Mathematical Modeling of Hypometabolism Process to Identify Peculiarities of Human Organism During the Work Under Condition of Highlands
The purpose of this work is to develop and to investigate the mathematical model of hypometabolism, and to develop the software for execution of computing experiments with the model. Предложена математическая модель динамики напряжений респираторных газов с учетом гипометаболизма, который развиваетс...
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Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України
2014
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| Цитувати: | Mathematical Modeling of Hypometabolism Process to Identify Peculiarities of Human Organism During the Work Under Condition of Highlands / I.L. Bobriakova// Кибернетика и вычислительная техника. — 2014. — Вип. 178. — С. 22-35. — Бібліогр.: 8 назв. — англ. |
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Bobriakova, I.L. 2015-07-09T21:12:58Z 2015-07-09T21:12:58Z 2014 Mathematical Modeling of Hypometabolism Process to Identify Peculiarities of Human Organism During the Work Under Condition of Highlands / I.L. Bobriakova// Кибернетика и вычислительная техника. — 2014. — Вип. 178. — С. 22-35. — Бібліогр.: 8 назв. — англ. 0452-9910 https://nasplib.isofts.kiev.ua/handle/123456789/84533 519.876 The purpose of this work is to develop and to investigate the mathematical model of hypometabolism, and to develop the software for execution of computing experiments with the model. Предложена математическая модель динамики напряжений респираторных газов с учетом гипометаболизма, который развивается в организме человека на высокогорье. Анализ вычислительных экспериментов позволил сделать выводы о характере изменений режимов функционирования организма при переходных процессах и в стационарных состояниях, влиянии систем внешнего дыхания и кровообращения на формирование уровней управляющих параметров, а также о роли гипометаболизма при воздействии на организм возмущений внутренней и внешней сред. Запропоновано математичну модель динаміки напружень респіраторних газів з урахуванням гіпометаболізму, який розвивається в організмі людини на високогір'ї. Аналіз обчислювальних експериментів дозволив зробити висновки щодо характеру змін режимів функціонування організму при перехідних процесах і в стаціонарних станах, впливу систем зовнішнього дихання та кровообігу на формування рівнів керуючих параметрів, а також ролі гіпометаболізму при впливі на організм збурень внутрішнього і зовнішнього середовищ. en Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України Кибернетика и вычислительная техника Системы и интеллектуальное управление Mathematical Modeling of Hypometabolism Process to Identify Peculiarities of Human Organism During the Work Under Condition of Highlands Математическое моделирование процесса гипометаболизма для выявления особенностей организма человека при работе в условиях высокогорья Математичне моделювання процесу гіпометаболізму для виявлення особливостей організму людини при роботі в умовах високогір'я Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Mathematical Modeling of Hypometabolism Process to Identify Peculiarities of Human Organism During the Work Under Condition of Highlands |
| spellingShingle |
Mathematical Modeling of Hypometabolism Process to Identify Peculiarities of Human Organism During the Work Under Condition of Highlands Bobriakova, I.L. Системы и интеллектуальное управление |
| title_short |
Mathematical Modeling of Hypometabolism Process to Identify Peculiarities of Human Organism During the Work Under Condition of Highlands |
| title_full |
Mathematical Modeling of Hypometabolism Process to Identify Peculiarities of Human Organism During the Work Under Condition of Highlands |
| title_fullStr |
Mathematical Modeling of Hypometabolism Process to Identify Peculiarities of Human Organism During the Work Under Condition of Highlands |
| title_full_unstemmed |
Mathematical Modeling of Hypometabolism Process to Identify Peculiarities of Human Organism During the Work Under Condition of Highlands |
| title_sort |
mathematical modeling of hypometabolism process to identify peculiarities of human organism during the work under condition of highlands |
| author |
Bobriakova, I.L. |
| author_facet |
Bobriakova, I.L. |
| topic |
Системы и интеллектуальное управление |
| topic_facet |
Системы и интеллектуальное управление |
| publishDate |
2014 |
| language |
English |
| container_title |
Кибернетика и вычислительная техника |
| publisher |
Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України |
| format |
Article |
| title_alt |
Математическое моделирование процесса гипометаболизма для выявления особенностей организма человека при работе в условиях высокогорья Математичне моделювання процесу гіпометаболізму для виявлення особливостей організму людини при роботі в умовах високогір'я |
| description |
The purpose of this work is to develop and to investigate the mathematical model of hypometabolism, and to develop the software for execution of computing experiments with the model.
Предложена математическая модель динамики напряжений респираторных газов с учетом гипометаболизма, который развивается в организме человека на высокогорье. Анализ вычислительных экспериментов позволил сделать выводы о характере изменений режимов функционирования организма при переходных процессах и в стационарных состояниях, влиянии систем внешнего дыхания и кровообращения на формирование уровней управляющих параметров, а также о роли гипометаболизма при воздействии на организм возмущений внутренней и внешней сред.
