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Energy-supply systems and
whole-body bioenergetic models
for cycling
Manuel Knitza (01/649991)
University of Konstanz
Multimedia Signal Processing
Prof. Dr. Dietmar Saupe
Contents
List of Figures
ii
1 Introduction
1
2 Types of energy supply
2.1 Anaerobic energy supply . . . . . . . . . . . . . . . . . . . . . . .
2
3
2.2
Aaerobic energy supply . . . . . . . . . . . . . . . . . . . . . . . .
4
3 Physiological performance parameters
6
4 Physiological parameters and hydraulic models
4.1 Two-component models . . . . . . . . . . . . . . . . . . . . . . .
8
8
4.2
Three-component models . . . . . . . . . . . . . . . . . . . . . . .
15
4.3
Applicability and development of the 3-component model . . . . .
20
References
21
i
List of Figures
2.1
Relative energy potential of each system . . . . . . . . . . . . . .
2
4.1
The CP concept as a hydraulic model . . . . . . . . . . . . . . . .
10
4.2
4.3
Wilkie’s correction to the CP model (1980) . . . . . . . . . . . . .
The effect of great values for t on τ -extended Eqn. (4.3) . . . . . .
11
12
4.4
Differences between Eqs. 4.1 and 4.3 . . . . . . . . . . . . . . . .
12
4.5
Wilkie’s correction to the CP hydraulic model (1980) . . . . . . .
13
4.6
4.7
Diagrammatic representation of the 3-parameter CP model equation 14
Margaria’s hydraulic bioenergetic model (1976) . . . . . . . . . . 16
4.8
The generalised M-M hydraulic model (1990) . . . . . . . . . . . .
17
4.9
Configurations of the M-M model (1986) . . . . . . . . . . . . . .
18
4.10 The generalised M-M hydraulic model (1990) . . . . . . . . . . . .
18
ii
1 Introduction
For the last 50 years attempts have been made to understand the interrelations
of power output and power supply of the human body. Hydraulic models have
been used successfully to depict the dynamics of physical strain and regeneration. Thus the availability of different body’s power supplies can be derived.
Hydraulic models represent the bioenergetic interrelations of power output and
regeneration. Furthermore, they explain how the output power is composed of
the several components of the body’s power supply systems.
Besides abstracting the review article “The critical power and related wholebody bioenergetic models”[8], this work intends to simplify comprehensibility of
interrelation between the presented hydraulic models and the body’s types of
energy supplies [9]. Due to the scope of the paper this work does not intend to
explain in greater detail how biochemical processes in the human body work.
Outline of this seminar paper
Muscles convert chemical energy into mechanical energy. Therefore, the first part
of this paper deals with the four basic metabolic components which muscles use
to produce force and perform work.
Physiological parameters, describing capacities and limitations for work and
power, are discussed in the second part, whereas the third part is dedicated to
the bioenergetic models based on those physiological parameters.
At the end applicability as well as benefits for the powerbike’ (ergometer Cyclus2, see http://www.cyclus2.com) are discussed.
1
2 Types of energy supply
The human body’s systems of energy supply are complex and interrelating. In
order to give a brief overview for a better comprehension of how the systems
interact, the following illustration was taken from a San Diego State University
course’s (“Physiology of Exercise”, Fall 2007) online material. As the source is
from 1976, this figure is used for schematic purposes only.
Fig. 2.1 is a schematic presentation showing how long each of the major energy systems can endure in supporting all-out-work. In this diagram the terms
oxidative and nonoxidative correspond to the aerobic (Chpt. 2.2) and anaerobic
(Chpt. 2.1) systems respectively. The solid line denoted immediate represents the
availability of phosphagenes and therefore is ranked among the anaerobic system.
Figure 2.1: Relative energy potential of each system - Energy sources for muscles
as a function of activity duration. Source: Edington and Edgerton,
1976, taken from Introduction to metabolism.
