Theoretical open-loop model of respiratory mechanics in the extremely preterm infant
Theoretical open-loop model of respiratory mechanics in the extremely preterm infant
Laura Ellwein Fix 0 1
Joseph Khoury 1
Russell R. Moores Jr. 1
Lauren Linkous 0 1
Matthew Brandes 1
Henry J. Rozycki 1
0 Department of Mathematics and Applied Mathematics, Virginia Commonwealth University , Richmond , Virginia, United States of America, 2 Division of Neonatal Medicine, Children's Hospital of Richmond, Virginia Commonwealth University , Richmond , Virginia, United States of America, 3 VCU School of Medicine, Virginia Commonwealth University , Richmond, Virginia , United States of America
1 Editor: Alexander Larcombe, Telethon Institute for Child Health Research , AUSTRALIA
Non-invasive ventilation is increasingly used for respiratory support in preterm infants, and is associated with a lower risk of chronic lung disease. However, this mode is often not successful in the extremely preterm infant in part due to their markedly increased chest wall compliance that does not provide enough structure against which the forces of inhalation can generate sufficient pressure. To address the continued challenge of studying treatments in this fragile population, we developed a nonlinear lumped-parameter respiratory system mechanics model of the extremely preterm infant that incorporates nonlinear lung and chest wall compliances and lung volume parameters tuned to this population. In particular we developed a novel empirical representation of progressive volume loss based on compensatory alveolar pressure increase resulting from collapsed alveoli. The model demonstrates increased rate of volume loss related to high chest wall compliance, and simulates laryngeal braking for elevation of end-expiratory lung volume and constant positive airway pressure (CPAP). The model predicts that low chest wall compliance (chest stiffening) in addition to laryngeal braking and CPAP enhance breathing and delay lung volume loss. These results motivate future data collection strategies and investigation into treatments for chest wall stiffening.
Data Availability Statement: All relevant
simulation parameters and equations are within the
paper. No experimental data was used in this
modeling study. Modeling and simulation code is
included as Supplementary Information, S1 Code.
Funding: This research was supported in part by
the Atlantic Pediatric Device Consortium via FDA
grant 5P50FD004193-07 (to HJR, LEF, MB). This
work was also supported in part by the VCU
College of Humanities and Sciences Faculty
Research Council award (to LEF, LL). The funders
The extremely preterm infant, born at < 28 weeks gestation and often < 1000g, is at risk of
developing chronic lung disease despite established treatments such as surfactant replacement
therapy. Currently the survival rate of this group ranges from 94% at 27 weeks to as low as 33%
at 23 weeks [
], with survivors living with varying degrees of morbidity. One risk factor for
lung disease remains the trauma associated with traditional mechanical ventilation including
endotracheal tube injury, high cyclic tidal volumes and pressures, and hyperoxia. Non-invasive
methods of ventilation such as continuous positive airway pressure (CPAP) are being used
with more frequency and have been successful with more mature infants but appear to fail in
the extremely preterm infant [2±4]. One hypothesis for the failure of non-invasive ventilation
had no role in study design, data collection and
analysis, decision to publish, or preparation of the
and the need for increasing invasive respiratory support is the markedly increased compliance
(floppiness) of the chest wall in the extremely preterm infant resulting from ribcage
undermineralization common at the start of the third trimester [5±7]. In the preterm infant, chest wall
compliance can be up to five times lung tissue compliance [
When the chest wall is not sufficiently rigid, the negative pressure within the pleural space
between the lung and chest wall generated from diaphragm contraction is diminished [
many cases this leads to progressive lung collapse (atelectasis) with each breath as the forces
needed to open airspaces after each exhalation become insurmountable [
], leading to
decreasing lung compliance and functional residual capacity (FRC) [
]. This progression of
events is observed clinically in X-rays and by symptoms of respiratory distress such as chest
retractions and rapid breathing. The clinical result is progressively reduced tidal volumes and
end-expiratory lung volume (EELV) as the forces needed to open airspaces after exhalation are
insufficient. Non-invasive ventilation has been observed to be become ineffective under these
conditions, necessitating placement of an endotracheal tube and positive pressure mechanical
ventilation and markedly increasing the risk of lung damage.
