Supplementary Data

Computer simulation clarifies mechanisms of carbon dioxide

clearance during apnoea

Marianna Laviola1, Anup Das2, Marc Chikhani1,3, Declan G. Bates2, Jonathan G. Hardman1,3

1 Anaesthesia and Critical Care, Division of Clinical Neuroscience, School of Medicine,

University of Nottingham, Nottingham NG7 2UH, UK

2 School of Engineering, University of Warwick CV4 7AL, UK

3 Nottingham University Hospitals NHS Trust, Nottingham NG7 2UH, UK

The online data supplement for this paper contains additional material that could not be

included in the main text due to space limitations. The following document describes the

simulation model employed in the paper and in detail the new modules added to the

simulator. The optimization strategy used in fitting the model to the healthy subject during

apnoea is also described.

Interdisciplinary Collaboration in Systems Medicine (ICSM)

simulator.

The Interdisciplinary Collaboration in Systems Medicine (ICSM) simulator is a highly

integrated computational model of the pulmonary and cardiovascular systems based upon the

Nottingham Physiology Simulator and it has been applied and validated on a number of

different studies 1-10.

Description of pulmonary model

The model is organized as a system of several components, each component representing

different sections of pulmonary dynamics and blood gas transport, e.g. the transport of air in

the mouth, the tidal flow in the airways, the gas exchange in the alveolar compartments and

their corresponding capillary compartment, the flow of blood in the arteries, the veins, the

cardiovascular compartment, and the gas exchange process in the peripheral tissue

compartments. Each component is described as several mass conserving functions and solved

as algebraic equations, obtained or approximated from the published literature, experimental

data and clinical observations. These equations are solved in series in an iterative manner, so

that solving one equation at current time instant (t k) determines the values of the independent

variables in the next equation. At the end of the iteration, the results of the solution of the

final equations determine the independent variables of the first equation for the next iteration.

The iterative process continues for a predetermined time, T, representing the total simulation

time, with each iteration representing a ‘time slice’ t of real physiological time. At the first

iteration(t k , k=0), an initial set of independent variables are chosen based on values selected

by the user. The user can alter these initial variables to investigate the response of the model

or to simulate different pathophysiological conditions. Subsequent iterations (t k=t k−1+ t)

update the model parameters based on the equations below.

The pulmonary model consists of an “anatomical” deadspace and multiple alveolar

compartments (N alv) in parallel.

The series deadspace (SD) is located between the mouth and the alveolar compartments

and consists of the trachea, bronchi and the bronchioles, representing the conducting zone

where no gas exchange occurs. Inhaled gases pass through the series deadspace during

inspiration and alveolar gases pass through the series deadspace during expiration. In the

model, the series deadspace is simulated as a series of stacked rigid layers (laminas) (N lam =

50) of equal volume. The total volume of the series deadspace (vSD) is set to 150 ml. Each

lamina, j, has a known fraction (f SD , jx ) of gas x. These gases comprise oxygen (O2), nitrogen,

carbon dioxide (CO2), water vapor and a 5th gas used to model additives such as helium or

anesthetic vapors. At each iteration (constituting a time step) of the model, the gases shift up

or down the stacked laminae. The pressure gradient across the series deadspace (between the

mouth/mechanical ventilator and the alveolar compartments) and rate of flow (set by the

ventilator rate, spontaneous breathing rate or insufflation rate) determine the amount of fresh

gas entering the series deadspace.

Fig. S1 describes the movement of a volume of gas through the series deadspace during

inhalation in one time step. At the beginning of a time step, the air within the deadspace is

split across the laminae. In Fig. S1, A refers to the volume of gas remaining in the first layer;

B refers to the extra volume of gas shifted into the final layer due to A, so that the total blue

area adds up to the vSD (volume of series deadspace) and C refers to the amount of fresh gas

entering the series deadspace. Due to the discretization of the flow, the volume of the gas

moving into the series deadspace might not be equal to a whole laminae. Therefore, white

laminae (or fraction of them) refer to empty laminae. During gas movement, the new f SD , jx of

a lamina is calculated proportionally from fractions of gases of the air entering the lamina and

any remaining air already within the lamina.