2
2.1 Anaerobic energy supply
2.1 Anaerobic energy supply
Anaerobically very high power output can be achieved. Being limited in capacity
this can lead, dependent on the power output, to exertion very fast. Although
regeneration of anaerobic energy reserves takes place, exercise’s power output
usually exceeds the amount of energy being replaced by the body. The anaerobic
system arises out of the alactic and lactic subsystems. Both subsystems were
discovered to yield the anaerobic working capacity (AWC, Chpt. 3) [8].
High energy phosphates (alactic)
Muscles can be provided with energy alactically anaerobic for approximately the
first 10 seconds of exercise, while neither production of lactic acid, nor need of
oxygen is involved. High energy phosphates (phosphagenes) allow maximal power
output rapidly, limited to short durations. Simplified the follwing happens during
depletion:
creatine phosphate
ADP GGGGGGGGGGGGGGGGGGGGGGGGA ATP
recovery:
ATP
creatine GGGGGGGGA creatine phosphate
The power results from adenosine triphosphate (ATP) and phosphocreatine (PCr)
available in muscles. ATP serves as a rapidly mobilizable reserve of energy. It
transports chemical energy within cells for metabolism. PCr is used to resynthesise (phosphorylate) ADP into ATP resulting in energy. While regenerating
ATP-value increases which leads to phosphorylation of creatine (cratine + ATP),
resulting in new PCr [4, p. 687f.].
After about 3 minutes of resting recovery is complete and the capacity of power
is available anew. If the capacity is depleted, anaerobic glycolysis intervenes to
keep up power supply.
Anaerobic glycolysis – Production of lactic acid (lactic)
The anaerobic glycolysis is the transformation of glucose to lactate. Because of
oxygen still not being available the muscles produce large amounts of lactic acid
3
2.2 Aaerobic energy supply
(pyruvate doesn’t completely get oxidised [4, p. 690]), which causes fatigue [6, p.
57] (inorganic phosphate as reason for fatigue also was discussed [11]).
High power usage can be met almost on demand by lactic anaerobic energy for
about 50 seconds after phosphagenes being used up. Compared to the replacement of energy stores in the anaerobic alactic system, resting time of over one
hour may be necessary for the body to get rid of all the lactic acid and return to
the pre-exercise level. Most reduction of lactic acid levels takes place within the
first 10 minutes of active recovery; removal of lactic acid is speeded up by light
activity.
2.2 Aaerobic energy supply
In contrast to the anaerobic energy sources oxygen is needed for the aerobic
chemical processes of producing energy. As well as the anaerobic energy system
it is comprised of two subsystems: aerobic glycolysis and lipolysis. For glycolysis
a delay of energy supply is implied due to the dependency on oxygen delivery via
the lungs and the circulation. Limitation in capacity is known not to be present
theoretically. Lipolysis has a delay of energy supply of over one hour and and is
limited to an even lower power output than oxidative glycolysis and has not been
extensively studied [9].
Aerobic glycolysis – Oxidation of glycogen
Known as ‘oxidative metabolism’ or ‘cell respiration’ – complementing anaerobic
energy – it is the most important (sub)system providing energy. Depending on
oxygen which is available indefinitely, the amount of producible energy is virtual
defined by the maximum amount of oxygen the lungs can absorb [8, p. 347f.]
(VO2max , Chpt. 3). For establishing 1–3 minutes are needed.
As the process cycle covers ten reactions, an explanation of the whole process
as well as the involved sub processes would be beyond the scope of this seminar
work. Simplified the process of aerobic glycolysis can be described as follows:
glycolysis −→ pyruvate decarboxylation −→ citric acid cycle
4
2.2 Aaerobic energy supply
The process of glycolysis is the conversion of glucose into pyruvate, but only
one sub process among others like pyruvate decarboxylation and citric acid cycle.
Effectively the citric acid cycle yields the most energy (approx. 63-66% [10]).
Nonetheless all the sub processes are of the same importance as they depend on
each other’s (end)products.
