text:
\section*{Abstract}
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 model of respiratory system mechanics 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.

\section*{Introduction}
The extremely  preterm infant, born at $<28$ weeks gestation and usually $<1000$g, 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~\cite{Stoll15}, 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 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~\cite{Manley13,Bhandari13,Siew15}. One hypothesis for the failure of non-invasive ventilation 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~\cite{Love53,Beltrand08,Kovacs15}. In the preterm infant chest wall compliance can be up to five times lung tissue compliance~\cite{Gerhardt80}.

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~\cite{Mortola83}. In many cases this leads to progressive lung collapse (atelectasis) with each breath as the forces needed to open airspaces after each exhalation become insurmountable~\cite{Frappell05}, leading to decreasing lung compliance and functional residual capacity (FRC)~\cite{Miller05}. 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 investigating the underlying mechanics 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~\cite{Liu98,Narusawa01,Athan00,Frazer13,Uzawa15}. 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 population.

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~\cite{Athan00} and modified for newborn lambs by LeRolle et al~\cite{LeRolle13}. 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~\cite{Smith76}. 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.

\begin{table}[!ht]
\caption{Glossary}
{\small
 \begin{tabular}{ll}
\toprule
	Parameter/State 								& Physiologic description									  							\\
\noalign {\smallskip} \hline \hline
\noalign {\smallskip}
TLC [ml]									& Total lung capacity 																\\
RV [ml]										& Residual volume  																	\\
FRC [ml] 									& Functional residual capacity 										\\
VC[ml]										& Vital capacity 																	\\
RR [br/min] 							& Respiratory rate																\\
$f$ [br/s]								& Respiratory frequency																\\
$T$ [s]										& Duration of respiratory cycle												\\
$V_T$ [ml]								& Tidal volume																	 \\
$\dot{V}_E$ [ml/min]			& Minute ventilation															\\
$\dot{V}_A$ [ml/s]				& Airflow																					\\
$A_{mus}$ [cm H$_2$O]			& Muscle pressure amplitude												\\
$P_{tm}$ [cm H$_2$O]				& Transmural pressure															\\
$P_{A}$ [cm H$_2$O]				& Alveolar pressure																\\
$P_{el}$ [cm H$_2$O]			& Lung elastic recoil (transpulmonary pressure)		\\
$P_{ve}$ [cm H$_2$O]			& Viscoelastic component of pressure									\\
$P_{l,dyn}$ [cm H$_2$O]		& Dynamic pulmonary pressure															\\
$P_{pl}$ [cm H$_2$O]			& Pleural pressure																\\
$P_{cw}$ [cm H$_2$O]			& Chest wall elastic recoil																\\
$P_{mus}$ [cm H$_2$O]			& Respiratory muscle pressure												\\
$C_A$ [ml/cm H$_2$O]			& Lung compliance																	\\
$C_w$ [ml/cm H$_2$O]			& Chest wall compliance														\\
$C_{rs}$ [ml/cm H$_2$O]		& Respiratory system compliance										\\
$R_{rs}$ [cm H$_2$O s/L] 	& Respiratory system resistance										\\
$\nu$								& Fraction of VC for chest wall relaxation volume 		\\
$V_0$ [ml] 								& Chest wall relaxation volume												\\
$\beta$										&	Baseline fraction of lung recruited at $P_{el}=0$	 \\
$\gamma$									&	Maximum recruitable function of lung							 \\
$\alpha$									&	Lower asymptote, fraction recruitment							  \\
$k$ [1/cm H$_2$O]					&	Characterizes slope, aggregate lung elasticity		 \\
$c_F$	[cm H$_2$O]					&	Pressure at maximum lung recruitment							  \\
$d_F$	[cm H$_2$O]					&	Characterizes slope at maximum lung recruitment		 \\
$a_w$ [ml]								& Lower asymptote, chest wall compliance							\\
$b_w$ [ml]								& Characterizes slope, $P_{cw}\rightarrow \infty$			\\
$c_w$ [cm H$_2$O]					& Transition point, chest wall compliance							\\
$d_w$ [cm H$_2$O]					& Characterizes slope, $P_{cw}\rightarrow \infty$		\\
$a_c$ [ml]								& Lower asymptote, collapsible airway									\\
$b_c$ [ml]								& Upper asymptote, collapsible airway										\\
$c_c$ [cm H$_2$O]					& Pressure at peak collapsible airway compliance			\\
$d_c$ [cm H$_2$O]					& Characterizes slope, peak coll. airway compliance	\\
$K_c$ 										& Collapsible airway resistance coefficient					\\
$V_{c,max}$ [ml]					& Peak collapsible airway volume											\\
$R_{s,m}$ [cm H$_2$O s/L]	& Minimum small airway resistance											\\
$R_{s,d}$ [cm H$_2$O s/L]	& Change in small airway resistance										\\
$K_s$ 										& Small airway resistance low pressure coefficient		\\
$I_u$ [cm H$_2$O s$^2$/L]	& Upper airway inertance														\\
$R_{u,m}$ [cm H$_2$O s/L]		& Laminar value, upper airway resistance						\\
$K_u$ [cm H$_2$O s/L]			& Turbulent coefficient, upper airway resistance		\\
$C_{ve}$ [L / cm H$_2$O] 	& Lung viscoelastic compliance														\\
$R_{ve}$ [cm H$_2$O s/L] 	& Lung viscoelastic resistance												\\
\bottomrule
\end{tabular}}
\label{tab:glossary}
\end{table}

\section*{Mathematical model}

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~\ref{fig:model}. As such, relevant states are in terms of pressure $P(t)$ [cm H$_2$O] and volume $V(t)$ [ml] in and between air compartments, with volumetric flow rate and rate of change represented as $\dot{V}(t)$ [ml/s] and $\frac{dV}{dt}$ respectively. Air pressure $P_i$ within a specific volume $i$ is defined as the difference between intra-airway pressure $P_{int}$ and pressure external to the body $P_{ext}$, i.e. $P_i=P_{int,i}-P_{ext,i}$. Since all pressures are relative to the same constant atmospheric pressure, all $P_{ext}=0$ and all intra-airway pressures $P_i=P_{int,i}$. The pressure $P_{ij}=P_i-P_j$ refers to the transmural pressure across a compliant boundary separating volumes $i$ and $j$.