The gas movement during inhalation as shown in Fig. S1 can be summarized as follows: 1)

the fractions of gases in B are combined proportionally to the flow of air leaving the series

deadspace; 2) all the fractions of gases are shifted down one lamina apart from the first

lamina; 3) a portion of C is added into the first lamina to make a complete lamina; 4) the

fractions of the gases in this lamina are updated and moved down into the empty lamina; 5)

the fractions of gases from any complete leftover laminas (tmp∫¿) and partial laminas (tmp¿¿)

due to be moved within this time step are added to the series deadspace layers from the top.

The same amount of gas is removed from the bottom lamina, adding to the flow out into the

lungs, such that the total volume of the series deadspace remains conserved.

During an iteration of the model, the flow (f) of air to or from an alveolar compartment i at

time t k is determined by the following equation:

f i(t k)=( pv (t k )−p i (t k ))

(Ru+RA, i ) for i=1 , …,Nalv (1)

where pv (t k ) is the pressure at mouth or supplied by the mechanical ventilator at (t k), pi (t k)

is the pressure in the alveolar compartment i at (t k), Ru is the constant upper airway resistance

and RA, i is the bronchial inlet resistances of the alveolar compartment i. N alv is the total

number of alveolar compartments (for the results in this paper, Nalv = 100). The total flow of

air entering the series deadspace at time t k is calculated by

f SD(t k )=∑i=1

N A

f i(t k) (2)

During the inhaling phase,f SD ≥ 0, while in the exhaling phasef SD<0.The volume of gas x, in

the ith alveolar compartment (v i , x), is given by:

vi , x (t k )={v i , x (t k−1 )−f i(t k) ∙v i , x( tk−1)

v i(t¿¿ k) Exhaling¿v i , x (t k−1)+ f i(t k) ∙FN SD

(t k )Inhaling

for i=1 , …,Nalv (3)

In (3), x is any of the five gases (O2, N2, CO2, H2O or α). The total volume of the ith alveolar

compartment, vi is the sum of the volume of the five gases in the compartment.

vi( tk )=v i ,O2(t k)+v i , N 2(t k)+v i ,CO2(t k )+v i , H 2O (t k)+vi , α (t k) (4)

For the alveolar compartments, the tension at the centre of the alveolus and at the alveolar

capillary border is assumed to be equal. The respiratory system has an intrinsic response to

low oxygen levels in blood which is to restrict the blood flow in the pulmonary blood vessels,

known as Hypoxic Pulmonary Vasoconstriction (HPV). The atmospheric pressure is fixed at

101.3kPa and the body temperature is fixed at 37.2°C.

At each t k, equilibration between the alveolar compartment and the corresponding capillary

compartment is achieved iteratively by moving small volumes of each gas between the

compartments until the partial pressures of these gases differ by <1% across the alveolar-

capillary boundary. The process includes the nonlinear movement of O2 and CO2 across the

alveolar capillary membrane during equilibration.

In blood, the total O2 content (CO2) is carried in two forms, as a solution and as

oxyhaemoglobin (saturated haemoglobin):

CO2(t k)=SO2(t k−1) ∙Huf ∙ Hb + PO2(t k−1) ∙O2 sol (5)

In this equation, SO2 is the hemoglobin saturation, Huf is the Hufner constant, Hb is the

hemoglobin content and O2sol is the O2 solubility constant. The following pressure-saturation

relation, as suggested by 11 to describe the O2 dissociation curve, is used in this model:

SO2(t k)=(( (PO23 (t k−1)+150 ∙PO2(tk −1))−1

×23400)+1)−1(6)

SO2 is the saturation of the hemoglobin in blood and PO2 is the partial pressure of oxygen in

the blood. As suggested by 12, PO2has been determined with appropriate correction factors in

base excess BE, temperature T and pH (7.5005168 = pressure conversion factor from kPa to

mm Hg):

PO2(t k )=7.5006 168 ∙ PO2(t k−1)∙10[ 0.48 (pH (tk−1)-7.4 )−0.024 (T-37 )−0.0013∙ BE ] (7)

The value of CCO2plasma is deduced using the Henderson-Hasselbach logarithmic equation for

plasma CCO2 13 :

CCO2(t k)=CCO2plasma(t k−1) ∙[1− 0.0289 ∙Hb(3.352−0.456 . SO2(t k)) ∙ (8.142−pH(t k−1)) ] (8)

where SO2 is the O2 saturation, Hb is the hemoglobin concentration and pH is the blood pH

level. The coefficients were determined as a standardized solution to the McHardy version of

Visser’s equation 14, by iteratively finding the best fit values to a given set of clinical data.