Lipolysis – Fuel stored as fat
As another aerobic source, energy may be available by stored fat. The resulting
power output is even lower than all other sources’ outputs mentioned before.
It therefore seems plausible that extensive studies haven’t been undertaken yet,
even more when paying attention to the fact that it may take up to an hour or
more of exercise until availability of energy [9, p. 491].
5
3 Physiological performance
parameters
Physiological parameters provide information about a person’s condition, fitness
and poerformance capability. This chapter sums up some phys. parameters that
are important for this seminar work and the models introduced in Chpt. 4.
Anaerobic working capacity (AWC)
The anaerobic working capacity is a component of anaerobic energy supply, limited in capacity and measured in joules. Utilisation is possible at slow or fast rate
as demanded. If totally depleted, exhaustion occurs [9]. Generally speaking it can
be understood as the sum of capacities as provided lactically as well alactically.
Critical power (CP)
The critical power, contrary to the AWC, is not limited in capacity, but has an
upper limit to the power output, which can (theoretically) be upheld indefinitely
without exhaustion. It is measured in watts and depends only on renewable
aerobic power [9], if the energy from regeneration (anaerobic lactic and alactic)
is neglected while a ‘steady state’ is reached. As this work progresses we will see
that investigation led to defining the amount of aerobic energy also is done by
VO2 and VO2max .
Maximum power (Pmax )
The maximum power describes a maximum instantaneous power. On the cycle
ergometer this means that a subject is unable to turn the pedals at a resistance
of Pmax . The relation between Pmax and CP is that at exhaustion the peak power
Pmax declines to CP [9, p. 494].
6
Oxygen consumption (VO2 and VO2max )
Both parameters define an absolute volumetric absorption rate of oxygen in the
lungs in litres per minute (l/min). The latter as well expresses a relative rate
in millilitres of oxygen per kilogram of bodyweight per minute (ml/kg·min) or a
percental value relative to the individual’s body. It reflects a physical fitness and
is also known as ‘maximal oxygen uptake’ or ‘aerobic capacity’.
Anaerobic Threshold (AT)
The anaerobic threshold (AT) is the power above which the muscles derive their
energy from nonoxygenic sources in addition to the oxygenic sources during exercise. Above this threshold the body can only operate for a short period of time,
before lactic acid builds up in the muscles. It is considered to be somewhere
between 90% and 95% of the individual’s maximum heart rate.
Aerobic Threshold (AeT)
The aerobic threshold (AeT) is a term to describe a level of exercise somewhat
below the AT. This tends to be at a heart rate of approximately 20 − 40 bpm
less than the anaerobic threshold and correlates with about 65% of the maximum
heart rate. The AeT is sometimes defined as the exercise intensity at which
anaerobic energy pathways start to operate and where blood lactate reaches a
concentration of 2 mmol/l (at rest it is around 1).
Estimating parameters
This section provides an insight into approximation of some of the above introduced physiological parameters. AWC, CP and Pmax for two- and three-parameter
models can be estimated by wingate anaerobic (WAnT) and weighted scoring test.
Within the scope of this work (and because D. Höckele already did [3]) neither
the procedures of the above tests, nor the derivation of the parameters will be
described.
7
4 Physiological parameters and
hydraulic models
This chapter is mainly dedicated to the review paper by R. H. Morton [8]. The
development of hydraulic models is investigated whereas the focus lies primarily
on the findings of Wilkie and Morton.
4.1 Two-component models
The hydraulic models in the following sections can be used to depict the dynamics
of physical strain and regeneration. They diagrammatically represent the critical
power model. Thus it is important to draw an analogy between the hydraulic
models and the human body bioenergetic processes during exercise and recovery.
The fluid contained in the vessels represents metabolic energy, the volumetric
flow of fluid therefore stands for the power.