Запропоновано математичну модель динаміки напружень респіраторних газів з урахуванням гіпометаболізму, який розвивається в організмі людини на високогір'ї. Аналіз обчислювальних експериментів дозволив зробити висновки щодо характеру змін режимів функціонування організму при перехідних процесах і в стаціонарних станах, впливу систем зовнішнього дихання та кровообігу на формування рівнів керуючих параметрів, а також ролі гіпометаболізму при впливі на організм збурень внутрішнього і зовнішнього середовищ.
|
| issn |
0452-9910 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/84533 |
| citation_txt |
Mathematical Modeling of Hypometabolism Process to Identify Peculiarities of Human Organism During the Work Under Condition of Highlands / I.L. Bobriakova// Кибернетика и вычислительная техника. — 2014. — Вип. 178. — С. 22-35. — Бібліогр.: 8 назв. — англ. |
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| first_indexed |
2025-11-25T22:42:34Z |
| last_indexed |
2025-11-25T22:42:34Z |
| _version_ |
1850569522849775616 |
| fulltext |
22
UDK 519.876
MATHEMATICAL MODELING OF
HYPOMETABOLISM PROCESS TO IDENTIFY
PECULIARITIES OF HUMAN ORGANISM DURING
THE WORK UNDER CONDITION OF HIGHLANDS
I.L. Bobriakova
V.M. Glushkov Institute of Cybernetics of NAS of Ukraine
Предложена математическая модель динамики
напряжений респираторных газов с учетом гипометаболизма, который
развивается в организме человека на высокогорье. Анализ
вычислительных экспериментов позволил сделать выводы о характере
изменений режимов функционирования организма при переходных
процессах и в стационарных состояниях, влиянии систем внешнего
дыхания и кровообращения на формирование уровней управляющих
параметров, а также о роли гипометаболизма при воздействии на организм
возмущений внутренней и внешней сред.
Ключевые слова: математическая модель, системы
дыхания и кровообращения, напряжение газов, гипоксия,
гипометоболизм, возбуждающие воздействия, вычислительные
эксперименты.
Запропоновано математичну модель динаміки напружень
респіраторних газів з урахуванням гіпометаболізму, який розвивається в
організмі людини на високогір'ї. Аналіз обчислювальних експериментів
дозволив зробити висновки щодо характеру змін режимів функціонування
організму при перехідних процесах і в стаціонарних станах, впливу
систем зовнішнього дихання та кровообігу на формування рівнів
керуючих параметрів, а також ролі гіпометаболізму при впливі на
організм збурень внутрішнього і зовнішнього середовищ.
Ключові слова: математична модель, система дихання і
кровообігу, напруга газів, гіпоксія, гіпометаболізм, збурюючі впливи,
обчислювальні експерименти.
INTRODUCTION
Hypometaboliс and hypoxic states are in the focus of attention of researchers
since organism is often impacted by hypoxic factors, particularly under conditions
of highlands, as well as during significant physical loads or some severe pathologic
process.
Hypoxic or exogenous hypoxia is developed during decreasing of partial
pressure of oxygen in inhaled air. During hypoxic hypoxia the oxygen tension in
arterial blood, the saturation of hemoglobin with oxygen and its total content in the
blood are being decreased. Negative impact can be caused also by hypocapnia,
developed as a result of compensatory hyperventilation of lungs. Extreme
hypocapnia causes worsening of blood supply to brain and heart (vasoconstriction)
and respiratory alkalosis. In this connection, it is interesting to investigate the
energy reserves of organism and the ways of its strengthening under hypoxic
conditions.
I.L. Bobriakova, 2014
ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 178
23
THE RELEVANCE OF THE TOPIC
Investigation of the processes of human organism breathing system self-
regulation represents particular scientific area that is being widely developed last
years as a result of success in mathematical modeling. However the regulation
mechanisms of functional state of human organism under conditions of highland
hypoxia are not sufficiently studied yet. Impossibility of such investigations was
related with the difficulties of experimental determination of the several most
important parameters and with the absence of adequate mathematical models for
dynamics of these processes.
PROBLEM DEFINITION
Mechanisms of formation of hypometabolic state of human being under
conditions of highlands are considered. Mathematical model of functional systems
of breathing and blood circulation has been used to analyze this phenomenon on
system level.
The purpose of this work is development and investigation of mathematical
model of hypometabolism, and development of software for execution of
computing experiments with the model.
RESULTS
1. Developed complex of mathematical model of the gas mass carry
process in organism and software is applied for assessment of the dynamics of
functional state of human being on conditions of work at highlands.
2. Implemented numerical analysis of the models of respiratory exchange
control enables to:
- follow dynamics of the main physiological parameters of the model during
transient processes and in stationary states;
- forecast and quantitatively assess regulatory reactions of the organism
under given disturbances;
- carry out individualization of model developments on condition of
availability of array of data on anatomic-physiological peculiarities of specific
human being.
