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South East Asian J. of Mathematics and Mathematical Sciences
Vol. 17, No. 2 (2021), pp. 367-384
ISSN (Online): 2582-0850
ISSN (Print): 0972-7752
A FOUR-COMPARTMENT MODEL TO ESTIMATE OXYGEN
AND CARBON DIOXIDE EXCHANGE CONCENTRATIONS VIA
BLOOD USING EIGENVALUE APPROACH
Ahsan Ul Haq Lone, M. A. Khanday and Saqib Mubarak
Department of Mathematics,
University of Kashmir, Srinagar - 190006, INDIA
E-mail : khanday@uok.edu.in
(Received: Feb. 01, 2021 Accepted: Jun. 29, 2021 Published: Aug. 30, 2021)
Abstract: A mathematical model of oxygen and carbon dioxide transport via
blood in the human body has been formulated. The model is represented by four
compartments: alveolar tissue, arterial blood, tissue and venous blood. The aim
of this study is to estimate the concentration profiles of oxygen and carbon dioxide
over alveolar tissue, arterial blood, tissue and venous blood compartments. The
formulation is based on the transport of oxygen from atmospheric air to alveo-
lar tissue and subsequently to capillary bed through inspiration and back flow of
carbon dioxide through expiration. Ordinary differential equations and balance
law have been employed to formulate compartment-wise transport phenomenon of
both oxygen and carbon dioxide in the respiratory tract via blood. The solution
of the model has been obtained using eigenvalue approach. The model provide the
information regarding absorption rate of oxygen and release rate of carbon dioxide
at the respective compartments. The results obtained in this study may help clin-
ical and bio-medical sciences to deal with respiratory ailments faced by the people
living at high altitudes. The results are in agreement with those arrived at by N. S.
Cherniack et al. (1968). In addition, these results may have utility in biomedical
engineering and physiological research problems.
Keywords and Phrases: Oxygen tension, Carbon dioxide tension, Blood; Com-
partment model, Balance law, Transfer rate, Eigenvalue method.
368 South East Asian J. of Mathematics and Mathematical Sciences
2020 Mathematics Subject Classification: 92-10, 92BXX, 92CXX.
1. Introduction
The respiratory system in human body plays a key role in transport and ex-
change of oxygen (O2) and carbon dioxide (CO2) between the surrounding atmo-
sphere and the tissues in the body. Oxygen is essential for life and it has been
verified by Nunn [9, 15, 16] that a human being consumes about 260 ml/min un-
der normal conditions. Oxygen is transported from atmospheric air into the lungs
via respiratory tract and is then carried in blood as dissolved in plasma and bound
to hemoglobin in RBC. It is transported to tissue via plasma (not directly from
Hb-O2). Hemoglobin acts as an O2storage to increase the oxygen carrying capacity
of blood. Carbon dioxide is carried by the blood as a waste product of oxidative
metabolism in opposite direction as that of oxygen transport, from tissues via blood
into the lungs, where it is removed by ventilation. The elimination rate of CO2is
approximately 160 ml/min under normal conditions as estimated by Nunn [9, 15,
16]. In addition to other functions the acid-base balance in the blood is maintained
by the exchange of O2and CO2in the body.
The respiration starts from the outside air towards lungs by inspiration. Air
enters the lungs connecting the atmospheric air with the alveoli - the air filled
sacs. From the alveoli O2diffuses across the wall of pulmonary capillaries into the
blood and transported in association with hemoglobin. The distribution of gases
throughout the body is accomplished by the blood circulation and it is important
to mention here that gas diffuses independently from the area where its partial
pressure is higher to the area where its partial pressure is lower. The larger partial
pressure differences accelerate the rates of gas diffusion. The oxygen leaves the
blood stream by diffusion and it enters the cells of the target tissue, where it is
used in respiratory metabolism. The oxygen transported through blood is utilized
by the cells in the metabolism that drives growth and other activities of the body.
The metabolic processes produce CO2that is transported into pulmonary capillar-
ies and finally diffuses across the lung membrane into alveoli. From the alveoli, it
is transported through the respiratory tract to the atmosphere.
Oxygen and carbon dioxide transport is known to play an important role in
cellular respiration operating in tissues. Oxygen is available free of cost in the
environment and therefore, humans need to be more conscious about the utility of
this valuable gas. Human tissues can survive for many days without food but not
without transport and exchange of oxygen and carbon dioxide. The first simple
mathematical model for oxygen transport to tissue was formulated by Krogh [13].
