MATH-MODEL DRIVEN APPROACH TO GAS-EXCHANGE MONITORING
Classic Approach to Gas-Exchange Monitoring
Normalized ratio /
difference
Equation / model
based monitoring
Dead Space Ventilation
-two compartment modeling-
FACO2/ET CO2
V T
VDana
water
FEO2
FECO2
FIO2
FICO2
VA1
FAO21
FACO21
VCO2 VO2
Q1
Q
Q2=0
Q
VA2
FAO22
FACO22
VA2/f =VDalv
ππ°CO2 = 0
FACO22 = 0
FECO2 = πππππ΄π½
PECO2
BTPS= FECO2 Γ (πππ β ππ)
VCO2 = FACO2 Γ ππ = π ππππ Γ ππ Γ π
FACO2 Γ ππ = FACO21 Γ VA
1 = VCO2
π ππππΓ ππ = FACO21 Γ VT β (VDana
+VDalv)
VDana +Vπalv = Vdphys = VTΓπ ππππ
πβπ ππππ
π πππππ
VDana + Vπalv = Vdphys = VT Γ πππππβπππππ
ππππ
πππππ
Bohr equation
VDanat/VT = π·π¨πͺπΆπβπ·π¬πͺπΆπ
π·π¨πͺπΆπ
Enghoff modification
VDphysiol/VT = π·ππͺπΆπβπ·π¬πͺπΆπ
π·ππͺπΆπ
Alveolar dead space by substraction VDalv/VT = Enghoff β Bohr
VDalv/VT alv = π·ππͺπΆπβπ·πππͺπΆπ
π·ππͺπΆπ
Dead Space -fractions-
PACO2-alv mixed-EtCO2(mean)
EtCO2=mean!-(min+max)/2
Dead Space Caveats
Shunt dependence ( βshunt dead spaceβ β Suter,1975)
because assuming that PaCO2=PACO2 is flawed
Shunt dependence will spuriously elevate VD
Regions with β Va/Q are poorly set apart from regions with
Va/Q = β(true dead space) because CO2 solubility is rather
modest in comparison to acetone solubility, which is used in
MIGET and distingushes VD as regions with Va/Q>100)
Severe V/Q mismatch => βsloping alveolar plateauβ
Severe heterogeneity in Ο => βsloping alveolar plateauβ
Sometimes PetCO2 > PaCO2
Shunt dependence of VD
Bull Eur Physiopathol Respir 1984
Effect of right-to-left shunting on alveolar dead space.
Mecikalski et al
Negative β CO2
Great heterogeneity in Rβ’C product or/and severe V/Q mismatch
-sloping alveolar plateau-
Negative β CO2
ETCO2 is continuously estimated while PaCO2 is a mean value.
ETCO2 can be regarded as a regional and temporal specific parameter while PaCO2 is a global, mean
parameter with no regional or temporal attributes
A low Ο CO2(FRC/VCO2) as in IACS or a non-homogeneous Ο lung(Rβ’C) will facilitate negative
differences
Slow alveoli are characterized by a high RC and this assigns them a constant,
moderate sloping.
Fast alveoli are characterized by a low RC and this gives them a 2 phase sloping, the
second being responsible for the overshoot ( high FRC/VCO2 )
Eg. Obese patients ( Ecw high )
Negative β CO2
Dead Space as risk factor -Enghoffβs dead space-
PULMONARY DEAD-SPACE FRACTION AS A
RISK FACTOR FOR DEATH IN THE ACUTE
RESPIRATORY DISTRESS SYNDROME, NEJM
2002, Nuckton et al
Dead Space as a PEEP setter
Optimum end-expiratory airway pressure in
patients with acute pulmonary failure, Suter et
al, NEJM 1975
Dead Space as a PEEP setter
OL-PEEP
OL-PEEP
Monitoring dead space during recruitment and PEEP titration in an
experimental model, ICM 2006, Suarez-Sipmann et al. Recruitment=β βEELV=βSTRAINst+dyn=βVD
Dead Space as a PEEP setter - VD as an image of respiratory mechanics more than of gas excahnge -
Best PEEP=lowest dynamic and static STRAIN
Best E=Best Vd
VD obeys Hickling model (1998 )
VD shows histeresis
VD is mechanics as well as E and is decoupled from gas
excange ( PaO2 )
Compliance and Dead Space Fraction Indicate an Optimal Level
of Positive End-Expiratory Pressure After Recruitment in
Anesthetized Patients, Anesth Analg 2008, Maisch and Tusman
Volumetric Capnography
Ξ²
Integrating the CO2 and volume signals
The abscissa is represented by volume
3 phases, 2 slopes, one inflection point-the curve changes sign-
on SII
Volumetric Capnography -phases and derived variables-
Phase I begins with the start of expiration and is completed after
βCO2>0.1% from baseline
Phase II starts at the end of phase I and ends at the intersection point of
slopes SII and SIII. Its inflection point (changes sign) is pretty much its
midpoint and likely represents the interface between Vdaw and alveolar gas,
that is the interface between convection and diffusion. It contains both
alveolar gas as well as Vdaw gas. RC influences phase II.