Obtained results are well correlated with the physiological experimental data.
Mathematical model of tension dynamics of respiratory gas is developed using
ideas of compartmental modeling [1, 2], i.e. describes the mass carry process of
respiratory gases among functionally connected but relatively autonomous
compartments.
Model represents the system of ordinary nonlinear differential equation whose
number depends on the degree of detailing of structural scheme of the object
(number of tissue regions, portions of blood washing the tissue etc.), describing the
dynamics of oxygen tension p (hereinafter the first index at variable — (1)), carbon
dioxide (2) and nitrogen (3) in structural compartments of the system —
respiratory tract (second index at variable — (rw)), alveolar space (А), blood of
pulmonary capillaries (LC), arterial blood (а), blood of tissue capillaries (cti), tissue
I.L. Bobriakova, 2014
ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 178
24
reservoirs (ti ) and mixed venous blood (
−
v ).
As distinct from existing models of gas mass transfer in the organism this
model presents dynamics of gas partial pressures in alveolar space during the
breathing cycle including phases of inhalation, exhalation and pause; process of
gas diffusion through air-blood barrier, as well as through capillary-tissue
membranes taking into account their structural and functional peculiarities have
been considered [3]. In addition tissue reservoirs are differentiated and peculiarities
of energy exchange of tissues of brain, heart muscle, liver, kidneys, skeletal
muscles, skin and other organism's tissues have been considered. Presented below
equations of mathematical model of object have been obtained on the basis of
continuity and material balance principles (conservation of mass) using known
empiric physiological dependencies among variables. Model equations are
described as follows.
Let p1, p2, p3 — partial pressures of oxygen, carbon dioxide and nitrogen in
inhaled air and p1+ p2+ p3= B, where В — total barometric pressure.
Then dynamics of partial pressures of gases in respiratory tract during
breathing cycle can be presented as follows:
],~~[
τ 11
1
1
rw
rw pp
Vn
V
d
dp
rw
−=
•
(1)
]~~[
τ 22
2
2
rw
rw pp
Vn
V
d
dp
rw
−=
•
, (2)
]~~[
τ 33
3
3
rw
rw pp
Vn
V
d
dp
rw
−=
•
, (3)
where rwV — volume of respiratory tract (generalized reservoir),
V& — ventilation depending of values of respiratory volume and duration of
breathing cycle; n1, n2, n3 — conversion factors for respiratory gases and nitrogen
respectively, and
3,1
),(
),(
,
~ , =
≤
>
= •
•
j
OV
OV
p
p
p
rwj
j
j ,
.3,1
),(
),(
,
,~ =
≤
>
= •
•
j
OV
OV
p
p
p
A
rw
rw j
j
j
(4)
During modeling of partial pressures of gases in alveolar space with volume
VL it is necessary to take into account gas flow through air-blood barrier
)( LCALC jjLCjj ppSDG −= , (5)
I.L. Bobriakova, 2014
ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 178
25
where Dj — coefficients characterizing permeability of gases through air-blood
barrier, SLC — area of gas mass transfer surface.
Then
]
τ
~[
)(
1
τ 11111
1
1
d
dVpnGpVn
VVnd
dp L
rwL
ALCrw
A −−
−
=
•
, (6)
]
τ
~[
)(
1
τ 22222
2
2
d
dVpnGpVn
VVnd
dp L
rwL
ALCrw
A −−
−
=
•
, (7)
]
τ
~[
)(
1
τ 33333
3
3
d
dVpnGpVn
VVnd
dp L
rwL
ALCrw
A −−
−
=
•
. (8)
It is assumed that lung volume during breathing cycle is changed according to
−
−+
=
pauseatV
exhaleandinhaleat
t
DVV
L
c
L
L
),τ(
,)π2ττcos1(
2
)τ(
0
0
0 (9)
where D — respiratory volume of lungs, τ0 — start of breathing cycle, tc — its
duration.
It is also assumed that
τd
dVV L=
•
(10)
(V& = 0 during pause).
As it can be seen from the equations (1)–(8), gas diffusion from respiratory
tract to alveolar space during pause is not considered.
Equations of blood gas tension dynamics are developed taking into account
biophysical and chemical properties of blood. It is known that oxygen and carbon
dioxide can be transported with blood flow both physically dissolved in blood
plasma, and attached to hemoglobin (and CO2 is bound also with buffer bases),
while nitrogen is transported only in dissolved form.
Changes of blood gas tension in pulmonary capillaries can be influenced by
gas flows from alveolar space LCjG (5), from mixed venous blood and gas flow
passing with circulating blood into arterial channel. In addition model takes into
account presence of pulmonary shunt having volumetric speed of blood flow Qs.