The model is based on the concept that oxygen from a capillary diffuses only into
a tissue cylinder concentric with the capillary, so that the O2flux at the external
A Four-compartment Model to Estimate Oxygen and Carbon Dioxide ... 369
surface of the cylinder vanishes. This model was further extended and modified
by Blum [1] by adding capillary wall resistance; Salathe et al. [19] introduced and
revised the model by adding flow conditions in the capillary; Reneau et al. [18]
refined the model in light of intra-capillary diffusion and convective effect of blood
and axial diffusion in the tissue. The mathematical study of O2uptake in spherical
tissues has also been studied by many researchers like Lin [14], Simpson and Ellery
[21] and Clark et al. [6]. Lin [14] studied diffusion of oxygen in a spherical cell with
non-linear uptake. Simpson and Ellery [21] studied steady state model of oxygen
transport in spherical tissue through Maclaurin’s series. This model was originally
reported by Lin [14] to represent the distribution of oxygen inside a cell and has
since been studied extensively by both the numerical analysis and formal analysis
communities. They extended these previous studies by deriving an analytic so-
lution to a generalized reaction-diffusion equation that encompasses Lin’s model
as a particular case. Clark et al. [6] developed a mathematical model of oxygen
diffusion across the bovine and murine cumulus-oocyte complex. They simulated
their results by MATLAB software and finally concluded that the model equations
cannot be solved analytically. However, Khanday and Najar [11] solved the model
equations analytically and the process of solution was based on the Maclaurin’s se-
ries method. Khanday and Najar [11, 12] computed the amount of oxygen uptake
in a spherical tissue through the capillary bed using analytical approach as well as
finite element approach.
Oxygen and carbon dioxide tensions of blood are important chemical determi-
nants of the level of ventilation. To a large extent, these tensions are determined
by the amount and distribution of oxygen and carbon dioxide stored in the body.
A mathematical model of carbon dioxide stores, reported by Cherniack et al. [5],
which closely predicted the changing carbon dioxide tensions during hyperventila-
tion and apnea at constant arterial oxygen tensions. In that model, carbon dioxide
was considered to be stored in multiple compartments representing several different
organs. Each compartment was assigned its own blood flow rate, metabolic rate,
and dissociation curve for carbon dioxide. In this paper, we formulate a mathemat-
ical model of oxygen and carbon dioxide transport via blood in the human body;
and estimate the concentration profiles of O2and CO2over alveolar tissue, arte-
rial blood, tissue and venous blood compartments. The model is represented by
four compartments: alveolar tissue, arterial blood, tissue and venous blood. The
transport and exchange mechanism of O2and CO2in the human body has partial
pressure as a main driving force. The concentration profiles of oxygen and carbon
dioxide at alveolar tissue, arterial blood, tissue and venous blood compartments
have been estimated with respect to time, initial O2and CO2concentration and
370 South East Asian J. of Mathematics and Mathematical Sciences
transfer rate from one compartment to another. The results obtained in this study
are in agreement with those arrived at by Cherniack et al. [5].
2. Mathematical Model and Solution
2.1. Postulated Conditions
Before setting up the O2and CO2transport model equations, following assump-
tions were made:
(i) Blood flowing through the capillaries is treated as a homogeneous mixture of
erythrocytes and plasma.
(ii) The interaction of O2and CO2in the blood is ignored.
(iii) The source and sink for both O2and CO2are taken constants.
(iv) The mass exchange between the compartments is represented by concentra-
tion gradient (diffusion).
(v) The production of carbon dioxide in the tissue is taken to be proportional to
the consumption of oxygen.
2.2. Nomenclature
The parameters used in the formulation of O2and CO2transport model equations
via alveolar tissue, arterial blood, tissue and venous blood are given in Table 1.
Table 1: Nomenclature of different parameters.
