Kawamura,Tsuyoshi

Sasahara,Hideo

Hiroshima University of Economics Academic Repository

Mathematical Solution and Numerical 
Simulation of Human Bioenergetics 
Hideo SASAHARA 
and Tsuyoshi KA W AMURA 
INTRODUCTION 
117 
Three component hydraulic model proposed by Margaria (1976) 
described the human energy process during exercise. Oxygen con-
sumption， lactate formation and phosphagen breakdown are modeled 
after the three interconnected vessels in the system. Morton (1986) 
solved this model mathematically on the basis of experimental data. It 
is what we cal M-M (Margaria-Morton) model， and is the generliza-
tion of乱1argaria'soriginal. 
The purpose of this study is to analyze the anaerobic power output 
during a strenuous exercIse by the use of M -M model. 
METHOD 
24 years male subject participated in our study. His weight and 
LBM were 55.0 kg and 48.7 kg. The maximal effort cycling bout were 
performed on a Monark bicycle ergometer with a resistive loading from 
2.0 to 3.0 kgω. A microswitch with dry battery was fixed on near the 
wheel of the ergometer. A pulse was generated with every rotation of 
wheel， and recorded by the data recorder (TEAC; R-40). By using the 
A/D converter (CONTEC; AD-12-16S-(98))， numerical data was 
118 第11巻 第2号(人文ー自然・社会科学編)
ca1culated on the 16 bits computer (NEC; PC-9801). 
M-M MODEL 
Figure 1 below gives a diagrammatical representation of the M-M 
model. 
w 
Fig. 1 Margaria-Morton model 
Three vessels 0， P， and L representing the oxidative source， the 
phosphagen， and the glycolytic source. The height of the base of 
vessel 0 above the base of vessel P is denoted by φ. The top of 
vessel L is at a level 8 below the fluid level in vessel O. ‘h' is a 
decreased fluid level of vessel P. And Wm represents the maximal 
energy expenditure. 
MATHEMATICAL SOLUTION 
In the first phase of activity， the fluid in P has dropped to the level 
h < 8. The form of equation is given by: 
Mathematical Solution and Numerical Simulation of Human Bioenergetics 119 
Wm=Mp (1-h) 
=Mo {h/ (1ー の}+ Ap (dh/dt) 
where: Mp=Vpmax， the maximal flow from vessel P 
Mo = V 02max， the maximal flow from vessel 0 
Ap=cross sectional area of vessel P 
This is a simple first order linear differential equation having as its 
general solution: 
h=Mp/{Mo/(1一φ)+Mp}[1+C.exp{一(1/Ap)(Mo/(1一φ)+Mp)t}] 
The paticular solution (h=O at t=O) is given by: 
h=Mp/{Mo/(1 φ)+ Mp} [1-exp {-1/Ap (Mo/(1φ)+ Mp)t}] 
Wm=Mp [{1-Mp/(Mo/(1φ)+ Mp)} + {Mp/(Mo/(1-φ)+Mp)} 
判 xp{-1/ Ap (Mo/(1-φ)+ Mp)}t] 
V02=Mo/(1-φ) [Mp/ {Mo/(1 φ)+Mp}] 
*f1-exp {-1/Ap (Mo/(1一φ)+Mp)t]
Vp=Wm-V02 
when h=θ(The end of first phase): 
t1=Ap/{Mo/(1一φ)+Mp} .ln[1-(1/Mp) {Mo/(1一φ)+Mp}O]
The general form is given by: 
Wm=α{β+(1β).exp (一δt)}
By using Runge-Kutta-Gill method， the numerical simulation was done 
after second phases. 
RESULTS and DISCUSSIONS 
Figure 2 showed， there was much difference between the simulation 
and the experimental data. Figure 3(a) showed the simulated total of 
羽Tm，Vo 2， Vp and V 1. Figure 3(b) Showed % of totals as such. 