Equations of dynamics of blood gas tension in pulmonary capillaries in connection
with above mentioned are as follows:
)],ηη)((γ
))((α[
)
∂
η∂γα(
1
τ
1
111
1
1
1
LCvs
s
LC
LC
QQHbG
ppQQ
p
HbVd
dp
LC
LCv
LC
LC
−−++
+−−
+
=
(11)
I.L. Bobriakova, 2014
ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 178
26
],
τ
η
γ)(α)(
)))((γ
γ))η1()η1[(((
)
∂
∂γα(
1
τ
2222
2
2
2
d
dHbVGQQpp
QQzzBH
Hbzz
p
zBHVd
dp
LC
LCS
SLCvBH
LCLCvvLC
BHLC
LCLCv
LC
LC
++−−+
+−−+
+−−−
+
=
(12)
)](α)(α[
α
1
τ 33333
3
3
ss
LC
QQpGQQp
Vd
dp
LCLCv
LC −−+−= . (13)
For more precise description of gas mass carry process in pulmonary
capillaries in the section «alveoli — pulmonary capillaries blood» the structure of
pulmonary capillaries can be differentiated, considering the elements of pulmonary
artery, capillary network itself and pulmonary vein.
In the equations (11)–(13) VLC — volume of blood of pulmonary capillaries,
Q — volume velocity of system blood flow, α1, α2, α3 — coefficients of gas
solubility in blood; γ, γBH — physiological constants, determined in Haldane and
Verigo-Bohr equations; η — hemoglobin saturation rate (its concentration will be
denoted as Hb) with oxygen, determined by empiric dependency
)12,0exp(75,0)052,0exp(75,11η 11 LCLC pmpm LCLCLC −+−−= . (14)
At mLC = const dependency of hemoglobin saturation rate Hb with oxygen has
a shape of S-curve, that is approximated by the expression (14). But it is known,
that alteration of pH value in blood causes shifting of the curve of oxyhemoglobin
dissociation (Bohr effect). Carbon dioxide facilitates displacement of oxygen from
oxyhemoglobin, and shape and location of dissociation curve are changed
depending on CO2 tension, namely, with its increasing affinity of hemoglobin with
oxygen is decreasing and dissociation curve is shifted to the right, i.e.
oxyhemoglobin dissociation is increased. Following equations serve as
mathematical interpretation of Bohr effect in the model
0, 25( 7,4) 1,LC LCm pH= − + (15)
α
lg1,6
22 LCp
BHpHLC += . (16)
Dependency (16) is Henderson-Hasselbach equation that is used for
determination of correlation among blood acidity (pH), tension of CO2 in blood
and concentration of hydrocarbonates that are buffer bases (BH). Saturation rate of
buffer bases of blood with carbon dioxide is expressed by the Michaelis-Menten
formula
352
2
+
=
LC
LC
P
p
ZLC . (17)
I.L. Bobriakova, 2014
ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 178
27
Equations (1)–(17) describe first three links of controlled system — change of
partial pressures of gases in respiratory tract, alveoli and blood of pulmonary
capillaries.
The blood saturated in pulmonary capillaries is flowing in the arterial channel
with volume velocity (Q–Qs), and, as a result of pulmonary shunt, — mixed venous
blood is flowing with volume velocity Qs, while the arterial blood is flowing out
with volume velocity Q. Taking into account conditions of continuity and material
balance the following can be written
],ηγηγη)(γ
αα
)(α[
)
∂
η∂γα(
1
τ
1111
11
1
1
1
avsLCs
s
s
a
a
HbQHbQQQHb
QppQ
pQQ
p
HbVd
dp
av
LC
a
a
−+−+
+−+
+−
+
=
(18)
],
τ
ηγ))(
(αγ))(
(γ))η1())(η1(
)η1[((
)
∂
∂γα(
1
τ
22
22
2
2
2
d
dHbVQppQQ
pQBHQzQQz
QzHbQzQQ
zQ
p
zBHVd
dp
a
as
sBHasLC
svzaLCsLC
vsva
BHa
aLC
v
a
a
a
+−−+
++−−+
++−−−−+
+−
+
=
(19)
],
τ
ηγ))(
(αγ))(
(γ))η1())(η1(
)η1[((
)
∂
∂γα(
1
τ
22
22
2
2
2
d
dHbVQppQQ
pQBHQzQQz
QzHbQzQQ
zQ
p
zBHVd
dp
a
as
sBHasLC
svzaLCsLC
vsva
BHa
aLC
v
a
a
a
+−−+
++−−+
++−−−−+
+−
+
=
]αα)(α[
α
1
τ 333333
3
3
avLC
a QppQpQQ
Vd
dp
ss
a
−+−= .
(20)
Let's express changes in organism gas tensions in tissue capillaries blood (cti)
and tissues (ti). Tissue reservoirs are considered on the level of organs and tissues,
namely, brain, heart, liver, kidneys, skeletal muscles (sk.m.), skin, adipose and
bone tissues (let m — number of tissue reservoirs).
Krogh model has been chosen as a model of tissue reservoir [4, 5], where
capillary network of tissues or organs is represented with one generalized
cylindrical capillary in the inlet of whose arterial blood is coming. During flowing
of blood along the capillary respiratory exchange between capillary blood and
tissue is occurring through its wall. Then blood is flowing in the vein.