\begin{figure}[!ht]
\includegraphics[scale=0.8]{Figure1_B.eps}
\caption{{\small {\bf Lumped-parameter respiratory mechanics model, in both volume-pressure (panel A) and electrical (panel B) system analogs.} Each non-rigid 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 $\dot{V}$ (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~(\ref{eq:Pcomp}). 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 $mus$.}}
\label{fig:model}
\end{figure}

\subsection*{State equations}

Each non-rigid compartment has an associated compliance $C_i$ [ml/cm H$_2$O], describing the change in compartmental volume $V_i$ given a change in transmural pressure $P_{ij}$ across its boundary with compartment $j$:
\begin{equation*}
C_i=\dfrac{dV_i}{dP_{ij}},
\label{eq:compliance}
\end{equation*}
The nature of $C_i$ does not change explicitly with time but instead is implicitly determined by the relationship between volume and pressure. This can be reformulated in terms of dynamic changes of state:
\begin{equation*}
\frac{dV_{i}}{dt}=C_i\left(\frac{dP_{ij}}{dt}\right).
\label{eq:volume}
\end{equation*}

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~\cite{Mead61},
\begin{equation}
\dot{V}_i=\dfrac{P_{i-1}-P_i}{R_i},
\label{eq:Ohms}
\end{equation}
where $R_i$ [cm H$_2$O$\cdot{}$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,
\begin{equation}
P_{i-1}-P_i=I_i\ddot{V}_{i}(t)
\label{eq:inertance}
\end{equation}
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~\cite{LeRolle13}.

The pressures $P_{ij}$ across each compliant compartment include transmural pressure between the compliant airways and the pleural space $P_{tm}=P_c-P_{pl}$, lung elastic recoil $P_{el}=P_A-P_T$, lung viscoelastic component $P_{ve}=P_T-P_{pl}$, and chest wall elastic recoil $P_{cw}=P_{pl}-P_{mus}$. Summing pressures over each of three loops according to Kirchhoff's mesh rule gives a system of time-varying algebraic equations:
\begin{eqnarray*}
0&=&P_{ao}-P_c+P_{tm}+P_{pl}\\
0&=&(P_c-P_A)+P_{el}+P_{ve}-P_{tm}\\
0&=&P_{ve}-R_{ve}(\dot{V}_A-\dot{V}_{ve})
\end{eqnarray*}
Using the additional relationships obtained from applying Eq~(\ref{eq:Ohms}-\ref{eq:inertance}), the system of loop equations can be rewritten as
\begin{eqnarray}
0&=&P_{ao}+R_c\dot{V}-R_u\dot{V}-I_u\ddot{V}+P_{tm}+P_{cw}-P_{mus} \label{eq:Pcomp}\\
0&=&R_s\dot{V}_A+P_{el}+P_{ve}-P_{tm} \nonumber\\
0&=&P_{ve}-R_{ve}(\dot{V}_A-\dot{V}_{ve}). \nonumber
\end{eqnarray}

Rearranging Eq~(\ref{eq:Pcomp}) and using Kirchhoff's current law along with Eq~(\ref{eq:Ohms}-\ref{eq:inertance}) produces the consolidated set of model differential equations:
\begin{eqnarray}
\ddot{V}&:&\frac{d\dot{V}}{dt}=\frac{1}{I_u}\left(P_{ao}-P_u-R_u\dot{V}\right) \label{eq:states}\\
\dot{V}_c&:&\frac{dV_c}{dt}=\dot{V}-\dot{V}_A \nonumber\\
\dot{P}_{el}&:&\frac{dP_{el}}{dt}=\frac{\dot{V}_A}{C_A} \nonumber\\
\dot{P}_{ve}&:&\frac{dP_{ve}}{dt}=\frac{\dot{V}_A-(P_{ve}/R_{ve})}{C_{ve}} \nonumber
\end{eqnarray}

Conservation laws also maintain that $V=V_{cw}=V_A+V_c$, 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 $C_i$ will be further described below.

\subsection*{Nonlinear resistance constitutive relations}

The airways begin with an upper rigid segment characterized by an inertance $I_u$ and a nonlinear Rohrer resistance $R_u$~\cite{LeRolle13,Rohrer15} that increases with airflow:
\begin{equation}
R_u=R_{u,m}+K_u|\dot{V}|
\label{eq:Ru}
\end{equation}
The constants $R_{u,m}$ and $K_u$ represent laminar and turbulent flow components.

A middle collapsible portion is modeled as a cylinder with constant length having nonlinear resistance $R_c$ that depends inversely on the 4th power of the radius according to Poiseuille's law. Therefore $R_c$ is formulated as~\cite{Olender76,Liu98}:
\begin{equation}
R_c=K_c\left(\frac{V_{c,max}}{V_c}\right)^2
\label{eq:Rc}
\end{equation}
where $R_c$ equals its minimum value $K_c$ when $V_c=V_{c,max}$, an estimate of dead space.

An inverse relationship between resistance in the smaller peripheral airways $R_s$ and lung volume $V_A$ reflects high resistance at low or near-zero volumes~\cite{Olender76}. To avoid $R_s\rightarrow \infty$ as $V_A\rightarrow 0$~\cite{Avanz01} from a strict exponential decay model, we adopt the formulation used by both Liu et al and Athanasiades et al~\cite{Liu98,Athan00}, a decaying exponential function of relative lung volume with finite $R_s$ at $V_A=0$:
\begin{sloppypar}
\begin{equation}
R_s=R_{s,d}\cdot e^{K_s(V_A-RV)/(TLC-RV)}+R_{s,m}
\label{eq:Rs}
\end{equation}
where $K_s<0$. This parameterization gives that ${R_s\approx R_{s,m}}$ when $V_A=TLC$, and $R_s=R_{s,d}+R_{s,m}$ when $V_A=RV$ (residual volume).
\end{sloppypar}

\subsection*{Nonlinear compliance constitutive relations}\label{sec:compliances}

The volumes $V_c$, $V_A$, and $V_{cw}$ representing physiological compartments are assumed to have nonlinear compliance which are modeled implicitly with a pressure-volume curve or explicitly by $C_i=\frac{dV_i}{dP_i}$.