Despite this being repeatedly observed clinically, there remains little quantification of the
impact of variable nonlinear chest wall compliance on tidal breathing dynamics, and even
fewer computational modeling efforts supporting these observations of progressive volume
loss. Most computational models of breathing address the extremes of lung capacity such as a
forced vital capacity maneuver, study a static, excised, or injured lung, or use an animal model
[12±16]. Existing computer models of tidal breathing have not fully accounted for the
physiology particular to premature infants and thus have limited applicability. Often, methods of
providing ventilator support have been developed in adults and children, then refined and scaled
for newborns and premature infants, limiting innovation aimed specifically at this vulnerable
In this work, we have developed a nonlinear computational model of respiratory mechanics
parameterized for the extremely preterm infant that demonstrates differential volume loss
under high vs low chest wall compliance conditions. We adapt a model first presented by
Athanasiades et al [
] and modified for newborn lambs by LeRolle et al [
]. In the latter,
differences such as smaller diameter airways, higher respiratory rates, higher lung resistance, and
higher chest wall compliances were considered, however many of the critical physiological
nonlinearities contributing to long-term dynamics were not included. The present model is
built upon the nonlinear compliance curves describing pressure-volume relationships specific
to preterm infants [
]. Dynamic alterations of compliance curves based on breath-to-breath
end-inspiratory lung volume (EILV) and peak inspiratory pressure (PIP) are shown to
influence tidal volume and EELV, thus simulating progressive lung volume loss. We also
demonstrate the effect of two simulated interventions that raise alveolar pressure and lung elastic
recoil: CPAP, which raises the pressure at the mouth; and laryngeal braking (grunting), which
increases upper airway resistance during expiration.
The lumped-parameter respiratory mechanics model describes dynamic volumes and
pressures in the airways, lungs, chest wall, and intrapleural space between lungs and chest. A signal
that represents diaphragm pressure generated during spontaneous breathing drives the model.
A compartment is assumed to display aggregate behavior, e.g. the alveolar compartment
represents the collective dynamics of the alveoli as a whole. The model is designed using the
volume-pressure analog of an electrical circuit, see Fig 1. As such, relevant states are in terms of
pressure P(t) [cm H2O] and volume V(t) [ml] in and between air compartments, with
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PLOS ONE | https://doi.org/10.1371/journal.pone.0198425
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Fig 1. Lumped-parameter respiratory mechanics model, in both volume-pressure (panel A) and electrical (panel B) system analogs. Each
nonrigid compartment has a volume V (black), pressure P, (black) and associated compliance C (green, for emphasis) that is a function of the transmural
pressures (purple) across the compartment boundaries. Air flows V_ (red) across resistances R and inertance I (blue) are positive in the direction of the
arrows. Circular yellow arrows indication direction of loop summations in Eq (3). Subscripts: airway opening ao, upper u, collapsible c, small peripheral
s, alveolar A, viscoelastic ve, lung elastic el, transmural tm, pleural pl, chest wall cw, muscle mus.
volumetric flow rate and rate of change represented as V_
t [ml/s] and dV respectively. Air
pressure Pi within a specific volume i is defined as the difference between intra-airway pressure
Pint and pressure external to the body Pext, i.e. Pi = Pint,i − Pext,i. Since all pressures are relative
to the same constant atmospheric pressure, all Pext = 0 and all intra-airway pressures Pi = Pint,i.
The pressure Pij = Pi − Pj refers to the transmural pressure across a compliant boundary
separating volumes i and j.
Each non-rigid compartment has an associated compliance Ci [ml/cm H2O], describing the
change in compartmental volume Vi given a change in transmural pressure Pij across its
boundary with compartment j:
The nature of Ci does not change explicitly with time but instead is implicitly determined by
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Bidirectional airflow through the trachea, bronchi, bronchioles, and to and from the lungs
results from contraction and relaxation of the diaphragm generating a pressure difference.
Airflow is opposed by the resistance of the airways as functions of their radaii or tissue properties.
This relationship is described by the flow-pressure analog of Ohm's law ,
where Ri [cm H2O s/ml] is the resistance to airflow prior to compartment i. If a compartment
includes inertial effects, the pressure gradient is also a function of the acceleration of flow,
where I is the inertance. Inertial effects are considered for the newborn upper rigid airway
because of its smaller radius, but neglected for the rest of the model tissues [
The pressures Pij across each compliant compartment include transmural pressure between
the compliant airways and the pleural space Ptm = Pc − Ppl, lung elastic recoil Pel = PA − PT,
lung viscoelastic component Pve = PT − Ppl, and chest wall elastic recoil Pcw = Ppl − Pmus.