In the same way as other equations of mathematical model, equations of
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28
dynamics pO2, pCO2, pN2 in blood of tissue capillaries and tissue are compiled on
the basis of principles of materiality of balance and flow continuity.
)]ηη(γ)(α[
)
∂
η∂
γα(
1
τ 1111
1
1
1
iiitictai
ict
i
i
ict
ctatt
ct
ct
HbQGppQ
p
HbV
d
dp
−+−−
+
=
(21)
],
τ
η
γγ)η1(γ)η1(
)(γ
)(α[
)
∂
∂
γα(
1
τ 2222
2
2
2
d
d
HbVzHbQzHbQ
zzBHQ
GppQ
p
z
BHVd
dp
i
iiiii
ii
itictai
ict
i
i
ict
ct
ctcttctata
ctaBHt
t
ct
BHct
+−−−+
+−+
+−−
+
=
(22)
].αα[
α
1
τ 33333
3
3
itiicti
i
ict GQpQp
Vd
dp
tta
ct
−−= (23)
Rate of intensity of metabolic process in tissue regions of this model is
characterized by the rate of oxygen consumption
it
q1 and rate of carbon dioxide
evolution
it
q2 . It is considered, that dependence of oxygen consumption rate
it
q1 in tissues of brain, kidneys and heart is determined by equation of Michaelis-
Menten:
it
it
itit pR
p
qq
1
10
11 )τ(
+
= , and in peripheral tissues, including skeletal
muscles, by correlation
2
0
0
11
η
)τ(η
)τ(
=
i
i
itit
ct
сtqq , (24)
where 0
1 it
q — consumption rate of O2 at given intensity of load under normal
conditions of external environment, R — const, 0η
ict — saturation rate of Hb in
blood with oxygen in these conditions, and )τ(η iсt — saturation rate of Hb in
changed conditions of experiment.
In addition, for definiteness it is considered that consumption rate of O2 in
heart muscles is linear function of the value of volume velocity of systemic blood
flow
,βα..1 += Qq msk
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29
,12 itiit
qnq t= (25)
where itn — respiratory coefficient.
In this connection dynamics of gas tension in tissue reservoirs in normal
physiological organism may be described by the equations
)(
)
∂
η∂
γα(
1
τ 11
1
1
1
itit
it
i
iiti
it qG
p
MbVd
dp
Mb
iMbt
−
+
= ,
(26)
),(
α
1
τ 22
2
2
itit
iti
it Gq
Vd
dp
t
+= (27)
it
iit
it G
Vd
dp
t
3
3
3
α
1
τ
= , (28)
where )12,0exp(1η 1 iti pMb −−= — saturation rate of myoglobin with oxygen,
1η0 ≤≤ iMb , iMbγ — coefficient characterizing maximal amount of O2, that can
be attached with 1 g of myoglobin (Mbi).
The link of breathing system «blood — tissue reservoirs» can be detailed at
the account of both differentiation of tissue reservoirs and modeling of transport
and exchange functions of blood in arterioles, tissue capillaries and venules.
To conclude the model it is necessary just to present equations representing
the dynamics of gas tension in mixed venous blood:
)),ηη(γ)(α[
)
∂
η∂
γα(
1
τ 1
1
1
11
1
1
1
∑∑
==
−+−
+
=
m
i
vctt
m
i
t
v
v
QQHbQppQ
p
HbVd
dp
iivicti
v
v
(29)
],
τ
η
γ)(γ
))η1()η1((γ
)(α[
)
∂
∂
γα(
1
τ
1
1
2
1
22
2
2
2
d
d
HbVQzzQBH
QzzQHb
QppQ
p
z
BHVd
dp
v
v
m
i
vcttBH
m
i
vvctctt
m
i
t
v
BHv
ii
iii
victi
v
v
+−+
+−−−+
+−
+
=
∑
∑
∑
=
=
=
(30)
∑
=
−=
m
i
t
v
QpQp
Vd
dp
viict
v
1
3333
3
3
).αα(
α
1
τ
(31)
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30
It is obvious that equations (1)–(31) represent dynamics of gas mass transfer
in organism in simplified form. However the principle of model development
allows the possibility to consider also arteriovenous anastomosis and model gas
tension on the walls of vessels — capillary, change of blood volume in tissue
capillaries, enabling to solve different tasks of theoretical and applied physiology.
The model is developed for average statistical (reference) person and it uses
the known experimental data on diffusion coefficients and gas permeability
through the membranes separating the medium, on other parameters characterizing
gas transport in the organism and metabolic processes taking place in tissues.
It would be important to clarify the role of these parameters in stabilization of
transients occurring during violation of equilibrium state caused by changes of
internal or external conditions. Transients caused by changes of composition of
inhaled air, transition from steady state to load and vice versa, during pressure
difference and process of the control of level of organism's gas homeostasis have
been studies. Data obtained in other studies [6, 7] have been accounted during the
work with the model.
