Sistema Respiratorio en Simulink

Section 3.7 • Chemical Regulation of Ventilation
3.7 CHEMICAL REGULATION OF VENTILATION
59
The final example that we will consider in this chapter is the chemoreflex regulation of
respiration. In nonnoxic conditions, breathing is controlled almostexclusively by the levelof
CO2 in the arterial blood. In fact, ventilation is highly sensitive to PaC0
2
, the partial pressure
of CO2 in arterial blood. A rise in PaC0 2 by 1mmHg from its normal level of approximately
40 nun Hg may increase the ventilatory output by a third of its resting level. However, upon
ascentto altitudeor duringinhalation of a gas mixturecontaining low O2 content, there is an
additional drive to breathedue to hypoxia. This hypoxic drive becomesnoticeable when the
partial pressure of O2 in arterial blood, Pa0 2, drops below 70mmHg. Since the metabolic
consumption rate of O2 and the metabolic elimination rate of CO2 are relatively constant in
the steadystate,a higherlevelof ventilation wouldlead to an increasein Pa02 and a decrease
in PaC0
2
, which in tum would lower ventilation. Therefore, the "negative" part of this
negative-feedback system is embedded in the gas exchange characteristics of the lungs. The
simple model that we will analyze is depicted in block diagram form in Figure 3.14a. The
ventilatory control system is divided into two components: the gas exchanging portion and
the respiratory controller. An important distinction between this model and the previous
models that we haveanalyzed is that the components here are eitherdual-input-single-output
(controller) or single-input-dual-output (lungs) systems.
3.7.1 The Gas Exchanger
The gas exchanging component involves a combination of many processes that take
place in the lungs, vasculature, and body tissues. However, as a first approximation, we will
restrict our attention only to gas exchange occurring in the lungs. The operating character-
isticsof the gas exchangerare obtained by deriving the mass balanceequations for CO2 and
O2. Webeginby considering CO2 exchange, whichis depicted schematically in Figure3.14b.
We assume the metabolic CO2 production rate to be VC0 2; this is the rate at which CO2 is
delivered to the lungsfromthe bloodthat is perfusing the pulmonarycirculation. In the steady
state,this must equalthe net flow of CO2 exitingthe lungs in gas phase.The latteris equalto
the difference in volumetric fraction (or concentration) of CO2 in the air entering (FICO) and• 2
leaving (FAC0
2
) the alveoli multiplied by the alveola.r ventilation, VA. The alveolar ventilation
represents that portion of the total ventilation, VE, that actually participates in the gas
exchange process. Part of VE is "wasted" on ventilating the non-gas-exchanging airways in
the lungs; this flow is known as "dead space ventilation", Vo. Thus, we have
(3.44)
and the CO2 massbalance:
(3.45)
In Equations (3.44) and (3.45), the ventilatory flow rates are generally measured in
BTPS (body temperaturepressure saturated) units, whilethe CO2 metabolic production rate is
usually expressed in STPD (standard temperature pressure dry, i.e., at 273K and 760nun Hg)
units. The constantk allows volumes and flows measured in BTPS units to be converted into
STPD units. This conversion is achieved by using the ideal gas equation:
VSTPD760 VBTPS(PB - 47)
=
273 310
(3.46a)

60
Respiratory
Controller
(a)
Chapter 3 • Static Analysis of Physiological Systems
Lungs
V
o
If
~ ~c~ !- -
~o2
(c)
(3.46b)
Figure 3.14 (a) Steady-state model of the chemical regulation of ventilation. (b) Model
of steady-state CO2 exchange in the lungs. (C) Model of steady-state O2
exchange in the lungs.
The above equation assumes body temperature to be 37°C or 310 K and a saturated water
vapor partial pressure of47 mmHg at that temperature. PB represents the barometric pressure
under which the gas exchange process is taking place; at sea level, this is 760 mmHg, but the
value decreases with ascent to high altitude. Upon rearranging Equation (3.46a), we obtain
the following expression for k:
k == VSTPD == Ps - 47
VBTPS 863
The volumetric fractions, F1C0 2 and FAC0
2
, can be converted into their corresponding partial
pressures, P 1C0 2 and P AC0
2
, using Dalton's law:
P IC0 2 == FIC02(PB - 47), P AC0 2 =FAC02(PS - 47) (3.47a,b)

