math modelled approach to gas-exchange monitoring

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

𝑭𝑰CO2 = 0

FACO22 = 0

FECO2 = 𝐕𝐂𝐎𝟐𝑴𝑽

PECO2

BTPS= FECO2 × (𝟕𝟔𝟎 − 𝟒𝟕)

VCO2 = FACO2 × 𝐕𝐀 = 𝐅𝐄𝐂𝐎𝟐 × 𝐕𝐓 × 𝐟

FACO2 × 𝐕𝐀 = FACO21 × VA

1 = VCO2

𝐅𝐄𝐂𝐎𝟐× 𝐕𝐓 = FACO21 × VT − (VDana

+VDalv)

VDana +V𝐝alv = Vdphys = VT×𝐅𝐀𝐂𝐎𝟐

𝟏−𝐅𝐄𝐂𝐎𝟐

𝐅𝐀𝐂𝐎𝟐𝟏

VDana + V𝐝alv = Vdphys = VT × 𝐏𝐚𝐂𝐎𝟐−𝐏𝐄𝐂𝐎𝟐

𝐁𝐓𝐏𝐒

𝐏𝐚𝐂𝐎𝟐

Bohr equation

VDanat/VT = 𝑷𝑨𝑪𝑶𝟐−𝑷𝑬𝑪𝑶𝟐

𝑷𝑨𝑪𝑶𝟐

Enghoff modification

VDphysiol/VT = 𝑷𝒂𝑪𝑶𝟐−𝑷𝑬𝑪𝑶𝟐

𝑷𝒂𝑪𝑶𝟐

Alveolar dead space by substraction VDalv/VT = Enghoff – Bohr

VDalv/VT alv = 𝑷𝒂𝑪𝑶𝟐−𝑷𝒆𝒕𝑪𝑶𝟐

𝑷𝒂𝑪𝑶𝟐

Dead Space -fractions-

PACO2-alv mixed-EtCO2(mean)

EtCO2=mean!-(min+max)/2

Dead Space Caveats

Shunt dependence ( “shunt dead space” – Suter,1975)

because assuming that PaCO2=PACO2 is flawed

Shunt dependence will spuriously elevate VD

Regions with ↑ Va/Q are poorly set apart from regions with

Va/Q = ∞(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 => “sloping alveolar plateau”

Severe heterogeneity in τ => “sloping alveolar plateau”

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 ∆ CO2

Great heterogeneity in R•C product or/and severe V/Q mismatch

-sloping alveolar plateau-

Negative ∆ 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 τ CO2(FRC/VCO2) as in IACS or a non-homogeneous τ lung(R•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 ∆ CO2

Dead Space as risk factor -Enghoff’s 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=↑ ∆EELV=↓STRAINst+dyn=↓VD

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

β

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

∆CO2>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

FOWLER≈

DRAGER

FLETCHER≈

NICO

TANG

Volumetric Capnography -the math behind the monitors-

Volumetric Capnography FOWLER 1948 - VDana

Ay=ABCD=PNCD

AMP=MNB=Ax

Ay=VTCO2

PNCD=PD×(PN+CD)/2=VTPD×meanCO2alv

VTCO2=mean expCO2×VT=mean expCO2(VTPD + VTOP)

( VT – VTOP) × mean CO2alv= mean expCO2×VT

VTOP/VT= (meanCO2alv-mean expCO2 )/meanCO2alv

D

o

A P

M

N B

O

C

Ax

Ax

Ay

Volumetric Capnography FLETCHER 1981 – all VDs

Az/Axyz = PaCO2 × VDanat/PaCO2 × VT = VDanat/VT

Ax = VTCO2 = EtCO2mean × Vtalv

Ay = Vtalv × ( PaCO2 - meanEtCO2 )

Vtalv × meanEtCO2 = Vtalveficient × PaCO2

Ay = Vtalv × PaCO2 – Vtalveficient × PaCO2

Ay = PaCO2 × Vdalv => Ay/Axyz = VDalv/VT

(Ay+Az)/Axyz = VDphys/VT

Volumetric Capnography TANG 2006 – all VDs

Vdana and Vdalv can be read simultaneously on the abscissa


=225 ml

=160 ml

=65 ml


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 × (1-PECO2/PaCO2) =

= VT × (1-AODC/ (PaCO2×VT))