Summing pressures over each of three loops according to Kirchhoff's mesh rule gives a system of
time-varying algebraic equations:
the relationship between volume and pressure. This can be reformulated in terms of dynamic
changes of state:
ddVt i Ci
V_ i Pi 1
Pi IiV i
Pc Ptm Ppl
PA Pel Pve
V_ c :
Rearranging Eq (3) and using Kirchhoff's current law along with Eqs (1) and (2) produces
the consolidated set of model differential equations:
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Conservation laws also maintain that V = Vcw = VA + Vc, in other words the total system
volume equals the chest wall volume, which is the sum of the alveolar and compressible airway
volumes. Pressure-volume relationships and compliances Ci will be further described below.
Nonlinear resistance constitutive relations
The airways begin with an upper rigid segment characterized by an inertance Iu and a
nonlinear Rohrer resistance Ru [
] that increases with airflow:
The constants Ru,m and Ku represent laminar and turbulent flow components.
A middle collapsible portion is modeled as a cylinder with constant length having nonlinear
resistance Rc that depends inversely on the 4th power of the radius according to Poiseuille's
law. Therefore Rc is formulated as [
where Rc equals its minimum value Kc when Vc = Vc,max, an estimate of dead space.
An inverse relationship between resistance in the smaller peripheral airways Rs and lung
volume VA reflects high resistance at low or near-zero volumes [
]. To avoid Rs ! 1 as
VA ! 0 [
] from a strict exponential decay model, we adopt the formulation used by both Liu
et al and Athanasiades et al [
], a decaying exponential function of relative lung volume
with finite Rs at VA = 0:
Rs Rs;d eKs
TLC RV Rs;m
where Ks < 0. This parameterization gives that Rs
when VA = RV (residual volume).
Rs,m when VA = TLC, and Rs = Rs,d + Rs,m
Ru Ru;m KujV_ j
affine occurs at Pcw = . The chest wall relaxation volume V0 = Vcw|Pcw = 0 is set using an
estimate from literature at 25% of VC (vital capacity) [
]. From this parameterization,
bw = (V0 − RV)/(ln 2). The single degree of freedom dw then characterizes the slope of the chest
wall compliance curve and is adjusted to produce a range of dynamic compliance values.
The volume of the lung compartment VA is modeled as the product of distention of lung
units Vel(Pel) and fraction of recruited alveoli Frec(Pel) [
]. To obtain VA RV near Pel = 0,
lung volume is given as
Pel RV :
Alveolar compliance CA as used in the system of differential Eq (4) is found with symbolic
computation as ddVPeAl .
The first term Vel represents the volume due to aggregate elasticity of the lung unit
structure, which is modeled here as a saturated exponential [26±28]
where k characterizes the lung stiffness. This representation has been found to suffice in cases
of a healthy or surfactant-treated lung. The second term of the lung compliance Frec represents
the contribution of recruitment and derecruitment of alveoli to compliance, which has been
modeled previously as dependent on both time and pressure [
25, 26, 29
]. It can be represented
by a sigmoid which resembles the probability density function of a Gaussian distribution
describing aggregate opening or closing pressures of individual alveolar sacs or ducts [
adopt the formulation of Hamlington et al [
Pel cF=dF ;
1 ecF=dF b
It follows that β is the baseline fraction of lung recruited at Pel = 0, γ represents the
maximum recruitable fraction of lung, cF is mean opening pressure at which recruitment is
maximum, and dF describes the transition to full recruitment capturing the heterogeneity of the
lung. Parameterization of Frec is based on the state of health being modeled and can change
breath-to-breath depending on conditions. For example, an increase in stiffness resulting from
derecruitment may manifest as higher mean opening pressure cF and move the VA curve to the
right. Likewise a lower maximum recruitable fraction γ would flatten the VA curve. Both
scenarios indicate a lower compliance and greater pressure required to increase the lung volume
in the region of operating pressure. In certain pathological situations such as ARDS, a
sigmoidal representation of VA(Pel) with a low compliance region at low Pel [28, 31±34] could be
captured in the parameterization of Frec.
