouagadougou, Burkina Faso
2Laboratoire LAIMA, Institut Africain d’Informatique, Libreville, Gabon
3Département de Mathématiques et Informatique, Université Antilles-Guyane,
Pointe-à-Pitre, France
4Laboratoire A.C.T.E.S., UPRES-EA 3596, U.F.R. S.T.A.P.S.-U.A.G.,
Faculté de Médecine, Pointe-à-Pitre, France
Received March 6, 2013; revised April 6, 2013; accepted April 13, 2013
Copyright © 2013 Elisée Gouba et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This study sought to determine the best method to quantify training based on heart rate data. It proposes a modification
of Banister’s original performance model to improve the accuracy of predicted performance. The new formulation in-
troduces a variable that accounts for changes in the subject’s initial performance as a result of the quantity of training.
The two systems models were applied to a well-trained female monofin swimmer over a 24-week training period. Each
model comprised a set of parameters unique to the individual and was estimated by fitting model-predicted performance
to measured performance. We used the Alienor method associated to Optimization-Preserving Operators to identify
these parameters. The quantification method based on training intensity zones gave a better estimation of predicted per-
formance in both models. Using the new model in sports in which performance is generally predicted (running, swim-
ming) will help us to define its real interest.
Keywords: Training Quantification; Banister’s Model
1. Introduction
In 1975, Banister et al. [1-4] proposed a systems model
to predict athletic performance. This model (and its ex-
tensions) is based on two antagonistic functions: the
positive function can be compared to a fitness impulse
resulting from the organism’s adaptation to training and
the negative function is similar to a fatiguing impulse.
Each function comprises a set of parameters that are in-
terpreted as the individual’s response profile, which can
be used for training prescription. Parameters include, for
example, time to recover performance and time to peak
performance after training completion.
Monofin swimming is a relatively new sport (recog-
nized by the IOC in 1986) that consists of propelling the
body at the water surface or underwater with an undula-
tion accentuated by the monofin [5,6]. Very high swim
velocities can thereby be reached (up to 14 km·h1), and
this sport is attracting a growing number of enthusiasts.
Although a few studies have investigated the techniques
of monofin swimming [5,7-9], little is actually known
about fin swimmers and the parameters that contribute to
performance. In 2005, O. Hue et al. found that both an-
thropometric and physiological (aerobic and anaerobic
components) factors contributed to the performances of
French West Indian monofin swimmers. The finding that
both maximal oxygen uptake and the second ventilatory
threshold were significantly correlated with performance
[10], suggested that monofin swimming performance,
like running [11,12], swimming [13,14] and cycling [15],
could be extrapolated using Banister’s model. However,
this model had been criticized by several authors [16-19]
who reported that the practical interpretation of the posi-
tive and negative influences might be difficult. For ex-
ample, Taha and Thomas [19] criticized the models stem-
ming from Banister’s original model [13], stressing their
inability to accurately predict future performance, the
difference between the estimated time course of perform-
ance changes and experimental observations, and the fact
that most of these models were poorly corroborated by
physiological mechanisms.
In Banister’s model, performance is mathematically
related to the training load using any one of the three
basic methods of training quantification: a method using
the mathematical formula developed by Banister and
Hamilton [20], another using the three intensity zones
defined by the aerobic and anaerobic ventilatory thresh-
olds, and the last using five intensity zones. Although
several authors have used one or another method to
quantify training without providing justification (for ex-
ample [21-23] used the mathematical formula; and [12,
14,24] used the training quantification method with five
intensity zones), we cannot rule out the possibility that
using various training quantification methods may affect
the accuracy of performance prediction with Banister’s
Our main objectives in this study of an internationally
ranked monofin swimmer were the followings: 1) to de-
termine the best method of quantifying training that pro-
vides a pertinent model over a 24-week training period
and for each time that performance is measured; and 2)
to propose a new model based on Banister’s original
model that would improve the accuracy of predicting
2. Methods
2.1. Subject
A 17-year-old female monofin swimmer, weighing 64 kg
for 164 cm, was recruited. The subject had previous
competitive swimming experience and at the time of the
study was among the top 15 monofin swimmers in the
Junior World ranking. The subject was fully informed of
the study conditions and gave written consent in accor-
dance with the regional ethics committee before partici-
2.2. The Progressive and Maximal Exercise Test
The ramp exercise test began with a 3-minute warm-up at
30 W, and the load was increased by 15 W every minute
until exhaustion. Pedaling speed remained constant at 70
rpm during the entire test.
Gas exchanges were measured during the first and
third repetitions using a breath-by-breath automated ex-
ercise metabolic system (Zan 680, Zan, Oberthulba, Ger-
many) and oxygen uptake was considered maximal (i.e.,
VO2max) if at least three of the following four criteria
were met:
1) a respiratory exchange ratio greater than 1.10,
2) attainment of age-predicted maximal heart rate
(HRmax) [210 (0.65 × age) ± 10%],
3) an increase in oxygen uptake (VO2) lower than 100