Quantity Symbol
Initial concentration of oxygen in alveolar tissue, mol/cm3X0
Initial concentration of carbon dioxide in tissue, mol/cm3Y0
Concentration of total oxygen (bounded and dissolved)
in the ith-compartment, mol/cm3Xi(t)
Concentration of total carbon dioxide (bounded and dissolved)
in the ith-compartment, mol/cm3Yi(t)
Partial pressure of oxygen, mmHg P O2
Partial pressure of carbon dioxide, mmHg P CO2
Solubility coefficient of oxygen in alveolar tissue, mol/(cm3mmHg)α
Solubility coefficient of carbon dioxide in tissue, mol/(cm3mmHg)β
A Four-compartment Model to Estimate Oxygen and Carbon Dioxide ... 371
2.3. Compartment Modelling
The compartments considered in the model can be generally represented as shown
in Figure 1. If the concentration of gas (O2or CO2) in the ith-compartment at
time t≥0 is Zi, then
dZi
dt = Input Rate −Output Rate.
=ki−1Zi−1(t)−kiZi(t) (1)
This is know as balance law [7]. The parameters ki−1and kidenotes the transfer
rates of gases (O2or CO2) from (i−1)th-compartment to ith-compartment and
from ith-compartment to (i+ 1)th-compartment, respectively.
-
ki−2(i−1)th
Zi−1
-
ki−1ith
Zi
-
ki(i+ 1)th
Zi+1
-
ki+1
Figure 1: Schematic representation of the compartment model.
Consider a four-compartment model consisting of alveolar tissue, arterial blood
and tissue with concentration of oxygen XL(t), XB(t) and XT(t), respectively; and
tissue, venous blood and alveolar tissue with concentration of carbon dioxide YT(t),
YB(t) and YL(t), respectively. The parameters a1and a2denotes, respectively, the
transfer rates of O2from alveolar tissue to arterial blood and from arterial blood
to tissue. The transfer rate of CO2from tissue to venous blood is denoted by
b1and from venous blood to alveolar tissue by b2. A schematic overview of the
four-compartment model of oxygen and carbon dioxide transport via blood in the
human body is shown in Figure 2.
X0
-
Alveolar Tissue
XL(t)
-
a1Arterial Blood
XB(t)
-
a2Tissue
XT(t)
---------
Tissue
YT(t)
Y0
b1
Venous Blood
YB(t)
b2
Alveolar Tissue
YL(t)
---------
Figure 2: A four-compartment model of oxygen and carbon dioxide transport via blood in the
human body with initial oxygen concentration X0and carbon dioxide concentration Y0.
372 South East Asian J. of Mathematics and Mathematical Sciences
The oxygen and carbon dioxide are carried by the blood to and from the tissues.
The O2(or CO2) that is carried by blood leaves one compartment and enters into
another at the rate proportional to the concentration of O2(or CO2) present in the
first compartment and so on. If the concentration of O2in the ith-compartment at
time t≥0 is Xi(t) (i=L, B, T ), then according to Balance Law [2]:
dXi
dt = (Amount of O2diffusion from (i−1)th-compartment to ith-compartment)
−(Amount of O2diffusion from ith-compartment to (i+ 1)th-compartment) (2)
Also, if the concentration of CO2in the ith-compartment at time t≥0 is
Yi(t) (i=L, B, T ), then again by Balance Law [2]:
dYi
dt = (Amount of CO2diffusion from (i−1)th -compartment to ith-compartment)
−(Amount of CO2diffusion from ith -compartment to (i+ 1)th-compartment) (3)
The partial pressure of a gas (O2or CO2), P, is related to the concentration,
C, through the coefficient of solubility, σ, according to Henry’s Law [9, 16]:
C=σP (4)
2.4. Oxygen Transport
Oxygen is continually being absorbed from the alveoli into the blood of the lungs,
and new oxygen is continually being breathed into the alveoli from the atmosphere.
The more rapidly oxygen is absorbed, the lower its concentration in the alveoli
becomes; conversely, the more rapidly new oxygen is breathed into the alveoli
from the atmosphere, the higher its concentration becomes. Therefore, oxygen
concentration in the alveoli, as well as its partial pressure, is controlled by the rate
of absorption of oxygen into the blood. The oxygen transport through arterial
blood admits the pattern shown in Figure 2.
The mathematical form of the compartment model describing oxygen transport
is given by the following set of ordinary differential equations:
dXL(t)
dt =−a1XL(t) + S;XL(0) = X0
dXB(t)
dt =a1XL(t)−a2XB(t); XB(0) = σ
dXT(t)
dt =a2XB(t)−S;XT(0) = δ
(5)
A Four-compartment Model to Estimate Oxygen and Carbon Dioxide ... 373
where σand δare constant oxygen concentrations in the blood and tissue com-
partments at t= 0, respectively; a1, a2(>0) are transfer rates that determines
how O2flux in the respective compartments (alveolar tissue/arterial blood/tissues)
decreases and the parameter Srepresents the source term or sink term for oxygen.