The compliance curve for the collapsible airway volume $V_c$ as a function of $P_{tm}$ represents data depicted in~\cite{Olender76} following a sigmoidal function~\cite{LeRolle13, Avanz01}
\begin{equation}
V_c=\frac{V_{c,max}}{1+e^{-(P_{tm}-c_c)/d_c}}
\label{eq:Vc}
\end{equation}
Maximal compliance occurs at the middle of the sigmoid $c_c$, with $d_c$ characterizing the slope of the sigmoid.

In newborns and infants an exponential-like chest-wall compliance curve is observed~\cite{Donn98,Goldsmith11} but with compliance being near infinite for $P_{cw}>0$. We chose to model the static compliance of the chest wall as a ``softplus'' function of the form $f(x)=\ln(1+e^x)$, the smooth approximation of the rectifier activation function $f(x)=\max(0,x)$. Accounting for translations and scaling, this is represented by
\begin{equation}
V_{cw}=RV+b_w \ln \left(1+e^{(P_{cw})/d_w}\right)
\label{eq:Vcw}
\end{equation}
The asymptotic volume at large negative pressure is thus assumed to equal RV. The ``transition point'' where the softplus function slope has the greatest rate of change from horizontal to affine occurs at $P_{cw}=$. The chest wall relaxation volume $V_0=V_{cw}|_{P_{cw}=0}$ is set using an estimate from literature at 25\% of VC (vital capacity)~\cite{Donn98,Goldsmith11}. From this parameterization, $b_w=(V_0-RV)/(\ln 2)$. The single degree of freedom $d_w$ 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 $V_A$ is modeled as the product of distention of lung units $V_{el}(P_{el})$ and fraction of recruited alveoli $F_{rec}(P_{el})$~\cite{Venegas94,Hamlington16}. To obtain $V_A\approx RV$ near $P_{el}=0$, lung volume is given as
\begin{equation}
V_A=V_{el}(P_{el})\cdot F_{rec}(P_{el})+RV.
\label{eq:VA}
\end{equation}
Alveolar compliance $C_A$ as used in the system of differential equations~\eqref{eq:states} is found with symbolic computation as $\frac{dV_A}{dP_{el}}$.

The first term $V_{el}$ represents the volume due to aggregate elasticity of the lung unit structure, which is modeled here as a saturated exponential ~\cite{Hamlington16,Colebatch79,Ferreira11}
\begin{equation}
V_{el}=VC\cdot(1-e^{(-kP_{el})})
\label{eq:Vel}
\end{equation}
 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 $F_{rec}$ represents the contribution of recruitment and derecruitment of alveoli to compliance, which has been modeled previously as dependent on both time and pressure~\cite{Venegas94, Hamlington16,Bates02}. 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~\cite{Sundaresan09}. We adopt the formulation of Hamlington et al~\cite{Hamlington16}:
\begin{eqnarray}
F_{rec}&=&\alpha+\frac{\gamma-\alpha}{1+e^{-(P_{el}-c_F)/d_F}}, \label{eq:Frec}\\
\mbox{where} \ \ \ \alpha&=&\frac{(1+e^{c_F/d_F})\beta-\gamma}{e^{c_F/d_F}}. \nonumber
\end{eqnarray}

It follows that $\beta$ is the baseline fraction of lung recruited at $P_{el}=0$, $\gamma$ represents the maximum recruitable fraction of lung, $c_F$ is mean opening pressure at which recruitment is maximum, and $d_F$ describes the transition to full recruitment capturing the heterogeneity of the lung. Parameterization of $F_{rec}$ 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 $c_F$ and move the $V_A$ curve to the right. Likewise a lower maximum recruitable fraction $\gamma$ would flatten the $V_A$ 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 $V_A(P_{el})$ with a low compliance region at low $P_{el}$~\cite{Ferreira11,Venegas98,Pereira03, Harris05, Oliveira16} could be captured in the parameterization of $F_{rec}$.

The viscoelastic properties of pulmonary tissue are represented with a linear Kelvin-Voigt model consisting of scalar compliance $C_{ve}$ and resistance $R_{ve}$, which contributes a viscoelastic pressure component $P_{ve}$ in series with lung elastic recoil $P_{el} $, see Fig~\ref{fig:model}. The sum of these two pressures is dynamic pressure $P_{l,dyn}$ which also equals $P_{pl}-P_A$.

\subsection*{Respiratory muscle driving pressure}

The pressure $P_{mus}$ describes the effective action of the respiratory muscles driving the model dynamics with $P_{mus}$ negative in the outward direction. We used a sinusoidal function  to describe tidal breathing, with maximum equaling zero at end-expiration:
\begin{equation}
P_{mus}=A_{mus}\cos (2\pi f t)-A_{mus},
\label{eq:Pmus}
\end{equation}
where $A_{mus}$ is the amplitude of the cosine wave and $f=RR/60$ is the frequency. The wave generates a negative pressure with total magnitude $2A_{mus}$ 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.~\cite{Athan00}). More sophisticated functions~\cite{Mecklen98, LeRolle13, Albanese16} 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.

\subsection*{Progressive volume loss}\label{sec: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 $F_{rec}$ (Eq~(\ref{eq:Frec})) as lung recruitment pressure parameters $c_F$ and $d_F$ increase with PIP and maximum recruitable fraction $\gamma$ decreases with EILV.