The critical power concept (1965/1981)
First observations of a hyperbolic relationship between the level of constant power
output P and corresponding time to exhaustion were made by Monod and Scherrer in the year 1965. Mathematically their model can be described by following
two equivalent formulas:
Wtot = AW C + CP · t = P · t
t = AW C/(P − CP )
(4.1)
(4.2)
The total work performed W is the sum of the amount of anaerobic working
capacity (AW C) and the product of critical power CP and the time t CP being
8
4.1 Two-component models
performed until exhaustion – equation (4.1). From an exercise performance viewpoint Eqn. (4.2) is the preferred one, as it allows predictions for time to exhaustion
depending on the power output P and the physiological parameters AW C and
CP . The idea for whole-body exercise on cycle ergometer were developed by
Moritani et al. (1981). This led to four basic but essential assumptions:
1. For human exercise there are only two components to power supply.
2. Anaerobic supply, unlimited in rate (but limited in capacity) ⇒ AW C.
3. Aerobic supply, unlimited in capacity, but limited in rate, by CP .
4. Exhaustion occurs on depletion of AW C.
From (4.2) follows t to be infinite for P < CP , which implicitly leads to the next
two assumptions:
5. power domain: CP < P < ∞
6. time domain: 0 < t < ∞
Furthermore the following results by implication as well:
7. Aerobic power is available from the beginning of the exercise and is available
at the same rate (CP ) until exercise ends.
8. AW C and CP are independent (of P and/or of t) constants.
9. Constant efficiency of transformation of metabolic energy into mechanical
energy across the whole power/time domain(s).
Figure 4.1 illustrates application of above assumptions: This schema of a hydraulic model is meant to be understood as follows: The vessels representing
AWC and CP, denoted An and Ae , are connected through a tube R1 . Both are
filled with a fluid; while An is ‘open’ which represents the infinite aerobic energy
supply (unlimited in capacity) An is ‘closed’ representing the anaerobic energy
supply being limited in capacity. The connection tube is limited in diameter to
illustrate the fact that CP is rate limited. The fluid’s output flow is regulated by
tap T which complies to power output .
9
4.1 Two-component models
Figure 4.1: The CP concept as a hydraulic model - 1976 by Margaria
When tap T is opened the fluid level of An drops and fluid from Ae starts
flowing and refilling An . In case power output P does not exceed CP, depletion
of vessel An does not take place. Otherwise exhaustion will occur as soon as An
is fully depleted.
Extended critical power models
Further considerations lead to questioning the initial assumptions [8, p. 343f.].
Reasons for that e.g. are:
• the distinction of the aerobic and anaerobic biochemical pathways to human
energy metabolism does not regard the existence of four energy sources, as they
are described in chapter 2.
• the assumption of unlimited available aerobic power supply is unrealistic; though
the available oxygen is not limited, the substrate to be oxidised definitely is.
There is an upper limit, which is not reached while exercise is of moderate duration.
• the AWC, said to be unlimited in rate, however disposes of a rate limitation. At
a certain resistance e.g. pedals cannot be turned. Without movement no physical
work is done.
• observations reveal the fact, that at exhaustion after short duration, due to very
high power output, significant amounts of glycogen still remain available.
• kinetics of oxygen uptake imply delayed but immediate establishment of aerobic
supply (⇒ Wilkie’s extension)
• whatever the amount of anaerobic energy is, human muscles are physiologically
limited to a maximum achievable power Pmax which leads to a new power domain:
CP < P < Pmax .
10
4.1 Two-component models
• despite the necessity of drinking and eating at least the need for sleep defines an
upper limit for exercise durations. This is contrary to the previous time domain.
• Bishop et al. 1999 have shown that AWC and CP estimates differ when different
sets of P are used in testing sessions. Therefore an independency of P for AWC
and CP can no longer be assumed.
Figure 4.2: Wilkie’s correction to the CP model (1980) - Adjusted capacity of
AWC
The assumption that establishment of aerobic supply is not instantaneously (at
any given rate) brought Wilkie to introduce a correction factor for oxygen uptake
kinetics. It is based on a single exponential with time constant τ without delay.