The value of CCO2plasma is deduced using the Henderson-Hasselbach logarithmic equation for

plasma CCO2 13 :

CCO2plasma( t¿¿k )=2.226∙ sCO 2∙PCO2(t k−1)(1+10 (pH (tk−1) – pK' ) )¿ (9)

where sCO 2 is the plasma CO2 solubility coefficient and pK' is the apparent pK (acid

dissociation constant of the CO2 bicarbonate relationship). PCO2 is the partial pressure of CO2

in plasma and ‘2.226’ refers to the conversion factor from miliMoles per liter to ml/100ml.

gives the equations for sCO 2 and pK' as:

sCO 2= 0.0307 + 0.0057 ∙ (37−T ) + 0.00002 ∙ (37−T )2 (10)

pK' = 6.086 +0.042 ∙ (7.4 - pH( tk−1) ) + (38−T ) ∙ (0.00472+ (0.00139−(7.4−pH (t k−1)) ))

(11)

PCO2 (t k )is determined by incorporating the standard Henry’s law and the sCO 2(the CO2

solubility coefficient above). For pH calculation, the Henderson Hasselbach and the Van

Slyke equation15 are combined. Below is the derivation of the relevant equation. The

Henderson-Hasselbach equation (governed by the mass action equation (acid dissociation))

states that:

pH = pK + log ( bicarbonate concentrationcarbonic acid concentration )

(12)

Substituting pK=6.1 (under normal conditions) and the denominator (0.225 ∙ PCO2) (acid

concentration being a function of CO2 solubility constant 0.225 and PCO2 (in kPa)) gives:

pH( t¿¿k ) = 6.1 + log(HCO3(t k−1)0.225 ∙PCO2 (tk)

)¿

(13)

For a given pH, base excess (BE), and hemoglobin content (Hb), HCO3 is calculated using

the Van-Slyke equation, as given by 15:

HCO3(t ¿¿k ) =((2.3 × Hb+7.7 )× (pH (t k )−7.4 ))+BE(1−0.023 × Hb)

+24.4¿ (14)

The capillary blood is mixed with arterial blood using the equation below which considers the

anatomical shunt (Sh¿ with the venous blood content of gas x (Cv, x ¿, the non-shunted blood

content from the pulmonary capillaries (Ccap, x), arterial blood content (Ca, x ¿, the arterial

volume (va ¿ and the cardiac output (CO).

Ca, x(t k )= CO (t k) ∙ (Sh ∙Cv, x(t k)+(1−Sh ) ∙Ccap, x( tk ))+Ca, x(t k )∙ (va(t k)−CO(t k ))va(t k)

(15)

The peripheral tissue model consists of a single tissue compartment, acting between the

peripheral capillary and the active tissue (undergoing respiration to produce energy). The

consumed O2 (VO2) is removed and the produced CO2 (VCO2) is added to this tissue

compartment. Similarly to alveolar equilibration, peripheral capillary gas partial pressures

reach equilibrium with the tissue compartment partial pressures, with respect to the nonlinear

movement of O2 and CO2. Metabolic production of acids, other than carbonic acid via CO2

production, is not modeled. After peripheral tissue equilibration of gases, the venous

calculations of partial pressures, concentrations and pH calculations are done using

comparable equations as above.

A simple equation of renal compensation for acid base disturbance is incorporated. The base

excess (BE) of blood under normal conditions is zero. BE increases by 0.1 per time slice if

pH falls below 7.36 (to compensate for acidosis) and decreases by 0.1 per time slice if pH

rises above 7.4 (under alkalosis).

Each alveolar compartment has an independent, configurable compliance, inlet resistance,

vascular resistance, extrinsic pressure and threshold opening pressure.