The viscoelastic properties of pulmonary tissue are represented with a linear Kelvin-Voigt
model consisting of scalar compliance Cve and resistance Rve, which contributes a viscoelastic
pressure component Pve in series with lung elastic recoil Pel, see Fig 1. The sum of these two
pressures is dynamic pressure Pl,dyn which also equals Ppl − PA.
Respiratory muscle driving pressure
The pressure Pmus describes the effective action of the respiratory muscles driving the model
dynamics with Pmus negative in the outward direction. We used a sinusoidal function to
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describe tidal breathing, with maximum equaling zero at end-expiration:
Pmus Amus cos
where Amus is the amplitude of the cosine wave and f = RR/60 is the frequency. The wave
generates a negative pressure with total magnitude 2Amus outward from the body. Though simple,
the sinusoidal function can admit time-varying frequency, show dynamics over multiple
breaths, is used in artificial ventilation, has compact support on the closed interval [0, T], and
has been used in previous modeling studies (see eg. [
]). More sophisticated functions [
] can model inhalation and exhalation with different durations or qualitative forms,
however the breath-to-breath dynamics displayed in this study can be captured sufficiently
with the sinusoidal function.
Progressive volume loss
The complete mechanism of interaction between inefficient inhalation resulting from high
chest wall compliance and the progressive nature of lung volume loss and respiratory distress
is not fully understood. Clinical X-ray evidence of delayed atelectasis and subsequent acute
respiratory distress in otherwise healthy lungs may suggest a process by which a lack of full
recruitment during a given breath lowers lung capacity and compliance for the following
breath, and continues to an unrecoverable level in the absence of neural modulation or
compensatory mechanisms such as sighing. As a first attempt at modeling progressive volume loss,
we empirically describe the breath-to-breath evolution of Frec (Eq (12)) as lung recruitment
pressure parameters cF and dF increase with PIP and maximum recruitable fraction γ decreases
The lung compliance curve shifts slightly with each breath via changes in mean threshold
opening pressure based on number of collapsed alveoli. A volume loss associated with
derecruited alveoli necessitates an increase in expanded volume of recruited alveoli relative to
the radius cubed, with an increased distending pressure proportional to the change in radius.
This is illustrated in [
] using a simple example of expansion of 3 alveoli that double in
volume with a 25% increase in radius; if 1 alveolus closes, the other two radaii must now increase
by 35% to achieve the same overall volume change and the required distending pressure
increases proportionally. This proportion applied to cF and dF shifts the compliance curve to
the right. In this way the compliance decreases approximately proportional to the amount of
]. If tidal breathing begins on the steepest part of the lung compliance curve,
compliance decreases monotonically until eventually tidal breathing occurs on the low
compliance tail on the left part of the curve and VT 0. Tidal breathing may begin at a higher
position towards the flatter upper part of the curve, in which case compliance will increase slightly
with this modification but will again eventually decrease in the manner described above.
Assuming constant amplitude of the sinusoidal muscle pressure pressure function and no
stochasticity, the maximum recruitable fraction of alveoli is achieved at end-inspiration (EI)
during steady-state oscillatory breathing and additional fraction will not be recruited under
a pressure of this same amplitude in subsequent breaths. The value for γ for subsequent breaths
is then dependent on Frec|EI and the percentage of alveoli assumed to be permanently collapsed /
no longer recruitable, represented by the calculation gnext gcurrent
where Fclosed is fraction closed. If all unrecruited alveoli remain as such, then Frec|EI becomes the
new γ for the next breath; likewise, if all alveoli remain recruitable, γ = 1 for the duration of the
simulation. Note that even for γ = 1, Frec < 1 for all Pel thus causing small changes in cF and dF
and shifts in the Frec(Pel) curve regardless of the % of alveoli permanently closed that still lead to
progressive volume loss. The rate at which volume loss progresses depends on where on the Frec
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curve tidal breathing occurs, and thus both the curve's intrinsic characterizing parameters and
extrinsic system variables.