The original Eqn. (4.1) becomes:
W = AW C + CP · t − CP · τ (1 − e−t/τ )
|
{z
}
(4.3)
*
∗
amount of energy released from anaerobic sources
before the attainment of an aerobic steady state at CP
Now underestimation of the “true” anaerobic capacity by AWC is taken into
consideration. The third term represents anaerobic energy before the attainment
of an anaerobic steady state. This is shown in Fig. 4.2 where the darker triangular
area within the interval [0,τ ] represents the amount of capacity neglected under
the CP model [8, p. 344f.].
Added to the area AWC the result is the “real” AWC. The correction does
apply for exercises leading to exhaustion, durating for times 50 s < t ≤ 600 s
as Wilkie showed. On closer inspection of Wilkie’s extended formula for the
11
4.1 Two-component models
corrected model one can see that for exercises of longer duration its effect is
lessened [8, p. 345].
Figure 4.3: The effect of great values for t on τ -extended Eqn. (4.3) - Resulting
values Wtotal (in joules) for Eqs. 4.1 and 4.3 for CP = 320.79 W and
AW C = 12081.3 J.
For Figures 4.3 and 4.4 values for AWC and CP were taken from D. Höckele’s
bachelor thesis [3, p. 19]. The subject is said to be a amateur racing cyclist [3,
p. 13]. Although, compared to another subject (untrained. spare time cyclist),
the values differ significantly, the plots for the other subject don’t. The plots are
for schematic purpose only and that is why the plots are not for a second subject
are not included.
Fig. 4.3 shows the resulting Wtotal of (4.1) and (4.3) for 0 s ≤ t ≤ 180 s and
τ = {10, 20, 30}. Whereas τ affects (4.3) during the first 80 seconds, the curves
seem to converge for every τ and t > 80 s.
Figure 4.4: Differences between Eqs. 4.1 and 4.3 - their percental deviation.
Fig. 4.4 supports illustration of the fact that for longer durations of exercise
the effect of the corrected equation is lessened. Percental deviations between Eqs.
12
4.1 Two-component models
4.1 and 4.3 are shown for τ = {10, 20, 30} again and 0 s ≤ t ≤ 1200 s.
Applying this to the first hydraulic model (Fig. 4.1) a new corrected model
results as shown in Fig. 4.5. The vessel Ae is lowered which creates a differential
h in level between An and Ae . This means that at the beginning of an exercise
the flow through R1 is low, rising with dropping of fluid level in vessel An . Once
the differential h equals the the height of Ae , the flow through R1 equates to the
amount of achievable CP.
Figure 4.5: Wilkie’s correction to the CP hydraulic model (1980) - Flow through
R1 now directly proportional to h (differential between Ae and An )
Three-parameter critical power model by Morton (1996)
This section focusses on the introduction of a third physical parameter (Pmax )
and a ‘feedback control system’ and as yet another extension.
A reachable maximum speed (for cycling, running. . . ), which cannot be exceeded, leads to the assumption that the power affordable has an upper limit.
Such an upper bound of power equals exactly the resistance which cannot be
coped with on an ergometer at the point when an individual is not able turn to
the pedals any further. This also leads to a correction of the previous powerdomain which now gets CP < P < Pmax . Fig. 4.6 illustrates the relationship
between the duration an individual is able to hold up an exercise and the power
necessary therefore. The vertical asymptote stands for a theoretically infinite
duration of exercise at P = CP . The hyperbola is cut by the power axis at
P = Pmax . The horizontal asymptote at t = k corresponds to the original power
13
4.1 Two-component models
axis of the 2-parameter hyperbolic model. Putting our subject’s parameters into
(4.2), setting P = Pmax , we get:
t=
AW C
12081.3 J
=
= 15.45 s
(P − CP )
(1102.75 W − 320.79 W)
As a matter of fact for a power output P > Pmax no work can be done and the
duration of exercise is not longer than 0 seconds which is contrary to the 15.45s
from above. By relaxing the time asymptote constraint of the original CP model
we now get a new power axis at t = 0 which in our case makes k = −15.45s in
Fig. 4.6.