ml with the last increase in work rate, and
4) an inability to maintain the required pedaling fre-
quency (70 rpm) despite maximum effort and verbal en-
couragement. The heart rate at rest (HRrest) and the maxi-
mal aerobic power (MAP) were also determined. A 10-
lead electrocardiogram (12-lead ECG, Del Mar Deynolds,
Spacelabs, Healthcare Inc., UK) was used to monitor
heart rate continuously. The results are shown in Table
2.3. Ventilatory Threshold Determination
The ventilatory thresholds were visually determined ac-
cording to the method of Wasserman et al. [25,26],
which describes the inflection points in pulmonary ven-
tilation during incremental exercise. The ventilatory
thresholds (VTs) were identified when breakpoints oc-
curred in the VE/VO2 and VE/VCO2 curves. The first
ventilatory threshold (VT1) was identified by the point of
non-linear increase in VE and a clear increase in VE/VO2.
At the same time, VE/VCO2 remained constant or slightly
decreased. VT1 was also determined by the V-slope me-
thod of [27], which consisted of plotting VCO2 against
VO2 to identify a steep rise in VCO2 compared with the
rise in VO2 with increasing intensity. The second venti-
latory threshold (VT2), which corresponds to the respi-
ratory decompensation for metabolic acidosis, was iden-
tified by a second non-linear increase in VE and a second
clear increase in VE/VO2. At this point, VE rose more
rapidly than VCO2, leading to a rise in VCO2. Three ex-
perienced investigators independently deter- mined these
thresholds (Table 1).
2.4. Experimental Design
During a 24-week period, the subject continued her re-
gular training regime and recorded details of each train-
ing session. In addition, she completed performance tests
and fatigue questionnaires. The performance tests were
not conducted during periods of national or international
2.5. Fatigue Questionnaire
Before each performance test, the subject completed the
original version of the Profile of Mood States (POMS)
questionnaire developed in 1971 by McNair, Lorr &
Droppleman [28]. The subject used a 5-point scale (0 =
“not at all” to 4 = “extremely”) to respond to each item.
In the present study, only the responses to the fatigue
subset of the POMS were considered.
2.6. Performance Test
Performance tests over a distance of 700 m were made
every three weeks. During training sessions and per-
formance tests, the subject wore a heart rate monitor
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Table 1. Physiological performance obtained for the subject.
VO2max VT1 VT2 MAP MAP (VT1)MAP (VT2)HRmax HR (VT1) HR (VT2)
ml·min1·kg1 % VO2max % VO2max watts watts watts bpm bpm bpm
45.5 61.5 82.1 232 151 202 194 155 181
VO2max, maximal oxygen uptake; VT1, ventilatory threshold; VT2, respiratory compensation threshold; MAP, maximal aerobic power (power attained at
VO2max); HRmax, maximal heart rate; MAP (VT1), maximal aerobic power at VT1; HR (VT1), heart rate corresponding to VT1.
(Polar S810i, Polar Electro, Kempele, Finland) that re-
corded heart rate every 5 s. The heart rate data for each
exercise bout was analyzed using Polar Precision soft-
3. Training Quantification
The exercise training intensity can be quantified using
several methods based on heart rate or lactate response,
the expression of intensity relative to maximum heart
rate or oxygen consumption, or the expression of inten-
sity relative to ventilation or the anaerobic threshold.
However, only the subset of methods generally used in
training-performance modeling were studied.
For each exercise session, training impulse (TRIMP)
was quantified three ways (Figure 1):
With the mathematical formula developed by Banister
and Hamilton [20],
Based on a repartition of heart rate into three zones of
intensity [29],
Based on a breakdown of heart rate into five zones of
intensity [14].
3.1. Mathematical Formula
The mathematical formula is given by:
wtd k x
where is the exercise duration expressed in minutes,
is defined by:
exercise rest
max rest
and is related to the subject’s gender; in our study,
the subject was female, thus
0.86 e
Here rest
R indicates the lowest measure of heart rate
recorded when the subject is awake and max
R, the
highest recorded during incremental testing on a cycle
3.2. Training Quantification with Three Intensity
The three intensity zones were defined taking into ac-
count the HR corresponding to the ventilatory thresholds.
Zone 1 includes all HR below the first ventilatory
threshold, zone 2 contains the HR between the two
thresholds, and zone 3 includes all HR above the second
The amount of training in each zone is equal to the ex-
ercise duration in minutes in this area, multiplied by the
physiological stress weighting factor for that intensity
zone (zone 1 1, zone 2 2, zone 3 3). The training
impulse for each exercise bout was thus recorded as the
sum of the training impulse scores for each intensity
3.3. Training Quantification with Five Intensity
The five intensity zones were defined according to the
study of Wood et al. [12]. The physiological stress
weighting factor used for each training intensity zone has
been reported elsewhere [14].
Figure 1 present training loads performed by the sub-
ject during the course of the study; training loads are
quantified with the three methods of training quantifica-
4. Modeling of the Response to Training
We used the model proposed by Banister et al. [13],
which considers the athlete as an open system with the
training impulse as the input and performance as the
output. This type of modeling models “cause” to “effect”
phenomena and usually leads directly to integral equa-
tions [30].
The time functions of performance and training
wt are mathematically related as
ptpp gt
where 0 is an additive term that depends on the initial
training status of the subject and * denotes the product of
is a transfer function that depends on
time and is defined by
exp exp
gt kkt
 
