Supplementary Data Computer simulation clarifies mechanisms of carbon dioxide clearance during apnoea Marianna Laviola 1 , Anup Das 2 , Marc Chikhani 1,3 , Declan G. Bates 2 , Jonathan G. Hardman 1,3 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
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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
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).