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~\cite{Brown73} 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 $c_F$ and $d_F$ shifts the compliance curve to the right. In this way the compliance decreases approximately proportional to the amount of derecruitment~\cite{Bates09}. 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 $V_T\approx 0$. 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 $\gamma$ for subsequent breaths is then dependent on $F_{rec}|_{EI}$ and the percentage of alveoli assumed to be permanently collapsed / no longer recruitable, represented by the calculation $\gamma_{next}=\gamma_{current}\dot(1-(\%_{permanent})\cdot(F_{closed}))$ where $F_{closed}$ is fraction closed. If all unrecruited alveoli remain as such, then $F_{rec}|_{EI}$  becomes the new $\gamma$ for the next breath; likewise, if all alveoli remain recruitable, $\gamma=1$ for the duration of the simulation. Note that even for $\gamma=1$, $F_{rec}<1$ for all $P_{el}$ thus causing small changes in $c_F$ and $d_F$ and shifts in the $F_{rec}(P_{el})$ curveregardless 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 $F_{rec}$ curve tidal breathing occurs, and thus both the curve's intrinsic characterizing parameters and extrinsic system variables.

\section*{Simulation conditions}

\subsection*{Parameterization}\label{sub:physiol}

 The lung curve was parameterized to obtain an approximate dynamic lung compliance $C_A$ of 2.3 ml/cm H$_2$O~\cite{Gerhardt80,Mortola87,Pandit00} calculated as the slope $(V_A|_{EI}-V_A|_{EE})/(P_{el}|_{EI}-P_{el}|_{EE})$ during normal breathing with no interventions. In particular, $k$ was tuned to produce a curve $V_{el}$ between RV and TLC with the calculated slope, and the parameters of $F_{rec}$ produced a curve that is $\approx 1$ for the whole range of normal breathing to represent a nearly fully recruited lung. High $C_w$ for a typical preterm infant was targeted at 8.5 ml/cm H$_2$O~\cite{Gerhardt80,Mortola87} and low $C_w$ about equal to lung compliance. The parameter $d_w$ characterized the approximate dynamic chest wall compliance, which was calculated as the slope $(V_{cw}|_{EI}-V_{cw}|_{EE})/(P_{cw}|_{EI}-P_{cw}|_{EE})$. Parameter values for $R_u$, $R_c$, $R_s$, and $V_c$ were estimated from previously published studies~\cite{Liu98,Athan00,Singh12}. The viscoelastic parameters $C_{ve}$ and $R_{ve}$ 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 $P_{resp}=P_{el}+P_{cw}=0$. The naturally high compliance of the healthy full-term and especially preterm infant (with even steeper $V_{cw}(P_{cw})$) lowers $P_{resp}$ and decreases FRC to about 20\% of vital capacity (VC), compared to at about 35-40\% of VC in the adult~\cite{Agostini86}. 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 H$_2$O. Lung and chest recoil pressure vectors are then added in the $P$ direction to obtain $P_{resp}$, and the index where $P_{resp}=0$ is used to determine FRC using either $V_{cw}$ or $V_A$. The lung, chest wall, and respiratory PV compliance curves for both high and low $C_w$ created from Eq~(\ref{eq:Vcw}-\ref{eq:VA}) are given in Fig~\ref{fig:PVcurves}. The value for $P_{el}|_{FRC}$ is then set as the initial condition for solving $dP_{el}/dt$. Note that the lung curve is identical between scenarios so the decreased slope in the low $C_w$ scenario with the same $\nu$ and $V_0$ raises FRC and thus EELV. Decreased lung compliance (flatter $V_A(P_{el})$) resulting from injury, disease, or progressive volume loss further reduces FRC and EELV. In our 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.

\begin{table}[!h]
\begin{adjustwidth}{0in}{1in}
\caption{Tuned steady-state and dynamic simulation parameters that remained unchanged during simulations}
{\small
 \begin{tabular}{llccc}
\toprule
	Parameter 							& Value	& Formula 								& References 				\\
\noalign {\smallskip} \hline \hline
\noalign {\smallskip}
TLC [ml]																							&  63		& \textemdash							& \cite{Smith76,Donn98}	\\
RV [ml]										&  23		& \textemdash							& \cite{Smith76}	\\
VC[ml]										&  40		&  TLC-RV									& \cite{Smith76,Donn98}	\\
RR [br/min] 							&  60		& \textemdash							& \cite{Donn98}	\\
$f$ [br/s]								& 	1		& RR/60										& \textemdash	\\
$T$ [s]										&  1		& $1/f$										& \textemdash	\\
$\nu	$							& 0.25	&	\textemdash							& \cite{Donn98,Goldsmith11}	\\
$V_0$ [ml] 								&  35		& $\nu\cdot$VC+RV	& \textemdash	\\
$\beta$										&	0.01	&	estimated								& \cite{Hamlington16}  \\
$\gamma$									&		1		&	estimated								& \cite{Hamlington16}  \\
$\alpha$									&	-0.76	&	$\frac{(1+e^{c_F/d_F})\beta-\gamma}{e^{c_F/d_F}}$	& \cite{Hamlington16}  \\
$k$ [1/cm H$_2$O]					&	0.07	&	estimated								& \cite{Ferreira11,Hamlington16}  \\
$c_F$	[cm H$_2$O]					&		0.1	&	estimated								& \cite{Hamlington16}  \\
$d_F$	[cm H$_2$O]					&	0.4		&	estimated								& \cite{Hamlington16}  \\
$a_w$ [ml]								&  23		& RV											& \cite{Donn98,Goldsmith11}	\\
$b_w$ [ml]								&  17.3	& $(V_0-RV)/\ln 2$	& \textemdash	\\
$c_w$ [cm H$_2$O]					&  0		& estimated								& \textemdash	\\
$a_c$ [ml]								&  0		& \textemdash							& \cite{Liu98}	\\
$b_c$ [ml]								&  2.5	& $V_{c,max}$							& \cite{Neumann15}		\\
$c_c$ [cm H$_2$O]					&  4.4	& estimated from adult		& \cite{Liu98}	\\
$d_c$ [cm H$_2$O]					&  4.4	& estimated from adult		& \cite{Liu98}		\\
$K_c$ 										&  0.1		& estimated from adult		& \cite{Athan00}	\\
$V_{c,max}$ [ml]					&  2.5	& estimated as dead space	& \cite{Donn98,Neumann15}	\\
$R_{s,m}$ [cm H$_2$O s/L]	&  12		& \textemdash							& \cite{Ratjen92,Singh12}	\\
$R_{s,d}$ [cm H$_2$O s/L]	&  20	& estimated from adult		& \cite{Athan00}	\\
$K_s$ 										&  -15	& estimated from adult		& \cite{Athan00}	\\
$I_u$ [cm H$_2$O s$^2$/L]	&  0.33	& \textemdash							& \cite{Singh12,LeRolle13}\\
$C_{ve}$ [L / cm H$_2$O] 	& 0.005 & estimated from adult 		& \cite{Athan00}				\\
$R_{ve}$ [cm H$_2$O s/L] 	& 20		& estimated from adult		& \cite{Athan00}	\\
\bottomrule
\end{tabular}}
	{\smallskip}
\begin{flushleft} See Table~\ref{tab:glossary} (Glossary) for variable definitions.
\end{flushleft}
\label{tab:SSparams}
\end{adjustwidth}
\end{table}