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 = VT×[1-ABCEF/(PaCO2×VT) ]=

= VT×[1-(PaCO2×BC)/(PaCO2×VT)]

= OB = VDphys Tang

Volumetric Capnography assessing recruitment/recruitability

Volumetric capnography for monitoring lung function during mechanical

ventilation, Yearbook of Intensive Care Medicine 2006, Suarez – 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 = 𝑷𝑨𝑪𝑶𝟐−𝑷𝑬𝑪𝑶𝟐

𝑷𝑨𝑪𝑶𝟐

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

𝒃𝒊𝒈𝒈𝒆𝒔𝒕 𝒂𝒔𝒔𝒖𝒎𝒑𝒕𝒊𝒐𝒏 𝑷𝑨𝑪𝑶𝟐 = 𝑬𝑻𝑪𝑶𝟐𝒎𝒆𝒂𝒏

PAO2=PIO2 - PACO2 × 𝑭𝑰𝑶𝟐 +𝟏−𝑭𝑰𝑶𝟐

𝑹

RDIFF = 𝟏

𝑫𝑳𝑶𝟐

RDIFF = 𝑷𝑨𝑶𝟐−𝑷𝒂𝑶𝟐

𝑽𝑶𝟐

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 × 𝑸𝒏𝑪𝒄𝑶𝟐𝒏𝟏

𝒏=𝟎

𝑪𝒂𝑶𝟐 = 𝟏/𝑸 𝑸𝒔𝒉𝒖𝒏𝒕× 𝑪𝒗𝑶𝟐+ 𝑸−𝑸𝒔𝒉𝒖𝒏𝒕 𝑪𝒄𝑶𝟐𝟏

𝒔𝒉𝒖𝒏𝒕 = 𝑪𝒄𝑶𝟐

𝟏−𝑪𝒂𝑶𝟐𝑪𝒄𝑶𝟐

𝟏−𝑪𝒗𝑶𝟐

PAO2=FIO2× 𝑷𝑩−𝑷𝑨𝑪𝑶𝟐

𝑹𝑸 (simplified alv.eq.)

𝑷𝑨𝑪𝑶𝟐 𝒎𝒖𝒔𝒕 𝒃𝒆, 𝒂𝒔 𝒊𝒎𝒑𝒍𝒊𝒆𝒅 𝒃𝒚 𝒕𝒉𝒆 𝒎𝒐𝒅𝒆𝒍, 𝑬𝑻𝑪𝑶𝟐𝒎𝒆𝒂𝒏 .

𝑺𝑨𝑶𝟐 𝒘𝒊𝒍𝒍 𝒃𝒆 𝒊𝒇𝒆𝒓𝒓𝒆𝒅 𝒇𝒓𝒐𝒎 𝑷𝑨𝑶𝟐 𝒂𝒄𝒄𝒐𝒓𝒅𝒊𝒏𝒈 𝒕𝒐 𝒂 𝒔𝒊𝒎𝒑𝒍𝒊𝒇𝒊𝒆𝒅 𝑶𝑫𝑪 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏

SO2= 𝑷𝑶𝟐𝟑+ 𝟏𝟓𝟎𝑷𝑶𝟐

− 𝟏 × 𝟐𝟑, 𝟒𝟎𝟎 + 𝟏 -1

𝑵𝒆𝒙𝒕, 𝑪𝒄𝑶𝟐 𝒘𝒊𝒍𝒍 𝒃𝒆 𝒊𝒏𝒇𝒆𝒓𝒓𝒆𝒅 𝒇𝒓𝒐𝒎: CcO2=(Hb× 𝑺𝑶𝟐 × 𝟏. 𝟑𝟒) + (𝑷𝑶𝟐 × 𝟎. 𝟎𝟎𝟑𝟏)

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 ∆PO2=PAO2-PcO2=PAO2-PaO2

𝑪𝒄𝑶𝟐𝟏 = 𝑷𝒄𝑶𝟐

𝟏𝜶𝑶𝟐+𝑯𝒃 𝑶𝑫𝑪 𝑷𝒄𝑶𝟐𝟏

CcO2

2 = PcO22αO2 + Hb ODC(PcO2

2)