The lung curve was parameterized to obtain an approximate dynamic lung compliance CA of
2.3 ml/cm H2O [
8, 39, 40
] calculated as the slope (VA|EI − VA|EE)/(Pel|EI − Pel|EE) during normal
breathing with no interventions. In particular, k was tuned to produce a curve Vel between RV
and TLC with the calculated slope, and the parameters of Frec produced a curve that is 1 for
the whole range of normal breathing to represent a nearly fully recruited lung. High Cw for a
typical preterm infant was targeted at 8.5 ml/cm H2O [
] and low Cw about equal to lung
compliance. The parameter dw characterized the approximate dynamic chest wall compliance,
which was calculated as the slope (Vcw|EI − Vcw|EE)/(Pcw|EI − Pcw|EE). Parameter values for Ru,
Rc, Rs, and Vc were estimated from previously published studies [
12, 14, 41
]. The viscoelastic
parameters Cve and Rve were manually tuned to obtain idealized tidal volume and
end-expiratory lung volume rather than the magnitude of the hysteresis.
FRC is the volume at the resting position of the respiratory system i.e. where Presp = Pel +
Pcw = 0. The naturally high compliance of the healthy full-term and especially preterm infant
(with even steeper Vcw(Pcw)) lowers Presp and decreases FRC to about 20% of vital capacity
(VC), compared to at about 35-40% of VC in the adult [
]. A nominal value for FRC for a
given set of static compliance curves is obtained by first computing volumes using a vector of
physiological pressures [-20. . .40] cm H2O. Lung and chest recoil pressure vectors are then
added in the P direction to obtain Presp, and the index where Presp = 0 is used to determine FRC
using either Vcw or VA. The lung, chest wall, and respiratory PV compliance curves for both
high and low Cw created from Eqs (9) and (10) are given in Fig 2. The value for Pel|FRC is then
Fig 2. Lung, chest wall, and total respiratory system compliance curves for high Cw (left) and low Cw (right). Curves are described by Eqs (9) and (10) and
parameterized using the procedures described in Parameterization. Tidal breathing loops with normal Ru (grey) and increased Ru (black) are superimposed for each
condition over the lung compliance curve and larger in each inset to display hysteresis.
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set as the initial condition for solving dPel/dt. Note that the lung curve is identical between
scenarios so the decreased slope in the low Cw scenario with the same ν and V0 raises FRC and
thus EELV. Decreased lung compliance (flatter VA(Pel)) resulting from injury, disease, or
progressive volume loss further reduces FRC and EELV. In this model we consider chest wall
compliance to be either high or low and unchanging for the duration of a simulation, but lung
compliance changes depending on breathing conditions.
Table 2 gives values and formulas / sources for parameters that remain unchanged between
simulations. These values as well as the FRC, respiratory pressure amplitude, chest wall
compliance, and upper airway resistance parameters in Table 3 that vary between simulation
conditions were manually tuned to best obtain the reported aggregate parameters and state
outputs as shown in Table 4. As an example, dynamic lung compliance CA is not an explicit
input into the model, but was determined as described above. For ease of computation and to
match the target demographic, we assumed the simulated subject weighed 1 kg.
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All simulations proceeded with an initial respiration rate of 60 breaths/min (f = 1), initial
minute ventilation V_ E 360, and initial tidal volume VT = 6 ml, with the expectation that tidal
volume changes with changes in dynamic lung compliance. The motivation for ths choice was
twofold: One, this is consistent with a physiological requirement of constant V_ E regardless of
chest or lung compliance; and two, this allowed for comparison of simulation results
originating from similar starting points. Distinct values for Amus were prescribed for each simulation
to achieve the initial V_ E 360, see Table 3.
Simulated conditions were chosen to demonstrate the model dynamics with high and low
chest wall compliance, under two interventions and two states of permanent alveolar closure.
An infant often exhibits compensatory mechanisms such as laryngeal braking (grunting) and
increased activity of diaphragm and intercostal muscles [
] to increase end-expiratory
pressure in order to keep EELV above the volume at which alveolar units start to collapse during
expiration. Laryngeal braking is simulated with a 10-fold increase in expiratory upper airway
resistance Ru. CPAP is simulated with an increase of Pao from 0 to 5 triggered at Frec,EI = 0.9,
0.95, 0.97 characterizing volume losses of 10%, 5%, and 3%. Simulations also include
assumptions of either no permanently closed alveoli, such that γ = 1 for all time, or10% permanently
closed alveoli per breath, such that gnext gcurrent
Fclosed. Each simulation was
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performed three times: constant f; variable f by breath according to V_ E VT;ave f where VT,ave
is a moving average of the previous 60 tidal volumes ( 1 minute of breathing); variable f
including a single 20 second apneic event.