Second attempt， the shape of P vessel was transformed into a funnel 
shape， there was then litle difference between them (fig. 4). Absolute 
120 第11巻第2号(人文・自然社会科学編)
500 
Wm (watt) 
3001-・
200 
100 
。ハ 2O 40 60 80 100 120 
'time (sec) 
Fig. 2 Wm of simulation and experimental data (first attempt) 
500 
Tota! Wm 
(watt) 300 
200 ??
(a) 
20 40 60 80 100 120 
time (sec) 
Fig.3(a) Simulated total of Wm， Vo 2 Vp， and Vl (a). and % of totals as such 
(b). (first attempt) 
Mathematical Solution and Numerical Simulation of Human Bioenergetics 121 
?????
• 屯
• 80卜司、
120 
。占 oftotal 
¥i'm 60 
40 
20 
1，])1 
time (sec) 
Fig.3(b) Simulated total of Wm， Vo 2 Vp， and Vl (a). and % of totals as such 
(b). (first attempt) 
500 
¥'ロ1 (watt) 
200 
。 20 40 60 80 100 120 
time (sec) 
Fig.4 Wm ofsimulation and experimental data (second attempt) 
122 第11巻 第2号(人文ー自然・社会科学編)
totals and % of totals were shown in the figures 5(a) and 5(b). 
Third attempt， the position of 0 vessel and L vessel was replaced 
each other， itwas reconnized that there existed no more difference bet-
ween them (fig. 6 fig 7(a) and 7(b). 
500 
Total Wm 
(wat) 
LEGEND 
200 
(al 
100 120 
time (secl 
、• 、?• • 
••  
? ?? ??????ょ
120 
% of total ・
Wm 
?? ??
time (sec) 
Fig.5 Simulated total of Wm， Vo 2， Vp and Vl (a). And % of totals as such (b). 
Mathematical Solution and Numerical Simulation of Human Bioenergetics 123 
500 
Wm (watt) 
200 
100 
?
??
20 40 60 80 100 120 
time (sec) 
Fig. 6 Wm of simulation and experimental data (third attempt) 
500 
LEGEND |、
400 
Wm (watt) 
3∞ 
200 
100 
(a) 。凶ー里 』ー
40 60 80 100 120 
time (sec) 
Fig.7(a) Soimulated total of Wm， Vo 2， Vp and Vl (a). And % of totals as such (b). 
124 第11巻第2号(人文 自然、・社会科学編)
80 
• % of total 
Wm c..1-・
????
120 
time (sec) 
Fig.7(b) Soimulated total of Wm， Vo 2， Vp and Vl (a). And % of totals as such 
(b). 
100 
80 
% of total 
Wm 60 
40 
20 
(a) 
LEGEND 
---High 
-一一ー・ Low
一一ー -ーl¥1oderate 
Iケヘ、、¥ 、
f¥¥¥ 、? 、、、、 、、
、 、
20 40 60 120 
time (sec) 
Fig. 8(a) Changed output level of L vessel (a). And Simulation of glycolytic 
power effect (b) 
Mathematical Solution and Numerical Simulation of Human Bioenergetics 125 
500 
400 
LEGEND 
一一 High
-ー ーー -Low 
-一一-Moderate 
Wm (watt) 
300 
200 
100 
?? ??
ω 。 20 40 60 80 100 120 
time (田c)
Fig. 8(b) Changed output level of L vessel (a). And Simulation of glycolytic 
power effect (b) 
Furhter more， an output level of L vessel could also be changed ar-
tificially (fig. 8(a) ). Figure 8(b) showed thete was simulation of a 
glycolytic power effect. It may be safely said， therefore， that Wingate 
test and the other anaerobic power test estimate the glycolytic energy 
flow accurately. 
References 
1) Margaria R (1976) Biomechanics and energetics of muscular 
exercise. Oxford University Press， Oxford. 
2) Morton R. H. (1986) On a model of human bioenergetics 2. 
Eur. J. Appl. Physiol.， 55， 413-418. 
3) Morton R. H. (1986) A three component model of human bioenergetics. 
J.乱1ath.Biol. 24， 451-466. 
(本研究は昭和62年度広島経済大学特定個人研究費のもとに遂行されたc)


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