 
 
where 1 and are gain terms and k2
and 2
time constants.
The definition of the convolution product leads to:
Wee ks
Mathematic alformula
Figure 1. Training loads performe d by the subject during the course of the study; tr aining loads are quantified with the three
methods of training quantification.
 
0d0,ptpgswt ss st
 
expexpif 0
 
 
 
0ifwt sst 
 
ptpgswt ss 
 
ptpwsgt ss 
Time discretization of (4) using the composite trape-
zoid rule gives an estimation of the model performance
on day , n, from the successive training loads
with varying from 1 to
. Thus
01 2
exp exp
ni i
ni ni
ppkw kw
  
 
 
 
In (5), 0 appears as a constant during the study, al-
though it is known that performance varies with the
quantity of training. For this, we here propose a modified
Banister model, with performance on day based on
previous performance. Mathematically, we have for
pp kw
 
where is the number of performances measured and
n, the number of training loads between performances
5. Fitting the Model
The model parameters are unique to the individual and
were evaluated by minimizing the residual sum of square
RSS between the real and modeled performances:
RSSp p
where is the number of real performances. is
a non-linear function; we first used the Alienor method
and OPO* [31] to identify 121 2
,, and kk
. Computa-
tions were completed using Maple 12 software.
Calculation of and
The time to recover performance n was calculated by
resolution of the equation we obtained
12 2
12 1
The time to peak performance after training comple-
t, was obtained by resolution of the equation
we obtained
12 12
12 21
 