The system of differential Eq (4), together with the constitutive relations (5±13), were
solved using MATLAB R2016b (MathWorks, Natick, MA) with the differential equations
solver ode15s (see S1 Code). Initial conditions were set at physiological values as given in
Table 5. The equations were solved for each new breath using the end conditions from the
previous breath as initial conditions. Parameter values as discussed earlier are given in Tables 2
and 3. The steady-state stability of the model was analyzed under constant non-oscillatory
muscle pressure by examining the eigenvalues of the Jacobian at the nominal parameter
set and varying parameters by multiples of 2 and 10. Results of this analysis are found in
Parameterized static compliance curves for Vcw(Pcw) and VA(Pel) are shown in Fig 2 for high
Cw (left) and low Cw (right). The hysteretic tidal breathing loops are superimposed on the
curve VA(Pel) for normal Ru in black and increased Ru in grey. Hysteresis is caused in the
model by the viscoelastic parameters Cve and Rve, which were tuned to maintain appropriately
valued lung volume outputs.
Fig 3 shows the impact of high vs low Cw and normal vs. high Ru on the five states PA, Pl,dyn,
Ppl, VA, and V_ . Increased Ru increases PA almost threefold, but Cw has very little impact.
However, decreased Cw increases Pl,dyn significantly, effectively raising it higher on the lung PV
curve. Increasing Ru even higher increases Pl,dyn but there is no difference with respect to Cw.
The opposite appears to occur with Ppl dynamics, in that decreasing Cw makes Ppl more
negative (ªincreasingº the magnitude of the pressure) and increasing Ru strengthens that effect.
Low Cw and subsequently high Ru increase VA, mimicking the effect for Pl,dyn. High Ru shifts V_
by 5 ml/s, with airflow more restricted during expiration. Tabulated magnitudes of the steady
states are gives in Table 4. These results compare favorably to the record reported in Abbasi
et al [
] in which esophageal (pleural) pressure, airflow, and tidal volume were approximately
-2 to -6 cm H2O, -30 to 30 ml/s, and 8 ml, respectively.
Table 6 presents the 14 simulations and their time to failure, defined for this study as 90%
volume loss. Dynamics were comparable between simulations with the major difference being
the timing, therefore only representative or significant results are presented in figures. Our
model consistently indicates a faster loss of end-expiratory lung volume in all simulations with
high Cw compared to the same with low Cw. Variable f did not significantly change TTF except
in the case of CPAP administered at 3% loss (S14), with TTF shortened by almost 2 hours.
Adding a single 20 second apnea shortened the TTF by 1-4 minutes in the shortest simulations
but by over an hour under high Cw and increased Ru (S8).
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Fig 3. Simulated periodic steady-state tracings of five breaths. Depicted are alveolar volume, airflow, alveolar pressure, dynamic elastic lung recoil, and pleural
pressure, under high and low Cw conditions, with normal vs. high Ru.
The breath-to-breath change in EELV and VT under high and low Cw conditions with no
interventions are given in Fig 4 (Simulations 1 and 3). The high Cw simulation reaches
accelerated loss of volume and eventually failure at 0.3 hours, much more quickly than the low Cw at
2.5 hours. This depicts a possible scenario in which lung volume loss and failure may appear to
onset suddenly after a long period of apparent steady conditions.
Fig 5 shows changes in dynamic lung compliance and tidal volume with high and low Cw
without changes in γ, then adding CPAP to the high Cw condition at three different levels (c.f.
Table 6, simulations 1,3,11,13-14). CPAP was simulated by an increase in mouth pressure Pao
to 5 cm H2O when Frec,max < 0.9, which happened when the lung volumes were already
decreasing quickly towards failure. However, CPAP triggering at Frec,max < 0.95 and 0.97
gained 3 and 9 hours of time, respectively. Note that regardless of timing, the administration
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CPAP, 10% loss
Increased Ru: A 10-fold increase in Ru was applied during expiration. CPAP: Simulated administration of Pao = 5 occurred when recruited fraction was down 10%, then
again at 5% and 3% with constant γ. AE: A single 20 second apneic event occurred at the 2 minute mark of the simulation.
of CPAP is correlated with reduced tidal volume (see also [
]). Increasing Pao moves the
resulting PV loop higher up on the lung compliance curve but does not change the nature of
the curve, thus eventually the influence of high Cw on dynamics induces the same lung volume
loss without other mitigating actions.