Phase III begins at the aforementioned intersection and ends with expiration.
This is gas inside the alveoli.
Slope II is an image of acini expiratory times. The more homogeneous the
expiration, the more the slope increases.
Slope III is again influenced by mechanical time constants but mostly by V/Q
mismatch. The slope increases with heterogeneity.
VD
FOWLERβ
DRAGER
FLETCHERβ
NICO
TANG
Volumetric Capnography -the math behind the monitors-
Volumetric Capnography FOWLER 1948 - VDana
Ay=ABCD=PNCD
AMP=MNB=Ax
Ay=VTCO2
PNCD=PDΓ(PN+CD)/2=VTPDΓmeanCO2alv
VTCO2=mean expCO2ΓVT=mean expCO2(VTPD + VTOP)
( VT β VTOP) Γ mean CO2alv= mean expCO2ΓVT
VTOP/VT= (meanCO2alv-mean expCO2 )/meanCO2alv
D
o
A P
M
N B
O
C
Ax
Ax
Ay
Volumetric Capnography FLETCHER 1981 β all VDs
Az/Axyz = PaCO2 Γ VDanat/PaCO2 Γ VT = VDanat/VT
Ax = VTCO2 = EtCO2mean Γ Vtalv
Ay = Vtalv Γ ( PaCO2 - meanEtCO2 )
Vtalv Γ meanEtCO2 = Vtalveficient Γ PaCO2
Ay = Vtalv Γ PaCO2 β Vtalveficient Γ PaCO2
Ay = PaCO2 Γ Vdalv => Ay/Axyz = VDalv/VT
(Ay+Az)/Axyz = VDphys/VT
Volumetric Capnography TANG 2006 β all VDs
Vdana and Vdalv can be read simultaneously on the abscissa
Volumetric Capnography TANG 2006 β all VDs
=225 ml
=160 ml
=65 ml
Volumetric Capnography TANG 2006 β all VDs
Vdana and Vdalv can be read simultaneously on the abscissa
We draw perpendiculars so that AOJA = AHJI (Fowler)
and AOKB = AFEDK (Tang)
VT = OC ; VDanat = OA (Fowler) ; VDphys = OB (Tang)
PECO2 = AODC /VT
VDphys Enghoff = VT Γ (1-PECO2/PaCO2) =
= VT Γ (1-AODC/ (PaCO2ΓVT))
G F E
D
H
I
K
J
C
J
A B C
D
E F G
H
I
K
AODC = AOKB + ABKDC = ABKDC+AFEDK = ABCEF
VDphys Enghoff = VTΓ[1-ABCEF/(PaCO2ΓVT) ]=
= VTΓ[1-(PaCO2ΓBC)/(PaCO2ΓVT)]
= OB = VDphys Tang
Volumetric Capnography assessing recruitment/recruitability
Volumetric capnography for monitoring lung function during mechanical
ventilation, Yearbook of Intensive Care Medicine 2006, Suarez β Sipmann et
al
Volumetric Capnography assessing recruitment/recruitability
Volumetric capnography for monitoring lung function during mechanical
ventilation, Yearbook of Intensive Care Medicine 2006, Suarez β Sipmann et
al
Volumetric Capnography assessing recruitment/recruitability
Lung Recruitment Improves the Efficiency of Ventilation and Gas
Exchange During One-Lung Ventilation Anesthesia, Anesth and Analg,
Tusman, Suarez Sipmann et al., 2004
How Tusman et al have confused Graf
Bohr equation
VDphysiol/VT = π·π¨πͺπΆπβπ·π¬πͺπΆπ
π·π¨πͺπΆπ
PACO2-alv mixed-
EtCO2(mean)
The answer lies in slopes
Diffusion Limitation - one compartment modeling -
V T
VDana water
FEO2
FECO2
FIO2
FICO2
CvO2
Q
CcO2
PcO2
CaO2
Q
VCO2 VO2
VA
PAO2
PACO2
πππππππ ππππππππππ π·π¨πͺπΆπ = π¬π»πͺπΆπππππ
PAO2=PIO2 - PACO2 Γ ππ°πΆπ +πβππ°πΆπ
πΉ
RDIFF = π
π«π³πΆπ
RDIFF = π·π¨πΆπβπ·ππΆπ
π½πΆπ
RDIFF, when computed through a one compartment model, is nothing but a global parameter,
It does not set apart any of the gas-exchange abnormalities.