The pressure of each alveolar compartment is described by a cubic function:

pi=((10 ∙ v i−300)¿¿3 /6600)−P ext ,i¿ vi>0 for i=0 , …, N alv (16)

pi=0 otherwise

Equation (16) determines the alveolar pressure pi(as the pressure above atmospheric in

cmH2O) for the ith of N alv alveolar compartments for the given volume of alveolar

compartment,vi in millilitres. Pext (per alveolar unit, in cm H2O) represents the effective net

pressure generated by the sum of the effects of factors outside each alveolus that act to

distend that alveolus; positive components include the outward pull of the chest wall, and

negative effects include the compressive effect of interstitial fluid in the alveolar wall.

Incorporating Pext in the model allows us to replicate the situation of alveolar units that have

less structural support or that have interstitial oedema, and thus have a greater tendency to

collapse. A negative value of Pext indicates a scenario where there is compression from

outside the alveolus causing collapse.

The pulmonary vascular resistance PVR is determined by

1PVR

= 1RV , 1

+ 1RV ,2

+⋯+ 1RV , N A

, for i=1 , …, N A (17)

where the resistance for each compartment RV ,iis defined as

RV ,i=δVi RV 0 (18)

RV 0 is the default vascular resistance for the compartment with a value of 160 ∙ N A dynes s

cm-5 min-1, and δVi is the vascular resistance coefficient, used to implement the effect of

hypoxic pulmonary vasoconstriction.

Description of cardiac model

The cardiac model consists of 19 compartments. Each compartmentx, is described with a

pressure P x, a volume V x and a flow leaving the compartment F x, which are iteratively

updated in a sampling interval. Furthermore, each compartment has the following fixed

parameters: a resistance Rx to the flow out of the compartment reflecting the viscosity of the

compartment, a coefficient λx governing the elastance of each compartment, a coefficient P x, c

andV x, u, depicting the unstressed volume of the compartment. The ventricles are modeled as

having time varying elastances over the duration of a cardiac cycle using different

exponential functions to describe the filling and emptying of the ventricles.16 The shift from

the systolic to diastolic relationship is governed by a pulsating activation function with period

T . For all the compartments, vascular elastance is assumed to be nonlinear and to have an

exponential relationship governed by the following equation,

P x=Px , c eλ x (V x−V x ,u)

(V x+V x,u ) (19)

where the subscript x represents the compartment number and the λx , P x ,c ,V x ,u are constants

that give flexibility in fitting specific shapes and peaks of pressure waveforms that could be

observed from clinical data. The model employs separate pressure volume relationships for

the systolic and diastolic behavior of ventricles. The left ventricular pressure calculation is

given by:

Plv=φPlv ,sys , c eλlv .sys ( V lv−V lv ,sys ,u)

(V lv+V lv ,sys ,u) + (1−φ ) Plv ,dys ,c eλlv .dys (V lv−V lv , dys,u)

( V lv+V lv , dys ,u) (20)

The right ventricular pressure calculation is given by:

Prv=φPrv , sys ,c eλrv .sys ( V rv−V rv ,sys ,u)

( V rv+V rv ,sys ,u) + (1−φ ) Plv , dys ,c eλrv .dys (V rv−V rv ,dys ,u )

(V rv+V rv,dys ,u) (21)

The function φ is a ventricle activation function which is assumed to attain the maximum

value of φ=1 at the peak of systolic contraction. The function φ attains its minimum value 0

at maximal diastole relaxation. A squared half-sine wave function 16, 17 is adopted for φ given

by

φ={(sin (πT uT sys ))

2

if u≥ 0∧u≤T sys

2

0 , if u>T sys

2∧u≤ 1

(22)

T=1/ HR ,T sys=(T sys , o−k sys )

Twhere T sys ,o=0.5∧ksys=0.075(23)

u is a real number ranging between 0 and 1 and it models the fraction of the cardiac cycle.

u=0 at the end of systole and u=1 at the end of diastole. T sysindicates the systolic period

which is proportional to the heart rate HR (in seconds).

The blood flow between compartments is determined by the pressure gradient between

compartments across a linear time invariant resistanceR x.