Fig 4. Breath-to-breath volumes. End-expiratory lung volume (left y-axis) and tidal volume (right y-axis) under high and low Cw conditions, no interventions.
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Fig 5. Breath-to-breath dynamic lung compliance and tidal volume. Depicted are high and low Cw conditions, with simulated CPAP triggered in the
high Cw condition when recruited fraction dropped 10%, 5%, and 3%.
In summary, we have developed a lumped-parameter respiratory mechanics model tuned with
parameters specific to the extremely preterm infant weighing 1 kg. The model includes a novel
representation of derecruitment based on alveolar pressure and volume expansion
compensating for collapsed alveoli. Model simulations suggest conditions under which volume loss may
result more quickly from higher vs lower chest wall compliance in the preterm infant,
indicating the plausibility of dynamics underlying the symptoms observed clinically. Given the fragile
nature of this population, it is extremely difficult to obtain non-pathological parameter or state
output values for a healthy or surfactant-treated infant during spontaneous breathing, and
even more so to obtain time series for model validation and eventual parameter estimation.
The much earlier study by Abbasi and Bhutani [
] and a later one by Pandit et al [
the best insight into the respiratory dynamics of an extremely preterm infant, making these the
standard against which our results were qualitatively validated. We therefore claim that this
effort is a ªproof of conceptº that will be further explored in future investigations using
pressure and airflow time series data in a parameter estimation / optimization procedure to
characterize parameter values specific to a particular patient dataset. Additional model modifications
will allow for hypothesis generation for future data testing and data collection.
As mentioned briefly in Nonlinear compliance constitutive relations, recruitment/decre
cruitment may have a time component [
], in that the time it takes for an airway or
alveolus to open may be a function of how far away its pressure is from its critical opening pressure.
Earlier studies have developed models that incorporate opening and closing pressures for
individual alveoli, contributing to the aggregate difference in inflation and deflation limbs of the
hysteretic PV curve [
]. These previous studies considered recruitment resulting from
one or two hyperinflations but not long-term derecruitment. In our model breath-to-breath
derecruitment is manifested as the change of the lung compliance curve during normal
spontaneous breathing as described in Progressive volume loss, and the hysteresis found in the
tidal breathing loop is accounted for by the viscoelastic component of the system of differential
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equations. It is clear from Table 6 that time to failure shortens if an assumption is made about
a non-zero percentage of alveoli permanently closing and being unavailable for recruitment.
As a topic for further study, the pulmonary tissue may be modeled by more complex
VoigtMaxwell models within the ªelectrical analogº model or other non-electrical analog
representations (e.g. those described in [
]). While such a modification may affect the overall trends
in observed states such as EELV, the differential impact between high and low Cw would be
expected to remain.
The noninvasive ventilatory intervention CPAP shifts the tidal volume loop to a higher
position on the lung compliance curve, operating with a higher EELV and end-expiratory lung
elastic recoil. Our model suggests that the timing of administration of CPAP and the
permanent closure or injury state of alveoli may impact its effectiveness. In our first simulation with
simulated CPAP triggered at 10% volume loss, the recruited volume fraction does not recover
fully to 1 and the use of CPAP only gains about a half hour of breathing before failure.
However, CPAP starting at 5% and 3% loss gained 3 and 9 hours of time, respectively. This
magnitude of loss may not be symptomatic at this point but would benefit from pressure support to
avoid the quick descent to failure. These results are reported for the case with all fully
recruitable alveoli. In the case of 10% permanent collapse of closed alveoli at each breath and
subsequent breath-by-breath decrease in γ, the function Frec can never reach 1 (full recruitment) for
the duration of breathing, tidal breathing occurs on a lower lung compliance curve, and CPAP
cannot recover the full volume loss in subsequent breaths. Results in Table 6 indicate that time
to failure is 10% faster with the permanent collapse. These simulated loss and collapse
percentages were arbitrarily chosen to demonstrate the capabilities of the model and possible
influences on breathing dynamics, but more investigation into actual loss values would add to the
model's usefulness. Starting from a lower Cw appears to be the optimal condition presented
here as hypothesized.