On the model described above the simulation of functional self-organization
of physiological breathing system under conditions of highlands has been carried
out.
It was assumed that before experiments the gas mass transfer system in
organism was in stationary state, breathing gas normoxic (21 % O2 and 79 % N2).
Calculations have been executed for normal physiological data of person weighting
75 kg, volume velocity of oxygen consumption under calm conditions q = 4,3
ml/sec, Q = 96 ml/sec, Hb=0,14 g/ml, BH=0,479 g/ml, D=550 ml, tc = 4 sec. To
determine initial status of the system in simulation of arbitrary extreme situation it
was necessary to simulate first calm conditions under normal external conditions
itQ ,
it
p1 , and
it
p2 since experimental determination of initial values is
complicated. Therefore starting from certain determined approximated state of the
system (1)–(31) trajectories
it
p1 and
it
p2 have been put in steady regime for time
T. Calculations have been executed during time interval Т = 3000 sec with
simulation time-step ∆τ = 0,01.
Air pressure at sea level everywhere on the globe is on average close to one
atmosphere. Going up from the see level air pressure is decreasing; respectively its
density is also decreasing: air becomes more and more rarefied, i.e. amount of
oxygen in inhaled air is decreased. Therefore for simulation of highlands
conditions known data of air pressure and oxygen content at different heights have
been taken [8].
In present work computer analysis of model at different heights has been
carried out: 1 km (Bо = 674 mm Hg), 2 km (Bо = 596 mm Hg), 3 km
(Bо = 526 mm Hg), 4 km (Bо = 462 mm Hg) with oxygen content in air
respectively 18,5 %, 16,2 %, 14,3 %, 12,6 %. At the time zero values of gas
tensions in arterial blood and skeletal muscles were taken at normal state, i.e. in
calm conditions at sea level.
For every height set of experiments have been carried out under following
conditions:
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31
І. Hypoxia in calm state with compensation — increasing of Qsk.m in 2 times,
Vinhale in 1,5 times: Vinhale = 800 ml; tc = 4,0 sec; tinhale = texhale = 1,5 sec;
Q = 117,1 ml/sec; Qbrain = 14,88; Qheart = 6,135; Q sk.m. = 38,45;
Qother tissues = 57,595; q = 4,44 ml/sec; qbrain = 0,632; qheart = 0,4725; qsk.m. = 1,488;
qother tissues = 1,849.
II. Load hypoxia with compensation — increasing of qsk.m. and Qsk.m. in 2
times, Vinhale in 1,5 times: Vinhale = 800 ml; tc = 4,0 sec; tinhale = texhale = 1,5 sec;
Q = 117,1 ml/sec; Qbrain = 14,88; Qheart = 6,135; Qsk.m. = 38,45; Qother tissues = 57,595:
q = 5,9292 ml/sec; qbrain = 0,6321; qheart = 0,4725; qsk.m. = 2,9756;
qother tissues = 1,849.
III. Load hypoxia with compensation — increasing of qsk.m. in 2 times,
Qsk.m. in 4 times, Vinhale up to 1000 ml: Vinhale = 1000 ml; tc = 3,0 sec;
tinhale = texhale = 1,5 sec; Q = 159,2 ml/sec; Qbrain = 14,88; Qheart = 9,84;
Qsk.m. = 76,90; Qother tissues = 57,595; q = 6,2146 ml/sec; qbrain = 0,6321;
qheart = 0,7579; qsk.m. = 2,9756; qother tissues = 1,8490.
Results of experiments are presented in the table and on figures 1–4.
On figures 1–4:
row 1 — Hypoxia in calm state with compensation — increasing of Qsk.m. in 2
times, Vinhale in 1,5 times:
row 2 — Load hypoxia with compensation — increasing of qsk.m. and Qsk.m. in 2
times, Vinhale in 1,5 times:
row 3 — Load hypoxia with compensation — increasing of qsk.m. in 2 times, Qsk.m.
in 4 times, Vinhale up to 1000 ml:
Fig. 1. Oxygen consumption at height 1 km
Fig. 2. Oxygen consumption at height 2 km
Comparative analysis of results demonstrates, that with the same values &V
and Q levels of paO2 are below normal, and levels of psk.m.O2 , paСО2, psk.m.СО2 are
higher, but later on significantly decreasing.
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32
To approximate gas tension levels to normal values, V& and Qsk.m. have to be
significantly higher.