Shunt Model - two compartments, one parameter -
FACO2/ET CO2
V T
VDana
water
FEO2
FECO2
FIO2
FICO2
VA1
FAO21
FACO21
VCO2 VO2
Q1
Q
Q0 or
Qshunt
Q=Q1+ Q0
VA0= 0
CvO2
CaO2 CvO2
CcO21 PcO2
1
CaO2= 1/Q Γ πΈππͺππΆπππ
π=π
πͺππΆπ = π/πΈ πΈπππππΓ πͺππΆπ+ πΈβπΈπππππ πͺππΆππ
πππππ = πͺππΆπ
πβπͺππΆππͺππΆπ
πβπͺππΆπ
PAO2=FIO2Γ π·π©βπ·π¨πͺπΆπ
πΉπΈ (simplified alv.eq.)
π·π¨πͺπΆπ ππππ ππ, ππ πππππππ ππ πππ πππ ππ, π¬π»πͺπΆπππππ .
πΊπ¨πΆπ ππππ ππ πππππππ ππππ π·π¨πΆπ ππππππ πππ ππ π ππππππππππ πΆπ«πͺ ππππππππ
SO2= π·πΆππ+ ππππ·πΆπ
β π Γ ππ, πππ + π -1
π΅πππ, πͺππΆπ ππππ ππ ππππππππ ππππ: CcO2=(HbΓ πΊπΆπ Γ π. ππ) + (π·πΆπ Γ π. ππππ)
True Shunt - FIO2 = 1 trial -
CcO2-CaO2/
CcO2-Cv02
True shunt(S1)
at FiO2=1, there is S2>S1 through
resorbtion atelectasis.
V/Q mismatch
V/Q Mismatch Model - two compartments, one parameter -
FAO2 VA
FACO2/ET CO2
V T
VO2
VDana
water
FEO2
FECO2
FIO2
FICO2
VA-VA2
PAO21
PACO21
VCO2
1 VO2
1
Q1
Q
Q2
Q=Q1+ Q2
CaO2=CcO2 CvO2
CcO21 PcO2
1
FAO21
FAO22
VA2
PAO22
PACO22
VCO2
2 VO2
2
PcO22 CcO2
2
PcO2n=PAO2
n βPO2=PAO2-PcO2=PAO2-PaO2
πͺππΆππ = π·ππΆπ
ππΆπΆπ+π―π πΆπ«πͺ π·ππΆππ
CcO2
2 = PcO22Ξ±O2 + Hb ODC(PcO2
2)
π½πΆππ = π β ππ¨π Γ π½π¨ Γ ππ°πΆπβ ππ¨πΆπ
π
VO21 = QΓ π. π Γ (πͺππΆπ
πβ πͺππΆπ)
π½πΆππ = ππ¨π Γ π½π¨ Γ ππ°πΆπβ ππ¨πΆπ
π
VO22 = QΓ π. π Γ (πͺππΆπ
πβ πͺππΆπ)
ππ¨π =π½π¨ππ½π¨
= ππ¨πΆπβ ππ¨πΆπ
π
ππ¨πΆππβ ππ¨πΆπ
π
Monoparametric
gas exchange monitoring
V/Q mismatch
Alveolar deadspace Rdiff
Shunt
Fitting one parameter models to data
84
86
88
90
92
94
96
98
100 SaO2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
FEO2
Shunt=21%
fA2=0.27
Shunt=11%
fA2=0.19
Shunt=10%
fA2=0.14
Shunt=8%
fA2=0.14
Dashed red line Alveolar dead space model
Vdalv/VT=53%
Diffusion limitation model
Rdiff=42kPa/(l/min)
V/Q model
fA2=0.27
Solid orange line Shunt model
Shunt=15.5%
stands for SaO2/FEO2 for the same
patient.