It represents source term with plus sign i.e., +Sand sink term with minus sign i.e.,
−S.
The solution to any initial value problem can be obtained using standard meth-
ods such as separation of variables, Laplace transform, Fourier transform etc. [3,
7, 17]. We use eigenvalue method to solve the model Equations (5) and (14). The
model Equations (5) and (14) have a solution on [0,∞) that is well behaved (i.e.,
bounded at any finite time and continuous for t > 0) and unique [3, 7, 17]. We
write Equations (5) in matrix form:
U0(t) = dU(t)
dt =A(t)U(t) + b(t), U(0) = U0(6)
where A(t) =
−a10 0
a1−a20
0a20
,
U(t) =
XL(t)
XB(t)
XT(t)
, b(t) =
S
0
−S
and U0=
X0
σ
δ
The eigenvectors u1,u2,u3corresponding to the eigenvalues −a1,−a2, 0 of the
matrix Aare:
u1=
a1−a2
a2
−a1
a2
1
, u2=
0
−1
1
, u3=
0
0
1
(7)
Therefore, the general solution of the system of Equations (5) is:
U(t) = c1u1e−a1t+c2u2e−a2t+c3u3+P.I (8)
where P.I =
S/a1
S/a2
−S(a1+a2)/a1a2
is the particular integral and c1,c2,c3are
arbitrary constants which can be determined from the initial condition. By using
the initial condition, we obtain c1,c2,c3:
c1=a2
a1−a2X0−S
a1, c2=−a1
a1−a2X0−S
a1+S
a2
−σ, c3=X0+δ(9)
374 South East Asian J. of Mathematics and Mathematical Sciences
Hence, the solution of the system of Equations (5) is:
XL(t) = X0−S
a1exp(−a1t) + S
a1
(10)
XB(t) = a1
a2−a1X0−S
a1(exp(−a1t)−exp(−a2t))−σ+S
a2exp(−a2t) + S
a2
(11)
XT(t) = X0−S
a1(1 + a1exp(−a2t)−a2exp(−a1t)
(a2−a1))+S
a2
−σexp(−a2t)−S
a2
+δ
(12)
Equations (10), (11) and (12) represent, respectively, the compartment-wise
concentration of oxygen in the alveolar tissue, arterial blood and tissues.
Flowing cases arise:
(A) If a1> a2, then from Equation (11) it follows that XB(t)>0.
(B) If a1< a2, we have XB(t)<0, which is not possible.
(C) If a1=a2, then there will be no pressure gradient and therefore, no flux of
O2.
Hence, a1> a2admits the flux between the compartments.
2.5. Carbon Dioxide Transport
The respiratory system is responsible for gas transfer between the tissues and the
outside air. The glucose oxidative metabolism reaction in the tissue cells is given
by [9]:
C6H12O6+ 6O2→6CO2+ 6H2O+AT P (13)
The carbon dioxide produced as a waste product by the metabolism in the
tissues must be moved by the venous blood to the outside air. Because an excessive
amount of CO2produces acidity that can be toxic to cells, excess CO2must be
eliminated quickly and efficiently. The carbon dioxide transport through venous
blood admits the reverse pattern shown in Figure 2.
The mathematical form of the compartment model describing carbon dioxide
transport is given by the following set of ordinary differential equations:
A Four-compartment Model to Estimate Oxygen and Carbon Dioxide ... 375
dYT(t)
dt =−b1YT(t) + M;YT(0) = Y0
dYB(t)
dt =b1YT(t)−b2YB(t); YB(0) = σ∗
dYL(t)
dt =b2YB(t)−M;YL(0) = δ∗
(14)
where σ∗and δ∗are constant carbon dioxide concentrations in the blood and lung
compartments at t= 0, respectively; b1, b2(>0) are CO2transfer rates, and the
terms +Mand −Mrepresents, respectively, the source and sink terms for carbon
dioxide.