F x=ηx ( P x−P y )

Rx

(22)

ηx={0 , if Px<P y

1, if Px ≥ Py(23)

The parameterηx allows the blood to flow in one direction but can be altered to investigate

flow backwards into a compartment, such as during aortic regurgitation. The volume of the

blood in each compartment is computed by applying conservation of mass as follows

V x=(V x 0+( F x−Fx ) Δt ) (24)

where F x is the flow entering the xth compartment (i.e. the flow leaving the upstream

compartment) and F x is the flow leaving the compartment x. V x0is the volume of

compartment x before the iteration, and Δt is the size of the time period in this iteration (set

to 1 ms). The total amount of blood in the whole body is obtained by V T=∑V x .

Cardio pulmonary interactions

The model includes the effect of radial compressive and axial stretching forces exerted onto

pulmonary capillaries as a result of increase in lung volume and pressure. The overall effect

on resistance to flow through each capillary is difficult to quantify, but we assume the

following: (i) at alveolar volumes above the functional residual capacity (FRC), the vessels

become compressed and raise the pulmonary vascular resistance (PVR), (ii) at alveolar

volumes below FRC, the vessels can collapse and thus result in an increase in PVR, while

closer to FRC the PVR remains unaffected. A separate mechanism called hypoxic

vasoconstriction, of the vessels contracting in response to hypoxia as a result of alveolar

collapse, is already present in the existing pulmonary model. The resultant ‘U’ shape change

in PVR at around the FRC has been suggested previously 18 and has been implemented in this

model as follows. The pulmonary vascular resistance PVR is determined as given in equation

17, but the vascular resistance for each alveolar compartment, RV ,i, is defined has been

modified from (18) to

RV ,i=pvrmult ,i δVi RV 0. for i=1,2 , …, Nalv (25)

N alv is the number of alveolar compartments (set to 100), RV 0 is the default vascular

resistance for the compartment with a value set to 160 ∙ N A dynes s cm-5 min-1, and δVi is the

vascular resistance coefficient. pvrmult ,i is calculated as follows:

pvrmult ,i=((1+0.5( (v i−v FRC )vFRC )

2

)(1+pi

q pvr ))npvr

(26)

where, pi is the pressure generated within the ith alveolar compartment, vi is the volume of the

ith alveolar compartment, vFRC is a constant representing the volume of the alveolar

compartment at rest (fixed to 30 ml). npvr and q pvr is used to adjust the effect on pulmonary

vascular resistance. npvr is set to 1 and q pvr has been set to 30 but they can be modified to fit

subject data.

Development of new simulator modules

The new modules in the extended cardio-pulmonary model implemented for this study are

as follows and we have preliminary validated of the new model against an experimental

animal study19.

1) Cardiogenic oscillations module

The effect of cardiac oscillations on alveolar compartments is described by following

equation:

Posc , i=Kosc ∙ φ for i=0 , …, Nosc where N osc ≤ N alv (27)

Posc , irepresents the pressure of the heart acting on the alveolar compartment i. This additional

pressure, in combination with existing pressure values of the alveolar compartments2, 10, 20

creates a pressure difference between the mouth and the alveolar compartments, across which

the flow of gas can occur. The parameter K osc ,iis a constant, representing the strength of the

effect of cardiogenic oscillations on alveolar ventilation, due to the alveoli being compressed

by the heart and/or trans-alveolar blood volume. N osc is the number of alveolar compartments

that are affected by cardiogenic oscillations. The function φ is the ventricle activation

function presented in the equations 22 and 23. Thus, the alveolar pressure is given by:

pi=((10 ∙ v i−300)¿¿3 /6600)−Pext ,i−Posc ,i ¿ vi>0 for i=0 , …, N alv (28)

pi=0 otherwise

2) Anatomical deadspace gas-mixing module

A new parameter (σ ) is added to the calculation of f SD , jx representing the proportion of gas

mixing between the laminas of the series deadspace. Previously, the f SD , jx was entirely based

on the movement of gas as determined by the amount of fresh gas entering the series

deadspace. σallows for various degrees of mixing between adjacent layers. σ=1 would

indicate a complete mixing of gases between layers, representing the effects of extreme

turbulent flow of air while σ= 0 would indicate no mixing of gases. Thus:

f SD , jx =¿ (29)

where f SD , jx is the fraction of gas x in lamina j and f SD, j+1

x is the fraction of gas x in the next

lamina.