using all necessary data, it is
calculated for each of these four
points shunt and fA2 according to
previous equations.
fitting parameter model to data =
finding the one parameter value that will
subsequently describe patientβs data
with utmost precision
one parameter models show dependence on inspired O2 fraction
they cannot appropriately describe gas-exchange
Beginnings of Two Parameter Models
The PIO2 vs. SpO2 diagram: A non-invasive measure of pulmonary oxygen
exchange, EUROPEAN JOURNAL OF ANAESTHESIOLOGY 1995, Sapsford and
Jones - Cambridge
The two parameters are shunt and V/Q mismatch +
PACO2/R effect measured as % and as P I O2-PcO2
(kPa) respectively
Mass balance for O2 in blood and air, ODC equation,
computer algorithm based on fitting the model
parameters to P I O2/SaO2 data pairs
Beginnings of Two Parameter Models
A noninvasive method for evaluating the effect of thoracotomy on shunt and
perfusion inequality, ANAESTHESIA 1997, Gray and Jones
Beginnings of Two Parameter Models - course of family of curves -
Noninvasive assessment of shunt and ventilation/perfusion ratio in neonates with
pulmonary failure, Arch Dis Child Fetal Neonatology Ed. 2001, J G Jones et al
We need the numbers
In A there is dependency of PIO2 vs SaO2 on aVDO2.
Given that aVDO2 is dependent on Q, we infer Q dependency.
Simply eyeballing might not be enough. We need the numbers.
In B there is dependency on Hb. Hb is nonetheless more
stable.
The PIO2 vs. SpO2 diagram: A non-invasive measure of pulmonary oxygen
exchange, EUROPEAN JOURNAL OF ANAESTHESIOLOGY 1995, Sapsford and
Jones - Cambridge
Reverse avDO2 dependency - monitoring cardiac output -
Cardiac output estimation using pulmonary mechanics in mechanically
ventilated patients, Biomedical Engineering Online 2010, Sundaresan et al
Refinement of the two parameter models
β’Rdiff(βPO2) Shunt
β’AlveolarDS(βPO2) Shunt
β’V/Q mismatch(βPO2) Shunt
Mathematical models of pulmonary gas exchange - validation and application to
postoperative hypoxaemia , Aalborg Hospital, Denmark, Soren Kjaergaard
V/Q mismatch and Shunt Model - three compartments, 2 parameters -
FAO2 VA
FACO2/ET CO2
V T
VO2
VDana
water
FEO2
FECO2
FIO2
FICO2
VA-VA2
PAO21
PACO21
VCO2
1 VO2
1
Q1
Q=Qc+ Qshunt
Q2
Qc=Q1+ Q2
CaO2
CvO2 Q
CcO21 PcO2
1
FAO21
FAO22
VA2
PAO22
PACO22
VCO2
2 VO2
2
PcO22 CcO2
2
PcO
2n=
PAO
2n
βPO
2=
PAO
2-P
cO2
Qshunt
CcO2 PcO2
π½πΆπ = π β ππ¨π Γ π½π¨ Γ ππ°πΆπβ ππ¨πΆππ + ππ¨π Γ π½π¨ Γ ππ°πΆπ β ππ¨πΆπ
fA2 = VA2/VA
FAO2 = (1-fA2)Γ ππ¨πΆπ
π+ ππ¨π Γ ππ¨πΆππ
π½πΆπ = π½πΆππ+ π½πΆπ
π = πΈ Γ (πͺππΆπ β πͺππΆπ)
VO2 = Q1 Γ πͺππΆππβ πͺππΆπ + πΈπ Γ (πͺππΆπ
πβ πͺππΆπ)
πͺππΆπ = Q1/ πΈπ Γ πͺππΆππ+ Q2/ πΈπ Γ πͺππΆπ
2
CaO2 = (1-shunt) Γ πͺππΆπ + shunt Γ πͺππΆπ
πͺππΆππ = π·ππΆπ
ππΆπΆπ+π―π πΆπ«πͺ π·ππΆππ
CcO2
2 = PcO22Ξ±O2 + Hb ODC(PcO2
2)
Fitting two parameter models to data
All three models are equivalent in assessing
shunt
All three models are equivalent in assessing
βPO2