Table~\ref{tab:SSparams} 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~\ref{tab:varparams} that vary between simulation conditions were manually tuned to best obtain the reported aggregate parameters and state outputs as shown in Table~\ref{tab:outputs}. As an example, dynamic lung compliance $C_A$ 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.

\begin{table}[!h]
\begin{adjustwidth}{-1.4in}{0in}
\centering
\caption{Parameters varying with chest wall compliance and simulation conditions}
{\small
 \begin{tabular}{lcccccc}
\toprule
\multirow{2}{*}{Parameter}& \multicolumn{2}{c}{High $C_w$}	& \multicolumn{2}{c}{Low $C_w$} 	& \multirow{2}{*}{Formula} 						& \multirow{2}{*}{References} 				\\
\noalign {\smallskip}
													&  normal $R_u$ & increased $R_u$		& normal $R_u$ & increased $R_u$	&  						& 									\\
\noalign {\smallskip} \hline \hline
\noalign {\smallskip}
FRC [ml] 									&  24.9	& 24.9 											& 28.1	& 28.1										& $P_{el}|_{FRC}+P_{cw}|_{FRC}=0$	& \cite{Smith76,Donn98,Thomas04}	\\
$A_{mus}$ [cm H$_2$O]			&  1.85	& 3.2 											& 2.78 	& 3.8 										& estimated								& \textemdash	\\
$d_w$ [cm H$_2$O]					&  0.48	& 0.48 											& 2.4 	& 2.4 										& estimated								& \textemdash	\\
$R_{u,m}$ [cm H$_2$O s/L]	&  20		& 200 											& 20		& 200 										& \textemdash							& \cite{Mortola87,Singh12}	\\
$K_u$ [cm H$_2$O s/L]			&  60		& 600 											& 60		& 600 										& estimated from adult		& \cite{Mortola87,Athan00,Singh12}	\\
\bottomrule
\end{tabular}}
	{\smallskip}
\begin{flushleft} See Table~\ref{tab:glossary} (Glossary) for variable definitions.
\end{flushleft}
\label{tab:varparams}
\end{adjustwidth}
\end{table}

\begin{table}[!ht]
\begin{adjustwidth}{-2.0in}{0in}
\centering
\caption{Aggregate parameters and output states targeted during simulations}
{\small
 \begin{tabular}{lccccccc}
\toprule
\multirow{2}{*}{Parameter}	& \multirow{2}{*}{Ref. Value} & \multicolumn{2}{c}{High $C_w$}	& \multicolumn{2}{c}{Low $C_w$} 	&  	\multirow{2}{*}{Formula} & \multirow{2}{*}{References} 				\\
\noalign {\smallskip}
									&  									& normal $R_u$ & increased $R_u$	& normal $R_u$ & increased $R_u$	&  									&	\\
\noalign {\smallskip} \hline \hline
\noalign {\smallskip}
$C_L$ [ml/cm H$_2$O]			& 2.3									& 2.7				& 2.1					& 2.3				&	2.1				&   $\frac{(V_A|_{EI}-V_A|_{EE})}{(P_{el}|_{EI}-P_{el}|_{EE})}$	& \cite{Gerhardt80,Mortola87,Pandit00}	\\
$C_w$ [ml/cm H$_2$O]			& 8.5									& 9.9				&	16.0				&	2.7			&	3.3				&   $\frac{(V_{cw}|_{EI}-V_{cw}|_{EE})}{(P_{cw}|_{EI}-P_{cw}|_{EE})}$	& \cite{Gerhardt80,Mortola87}	\\
$C_{rs}$ [ml/cm H$_2$O]		& 1.8									& 2.1				&	1.9				&	1.2			&	1.3				&   $(1/C_L+1/C_w)^{-1}$	& \cite{Mortola87,Neto95}	\\
$R_{rs}$ [cm H$_2$O s/L] 	& 40									& 34 to 41		&	32 to 223		&	33 to 36		&	32 to 223		&   $R_u+R_c+R_s$ 			& \cite{Goetz01}	\\
$\dot{V}_E$ [ml/min]			& 360									& 359.2			&	358.2				& 360.0			& 360.0				&   \textemdash				& \cite{Donn98}	\\
$V_T$ [ml]						& 6									& 5.99			& 5.97				& 6.00 			& 6.00				&  \textemdash					& \cite{Pandit00,Habib03,Schmalisch05} \\
$\dot{V}_A$ [ml/s]			& $\pm 20$							& -19.4 to 20.8& -16.4 to 28.9 	& -20.3 to 20.4& -17.1 to 26.4	&   \textemdash				& \cite{Pandit00,Habib03}	\\
$P_{A}$ [cm H$_2$O]			& $\pm 1-2$							&-0.75 to 0.84 & -0.96 to 3.63 	& -0.69 to 0.69& -0.89 to 3.80	&   \textemdash				& \cite{Donn98}	\\
$P_{el}$ [cm H$_2$O]			& 1 to 6								& 0.9 to 3.6	& 2.7 to 5.9	 	& 1.8 to 4.8	& 2.8 to 6.1		&   \textemdash				& \cite{Pandit00}	\\
$P_{pl}$ [cm H$_2$O]			& -3 to -6							&-0.6 to -3.8	& 0 to -6.2 		& -1.5 to -4.9 & 0 to 6.4			&   \textemdash				& \cite{Donn98}	\\
\bottomrule
\end{tabular}}
	{\smallskip}
	\begin{flushleft} EI: end-inspiratory. EE: end-expiratory. See Table~\ref{tab:glossary} (Glossary) for variable definitions.
\end{flushleft}
\label{tab:outputs}
\end{adjustwidth}
\end{table}

\subsection*{Computational procedures}

All simulations proceeded with an initial respiration rate of 60 breaths/min ($f=1$), initial minute ventilation $\dot{V}_E=360$, and initial tidal volume $V_T=6$ 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 $\dot{V}_E$ regardless of chest or lung compliance; and two, this allowed for comparison of simulation results originating from similar starting points. Distinct values for $A_{mus}$ were prescribed for each simulation to achieve the initial $\dot{V}_E=360$, see Table~\ref{tab:varparams}.