The matrix formulation of the system of Equations (14) is:
V0(t) = dV (t)
dt =B(t)V(t) + b∗(t), V (0) = V0(15)
where B(t) =
−b10 0
b1−b20
0b20
,
V(t) =
YT(t)
YB(t)
YL(t)
, b∗(t) =
M
0
−M
,and V0=
Y0
σ∗
δ∗
By similar procedure as we did above in case of oxygen transport, we obtain
the general solution of the system of Equations (14):
V(t) = d1v1e−b1t+d2v2e−b2t+d3v3+P.I (16)
where v1,v2,v3are the eigenvectors corresponding to the eigenvalues −b1,−b2, 0
of the matrix B;d1,d2,d3are arbitrary constants which can be determined from
the initial condition; and P.I is the particular integral and are given in (17).
v1=
b1−b2
b2
−b1
b2
1
;v2=
0
−1
1
;v3=
0
0
1
;
d1=b2
b1−b2Y0−M
b1;d2=−b1
b1−b2Y0−M
b1+M
b2
−σ∗;d3=Y0+δ∗;
P.I =
M/b1
M/b2
−M(b1+b2)/b1b2
(17)
376 South East Asian J. of Mathematics and Mathematical Sciences
Hence, the solution of the system of Equations (14) is:
YL(t) = Y0−M
b1exp(−b1t) + M
b1
(18)
YB(t) = b1
b2−b1Y0−M
b1(exp(−b1t)−exp(−b2t))−σ∗+M
b2exp(−b2t) + M
b2
(19)
YT(t) = Y0−M
b1(1 + b1exp(−b2t)−b2exp(−b1t)
(b2−b1))+M
b2
−σ∗exp(−b2t)−M
b2
+δ∗
(20)
Equations (18), (19) and (20) represent, respectively, the compartment-wise
concentration of carbon dioxide in the tissues, venous blood and alveolar tissue.
Flowing cases arise:
(I) From Equation (19), we have YB(t)>0 if b1> b2, and YB(t)<0 if b1< b2,
which is not possible.
(II) If b1=b2, then there will be no pressure gradient and therefore, no flux of
CO2.
Hence, we have b1> b2.
3. Results and Discussion
A mathematical model based on ordinary differential equations and balance law has
been formulated to estimate the concentration profiles of oxygen and carbon dioxide
at alveolar tissue, arterial blood, tissue and venous blood compartments using four-
compartment model scheme. We validate our model to induce respiratory changes
such as apnea using the transfer rates a1, a2, b1, b2by comparing computed results
with the reference results [4], as depicted in Figure 3. Apnea denotes the cessation
of breading and ensuring hypoxia. Apnea can affect people of all ages and the
cause depends on the type of apnea we have. Apnea usually occurs while we are
sleeping. For this reason, it is often called sleep apnea.
Table 2: Numerical values of different parameters.
Parameter Numerical Value
α5.18 ×10−8mol cm−3mmHg−1[10]
β2.59 ×10−8mol cm−3mmHg−1[20]
A Four-compartment Model to Estimate Oxygen and Carbon Dioxide ... 377
The transfer rate a2= 0.625/sec determines how oxygen flux in the tissues
decreases for P O2>0 [8]. By cases-(A),(B),(C), we have a1> a2and we take
a1= 0.9375/sec. Thus, we have O2transfer rates N:= {a1, a2:a1= 0.9375, a2=
0.625 (/sec)}. By assumption-(v), we have ai=bi, i = 1,2 and by cases-(I),(II),
we have CO2transfer rates N∗:= {b1, b2:b1= 0.9375, b2= 0.625 (/sec)}. The
numerical value of the parameter S(= M) is taken to be 3.72×10−8mol cm−3sec−1
[19].
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
0
50
100
150
200
250
Time (min)
O2 Concentration (10−8 mol cm−3)
Model
Cherniack et al. (1968)
(a)
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
50
100
150
200
250
Time (min)
CO2 Concentration (10−8 mol cm−3)
Model
Cherniack et al. (1968)
(b)
Figure 3: Comparison of model result and the result reported by Cherniack et al. (1968) [4]
during apnea: (a) change in oxygen concentration in arterial blood, and (b) change in carbon
dioxide concentration in venous blood.