3) Tracheal insufflation module

For tracheal insufflation, the flow of air entering the series deadspace (Fnew) is determined by

the parameter r insuff , representing the rate of insufflation in L·min-1. A further variable called

linsuff allows the user to change the location of insufflation to different positions across the

series deadspace. linsuff =1 allows for insufflation in the glottis, whereas linsuff=5represents

insufflation in the carina.

During the oxygen insufflation, f SD , kO2 at laminak is given as follows:

f SD , kO2 =

(vk∗f SD, kO2 )+( Fnew∗f insuff )

vk+Fnew

(30)

Fnew=f insuff∗r insuff (31)

where f insuff is the fraction of O2 in the insufflated air (accounting for humidified air) and vk is

the volume of air in lamina k . Following this, the fractions of the gases in all the layers are

adjusted taking in account proportion of gas mixing between the laminas of the series

deadspace.

4) Pharyngeal pressure

To represent the delivery of high-flow nasal oxygen (HFNO), an oscillatory pressure Ppharhas

been added to the tracheal pressure (equal to the atmospheric pressure):

Pphar=A ∙ sin ( πf)

2

(32)

where, A is the amplitude of the pressure in cmH2O (values 0-2 cm H2O)21 and f is the

frequency in Hz equal to 70.

Model parameters for simulated healthy state

Table S1 reports the cardiovascular model default parameters.

Model calibration to clinical data

The pulmonary model was fitted as virtual subject to dynamic data reported in Frumin and

co-workers22. We tuned model parameters by hand to find the model configuration that would

generate outputs that most closely fitted the clinical data available. The model parameters to

be optimized for each of the 100 alveolar compartments were: K osc, representing the strength

of the effect of cardiogenic oscillations on alveolar ventilation,N osc representing the number

of alveolar compartments that are affected by cardiogenic oscillations and σ representing the

degree of mixing of gases between layers of series deadspace.

Once a satisfactory fit and behaviour of the model was found for this dataset, the model was

validated against the other 4 studies Fraioli and co-workers,23 Berthoud and co-workers,24

Baraka and co-workers,25 and Gustasfsson and co-workers.26 For the study of Gustafsson

and co-workers,26 the model parameters to be optimized for each of the 100 alveolar

compartments were Kosc, Nosc, σ and A , representing the amplitude of an oscillatory

pressure for the delivery of HFNO.

Ranges and units of parameters are presented in Table S2.

The model fitting problem was formulated to find the minimum value of an objective

function J in the equation below:

minx

J=2√∑i=1

4

r i2where r i=

y i− yi 'y i '

(36)

where y = [PaCO2, arterial pH] are the model outputs and y ' = [PaCO2, arterial pH]. PaO2

and PaCO2 are measurements obtained from clinical data.

References

1 Hardman JG, Bedforth NM. Estimating venous admixture using a physiological simulator.

Br J Anaesth 1999; 82: 346-9

2 Hardman JG, Bedforth NM, Ahmed AB, Mahajan RP, Aitkenhead AR. A physiology

simulator: validation of its respiratory components and its ability to predict the patient's

response to changes in mechanical ventilation. Br J Anaesth 1998; 81: 327-32

3 Hardman JG, Wills JS. The development of hypoxaemia during apnoea in children: a

computational modelling investigation. Br J Anaesth 2006; 97: 564-70

4 McCahon RA, Columb MO, Mahajan RP, Hardman JG. Validation and application of a

high-fidelity, computational model of acute respiratory distress syndrome to the examination

of the indices of oxygenation at constant lung-state. Br J Anaesth 2008; 101: 358-65

5 Das A, Haque M, Chikhani M, et al. Hemodynamic effects of lung recruitment maneuvers

in acute respiratory distress syndrome. BMC Pulmonary Medicine 2017; 17: 34

6 Hardman JG, Aitkenhead AR. Estimation of alveolar deadspace fraction using arterial and

end-tidal CO2: a factor analysis using a physiological simulation. Anaesth Intensive Care