6. Statistical Analysis
The coefficient of determination gave the percent
variation explained by the model and was calculated to
establish the goodness of fit for the model. The statistical
significance of this fit was assessed by an analysis of
variance on the residuals with associated degrees of
freedom. Statistical significance was accepted if
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E. GOUBA ET AL. 1677
0.05p. The correlations between modeled fatigue and
the corresponding POMS fatigue subset scores were
analyzed using the Pearson product-moment correlation
coefficient. All statistical analysis was completed using
statistical software R.
7. Results
The coefficient of determination was equal to 0.28
for Banister’s model and 0.38 for the alternative model
(Figure 2). The values for 12
were 42.25 days
and 15.29 days, respectively. The weighting factors,
, were 0.00001083 and 0.000015 respectively.
The time needed after training impulse for the effects of
fatigue to be dissipated sufficiently to allow the effects of
training to return performance to the pretraining level n
was 7.81 days, and the time needed to reach maximal
performance after training impulse
and kk
t was 32.17 days.
Using the fatigue subset of the POMS questionnaire, the
athlete reported substantial fluctuations in fatigue status
across the training period [12]. The fatigue scores in this
study ranged from 2 to 13 on the 28-point scale. The fit
between the fatigue component of the model and the
POMS fatigue subset score gave (
Figure 3).
Figure 2 shows the relationship between model-pre-
dicted and actual performance.
Figure 3 illustrates the relationship between the fa-
tigue component of the model in arbitrary units (AU),
and the score from the fatigue subset of the POMS ques-
7.1. Comparison of the Three Methods of
RSS measures the gap between calculated and meas-
ured performance; the smaller this difference is, the bet-
ter the model. Using the three methods of training quan-
tification indiscriminately, we found that they gave the
same minimizers, 121 2
,, and kk
but the minimum of
differed depending on the method. We present the
results for Banister’s model in Table 2.
Mathematically, we can deduce that quantification us-
ing three intensity zones was the best method for our
study using Banister’s model.
With the modified model of Banister, we have the re-
sults in Table 3.
We eks
Modifiedmode l
Figure 2. The relationship between mode l-pr edic ted and actual performance.
POMSfatiguesubsetscor e
Fat igue(AU)
Fatigue poms
Figure 3. The relationship between the fatigue component of the model in arbitrary units (AU), and the score from the fatigue
subset of the POMS questionnaire.
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Table 2. Minimum of RSS assessed with Banister’s model
from the three methods of training quantification.
Quantification method Minimum of RSS
Mathematical formula 0.035645
Three intensity zones 0.035370
Five intensity zones 0.03761
Table 3. Minimum of RSS assessed with the modified
model of Banister from the three methods of training quan-
Quantification method Minimum of RSS
Mathematical formula 0.024020
Three intensity zones 0.025069
Five intensity zones 0.018657
From a mathematical point of view, training quantify-
cation using five intensity zones was the best method for
this study using the modified model of Banister.
7.2. Training Quantification and Changes over
In Table 4 , we present the results using Banister’s model
and the modified model according to the number of ac-
tual performances measured over time.
8. Discussion
The main findings of the present study were:
The modified model of Banister explains performance
better than Banister’s model using the three methods
of training quantification.
The method of training quantification plays an im-
portant role in predicting performance.
The choice of training quantification method should
be made on the basis of the number of performances
that will be measured.
The model parameters in the present study were simi-
lar to those reported previously [11,14,15,18,24]. The
weighting factors (1 and 2), giving a k k21
kk ratio of
1.36, and the positive and negative time constants
and 215.29 days
) of the models
were within the ranges previously reported. Values for
n and t
t were thus also within the reported ranges [14,
The systems model proposed by Banister et al. [13]
was reported to account for up to 94% of the variance in
actual performance [16], and 92% in the study of [12]. In
the present study, the variance in modeled performance
explained 28% of the variance in actual performance
over a 24-week training period using Banister’s model
Table 4. Modified model and Banister model parameters
and the minimum of RSS assessed according to the number
of performances measured with the quantification of train-
ing using five intensity zones.
N 1
k 1
k 2
1 0.0016050354.26 0.0017181 22.28 0.00125766
2 0.0016050354.26 0.0017181 22.28 0.0013061
3 0.0000108342.25 0. 000015 15.29 0.0049432
4 “ “ “ “ 0.0057478
5 “ “ “ “ 0.0186574
N 1
k 1
k 2
1 0.0016050354.26 0.0017181 22.28 0.00125766
2 0.0000108342.25 0. 000015 15.29 0.0027070
3 “ “ “ “ 0.0039183
4 “ “ “ “ 0.00833458
5 “ “ “ “ 0.0376192
N= the number of performances measured; and 2 = the fitness and
fatigue magnitude factors, respectively;
and 2
= the fitness and
fatigue decay time constants, respectively.
and 38% using the alternative model. These relatively
poor correlations may be due to the following:
First, we quantified training during a 24-week train-
ing period with only the parameters of the first test of
VO max
; thus the ventilatory thresholds were not
adjusted with a second test of VO2max (which could
have been part of the experimental design). In some
studies, ventilatory thresholds [12] or blood lactate