The partial pressure of oxygen in alveolar air is assumed to be 104 mmHg, it
decreases gradually due to flux once it enters inside alveolar tissue (lungs). The
initial oxygen concentration in alveolar tissue is taken to be X0= 104 mmH g ×α
and based on this input, the oxygen transport patterns through alveolar tissue,
arterial blood and tissues are presented in Figures 4. It has been observed from
the graphs that O2has a rapid decay in lungs, a hyperbolic behaviour in blood
and a sigmoid characteristic in tissue. The dash-dotted (−.−.) curves shown in
Figure 4 show efficient transport of O2in human body. These dash-dotted curves
are drawn against the O2transfer rates Nfor alveolar tissue, arterial blood and
tissue. The neighbourhood values of these O2transfer rates Nrepresent normal
rates of respiration taking place in the respective compartments. The results, how-
ever, reflect that this neighbourhood of O2transfer rates Nis very small defined
378 South East Asian J. of Mathematics and Mathematical Sciences
0 1 2 3 4 5 6 7 8 9 10
0
100
200
300
400
500
600 Alveolar Tissue
Time (min)
O2 Concentration (10−8 mol cm−3)
(A)
(B)
(N)
(C)
(D)
(a)
0 1 2 3 4 5 6 7 8 9 10
0
50
100
150
200
250 Arterial Blood
Time (min)
O2 Concentration (10−8 mol cm−3)
(A)
(B)
(N)
(C)
(D)
(b)
0 1 2 3 4 5 6 7 8 9 10
0
100
200
300
400
500
600 Tissue
Time (min)
O2 Concentration (10−8 mol cm−3)
(A)
(B)
(N)
(C)
(D)
(c)
Figure 4: Temporal variation of oxygen concentration in the respective compartments (alveolar
tissue/arterial blood/tissue) with X0=α×P O2and at different transfer rates /sec : (A) =
{a1= 1.98, a2= 1.32},(B) = {a1= 1.5, a2= 1.0},(N) = {a1= 0.9375, a2= 0.625},(C) =
{a1= 0.4, a2= 0.27}and (D) = {a1= 0.1, a2= 0.07}.
by {a:ai−0.25 < a < ai+ 0.25, i = 1,2}. Any increase or decrease in values of
O2transfer rates outside the neighbourhood values of Naffects respiration.
From our results, it follows that more the value of transfer rates (greater than
the neighbourhood values of N), greater will be the decrease in O2concentration
as depicted in Figure 4(a) by solid (−) curves and hence, faster will be the absorp-
tion of O2within the alveolar tissue. The cases in the point stem from different
situations in daily life like running, heavy exercise, etc. For values of O2transfer
A Four-compartment Model to Estimate Oxygen and Carbon Dioxide ... 379
rates lesser than the neighbourhood values of N, absorption of O2will be slow as
depicted in Figure 4(a) by dashed (− − −) curves. As a result less molecules of O2
are available for oxy-hemoglobin formation in the blood compartment and hence,
respiratory ailments like hypoxia happen to occur.
The plots in Figure 4(b) shows O2concentration increases rapidly inside the
blood compartment up to a certain (peak) value, following which it abruptly falls
down and plateaus thenceforth. The initial increasing trend can be attributed
to increase in influx of oxygen from alveolar tissue to arterial blood wherein it
combines with available hemoglobin molecules to yield oxy-hemoglobin, carrier of
oxygen to tissues. The oxy-hemoglobin molecules unload the oxygen in the tis-
sues, decreasing the concentration of O2in the blood as depicted by the decreasing
trend in the graph. The plateau part of the graph indicates a steady state in the
formation of oxy-hemoglobin in the arterial blood (determined by a1) and oxygen
unloading into tissues (determined by a2). The plots also show that the decrease
in the value of transfer rates (a1and a2) is associated with the dampening of peaks
of concentration plots; the higher values of transfer rates result in higher peaks
of oxygen concentration in blood. The graph is in confirmation with the results
arrived at by Cherniack et al. (1968) [4], attesting to the credibility of our results.
The curves in Figure 4(c) shows logistic patterns of the O2concentration in
the tissues with increase in the value of transfer rates and almost linear pattern
with decrease in the value of transfer rates. The O2concentration increases to a
certain level and then, after a definite time period, the consumption level reaches
saturation point resulting in the horizontal curve. This may be attributed to the
fact that inside the tissue, oxygen consumption in the respiratory metabolism keeps
the gradient in favour of oxygen diffusion into the tissues.