 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~\cite{Smith76} 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 $R_u$. CPAP is simulated with an increase of $P_{ao}$ from 0 to 5 triggered at $F_{rec,EI}=0.9,0.95,0.97$ characterizing volume losses of 10\%, 5\%, and 3\%. Simulations also include assumptions of either no permanently closed alveoli, such that $\gamma=1$ for all time, or10\% permanently closed alveoli per breath, such that $\gamma_{next}=\gamma_{current}\dot(1-0.1(F_{closed}))$.  Each simulation was performed three times: constant $f$; variable $f$ by breath according to $\dot{V}_E=V_{T,ave}\cdot f$ where $V_{T,ave}$ is a moving average of the previous 60 tidal volumes ($\approx 1$ minute of breathing); variable $f$ including a single 20 second apneic event.

The system of differential equations~\eqref{eq:states}, together with the constitutive relations~(\ref{eq:Ru}-\ref{eq:Pmus}), were solved using MATLAB R2016b (MathWorks, Natick, MA) with the differential equations solver \verb|ode15s|. Initial conditions were set at physiological values as given in Table~\ref{tab:ICs}. 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~\ref{tab:varparams} and~\ref{tab:SSparams}. 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~\nameref{sec:appendix}.

\begin{table}[h]
\begin{adjustwidth}{0in}{1in}
\caption{Initial conditions}
{\small
\begin{tabular}{lcccc}
\toprule
Initial Condition 			& High $C_w$ 	& Low $C_w$ & Formula 			 				& References		\\ \hline \hline
\noalign {\smallskip}
$\dot{V}(0)$						& 0 					& 0 				&	\textemdash						& \textemdash		\\
$V_c(0)$								& 0.0001			& 0.0001 		&	estimated from adult 	&~\cite{Liu98}	\\
$P_{el}(0)$							& 0.954				& 2.015   	& $P_{el}|_{FRC}$				&~\cite{Smith76}\\
$P_{ve}(0)$							& 0 					& 0 				& \textemdash						& \textemdash		\\
\bottomrule
\end{tabular}}
\label{tab:ICs}
\end{adjustwidth}
\end{table}

\section*{Results}

\begin{figure}[!h]
\begin{adjustwidth}{-1.53in}{0in}
\centering
\includegraphics[scale=0.7]{Figure2_B.eps}
\caption{{\small {\bf Lung, chest wall, and total respiratory system compliance curves for high $C_w$ (left) and low $C_w$ (right).} Curves are described by Eq~(\ref{eq:Vcw}) and (\ref{eq:VA}) and parameterized using the procedures described in~\nameref{sub:physiol}. Tidal breathing loops with normal $R_u$ (grey) and increased $R_u$ (black) are superimposed for each condition over the lung compliance curve and larger in each inset to display hysteresis.}}
\label{fig:PVcurves}
\end{adjustwidth}
\end{figure}

\begin{table}[!ht]
\begin{adjustwidth}{-0.9in}{0in}
\centering
		\caption{Simulations and time to failure (TTF, in hours), defined as 90\% volume loss. }
	{\small
		\begin{tabular}{ccccccc}
\toprule
\multirow{2}{*}{Intervention}& \multirow{2}{*}{$C_w$ condition}	& \multirow{2}{*}{Variable $\gamma$} 	& \multirow{2}{*}{Simulation} & 	\multicolumn{3}{c}{TTF, hours} 						\\
\noalign {\smallskip}
									&  										 & 													& 										& constant $f$	& variable $f$ 	& variable $f$ + AE 	\\
\noalign {\smallskip} \hline \hline
\noalign {\smallskip}
    \multirow{4}{*}{None} 					& {\multirow{2}{*}{Low}} 		& No 										& 1 					& 2.49 				&	2.53 & 2.51\\
		\cmidrule{3-7}          				&	       						& Yes										& 2 					&	2.25 				&	2.28 & 2.26 \\
		\cmidrule{2-7}         					& {\multirow{2}{*}{High}} 	& No 			 							& 3 					&	0.30 				&	0.32 & 0.30 \\
		\cmidrule{3-7}          				&       							& Yes										& 4 					&	0.27 				& 0.29 & 0.27 \\
		\midrule
    \multirow{4}{*}{Increased $R_u$}& {\multirow{2}{*}{Low}} 		& No  										& 5 					&	24.7 				&	24.7 & 24.5\\
		\cmidrule{3-7}          				&	       						& Yes										& 6 					&	22.2 				&	22.2 & 21.9 \\
		\cmidrule{2-7}         					& {\multirow{2}{*}{High}} 	& No 			 							& 7 					&	18.5 				&	18.4 & 17.3\\
		\cmidrule{3-7}          				&       							& Yes										& 8 					&	16.6 				& 16.5 & 15.0 \\
    \midrule
     \multirow{4}{*}{CPAP, 10\% loss}	& {\multirow{2}{*}{Low}}	& No  										& 9 					&	2.56 				&	2.61 & 2.60 \\
		\cmidrule{3-7}          				&	       						& Yes										& 10 					&	2.32 				&	2.36 & 2.34 \\
		\cmidrule{2-7}         					& {\multirow{2}{*}{Yes}} 	& No 			 							& 11 					&	0.83 				&	0.79 & 0.76 \\
		\cmidrule{3-7}          				&       							& Yes										& 12 					&	0.83 				& 0.80 & 0.77 \\
		\midrule
		CPAP, 5\% loss									& {\multirow{2}{*}{High}} 	& {\multirow{2}{*}{No}} 		& 13 					&	2.94 				& 2.50 & 2.46 \\
		\cmidrule{1-1} \cmidrule{4-7}
		CPAP, 3\% loss									&										&									&	14				  & 8.57  				& 6.89 & 6.83 \\
		\bottomrule
  \end{tabular}}
	{\smallskip}
\begin{flushleft} Increased $R_u$: A 10-fold increase in $R_u$ was applied during expiration. CPAP: Simulated administration of $P_{ao}=5$ occurred when recruited fraction was down 10\%, then again at 5\% and 3\% with constant $\gamma$. AE: A single 20 second apneic event occurred at the 2 minute mark of the simulation.
\end{flushleft}
\label{tab:simulations}
\end{adjustwidth}
\end{table}