Prolonged shallow breathing has been associated with increased surface tension and
decreased surface area that further hinders breathing [
]. We safely assume in our model that
derecruitment is a continuous process that will eventually induce loss of lung volume if left
]. In a healthy lung in the absence of fatigue, permanent alveolar
collapse (due to injury or disease), and/or high chest wall compliance, this process is on a much
longer time scale than the natural compensation mechanisms that compensate for and recoup
volume loss (such as grunting in the infant). One such mechanism is spontaneous deep
breathing, or ªsighingº, which may help prevent atelectasis [58±60] by re-opening air spaces that
collapse naturally under tidal breathing  via increased pressure and surfactant activation and
possibly affect neurorespiratory control. Sighing occurs more frequently and at relatively larger
magnitude in the infant vs adults [
]. A natural extension of our model would be
incorporating the restorative actions of sighing and testing the hypothesis that spontaneous deep breaths
mitigate or reverse volume loss.
Several features of the physiology of preterm infants are not currently addressed in this
model but should be considered in future model enhancements for further investigations.
Preterm infants commonly exhibit diaphragm weakness and dysfunction and paradoxical
breathing. While a sinusoidal waveform is used in the clinic under some mechanical
ventilation protocols, the sinusoidal pressure function used here is an elementary representation for
spontaneous breathing and does not capture dynamics related to diaphragm dysfunction or
possible expiratory flow limitation. Modifications reflecting such dynamics may include
adjustments to the pressure amplitude, varying fractions of time spent in inspiration vs
expiration, and the use of a model that combines functional forms such as polynomials or
exponentials (see e.g. [
]). Components that differentiate between abdominal and rib cage
movements (see e.g. [
]) may model the paradoxical chest movement.
16 / 21
Another limitation of this model is the absence of any feedback mechanisms compensating
for loss of volume. More sophisticated models of central pattern generators have been
developed in conjunction with simple lung mechanics [
] that could potentially be
incorporated with ours. A chemoreflex model, see for example [
], may also augment our model.
Despite these limitation, we expect that the timing of dynamics of individual simulations may
change with model enhancements but that time to failure would still be extended under low
chest wall compliance conditions as observed in this study.
Respiratory mechanics models have been investigated for several years and many formulations
exist; the challenge to be appreciated is the customization to the preterm infant with
significantly different physiological features than adults and even term infants. Hence future model
modifications must always keep this at the forefront of any investigation. The
lumped-parameter respiratory mechanics model developed in this study will be used in future studies with
data currently being collected in the NICU to estimate patient-specific parameters, which may
shed light on factors influencing volume loss dynamics. This process may help generate
hypotheses about predicting volume loss and recovery to motivate future data collection
strategies. Our hope is that these investigations lead to a chest-stiffening treatment that can target an
infant's specific physiological characteristics and prevent volume loss in this vulnerable
S1 Appendix. Model stability analysis. The inherent stability of the model was analyzed
under constant non-oscillatory muscle pressure by examining the eigenvalues of the Jacobian
at the nominal parameter set and varying parameters by integer multiples.
S1 Code. Minimal representative code. Attached MATLAB code runs Simulation 3 (high Cw,
normal Ru) with constant frequency for 6 periods.
This research was supported in part by the Atlantic Pediatric Device Consortium / FDA (HJR,
LEF, MB) and the VCU College of Humanities and Sciences Faculty Research Council (LEF,
Conceptualization: Joseph Khoury, Russell R. Moores, Jr., Matthew Brandes, Henry J.
Data curation: Laura Ellwein Fix.
Formal analysis: Laura Ellwein Fix, Lauren Linkous.
Funding acquisition: Henry J. Rozycki.
Investigation: Laura Ellwein Fix, Lauren Linkous.
Methodology: Laura Ellwein Fix, Joseph Khoury, Russell R. Moores, Jr., Henry J. Rozycki.
Project administration: Laura Ellwein Fix.
17 / 21
Resources: Laura Ellwein Fix.
Software: Laura Ellwein Fix, Lauren Linkous.
Supervision: Laura Ellwein Fix.
Validation: Laura Ellwein Fix, Lauren Linkous.
Visualization: Laura Ellwein Fix.
Writing ± original draft: Laura Ellwein Fix.
Writing ± review & editing: Joseph Khoury, Russell R. Moores, Jr., Lauren Linkous, Matthew
Brandes, Henry J. Rozycki.
18 / 21
19 / 21
20 / 21
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