Section 3.7 • Chemical Regulation of Ventilation 61
Therefore, using Equations (3.46b) and (3.47a,b) in Equation (3.45) yields the following
result:
863Vc 0 2
PACO = PICO +--.-~
2 2 VA
(3.48)
Equation (3.48) shows a hyperbolic relation between PAC0
2
and VA' and for this reason, is
commonly referred to as the metabolic hyperbola. By employing the same kind of mass
balance analysis (see Figure3.14c),a similar "metabolichyperbola" can be deduced for O2:
(3.49)
The negative sign in Equation (3.49)accounts for the fact that O2 is removed from the lungs
by the perfusing blood and, therefore, the alveolar O2 content (PAC0
2
) will always be lower
than the inhaled O2 content (PI0 2).
A further assumption that we will make in this model is that the alveolar partial
pressures are completely equilibrated with the corresponding arterial blood gas partial
pressures, i.e.,
(3.50a,b)
This is approximately true in normals, although for O2, there is an alveolar-arterial gradient
of 5mmHg or more. However, in patients with lung disease, ventilation-perfusion mismatch
can giverise to rathersubstantial gradients between the alveolarand arterial partialpressures.
Apart from the shared value of VA' Equation (3.48) and Equation (3.49) appear to
suggest that CO2 and O2 exchange are independent of each other. This, however, is a
consequence of limiting our considerations only to the exchange processes that occur in gas
phase. For more realistic modeling, it is essential to incorporate the blood-gas dissociation
relationships for CO2 and 02, as well as considerations of gas exchange at the level of the
body tissues. For instance, CO2 affects the affinity with which O2 is bound to hemoglobin
(Bohr effect), and the levelof oxygenation affects the blood CO2 concentration at any given
partial pressure (Haldane effect). At the level of cellularmetabolism, the rate at whichCO2 is
produced for a given O2 consumption rate depends on the type of nutrient being oxidized.
Fortunately, the effects of these complications on the final predictions of the alveolar or
arterial partial pressures are not very large.
3.7.2 The Respiratory Controller
The controller part of the system includes the chemoreceptors, the neuronal circuits in
the lowerbraininvolved in the generation of the respiratory rhythm as wellas the neuraldrive
to breathe, and the respiratory muscles. The controller response to CO2 has been shownto be
linear over the physiological range. In the absence of vigilance, such as during sleep, the
controlleroutputfallsrapidly to zero(i.e.,centralapneaoccurs)whenPaC0 2 decreases slightly
below normal awake resting levels. Exposure to hypoxia (i.e., when Pa0 2 decreases below
100mmHg) leads to an increase in the CO2 response slope as well as the ventilatory
controlleroutput. Hence, there is a stronginteraction between CO2 and O2 at the level of the
controller. Cunningham (1974) has modeled the ventilatory controller output(Vc) as the sum

62 Chapter3 • Static Analysis of Physiological Systems
of an 02-independent term and a term in which there is a multiplicative interaction between
hypoxia and hypercapnia:
. ( 32)Vc = 1.46+ Pa02 _ 38.6 (PaC02 - 37),
=0,
(3.51a)
(3.51b)
Note that the above expression becomes progressively less valid as Pan approaches the• 2
asymptotic value of 38.6, in which case Vc would become infinitely large. As pointed out
below, precautions have to be taken to ensure that Pa02 does not fall belowa physiologically
realistic range.
3.7.3 Closed-Loop Analysis: Lungs and Controller Combined
In the closed-loop situation, the controlleroutput, r-. wouldequal the ventilation, VE'
drivingthe gas exchange processes for CO2and O2,as shown in Figure3.14a. To obtain the
steady-state operating point for the closed-loop system,Equations (3.48) through(3.51)must
be solved simultaneously. As we have done previously, it is possibleto arrive at the solution
throughgraphical analysis. However, sincethreevariables (VE, PAC0
2
and PA0
2
) are involved,
both graphical and algebraic methods of solution can be quite laborious. Thus, in this case,
we resort to a numerical approach using SIMULINK.
Figure 3.15 displays the layout of the SIMULINK model file "respss . mdl " that
allows the solution of the steady-state ventilatory control equations. Basically, the program
simulates the closed-loop system in "open-loop mode." A repea ting sequence block
(labeled"VdotEin Input Ramp") is used to generate a linearly increasing sequenceof VE
values. Each VE value is fed into Equation(3.48) and Equation (3.49) so that corresponding
PAC0
2
and PA0
2
values are generated. Each pair of PAC0
2
and PA0
2
values is subsequently
used in Equation (3.50) and Equation (3.51) to generate the corresponding ventilatory
controller output, Vc (labeled "VdotEout" in Figure 3.15). The initially low VE values
wouldproducehigh PACO and lowPAO levels, whichwouldact on the controllerto produce• 2 • 2
high Vc values. However, as VE increases, chemical drive levels would decrease, in tum,
decreasing r: The steady-state equilibrium point is established at that combination of PACO
• • 2
and PA0
2
values where VE level becomes equal to Vc . A relational operator block is
incorporated to check for this condition and to stop the simulation when the condition is
satisfied. The steady-state valuesof VE, PAC0
2
andPA0
2
are savedto the Matlabworkspace in
the scalar variables "vent", "paco2" and "pao2", respectively. An important point to
note is that we included a satura tion block to limit the allowable range for PA0
2
' This
ensures that PA0
2
would not fall to a point where the 02-dependent term in the controller
became infinite or negative.
The resultsof twoSIMULINK simulations are shownin Figure3.16. In case(a),PI0 2 is
set equal to 150mmHg (i.e., 21o~ room air) while PIca is set equal to zero. Due to the• 2
initially low VE value,PACO and PAO are initially "V 67 and "V 65mmHg, respectively, while
Vc is higherthan 20Lmin- r.As VE i~creases, PACO decreases whilePAO risesand Vc falls.• 2 • 2
The simulation is terminated when Vc becomes equal to VEe This occurs at
VE = Vc = 6Lmin-I, PAC0
2
= 40mmHg and PA0
2
= 100mmHg. In case (b), we simulate
a subjectinhalinga gas mixture containing only 15%O2 or, equivalently, a subjectascending
to an altitudeof 8500ft. Thus, PI02
is set equal to 107mmHg whileP,co2
is left at zero. As