Parameterized static compliance curves for $V_{cw}(P_{cw})$ and $V_A(P_{el})$ are shown in Fig~\ref{fig:PVcurves} for high $C_w$ (left) and low $C_w$ (right). The hysteretic tidal breathing loops are superimposed on the curve $V_A(P_{el})$ for normal $R_u$ in black and increased $R_u$ in grey. Hysteresis is caused in the model by the viscoelastic parameters $C_{ve}$ and $R_{ve}$, which were tuned to maintain appropriately valued lung volume outputs.

Fig~\ref{fig:SSstates} shows the impact of high vs low $C_w$ and normal vs. high $R_u$ on the five states $P_A, P_{l,dyn}, P_{pl}, V_A$, and $\dot{V}$. Increased $R_u$ increases $P_A$ almost threefold, but $C_w$ has very little impact. However, decreased $C_w$ increases $P_{l,dyn}$ significantly, effectively raising it higher on the lung PV curve. Increasing $R_u$ even higher increases $P_{l,dyn}$ but there is no difference with respect to $C_w$. The opposite appears to occur with $P_{pl}$ dynamics, in that decreasing $C_w$ makes $P_{pl}$ more negative (``increasing'' the magnitude of the pressure) and increasing $R_u$ strengthens that effect. Low $C_w$ and subsequently high $R_u$ increase $V_A$, mimicking the effect for $P_{l,dyn}$. High $R_u$ shifts $\dot{V}$ by ~5 ml/s, with airflow more restricted during expiration. Tabulated magnitudes of the steady states are gives in Table~\ref{tab:outputs}. These results compare favorably to the record reported in Abbasi et al~\cite{Abbasi90} in which esophageal (pleural) pressure, airflow, and tidal volume were approximately -2 to -6 cm H$_2$O, -30 to 30 ml/s, and 8 ml, respectively.

\begin{figure}[!h]
\begin{adjustwidth}{-2in}{-0in}
\centering
 \includegraphics[scale=0.7]{Figure3_B.eps}
\caption{{\small {\bf 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 $C_w$ conditions, with normal vs. high $R_u$.}}
\label{fig:SSstates}
\end{adjustwidth}
\end{figure}

 Table~\ref{tab:simulations} 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 $C_w$ compared to the same with low $C_w$. 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 $C_w$ and increased $R_u$ (S8).

The breath-to-breath change in EELV and $V_T$ under high and low $C_w$ conditions with no interventions are given in Fig~\ref{fig:lungvolumes} (Simulations 1 and 3). The high $C_w$ simulation reaches accelerated loss of volume and eventually failure at ~0.3 hours, much more quickly than the low $C_w$ 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.

\begin{figure}[!h]
\centering
\centerline{\includegraphics[scale=0.7]{Figure4_B.eps}}
\caption{{\small {\bf Breath-to-breath volumes.} End-expiratory lung volume (left y-axis) and tidal volume (right y-axis) under high and low $C_w$ conditions, no interventions.}}
\label{fig:lungvolumes}
\end{figure}

Fig~\ref{fig:Cdyn} shows changes in dynamic lung compliance and tidal volume with high and low $C_w$ without changes in $\gamma$, then adding CPAP to the high $C_w$ condition at three different levels (c.f. Table~\ref{tab:simulations}, simulations 1,3,11,13-14). CPAP was simulated by an increase in mouth pressure $P_{ao}$ to 5 cm H$_2$O when $F_{rec,max}<0.9$, which happened when the lung volumes were already decreasing quickly towards failure. However, CPAP triggering at $F_{rec,max}<0.95$ and 0.97 gained ~3 and ~9 hours of time, respectively. Note that regardless of timing, the  administration of CPAP is correlated with reduced tidal volume (see also ~\cite{Waugh99}. Increasing $P_{ao}$ 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 $C_w$ on dynamics induces the same lung volume loss without other mitigating actions.

\begin{figure}[!h]
\centering
\centerline{\includegraphics[scale=0.65]{Figure5.eps}}
\caption{{\small {\bf Breath-to-breath dynamic lung compliance and tidal volume}. Depicted are high and low $C_w$ conditions, with simulated CPAP triggered in the high $C_w$ condition when recruited fraction dropped 10\%, 5\%, and 3\%.}}
\label{fig:Cdyn}
\end{figure}

\section*{Discussion}

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~\cite{Abbasi90} and a later one by Pandit et al~\cite{Pandit00} gave 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 ``proof of concept'' 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~\nameref{sec:compliances}, recruitment/decrecruitment may have a time component~\cite{Bates02,Albert09}, 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~\cite{Crotti01,Markhorst04}. 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~\nameref{sec:loss}, and the hysteresis found in the tidal breathing loop is accounted for by the viscoelastic component of the system of differential equations. It is clear from Table~\ref{tab:simulations} 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 Voigt-Maxwell models within the "electrical analog" model or other non-electrical analog representations (e.g. those described in~\cite{Bates02}). While such a modification may affect the overall trends in observed states such as EELV, the differential impact between high and low $C_w$ 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 $\gamma$, the function $F_{rec}$ 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~\ref{tab:simulations} 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 $C_w$ 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~\cite{Williams66}. We safely assume in our model that derecruitment is a continuous process that will eventually induce loss of lung volume if left uncompensated~\cite{Mead59,Ferris60}. 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 ``sighing'', which may help prevent atelectasis~\cite{Bartlett71,Duggan05,Qureshi08} by re-opening air spaces that collapse naturally under tidal breathing~\cite{Bendixen63} 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~\cite{Davis94}. 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.~\cite{Mecklen98}). Components that differentiate between abdominal and rib cage movements (see e.g.~\cite{Primiano82}) may model the paradoxical chest movement.

