EFFECTS OF NEURAL DRIVES ON BREATHING IN THE AWAKE STATE IN HUMANS
APPENDIX
System
Equations
The Controller
Nomenclature
Symbol Value
AF Alertness
Factor, lit/minute
DIST Incremantal
ventilation above normal after a disturbance, lit/min
Dp Peripheral
ventilatory drive, lit/min
Dc Central
ventilatory drive, lit/min
Dchem Chemical
drive, Dp+Dc, lit/min
HVD Hypoxic Ventilatory Depression,
lit/min
HVDEQ HVD
equilibrium value, lit/min
PSPP PSP
equilibrium value, lit/min
PSP_Th PSP
Threshold, lit/min
PSP Post-stimulus
potential, lit/min
PaCO2 Partial
pressure of CO2 in the arterial blood, mm Hg
PvBCO2 Partial
pressure of CO2 in and leaving the brain, mm Hg
SaO2 Arterial
oxygen saturation , %
tauHVD Hypoxic
Ventilatory Depression time constant, min 2.5 minutes
tauPSPC PSP module charging time
constant, min 0.125
minutes
tauPSPD PSP module discharging time
constant, min 0.25 minutes
TPCO2 Threshold
for arterial PCO2, mm Hg
TPCO2,B Threshold
for brain PCO2, PvBCO2, mm Hg
A.1. The Controller
The
controller input is the sum of the peripheral and central chemical drives from
the Chemical Controller, the Alertness Factor, Hypoxic Ventilatory Depression,
and Post-Stimulus Potential.
A.1.1 The Chemical Controller
The
chemical controller is a proportional controller whose inputs are arterial
blood carbon dioxide tension and arterial oxygen tension measured at the
peripheral sensor, and brain carbon dioxide tension, which is taken as the tension
of the venous blood leaving the brain. The gas tension values at the chemical
controller are delayed about 1 ˝ to 2 breaths in time from when they appeared
at the lung, about 6 to 8 seconds. The output of the chemical controller is
ventilation, in liters per minute. The output of the chemical controller can be
augmented or diminished by the neural inputs such as alertness drives,
post-stimulus potentiation, and also by hypoxic ventilatory depression all of
which which are assumed to act centrally.
The
equations immediately following define the chemical controller characteristic,
the steady state ventilatory characteristic for chemical ventilation as a
function of carbon dioxide tensions and oxygen saturation.
The
responses of the chemical controller are defined in two regions separated by a
“transition” PCO2 threshold, TPCO2. Controller carbon dioxide and
oxygen gains are higher above the transition threshold and lower below the
threshold.
For
PaCO2 > TPCO2 ,
Dp, the
peripheral ventilatory drive is
Dp=0.124*(101.72-SaO2)*(PaCO2-31.123)-1.43
liters per minute
Dc the
central ventilatory drive is
Dc=1.573*(PvBCO2-44.35)
liters/minute
For
PaCO2 <TPCO2,
Dp, the
peripheral ventilatory drive is
Dp=[(0.124*(101.72-SaO2)*(TPCO2
-31.123)-1.43)/ TPCO2]*PaCO2>
Dc, the
central drive is
Dc=[1.57*(TPCO2,B-44.35)/ TPCO2,B]*PvBCO2
TPCO2,B is
the brain carbon dioxide tension at TPCO2 in the normoxic steady state.
The total
chemical ventilatory drive, Dchem is
Dchem=Dp+Dc
Above TPCO2
the controller gain for carbon dioxide is
CO2_Gain=0.124*(101.72-SaO2)+1.57
liters/minute/mm Hg increase in PaCO2
Below TPCO2
the controller gain for carbon dioxide is
CO2_Gain=0.124*(101.72-SaO2)*(TPCO2
-31.123)-1.43)/ TPCO2 +
1.57*(TPCO2 -44.35)/ TPCO2
A.1.2. The Alertness Factor
The
alertness factor (AF) includes all the internal and external stimuli that
affect ventilation independent of PCO2, PO2, and
ventilation level. It includes the wakefulness drive described by other
authors, and as such it can vary with time as during transition states between
wakefulness and sleep and during sleep.
An
alertness factor of –11.5 liters/minute, which we consider to correspond to
deep NREM sleep, results in an equilibrium PaCO2 of 44.24 mm Hg, a
change of about 5 mm Hg from the awake state (AF= -0). We recognize that alertness factors can be
less than that, as in anesthesia, coma or death. For the current model an alertness factor equivalent to -2
liters/minute with a TPCO2 of 36.75 mm Hg. yields hyperventilation results
consistent with the work of Meah and Gardner.
This value is 1/2 mm Hg above where the peripheral and brain threshold
at normal PaO2 (relative to arterial pressure) would be without a transition
TPCO2. Figure 2 shows the effect on
ventilation for different alertness drives.
Shown
in Table I are
the equilibrium values resulting from the interaction between the chemical
controller and alertness drive. The ventilation can't be set without assuming
some value for the wakefulness drive. The choice is arbitrary, and we have
chosen the zero alertness drive as the awake state where the total ventilation
axis is the same as the zero chemical ventilation axis.