Section 3.7 • Chemical Regulation of Ventilation 63
PAC02
CIOlEout
•dote I"
R.a-1Ionel
Ope.tor
LlMG 02 EXCHANGE
Figure 3.15 SIMULINK program "respss.mdl" to determine the steady-state
operating point of the ventilatory control system.
before, the initial value of PAC0
2
is in the high 60s, while PA0
2
is consistent with a value
lower than 40 mrn Hg. However, due to the effect of the saturation block, PA0
2
is not allowed
to fall below 40. The final equilibrium point is established at VE =: 6.1 Lrnin", PAC0 2 = 39
and PA0
2
=: 58.3. These two examples demonstrate quite clearly the negative feedback nature
of respiratory control. Although exposure to hypoxia tends to produce an additional drive to
breathe, the added ventilation blows off CO2, and consequently, the lower PAC0
2
acts to offset
the hypoxic-induced drive. As a result, ventilation remains close to its original nonnoxic
level.
The equivalent graphical analyses of cases (a) and (b) are presented in Figure 3.17a and
b. The controller responses are depicted as bold curves, while the gas exchange responses are
shown as light curves. The steady-state operating points for normoxia and hypoxia are labeled
Nand H, respectively. The two-dimensional plots do not provide a good sense of the three-
dimensional nature of the problem. For instance, the controller plot shown in the ventilation-
pAC0
2
graph in case (a) represents only the PA0
2
value of 100 mmHg. Similarly, the controller
plot shown in the ventilation-PAo 2 graph in case (a) assumes PAC0
2
to be 40mmHg. The
same comments apply to the graphs in case (b).

64 Chapter 3 • Static Analysis of Physiological Systems
(a)
xY Plot X Y Plot
20

150
15
(It
.!B
~ 10 ~ 100
>

>
0 50
0 10 15 20 30 40 50 60 70
X Axis X Axis
x Y Plot
(b)
xY Plot
20 80
70
15
60
(It en
~ 10
)(
<C:
>

> 50
40
30
0
0 10 15 20 30 40 50 60 70
X Axis
X Axis
Figure 3.16 Results of SIMULINK simulations to determine the steady-state operating
point during (a) nonnoxia (PI0 2 = 150nunHg) and (b) inhalation of 15%
02 mixture (P01
2
= 107nun Hg). Left panels: Ventilatory controller output
vs. ventilation in L min- t (simulation is terminated when they become
equal); Right panels: corresponding trajectory ofPA0
2
vs. PAC0
2
in nun Hg.

Section Bibliography 65
4540
PAC0
2
:(mm Hg)
35
o
10
2 2
10
'I
c
6 'E 6
~
c
.2
4 ~ 4
E
~
8 8
10510095
r-'-~~""""""'---L.....~~-L--L~'----&.-.l-.L--+- 0
11090
(a)
454035
o
10
'I
c
6 'E 6
~
c
.2
4 ~ 4
E
~
8 8
2 2
6055
,............~~'-r--A--~L.-.L--r--4-.JL.......L.--'--~ 0
6550
(b)
Figure 3.17 Graphical analysis of the steady-state regulation of ventilation during (a)
normoxia (PI02 == 150mm Hg) and (b) exposure to mild hypoxia through
inhalation of 15% O2 mixture or ascent to altitude (~ 8500 ft). The steady-
state operating points are labeled N in case (a) and H in case (b).
BIBLIOGRAPHY
Cunningham, DJ.C. Integrative aspects of the regulation of breathing: A personal view. In: MTP
International Review ofScience: Physiology Series One, Vol. 2, Respiratory Physiology (edited
by J.G. Widdicombe). University Park Press, Baltimore, 1974; pp. 303-369.
Guyton, A.C., C.E. Jones, and T.G. Coleman. Circulatory Physiology: Cardiac Output and Its
Regulation, 2d ed. W.B. Saunders, Philadelphia, 1973.

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