Figure 4.6: Diagrammatic representation of the 3-parameter CP model equation - It is represented as a generalised rectangular hyperbola, with
vertical power asymptote at P = CP , horizontal asymptote at
C
t = k = CPAW
−Pmax , and cutting the power axis at P = Pmax .
The idea of a ‘feedback control system’ results from considering the circumstance
that e.g. the last 100 m of a 5000 m race cannot be run at the same speed as at
the distance of a 100 m sprint. The anaerobic reserve, denoted An therefore is
used to take account of that. The function
P̂max = CP + An (Pmax − CP )/AW C
(4.4)
minds that and pays regard to the remaining anaerobic capacity and provides
maximal achievable power P̂max (relative to An ) at any time. An = AW C for fully
rested and repleted individual. In that case Pmax can be achieved. Otherwise, if
An is empty, only CP can be achieved [8, p. 345].
14
4.2 Three-component models
Based on that the 3-parameter CP model, fully described by Morton [7], can be
exemplified. Therefore Eqn. (4.3), respectively an adapted version thereof, given
by di Prampero (1999), and Eqn. (4.4) can be combined which results in:
and
(AW C − (P − CP )t − CP τ (1 − e−t/τ ))
P = CP + (Pmax − CP )
AW C
AW C
AW C
CP τ
t=
+
−
P − CP
CP − Pmax (P − CP )
(4.5)
(4.6)
Assuming τ = 0, as t usually is greater by a factor of > 4 [8, p. 346], Eqn. (4.6)
reverts exactly to the 3-parameter model equation
t=
AW C
AW C
+
P − CP
CP − Pmax
(4.7)
where t is time until exhaustion. This model seems to correct overestimation
bias in estimation of CP and the underestimation bias in estimation of AW C by
adding Pmax to the parameters of the standard CP model [7, p. 618].
4.2 Three-component models
Margarias hydraulic bioenergetic model
Margaria was the first who proposed a 3-component hydraulic model of human
bioenergetics [8, p. 346f.]. Fig. 4.7 illustrates Margaria’s idea of extending the
existing models by an additional component. For the first time attention was
payed to the distinction of the alactic and lactic anaerobic supply systems.
As seen in the precedent hydraulic models, vessel O, which represents the oxidative or aerobic source, is of infinite capacity. The flow through R1 analogously
stands for VO2 which we will come back to later on. Fig. 4.7 represents the
original hydraulic model extended by an additional vessel. The original vessel
An is replaced by two individual ones, denoted P and L. While L represents the
lactic energy supply, P stands for energy supplied alactically by high energy phosphates. As the phosphagenes can only afford short-time supply, it’s reasonable
for P to be put in between L and O (which stands for oxidative energy supply).
O as well as L are connected to P by tubes R1 and R2 , which both only allow
mono-directional flow towards vessel P. Additionally P is connected to L by a
15
4.2 Three-component models
Figure 4.7: Margaria’s hydraulic bioenergetic model (1976) - Anaerobic energy
supply represented by two separate vessels
tube mono-directionally as well, denoted R3 . Again two vessels, O and P, are
arranged so that a differential h between them can be created by fluid flowing
through tube R1 . L possesses an extra narrow extension tube B, representing the
resting blood and tissue lactate [5, p. 453]. As seen before this model likewise
has a tap T which regulates the flow of fluid, respectively the amount a power
output, delivered by three energy supply systems.