𝑽𝑶𝟐𝟏 = 𝟏 − 𝒇𝑨𝟐 × 𝑽𝑨 × 𝑭𝑰𝑶𝟐− 𝑭𝑨𝑶𝟐

𝟏

VO21 = Q× 𝟎. 𝟏 × (𝑪𝒄𝑶𝟐

𝟏− 𝑪𝒗𝑶𝟐)

𝑽𝑶𝟐𝟐 = 𝒇𝑨𝟐 × 𝑽𝑨 × 𝑭𝑰𝑶𝟐− 𝑭𝑨𝑶𝟐

𝟐

VO22 = Q× 𝟎. 𝟗 × (𝑪𝒄𝑶𝟐

𝟐− 𝑪𝒗𝑶𝟐)

𝒇𝑨𝟐 =𝑽𝑨𝟐𝑽𝑨

= 𝑭𝑨𝑶𝟐− 𝑭𝑨𝑶𝟐

𝟏

𝑭𝑨𝑶𝟐𝟐− 𝑭𝑨𝑶𝟐

𝟏

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 patient’s 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

•Rdiff(∆PO2) Shunt

•AlveolarDS(∆PO2) Shunt

•V/Q mismatch(∆PO2) 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

∆PO

2=

PAO

2-P

cO2

Qshunt

CcO2 PcO2

𝑽𝑶𝟐 = 𝟏 − 𝒇𝑨𝟐 × 𝑽𝑨 × 𝑭𝑰𝑶𝟐− 𝑭𝑨𝑶𝟐𝟏 + 𝒇𝑨𝟐 × 𝑽𝑨 × 𝑭𝑰𝑶𝟐 − 𝑭𝑨𝑶𝟐

fA2 = VA2/VA

FAO2 = (1-fA2)× 𝑭𝑨𝑶𝟐

𝟏+ 𝒇𝑨𝟐 × 𝑭𝑨𝑶𝟐𝟐

𝑽𝑶𝟐 = 𝑽𝑶𝟐𝟏+ 𝑽𝑶𝟐

𝟐 = 𝑸 × (𝑪𝒂𝑶𝟐 − 𝑪𝒗𝑶𝟐)

VO2 = Q1 × 𝑪𝒄𝑶𝟐𝟏− 𝑪𝒗𝑶𝟐 + 𝑸𝟐 × (𝑪𝒄𝑶𝟐

𝟐− 𝑪𝒗𝑶𝟐)

𝑪𝒄𝑶𝟐 = Q1/ 𝑸𝒄 × 𝑪𝒄𝑶𝟐𝟏+ Q2/ 𝑸𝒄 × 𝑪𝒄𝑶𝟐

2

CaO2 = (1-shunt) × 𝑪𝒄𝑶𝟐 + shunt × 𝑪𝒗𝑶𝟐

𝑪𝒄𝑶𝟐𝟏 = 𝑷𝒄𝑶𝟐

𝟏𝜶𝑶𝟐+𝑯𝒃 𝑶𝑫𝑪 𝑷𝒄𝑶𝟐𝟏

CcO2

2 = PcO22αO2 + 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

∆PO2

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

∆PO2 ( image of V/Q ) varies from 0 kPa to 30 kPa

Thick lines stand for clinically important SaO2 (92%-98%)

At each ∆PO2 value, PaO2/FIO2 shows FIO2 dependence

PaO2/FIO2 FIO2 dependency – switching risk groups

Six pacients, graphs with SaO2/FIO2 and PaO2/FIO2 FIO2 dependency, two models are used – 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.

FiO2↓ FiO2↑

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


N > 350 ; mild hypoxemia = 300 –350 ; ALI = 201-300 ; ARDS < 200


risk group “switching’’ is 50% for shunt model and 38% for two

parameter model

by ↑ FiO2 (SpO2=92-98%)

- shunt model ALI 14→40

- shunt model ARDS 18→38

- two parameter model ALI 23→31

- two parameter model ARDS 18→24

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

“Perhaps 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.”

Kjaergaard and Rees, Critical Care 2007

math modelled approach to gas-exchange monitoring

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