Fig. 3. Oxygen consumption at height 3 km
Fig. 4. Oxygen consumption at height 4 km
Table Results of experiments
Vinh=800 ml, tc=4 s, V& =12 l/min, Q=117 ,1 ml/sec, q=4.44 ml/sec
I H=1 km
B0=674 mm Hg
H=2 km
B0=596 mm Hg
H=3 km
B0=526 mm Hg
H=4 km
B0=462 mm Hg
t, c 0 100 2800 0 100 2800 0 100 2800 0 100 2800
q, ml/sec 4.4414 4.6629 4.6912 4.4414 4.2340 4.2052 4.4414 3.7204 3.5163 4.4414 3.2325 2.8227
PaО2, mm Hg 92.9 70.33 73.76 92.9 46.53 45.25 92.9 34.28 30.35 92.9 26.82 21.58
Рsk.m.О2, mm
Hg 26.71 35.95 33.46 26.71 30.81 27.71 26.71 25.86 21.41 26.71 21.92 16.33
РaСО2, mm Hg 26.17 34.73 18.40 26.17 34.22 16.61 26.17 33.54 13.93 26.17 32.89 11.21
Рsk.m.СО2, mm
Hg 32.03 47.53 22.07 32.03 47.13 19.69 32.03 46.65 16.30 32.03 46.19 12.94
Vinh=800 ml, tc=4 s, V& =12 l/min, Q=117,1 ml/sec, q=5,93 ml/sec
II H=1 km
B0=674 mm Hg
H=2 km
B0=596 mm Hg
H=3 km
B0=526 mm Hg
H=4 km
B0=462 mm Hg
t, c 0 100 2800 0 100 2800 0 100 2800 0 100 2800
q, ml/sec 5.9292 5.5470 5.5540 5.9292 4.8573 4.7243 5.9292 4.1594 3.8328 5.9292 3.5410 3.0263
PaО2, mm Hg 92.9 58.16 60.08 92.9 39.28 37.17 92.9 29.41 25.79 92.9 23.17 18.78
Рsk.m.О2, mm
Hg 26.71 25.66 24.08 26.71 21.96 19.61 26.71 18.68 15.48 26.71 16.03 12.13
РaСО2, mm Hg 26.17 36.14 21.81 26.17 35.26 18.66 26.17 34.36 15.18 26.17 33.58 12.01
Рsk.m.СО2, mm
Hg 32.03 48.92 27.24 32.03 48.29 22.94 32.03 47.65 18.38 32.03 47.09 14.31
Vinh=1000 ml, tc=3 s, V& =20 l/min, Q=159,2 ml/sec, q=6,21 ml/sec
III H=1 km
B0=674 mm Hg
H=2 km
B0=596 mm Hg
H=3 km
B0=526 mm Hg
H=4 km
B0=462 mm Hg
t, c 0 100 2800 0 100 2800 0 100 2800 0 100 2800
q, ml/sec 6.2146 6.7337 6.7791 6.2146 6.0453 6.0920 6.2146 5.1688 5.0975 6.2146 4.3299 4.0948
PaО2, mm Hg 92.9 67.67 70.52 92.9 43.58 43.83 92.9 31.26 29.45 92.9 23.74 20.91
Рsk.m.О2, mm
Hg 26.71 35.12 32.75 26.71 29.29 27.05 26.71 23.53 20.74 26.71 19.04 15.72
РaСО2, mm Hg 26.17 31.00 16.01 26.17 30.43 14.50 26.17 29.70 12.20 26.17 29.00 9.94
Рsk.m.СО2, mm
Hg 32.03 40.79 19.13 32.03 39.98 17.09 32.03 38.98 14.13 32.03 38.03 11.20
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33
CONCLUSIONS
Presented results of computer analysis of the model show that
hypometabolism is a necessary condition for stabilization of the status of organism
during highland hypoxia.
Calculations using mathematical model (1)-(33) with hypometabolism
mechanism (24) demonstrates that stabilization of the status of the breathing and
blood circulation system require less metabolic cost of regulatory mechanism
during the work under conditions of highlands.
1. Gray J.S. The multiple factor theory of respiratory regulation. Science. 1946. V.103.
P.739–744.
2. Grodins F. Mathematical analysis and digital simulation of the respiratory control system.
J. Appl. Physiol. 1967. Vol. 22, № 2. P.260–276.
3. Bioecomedicine. United information space / edited by V.I. Grytsenko. Kiev: Naukova
dumka, 2001. 318 p. (in Russian).
4. Lauer N.V., Kolchinskaya A.Z., Kulikov M.A. Calculation of parameters of organism's
oxygen regimes and development of oxygen cascades. Organism's oxygen regime and its
regulation. - Kiev: Naukova dumka, 1966, p.16–22 (in Russian).
5. Kolchinskaya A.Z., Man'kovskaya I.N., Misiura A.G. Breathing and oxygen regimes of
dolphins' organisms. — Kiev: Naukova dumka, 1980. 332 p. (in Russian).
6. Biloshyts'kyi P. V., Kliuchko O. M., Onopchuk Yu. M. Study of hypoxia issues by Ukrainian
scientists in the area of Elbrus mountain. Bulletin of NASU. 2007, no 3–4. P. 44–50
(in Russian).