A.1.3 Hypoxic Ventilatory Depression
The
magnitude of hypoxic ventilatory depression, HVD is saturation and time
dependent.
To describe
HVD a third order response to hypoxia was used with three equal time constants
of 2.5 minutes, with an equilibrium value of
HVDEQ=35* (normoxic_saturation-SaO2). For normoxic saturation values of
99.3%, and saturation during hypoxia of 78 % this amounts to an equilibrium
depression of about 7.5 liters/minute. In this model,
with 10% fiO2, saturation was 78%, PaO2 37.5 mm Hg and PaCO2, 29 mmHg the loss
is 55% of the initial increase.
The
equations for the development of HVD are
Dhvd1=(1./(tau))*((HVDEQ)-hvd1);
Dhvd2=(1./(tau))*(hvd1-hvd2);
DHVD=delt*(1.0/(tau))*(hvd2-HVD);
D
signifies the time derivative.
A.1.4 Post-Stimulus Potentiation
Post-stimulus potentiation, PSP is simulated using the
equations for an R-C circuit. As a disturbance increases ventilation, the
module charges toward an equilibrium potential equal to the incremental
ventilation over normal caused by the disturbance. This charging occurs
exponentially in time. During charging there is no effect on ventilation. The module is gated however so that if there
is a sudden drop in ventilation, the module "discharges”, again
exponentially, allowing ventilation to fall gradually instead.
The
equilibrium potential for PSP at any time is
during
charging
DPSPP=
(1/tauPSPC)*(DIST-PSPP)
and during
discharging is
DPSPP=(1/tauPSPD)*(-PSPP)
PSP is made to disappear after about five minutes of
hypoxia by the introduction of a PSP threshold, which increases with hypoxia
and time. The threshold increases slowly during the first 2.5 minutes of
hypoxia, and then rises rapidly during the next several minutes so that the PSP
disappears by about 5 minutes.
PSP=PSPP-PSP_Th
Figure
3 shows the behavior of the PSP potential, the PSP threshold, and the PSP
available over time when ventilation is disturbed.
A.1.5 Total Ventilation
Total ventilation is the sum of
the chemical drives, alertness factor, post stimulus potentiation less hypoxic
ventilatory depression.
Total
Ventilation=Vchem+AF+PSP-HVD
System Equations
The Brain
and Muscle
Nomenclature
Symbol Value
betavBCO2 dissociation slope of CO2 of
blood leaving the brain, lit/lit/mmHg
betaBCO2 dissociation slope of CO2 in
brain tissue, lit/lit/mmHg 2/3*betaBCO2
betaMCO2 dissociation slope of CO2 in
muscle tissue, lit/lit/mmHg
betaMO2 dissociation slope of O2 in
muscle tissue, lit/lit/mmHg
CaBCO2 concentration of CO2 in blood
entering the brain, lit/lit
CaBO2 concentration
of O2 in blood entering the brain, lit/lit
CvBCO2 concentration of CO2 in blood
within and leaving the brain, lit/lit
CvBO2 concentration
of O2 in blood within and leaving the brain, lit/lit
CaMCO2 concentration of CO2 in blood
entering the muscle, lit/lit
CaMO2 concentration
of O2 in blood entering the muscle, lit/lit
CvMCO2 concentration of CO2 in blood
within and leaving the muscle, lit/lit
CvMO2 concentration
of O2 in blood within and leaving the muscle, lit/lit
MRBCO2 rate of production of CO2 in the
brain, lit/min 0.04 lit/min
MRBO2 rate
of consumption of O2 in the brain, lit/min 0.04 lit/min
MRMCO2 rate of production of CO2 in the
muscle, lit/min 0.15
lit/min
MRMO2 rate of consumption of O2 in the
muscle, lit/min 0.18 lit/min
PaBCO2 partial
pressure of CO2 in the blood entering the brain, mm Hg
Qdot cardiac
output; Qdot=QdotB+QdotM, lit/min 6
liters/min
QdotB cerebral
blood flow =0.038*PaBCO2+1/SaO2-1.42, lit/min
QdotM rate
of blood flow through the muscle; Qdot-QdotB, lit/min
VbB volume
of blood in the brain, lit 0.5 liters
VtiB volume
of tissue in the brain,lit 1.0 liter
VbM volume
of blood in the muscle,lit 4.3
liters
VtiM volume
of tissue in the muscle,lit
VtiM*
betaMCO2 0.00525 lit/mmHg
QBCO2 quantity
of CO2 in the brain, lit
QBO2 quantity
of O2 in the brain, lit
QMCO2 quantity
of CO2 in the muscle, lit
QMO2 quantity
of O2 in the muscle, lit
A.2 The Body Stores
A.2.1 Brain and Muscle
The body stores of carbon dioxide and oxygen are divided into
compartments; the brain, the muscle and the lung compartment, which includes
the anatomical dead space.