The amount of flow after opening tap T corresponds to workload W. Oxygen
consumption (VO2 ) is connected to the flow through R1 , induced by falling level in
P. That in turn accords to the differential h, which regulates the rate at which level
P falls. Depending of the amount of W the levels in O an P will be balanced above
R1 . As soon as the output flow equals the maximum amount of energy suppliable
oxidatively (VO2max ), level of P will reach the height of R1 ’s output. The empty
volume is referred to as the alactic oxygen debt. If workload increases, fluid starts
flowing from vessel L through tube R2 . Though the so called ‘early lactate’ (B)
is measurable, it isn’t contributing to the system’s flows in any significant degree
and therefore is incorporated into the model for the sake of completeness. The
maximum power achievable Pmax in this model is composed of:
• oxygenetic power producable (VO2max )
• the amount of flow through R2 , limited by diameter, if L isn’t depleted
16
4.2 Three-component models
– else: the amount of permanently regenerated lactic energy
• the amount of alactic power without lactic power, supplementing oxygen
power until W = Pmax , while P isn’t depleted
– else: the amount of permanently regenerated alactic energy
As soon as tap T is closed, refilling of the system begins, initially through R2 in
case of foregoing exhaustion due to severe exercise (P depleted). Simultaneously
P will be refilled through R1 . Having reached the level of tube R1 , refilling ceases
due to the differential h and the subject is said to have repaid his oxygen debt.
At the same time L starts getting refilled through R3 if it was fully depleted
too, but at a much slower rate than depletion is possible. Because of observed
discrepancies this model had to be rejected in it’s original form [8, p. 347]. It
has been reformulated by Morton (1986) and has led to the formulation of the
3-component M-M-model.
The generalised M-M hydraulic model
Figure 4.8: The generalised M-M hydraulic model (1990) - This illustrates the
point of origin for the 16 possible configurations
Taking a look at Fig. 4.8 gives us the opportunity to envision the sixteen possible configurations for this generalised M-M model (explained in detail by R. H.
17
4.2 Three-component models
Morton, 1986 [5]). Twelve inconsistent configurations can be eliminated by checking them against known physiological facts. Fig. 4.9 shows the four remaining
possible configurations which were reduced by another three models [6].
Figure 4.9: Configurations of the M-M model (1986) - Due to consistency of physiological facts 4 out of 16 possible configurations remain
As discussing those facts and inconsistent configurations would go too far beyond the scope of Morton’s review article this seminar work is based on, the
discussion is continued with the one and only configuration left at this point, as
shown in Fig. 4.10.
Figure 4.10: The generalised M-M hydraulic model (1990) - This model corresponds to configuration A in Fig. 4.9
Compared to the previous hydraulic model by Margaria (Fig. 4.7) the generalised M-M-model attracts attention not only by the fact that the refilling of the
18
4.2 Three-component models
modelled system has two bi-exponential phases. The top of vessel An L resides
between O’s level of fluid and its inlet tube R1 . This implies that anaerobic threshold (AT) is reached at a approx. 40% of VO2max . This additionally is illustrated
by commencement of lactic acid production, represented by a flow through tube
R2 . This commencement has influence on flow through R1 which in turn changes
VO2 dynamics. This allows crossing the anaerobic threshold to be modelled by
a hydraulic model. At this point VO2 dynamics become bi-exponentially dependent. Summarising for fluid level from bottom up to the level of tube R1 , the flow
through R1 is independent. Upwards from that level the VO2 dynamics become
mono-exponential, as the flow through R1 is dependent to differential between O
and An A. If the fluid level reaches R2 , VO2 dynamics change to bi-exponential
[8, p. 348].
At a certain consumption of W a ‘steady state’ can be reached, at which the
level in An A does not drop below R2 . The so called ‘maximal steady state’ is
reached when the level in O equals the level of R2 . This is the maximal attainable
amount of of power not leading to exhaustion immediately. If W increases, An L
gets empty an exhaustion will occur. Until then the flow through R1 equals P,
the true maximal sustainable aerobic power. That is said to be at approx. 85%
of VO2max [8, p. 348].
The system refills after tap T is closed, initially through R1 and R2 . If the fluid
in An A reaches the bottom of An L, refilling continues through R1 and R3 . Again
bi-exponentially behaviour in recovery VO2 kinetics, divided into two phases, can
be observed. The second phase begins with the reversal flow from R2 to R3 .