VDalv inferred from an O2 based 2 parameter
model is NOT equivalent to the one determined
from a CO2 based model
Rdiff is not supported by MIGET as an
important constituent of gasexchange
disturbances
VDalv O2 based has no meaning in day to day
clinical practice
V/Q mismatch and Shunt Model - shunt and fA2 impact on ODC -
SHUNT V/Q or fA2
Predicting risk of hypoxemia
V/Q mismatch vs Shunt
Predicting risk of hypoxemia
DISCRIMINATING BETWEEN THE EFFECT OF SHUNT AND REDUCED VA/Q ON ARTERIAL OXYGEN
SATURATION IS PARTICULARLY USEFUL IN CLINICAL PRACTICE, J Clin Monit and Comp 2000, Jones et al
MIGET at the bedside
PaO2/FIO2
Risk indicator as in Berlin ARDS definition
Global gas - exchange parameter
Non independent behavior with respect to shunt,
avDO2, PaCO2, RQ, Hb
Non independent parameter when FIO2 is varied
PaO2/FIO2 FIO2 dependency according to shunt
avD02 is constant, that is constant metabolism
Three shunt values
At each shunt value, PaO2/FIO2 shows FIO2 dependence
PaO2/FIO2 FIO2 dependency according to avDO2
avD02 varies, that is changing CO for a constant VO2
Same shunt value
At each avDo2 value, PaO2/FIO2 shows FIO2 dependence
PaO2/FIO2 FIO2 dependency according to shunt
Shunt varies from 0% to 30%
Thick lines stand for clinically important SaO2 (92%-98%)
At each shunt value, PaO2/FIO2 shows FIO2 dependence
PaO2/FIO2 FIO2 dependency according to V/Q
βPO2 ( image of V/Q ) varies from 0 kPa to 30 kPa
Thick lines stand for clinically important SaO2 (92%-98%)
At each βPO2 value, PaO2/FIO2 shows FIO2 dependence
PaO2/FIO2 FIO2 dependency β switching risk groups
Six pacients, graphs with SaO2/FIO2 and PaO2/FIO2 FIO2 dependency, two models are used β shunt and shunt+V/Q,
thick lines pertain to SaO2 = 92%-98%, dashed line is shunt model whereas solid line is the other PaO2/FIO2 FIO2
dependency brings about different risk groups even though shunt or V/Q do not really change.
FiO2β FiO2β
normal Mild hO2 ALI ARDS
Shunt model Nr =23
Nr=15
Nr=40
Nr=38
Normal =64 23 14 27 0
Mild hO2 =20 0 1 13 6
ALI=14 0 0 0 14
ARDS=18 0 0 0 18
Shunt+V/Q mism Nr=42 Nr=19 Nr=31 Nr=24
Normal=56 39 12 5 0
Mild hO2 =19 3 6 9 1
ALI=23 0 1 16 6
ARDS=18 0 0 1 17
PaO2/FIO2 FIO2 dependency β switching risk groups
N > 350 ; mild hypoxemia = 300 β350 ; ALI = 201-300 ; ARDS < 200
PaO2/FIO2 FIO2 dependency β switching risk groups
risk group βswitchingββ is 50% for shunt model and 38% for two
parameter model
by β FiO2 (SpO2=92-98%)
- shunt model ALI 14β40
- shunt model ARDS 18β38
- two parameter model ALI 23β31
- two parameter model ARDS 18β24
The shunt model has a poor fit to the data
PaO2/FiO2 is FIO2 dependent (use the same FIO2 when tracking
evolution)
PaO2/FiO2 is a poor gas exchange tracker
βPerhaps more appropriate would be to replace the PaO2/FiO2 ratio
with two parameters, a parameter to describe the oxygenation
problems due to V/Q mismatch and one to describe oxygenation
problems due to shunt.β
Kjaergaard and Rees, Critical Care 2007