The compartments for the brain and for the muscle each consist of
a pool of blood and tissue. In the compartment oxygen is removed from the blood
and carbon dioxide produced. The CO2
tension of venous blood leaving a compartment is assumed to be in chemical
equilibrium with the tissue. The equations describing these compartments are
first-order differential equations for the conservation of carbon dioxide and
oxygen. For either brain or muscle tissue
The
quantity of CO2 in the brain is
QBCO2=VbB*CvBCO2 +VtiB*2/3*betaBCO2
The
quantity of O2 in the brain
QBO2=VbB*CvBO2
The
quantity of O2 in the muscle is
QMO2=VbM*CvMO2
The
quantity of CO2 in the muscle is
QBCO2=VbM*CvMCO2
+(VtiM*betavMCO2)
Cerebral
blood flow is
QdotB=0.038*PaBCO2+1/SaO2-1.42
This formulation delivers a constant supply of oxygen to the brain
when arterial saturation varies. At normal conditions cerebral blood flow is
1.096 liters per minute.
For brain, the conservation equations are
DQBO2=QdotB*(CaBO2-CvBO2)-MRBO2
DQBCO2=QdotB*(CaBCO2-CvBCO2)+MRBCO2
For muscle, the conservation equations are
DQMO2=QdotM*(CaMO2-CvMO2)-MRMO2
DQMCO2=QdotM*(CaMCO2-CvMCO2)+MRMCO2
The brain time constant with
rising arterial PCO2 at normal CO2 and oxygen levels is
70
seconds. The rise of PaCO2 during apnea is 6.4 mm Hg/min, which matches
experiment.
System Equations
The Lung
and Dead Space
Nomenclature
Symbol Value
betaLO2 dissociation
slope of CO2 and O2 in lung gas, lit/lit/mm Hg
CaLCO2 concentration of CO2 in blood
within and leaving the lung, lit/lit
CaLO2 concentration
of O2 in blood within and leaving the lung, lit/lit
CvLCO2 concentration of O2 in blood
entering the lung, lit/lit
CvLO2 concentration
of CO2 in blood entering the lung, lit/lit
CvBO2 concentration of CO2 in blood within and
leaving the brain, lit/lit
CvBCO2 concentration of CO2 in blood within and
leaving the brain, lit/lit
CvMO2 concentration
of CO2 in blood within and leaving the muscle, lit/lit
CvMCO2 concentration
of CO2 in blood within and leaving the muscle, lit/lit
DV dV/dt,
rate of change of lung volume, ventilation, lit/min
FRC Functional
residual capacity of the lung 2.5
liters
PaLCO2 partial
pressure of CO2 in and leaving the lung, mmHg
PaLO2 partial
pressure of O2 in and leaving the lung, mmHg
Qdot cardiac
output; Qdot=QdotB+QdotM, lit/min
QdotB cerebral
blood flow, lit/min
QdotB=QdotB1*PaBCO2+(QdotB2/CaBO2)-QdotB3
QdotM rate
of blood flow through the muscle, lit/min
QLCO2 quantity
of CO2 in the lung, lit
QLO2 quantity
of O2 in the lung,lit
VbL volume
of blood in the lung, including lung tissue volume, lit 0.22
liters
V volume
of gas in the lung, lit
A.2.2 The Lung and Dead Space
The lung compartment has two regions; blood and gas. The time
varying components are the total quantities of O2 and CO2 in the compartment
and quantities of O2 and CO2 in the incoming venous blood and air. The outputs
are the concentrations in the blood of oxygen and carbon dioxide in the
arterial blood leaving and the corresponding partial pressures. The model
assumes equality of partial pressure in the two regions. The average lung volume is 2.5 liters, and
the anatomical dead space is 0.15 liters.
The
concentrations of oxygen and carbon dioxide in the venous blood are the merged
outputs from the muscle and the brain.
CvLO2=(QdotB*CvBO2+QdotM*CvMO2)/Qdot;
CvLCO2=(QdotB*CvBCO2+QdotM*CvMCO2)/Qdot;
To
calculate the rates of change needed to conserve mass
during
inspiration,
DQLCO2=Qdot*CvLCO2-Qdot*CaLCO2+DV*fiCO2*f2c+DV*PaLCO2*betaLCO2+DV*PaLEXCO2*betaLCO2*f1c;
DQLCO2=Qdot*CvLCO2-Qdot*CaLCO2+DV*betaLCO2*PaLCO2+
DV*fiCO2*f2c+PaLCO2EX/betaLCO2;
Switches f1c and f2c are set and changed as the dead space volume is reached.
and
during expiration
DQLO2=Qdot*CvLO2-Qdot*CaLO2+DV*betaLO2*PaLO2;
DQLCO2=Qdot*CvLCO2-Qdot*CaLCO2+DV*betaLCO2*PaLCO2;
A.3. Dissociation Curves
To relate a concentration to gas tensions the dissociation
equations of Gomez are used. In these equations PO2 and PCO2 are the
appropriate gas tensions, e.g. PvbCO2 and PaBCO2, and CO2 and CCO2 are the
related concentrations, e.g., CvBO2 and CvBCO2.
v=(0.004273+0.04326*pow(PCO2,-.532))*PO2;
u=((30.0*v+2.8)*v+0.925)*v;
CO2=0.200*u/(1+u)+3e-5*PO2;
CCO2=(0.149-0.014*u/(1+u))*pow(PCO2,0.35);