Because of R3 being very narrow it is implicated that An L takes a considerable
time to refill, which already was describe by Hill in the 1920s and later reviewed
in di Prampero (1981).
Although this model depicts very well the dynamics during exercise and recovery, a criterion for exhaustion as described as ‘feedback control system’ in the
3-parameter model is not incorporated. Also it has not yet been investigated to
determine an optimal strategy for completing a given amount of work in minimum
time [8, p. 349].
19
4.3 Applicability and development of the 3-component model
4.3 Applicability and development of the
3-component model
The 3-component hydraulic can be used in different manners. In general this
model is said to be an obvious possibility for giving a macroscopic overview
of the operation of the human bioenergetic system during physical strain and
regeneration. It very intuitively shows the concept by the flows of a fluid between
vessels being interconnected, as well as the analogies between hydraulic system’s
attributes and the corresponding human bioenergetic attributes: capacities, flow
rates, height differentials etc. ←→ energy stores, power etc.
On a ‘higher’ level this model can be used to provide more insight by graphical
illustrations of changes in various components. Existing “real data” can be chosen
to verify the accuracy of the workings of the model. Also deeper comprehension
can be gained by working through the mathematics to obtain solutions to the
system corresponding to a variety of conditions. In terms of ‘scientific research’
the model can be verified in a much more exact context by data retrieved by
appropriate designed experiments [8, p. 349f.]. In ‘competitive running’ this
model for example was used as a submodel in a larger mathematical model. [1]
Conclusions and further considerations
In general it can be said that over the last 50 years of development significant
progress has been made on all counts. However still no model is existent incorporating all the developments, which may be regarded as a challenge. All the
models known only have been tested against exercises which ultimately lead to
exhaustion. So they are all yet to be tested against data from submaximal work
which does not explicitly lead to exhaustion. This also may be a task to be
challenged [8, p. 352].
This leads to the fact, that application for powerbike is not yet known to lead to
successful results being better than ones that already exist. Tests can be designed
for the powerbike to retrieve data which can be tested against the mathematical
equations behind the hydraulic models. As stated in Gordon (2005), this “could
be a useful starting point for further investigation” [2, p. 88].
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References
[1] H. Behncke. A mathematical model for the force and energetics in competitive
running. Journal of Mathematical Biology, 31:853–878, 1993. 20
[2] S. Gordon. Optimising distribution of power during a cycling time trial. European Journal of Applied Physiology, 8:81–90, 2005. 20
[3] D. Höckele. Trainingsanalyse von Ergometer- und Fahrradcomputerdaten. PhD in information engineering, Universität Konstanz, 2009. 7, 12
[4] G. Löffler. Basiswissen Biochemie: mit Pathobiochemie (Springer-Lehrbuch) (German
Edition). Springer, 5. edition, 2003. 3, 4
[5] R. H. Morton. A three component model of human bioenergetics. Journal of
Mathematical Biology, 24:451–466, 1986. 16, 18
[6] R. H. Morton. Modelling human power and endurance. Journal of Mathematical
Biology, 28:49–64, 1990. 4, 18
[7] R. H. Morton. A 3-parameter critical power model. Ergonomics, 39(4):611–619,
1996. 15
[8] R. H. Morton. The critical power and related whole-body bioenergetic models.
European Journal of Applied Physiology, 96:339–354, 2006. 1, 3, 4, 8, 10, 11, 12, 14, 15,
17, 19, 20
[9] R. H. Morton and D. J. Hodgson. The relationship between power output and
endurance: a brief review. European Journal of Applied Physiology, 73:491–502, 1996.
1, 4, 5, 6
[10] P. R. Rich. The molecular machinery of Keilin’s respiratory chain. Biochemical
Society Transactions, 31(6):1095–1105, 2003. 5
[11] Håkan Westerblad, David G. Allen, and Jan Lännergren. Muscle Fatigue:
Lactic Acid or Inorganic Phosphate the Major Cause? News Physiological Science,
17(1):17–21, 2002. 4
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