1999; 27: 452-8

7 Hardman JG, Aitkenhead AR. Validation of an original mathematical model of CO2

elimination and dead space ventilation. Anesth Analg 2003; 97: 1840-5

8 Das A, Gao Z, Menon PP, Hardman JG, Bates DG. A systems engineering approach to

validation of a pulmonary physiology simulator for clinical applications. J R Soc Interface

2011; 8: 44-55

9 Hardman JG, Al-Otaibi HM. Prediction of arterial oxygen tension: validation of a novel

formula. Am J Respir Crit Care Med 2010; 182: 435-6

10 Chikhani M, Das A, Haque M, Wang W, Bates DG, Hardman JG. High PEEP in acute

respiratory distress syndrome: quantitative evaluation between improved oxygenation and

decreased oxygen delivery. Br J Anaesth 2016; 117: 9

11 Severinghaus JW. Simple, accurate equations for human blood O2 dissociation

computations. J Appl Physiol Respir Environ Exerc Physiol 1979; 46: 599-602

12 Severinghaus JW. Blood gas calculator. J Appl Physiol 1966; 21: 1108-16

13 Kelman GR, Nunn JF. Nomograms for correction of blood Po2, Pco2, pH, and base excess

for time and temperature. J Appl Physiol 1966; 21: 1484-90

14 McHardy GJ. The relationship between the differences in pressure and content of carbon

dioxide in arterial and venous blood. Clin Sci 1967; 32: 299-309

15 Siggaard-Andersen O. The van Slyke equation. Scand J Clin Lab Invest Suppl 1977; 146:

15-20

16 Ursino M. Interaction between carotid baroregulation and the pulsating heart: a

mathematical model. Am J Physiol 1998; 275: 1733-47

17 Piene H. Impedance matching between ventricle and load. Ann Biomed Eng 1984; 12:

191-207

18 Shekerdemian L, Bohn D. Cardiovascular effects of mechanical ventilation. Archives of

disease in childhood 1999; 80: 475-80

19 Laviola M, Das A, Chikhani M, Bates D, Hardman JG. Investigating the effect of cardiac

oscillations and deadspace gas mixing during apnea using computer simulation. Conf Proc

IEEE Eng Med Biol Soc 2017: 337-40

20 Das A, Cole O, Chikhani M, et al. Evaluation of lung recruitment maneuvers in acute

respiratory distress syndrome using computer simulation. Crit Care 2015; 19: 8

21 Parke RL, Bloch A, McGuinness SP. Effect of Very-High-Flow Nasal Therapy on Airway

Pressure and End-Expiratory Lung Impedance in Healthy Volunteers. Resp Care 2015; 60:

1397-403

22 Frumin J, Epstein RM, Cohen G. Apneic oxygenation in man. Anesthesiology 1959; 20:

789-98

23 Fraioli RL, Sheffer LA, Steffenson JL. Pulmonary and Cardiovascular Effects of Apneic

Oxygenation in Man. Anesthesiology 1973; 39: 588-96

24 Berthoud MC, Peacock JE, Reilly CS. Effectiveness of preoxygenation in morbidly obese

patients. Br J Anaesth 1991; 67: 464-6

25 Baraka AS, Taha SK, Siddik‐Sayyid SM, et al. Supplementation of pre‐oxygenation in

morbidly obese patients using nasopharyngeal oxygen insufflation. Anaesthesia 2007; 62:

769-73

26 Gustafsson I-M, Lodenius Å, Tunelli J, Ullman J, Jonsson Fagerlund M. Apnoeic

oxygenation in adults under general anaesthesia using Transnasal Humidified Rapid-

Insufflation Ventilatory Exchange (THRIVE)–a physiological study. Br J Anaesth 2017; 118:

610-7

Table S1. Cardiovascular model parameters and their default values for a healthy heart model.