testing [14] were repeated during the testing protocol.
It has been demonstrated that values of target HR for
training orientation generally remain stable in elite
endurance athletes [32]. However, because our study
was somewhat long, we cannot rule out the possibility
that the ventilatory thresholds may have changed dur-
ing the training period, inducing changes in the per-
formance modeling.
Second, the number of actual performances (6 for this
study) was very low. It takes at least ten actual per-
formances to ensure that the adjustment of perform-
ance calculated from the model is statistically signi-
ficant [17]. However, in real conditions, more than
one maximal performance every three weeks seemed
unrealistic to us, at least for internationally-ranked
athletes who do a lot of traveling for international
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E. GOUBA ET AL. 1679
8.1. Modified Model of Banister
Banister and al. [13] initially proposed modeling the ef-
fects of training by quantifying the training and its effects
on performance. This model assumed that performance
results from a balance between the benefits of work (fit-
ness) and the associated risks (fatigue). As the relation-
ship between the amount of training and performance
seems more complex than the relationship assumed by
this model, many authors have proposed modified ver-
sions of the model. The model we propose here, unlike
the original model of Banister, lets the initial perform-
ance of the subject vary over time. This model and the
original model were applied to the same heart rate data
from the subject so that we could compare the two mod-
els using the minimum of RSS. Of course, in this case the
model that provides the lower value of the minimum of
RSS is the better model. Tables 2 and 3 showed that our
model gave the lower value. For example, with the train-
ing quantification method using five zones of intensity,
Banister’s model gave a minimum RSS value of 0.03761
versus 0.01865 for the modified model; this model pro-
vides the same benefits as Banister’s model but it signi-
fycantly reduces the residual error. With a good method
to quantify training, it could objectify the subject’s over-
all reaction to training.
8.2. Best Method of Training Quantification
Training load, which includes the duration and intensity
of exercise, can be seen as a reflection of the constraints
imposed on the athlete’s body. Its calculation is particu-
larly important in performance assessment. The limita-
tions of training load quantification come from the fact
that exercise intensity is assessed by heart rate. But the
relationship between the actual exercise intensity and HR
may be affected by body position [33], fatigue, heat, al-
titude [34], and the psycho-emotional state of the athlete.
Despite these limitations, the use of HR in general re-
mains the best way to quantify training.
We found that training quantification using intensity
zones allowed the two models of performance to ap-
proach measured performance more closely than with the
mathematical formula.
Training quantification using the mathematical for-
mula developed by Banister and Hamilton (1985) is easy
(logiperf software R2D2, Paris) and precise. This for-
mula could be improved by taking into account changes
in HR at thresholds over the course of training.
8.3. Best Method of Training Quantification over
The fit of our subject’s performances measured through-
out the study to the two models is illustrated in Table 4.
Table 4 shows once again that the method of training
quantification using intensity zones is better; depending
on the number of performances measured, three or five
intensity zones can be used.
Moreover, this study underlined the importance of
performance measurement that is regular and in suffi-
cient numbers (at least 3) so that the model parameters
unique to the individual reach a certain stability. For
example, in Table 4 with the modified model, the pa-
rameter 1
had a value of 54.26 days for the data from
one and two measurements of performance and then sta-
bilized at 42.25 days from the third measured perform-
8.4. Relationship between the Fatigue
Component of the Model and the Score from
the Fatigue Subset of the Profile of Mood
States (POMS) Questionnaire
The fatigue questionnaire allowed us to closely follow
the athlete’s reactions to work loads. Previous studies
used the POMS questionnaire to track mood changes in
athletic populations [35] or to attempt to validate the fa-
tigue component of a model [12]. In the study of Wood
et al. [12], the fatigue component of their model ac-
counted for only 56% of the variance in POMS-measured
fatigue during the 12-week training period. Similarly, in
the present study the fatigue component of the model
accounted for 58% of the variance in POMS-measured
fatigue during the 24-week training period (Figure 3).
This relatively poor correlation could be due to several
factors: the fatigue component of the model depends only
on the training impulse but in reality fatigue is deter-
mined by other life factors [12].
9. Conclusion
The aim of this study was to determine the best method
of training quantification using Banister’s model and a
modified model of Banister. The results show that the
methods using intensity zones provide a good perform-
ance fit. The modified version of Banister’s original
model that is proposed in this study reduced the gap be-
tween measured and calculated performance. But, like
the original model, it had limitations, indicated by the
poor correlations between predicted and real perform-
ance. Using this new model in sports in which perform-
ance is usually predicted (i.e., running, cycling or swim-
ming) will help us to define its real interest.
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