7. Onopchuk Yu.N., Misiura A.G. Methods of mathematical modeling and control in theoretical
research and applied problem solving in sports medicine and physiology. Sports medicine.
2008, no 1. P. 181–188 (in Russian).
8. Table of International Standard Atmosphere. Available at: http://www.vsetabl.ru/154.htm.
UDK 519.876
MATHEMATICAL MODELING OF
HYPOMETABOLISM PROCESS TO IDENTIFY
PECULIARITIES OF HUMAN ORGANISM DURING
THE WORK UNDER CONDITION OF HIGHLANDS
I.L. Bobriakova
V.M. Glushkov Institute of Cybernetics of National Academy of Sciences of Ukraine
Introduction: Hypometaboliс and hypoxic states are in the focus of attention
of researchers since organism is often impacted by hypoxic factors, particularly
under conditions of highlands, as well as during significant physical loads or some
severe pathologic process.
Hypoxic or exogenous hypoxia is developed during decreasing of partial
pressure of oxygen in inhaled air. During hypoxic hypoxia the arterial oxygen
tension, the saturation of hemoglobin with oxygen and its total content in the blood
are being decreased. Negative impact can be caused also by hypocapnia, developed
as a result of compensatory hyperventilation of lungs. Extreme hypocapnia causes
worsening of blood supply to brain and heart (vasoconstriction) and respiratory
alkalosis. In this connection, it is interesting to investigate the energy reserves of
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34
organism and the ways of its strengthening under hypoxic conditions.
Problem definition: Investigation of the processes of human organism
breathing system self-regulation represents particular scientific area that is being
widely developed last years as a result of success in mathematical modelling.
However the regulation mechanisms of functional state of human organism under
conditions of highland hypoxia are not sufficiently studied yet. Infeasibility of such
investigations was related with the difficulties of experimental determination of the
several most important parameters and with the absence of adequate mathematical
models for dynamics of these processes.
The main task of this work consists in investigation of the mechanism of
formation of hypometabolic state of human being under conditions of highlands.
Mathematical model of functional systems of breathing and blood circulation has
been used to analyze this phenomenon on system level.
The purpose of this work is to develop and to investigate the mathematical
model of hypometabolism, and to develop the software for execution of computing
experiments with the model.
Results:
1. Developed complex of mathematical model of the gas mass carry process
in organism and software is applied for assessment of the dynamics of functional
state of human being on conditions of work at highlands.
2. Implemented numerical analysis of the models of respiratory exchange
control enables to:
• follow dynamics of the main physiological parameters of the model during
transient processes and in stationary states;
• forecast and quantitatively assess regulatory reactions of the organism
under given disturbances;
• carry out individualization of model developments on condition of
availability of array of data on anatomic-physiological peculiarities of specific
human being.
Obtained results are well correlated with the physiological experimental data.
Conclusions: Presented results of computer analysis of the model demonstrate
that hypometabolism is a pre-requisite for stabilization of organism's state under
highland hypoxia.
Calculations on mathematical model of the process of gas mass transfer with
hypometabolism mechanism demonstrate that stabilization of breathing and blood
circulation system state require less metabolic cost of regulatory mechanisms under
highland conditions.
Keywords: mathematical model, breathing and blood circulation system, gas
tensions, hypoxia, hypometabolism, disturbing impacts, computing experiments.
1. Gray J.S. The multiple factor theory of respiratory regulation. Science. 1946. V.103.
P.739–744.
2. Grodins F. Mathematical analysis and digital simulation of the respiratory control system.
J. Appl. Physiol. 1967. Vol. 22, № 2. P.260–276.
3. Bioecomedicine. United information space / edited by V.I. Grytsenko. Kiev: Naukova
dumka, 2001. 318 p. (in Russian).
I.L. Bobriakova, 2014
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35
4. Lauer N.V., Kolchinskaya A.Z., Kulikov M.A. Calculation of parameters of organism's
oxygen regimes and development of oxygen cascades. Organism's oxygen regime and its
regulation. - Kiev: Naukova dumka, 1966, p.16–22 (in Russian).
5. Kolchinskaya A.Z., Man'kovskaya I.N., Misiura A.G. Breathing and oxygen regimes of
dolphins' organisms. — Kiev: Naukova dumka, 1980. 332 p. (in Russian).
6. Biloshyts'kyi P. V., Kliuchko O. M., Onopchuk Yu. M. Study of hypoxia issues by Ukrainian
scientists in the area of Elbrus mountain. Bulletin of NASU. 2007, no 3–4. P. 44–50
(in Russian).
7. Onopchuk Yu.N., Misiura A.G. Methods of mathematical modeling and control in theoretical
research and applied problem solving in sports medicine and physiology. Sports medicine.
2008, no 1. P. 181–188 (in Russian).
8. Table of International Standard Atmosphere. Available at: http://www.vsetabl.ru/154.htm.
Получено 29.09.2014
I.L. Bobriakova, 2014
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