Parameter Symbol Units Parameter

Pulmonary Vein Resistance Rpv mm Hg.s. ml-1 0.0056

Mitral Valve Resistance Rla mm Hg.s. ml-1 0.008

Aortic Valve Resistance Rlv mm Hg.s. ml-1 0.01

Systemic Artery Resistance R sa mm Hg.s. ml-1 0.14

Systemic Vein Resistance R sv mm Hg.s. ml-1 0.0007

Tricuspid Valve Resistance Rra mm Hg.s. ml-1 0.001

Pulmonary Valve Resistance Rrv mm Hg.s. ml-1 0.015

Pulmonary Arterial Resistance Rpa mm Hg.s. ml-1 0.005

Total blood Volume V T ml 5050

Volume in Cardiac Chambers V H ml 0.066V T

Left ventricle systolic volume constant V lv ,sys , u ml 0.32 V H

Left ventricle diastolic volume constant V lv ,dys ,u ml V lv ,sys , u−40Right ventricle systolic volume constant V rv ,dys , u ml 0.38 V H

Right ventricle diastolic volume constant V rv ,sys , u ml V rv ,sys , u−40Right Atrium, unstressed volume V ra ,u ml 0.15 V H

Left Atrium, unstressed volume V la ,u ml 0.15 V H

Total Systemic Arterial volume VSA ml 0.24 V T

Systemic Artery unstressed volume V sa , u ml 0.5VSASystemic Arterioles unstressed volume V sai ,u ml 0.1 VSA

Total Systemic Venous volume VSV ml 0. 60 V T

Systemic Vein unstressed volume V svi . u ml 0.65 VSVSystemic Venules unstressed volume V sv ,u ml 0.07 VSVPulmonary Artery unstressed volume V pa ,u ml 0.023 V T

Pulmonary compartment unstressed volume V lungs ,u ml 0.013 V T

Pulmonary Vein unstressed volume V pv ,u ml 0.054 V T

Coefficient for end diastolic pressure in left ventricle Plv ,dys , c mm Hg 2

Coefficient for end systolic pressure in left ventricle Plv ,sys , c mm Hg 110

Coefficient for end diastolic pressure in right ventricle Prv ,dys , c mm Hg 2

Coefficient for end systolic pressure in right ventricle Prv ,sys , c mm Hg 12

Coefficient for relaxed left atrium Pla ,c mm Hg 5

Coefficient for relaxed right atrium Pra ,c mm Hg 5

Coefficient for relaxed systemic artery Psa , c mm Hg 110

Coefficient for relaxed systemic vein Psv ,c mm Hg 8

Coefficient for relaxed systemic arterioles Psai ,c mm Hg 20

Coefficient for relaxed systemic venules Psvi . c mm Hg 1.6

Coefficient for relaxed pulmonary artery Ppa ,c mm Hg 15

Coefficient for relaxed lung compartment Plungs , c mm Hg 13

Coefficient for relaxed pulmonary vein Ppv ,c mm Hg 12

Coefficient for elastance of left ventricle diastole λ lv , dys ,u - 10

Coefficient for elastance of left ventricle systole λ lv , sys ,u - 5

Coefficient for elastance of right ventricle diastole λrv , dys ,u - 10

Coefficient for elastance of right ventricle systole λ rv, sys ,u - 5

Coefficient for elastance of left atrium λ la, u - 9

Coefficient for elastance of right atrium λra, u - 9

Coefficient for elastance of systemic artery λ sa ,u - 10

Coefficient for elastance of systemic arterioles λsai, u - 10

Coefficient for elastance of systemic venules λ svi .u - 10

Coefficient for elastance of systemic vein λsv , u - 10

Coefficient for elastance of pulmonary artery λ pa, u - 5

Coefficient for elastance of vascular lung compartment λ lungs ,,u - 5

Coefficient for elastance of pulmonary vein λ pv, u - 11

Splinting coefficient γ pvr mm Hg 0.5

PVR exponent npvr - 1

PVR coefficent q pvr - 60

Table S2. Configured new modules parameters.Parameter Size Range Units

Kosc 100 1-10 cmH2O

Nosc 1 0-100 %

σ 1 0-1 -A 100 1-3 cmH2O

Figure

3

1

A

B

array (solid black): holds the fraction of gas X in each laminae.

C

vSD

2

4

tmp_inttmp_frac

5

6

Fig S1. Diagrammatic explanation of how serial deadspace gas movement under inhalation is

implemented in the model (see text for full explanation).