The carbon dioxide flow in backward direction, from tissues via blood into the
alveolar tissue, with respect to P C O2= 45 mmHg assumed to be produced in
the tissue is depicted in Figures 5. The graphs with different transfer rates shows
CO2behaviour in tissues, venous blood and alveolar tissue compartments. The
dash-dotted (−.−.) curves shown in Figures 5 drawn against the CO2transfer
rates N∗show efficient expulsion of CO2from human body. The neighbourhood
values of these CO2transfer rates N∗show normal elimination of CO2, without
occurrence of any CO2related respiratory problems. This neighbourhood of CO2
transfer rates N∗is similarly defined by {b:bi−0.25 < b < bi+ 0.25, i = 1,2}.
The increase in CO2transfer rates above the neighbourhood values of N∗re-
sults in faster elimination of CO2from the body and hence, results in steep decrease
in CO2concentration as depicted in Figure 5(a) by solid (−) curves. This rapid
expulsion of CO2from the body results in less availability of CO2for carbonic acid
380 South East Asian J. of Mathematics and Mathematical Sciences
0 1 2 3 4 5 6 7 8 9 10
0
20
40
60
80
100
120
140 Tissue
Time (min)
CO2 Concentration (10−8 mol cm−3)
(A*)
(B*)
(N*)
(C*)
(D*)
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
10
20
30
40
50
60 Venous Blood
Time (min)
CO2 Concentration (10−8 mol cm−3)
(A*)
(B*)
(N*)
(C*)
(D*)
(b)
0 1 2 3 4 5 6 7 8 9 10
0
20
40
60
80
100
120
140 Alveolar Tissue
Time (min)
CO2 Concentration (10−8 mol cm−3)
(A*)
(B*)
(N*)
(C*)
(D*)
(c)
Figure 5: Temporal variation of carbon dioxide concentration in the respective compartments
(tissue/venous blood/alveolar tissue) with Y0=β×P CO2and at different transfer rates /sec :
(A∗) = {b1= 1.98, b2= 1.32},(B∗) = {b1= 1.5, b2= 1.0},(N∗) = {b1= 0.9375, b2=
0.625},(C∗) = {b1= 0.4, b2= 0.27}and (D∗) = {b1= 0.1, b2= 0.07}.
formation in the blood compartment and consequently increases blood pH above
normal level. The increase in pH of blood is attributed to respiratory alkalosis in
human body. Conversely, if the CO2transfer rates decreases below the neighbour-
hood values of N∗, the expulsion of CO2will be slow and hence, it would lead to
increase in CO2concentration as shown in Figure 5(a) by dashed (−−−) curves.
This increase in CO2content in the blood increases carbonic acid formation and
decreases blood pH below normal level, resulting respiratory acidosis in human
A Four-compartment Model to Estimate Oxygen and Carbon Dioxide ... 381
body.
The graphs in Figure 5(b) represent change in concentration of CO2in the ve-
nous blood compartment in relation to time. The graph is in agreement with the
results arrived at by Cherniack et al. (1968) [4]. The plots reflect that the CO2
concentration increases initially to a peak value followed by steep decrease to a
certain value before it gets plateaued. The trend is similar to that shown by O2
concentration in the blood except that the path followed by CO2is opposite to
that of O2: oxygen flows from alveolar tissue to tissues via arterial blood, while
as CO2flows back from tissues to alveolar tissue via venous blood. The plot of
CO2concentration versus time shows similar trend in alveolar tissue as that of O2
concentration in the tissue, shown in Figure 5(c).
4. Conclusion
A mathematical model of O2and CO2transport via blood between the lung and
the tissue in the human body represented by four compartments: alveolar tissue,
arterial blood, tissue and venous blood was formulated. The aim of this study was
to estimate the concentration profiles of O2and CO2over alveolar tissue, arterial
blood, tissue and venous blood compartments. From the above discussion, it can
be concluded that our model provides useful information regarding absorption rate
of O2at alveolar tissue, arterial blood and tissue compartments and release rate
of CO2at tissue, venous blood and alveolar tissue compartments. The results
obtained in this study have applications in medical sciences in general and in the
field of biomedical engineering in particular to deal with respiratory ailments faced
by the people living at high altitudes. Hypoxia, pH level increase or decrease and
other respiratory ailment conditions can be handled by using suitable dataset in
the present model. This work can be further extended by incorporating hypoxia,
the interaction of O2and CO2in the blood and other environmental issues as
parameters in the model.
Acknowledgements
This work was supported by the SERB-DST, Government of India, under award
No.-EMR/2015/002487. We are highly thankful to the funding agency. The au-
thors would also like to thank Mr. Feroze A. Reshi for useful comments and
discussions.
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