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~\cite{Jallon09,Diekman17} that could potentially be incorporated with ours. A chemoreflex model, see for example~\cite{Chiari95,Diekman17}, 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.

\section*{Conclusion}

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 population.

\section*{Acknowledgments}
This research was supported in part by the Atlantic Pediatric Device Consortium via FDA grant 5P50FD004193-07 (H. Rozycki, L. Ellwein, M. Brandes) and the VCU College of Humanities and Sciences Faculty Research Council (L. Ellwein, L. Linkous).

\section*{Appendix}\label{sec:appendix}

We analyze the inherent stability of the model 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. To obtain steady-states, $P_{mus}$ is set at a constant called $P_{mus,C}$ and the system of ODE's is set to equal 0:
\begin{eqnarray*}
0&=&\frac{1}{I_u}\left(P_{ao}-P_u-R_u\dot{V}\right)\\
0&=&\dot{V}-\dot{V}_A\\
0&=&\frac{\dot{V}_A}{C_A}\\
0&=&\frac{\dot{V}_A-(P_{ve}/R_{ve})}{C_{ve}}
\end{eqnarray*}
It follows that $\dot{V}=\dot{V}_C=\dot{V}_A=0$ and $P_u=P_{ve}=P_c=P_A=0$. Given the relations $P_{el}=f(V_A)$, $P_{tm}=f(V_c)$, and $P_{cw}=f(V_{cw})$,  we have 3 equations with 6 unknowns $(V_A,V_c,V_{cw},P_{el},P_{tm},P_{cw}$). Three additional equations come from incorporating loop summations such that $P_{el}(V_A)=P_{tm}(V_c)$, $P_{tm}=-(P_{cw}+P_{mus,C})$, and $V=V_{cw}=V_A+V_c$. Noting that $P_{el}$ is defined implicitly by Eq.~(10) and using the compliance curve functions
\begin{eqnarray*}
P_{cw}&=&c_w+d_w\ln \left(e^{(V_A+V_c-a_w)/b_w}-1\right)\\
P_{tm}&=&c_c-d_c \ln \left(\frac{V_{c,max}}{V_c}-1\right)
\end{eqnarray*}
 we obtain two equations
\begin{eqnarray*}
0&=&P_{cw}(V_A(P_{el}),V_c)+P_{mus,C}+P_{tm}(V_c)\\
0&=&P_{cw}(V_A(P_{el}),V_c)+P_{mus,C}+P_{el}
\end{eqnarray*}
that are solved numerically for two unknowns $P_{el},V_c$ for each modified parameter set using an iterative algorithm. $P_{el}$ From these steady-state values we can calculate the remaining state variables.

In order to linearize the system and determine asympototic behavior we rewrite the system in terms of the state variables $\dot{V},V_c,P_{el},P_{ve}$ using previously described relationships:
\begin{eqnarray*}
\ddot{V}&=&\frac{1}{I_u}\left(P_{ao}-R_c(V_c)\dot{V}-P_{tm}(V_c)-P_{cw}(V_A(P_{el})+V_c)-P_{mus}-R_u(\dot{V})\dot{V}\right)\\
\dot{V}_c&=&\dot{V}-\frac{P_{tm}(V_c)-P_{el}-P_{ve}}{R_s(V_A(P_{el}))}\\
\dot{P}_{el}&=&\frac{1}{C_A(P_{el})}\left[\frac{P_{tm}(V_c)-P_{el}-P_{ve}}{R_s(V_A(P_{el}))}\right]\\
\dot{P}_{ve}&=&\frac{1}{C_{ve}}\left[\frac{P_{tm}(V_c)-P_{el}-P_{ve}}{R_s(V_A(P_{el}))}-\frac{P_{ve}}{R_{ve}}\right]
\end{eqnarray*}
where the quantity $\frac{P_{tm}(V_c)-P_{el}-P_{ve}}{R_s(V_A(P_{el}))}=\dot{V}_A.$ The Jacobian
\[ J=
\begin{bmatrix}
\ffrac{\partial\ddot{V}}{\partial\dot{V}} & \ffrac{\partial\ddot{V}}{\partial V_c} & \ffrac{\partial\ddot{V}}{\partial P_{el}} & \ffrac{\partial\ddot{V}}{\partial P_{ve}} \\
\ffrac{\partial\dot{V}_c}{\partial\dot{V}} & \ffrac{\partial\dot{V}_c}{\partial V_c} & \ffrac{\partial\dot{V}_c}{\partial P_{el}} & \ffrac{\partial\dot{V}_c}{\partial P_{ve}} \\
\ffrac{\partial\dot{P}_{el}}{\partial\dot{V}} & \ffrac{\partial\dot{P}_{el}}{\partial V_c} & \ffrac{\partial\dot{P}_{el}}{\partial P_{el}} & \ffrac{\partial\dot{P}_{el}}{\partial P_{ve}} \\
\ffrac{\partial\dot{P}_{ve}}{\partial\dot{V}} & \ffrac{\partial\dot{P}_{ve}}{\partial V_c} & \ffrac{\partial\dot{P}_{ve}}{\partial P_{el}} & \ffrac{\partial\dot{P}_{ve}}{\partial P_{ve}} \\
\end{bmatrix}
\]
was found using symbolic computation then used numerically to calculate eigenvalues. All parameter variations gave stable solutions except $c_c, d_c$, and $V_{c,max}$. However, varying these parameters by 2 and 10 is not actually physiological and the instability comes from these modifications making the parameter values inconsistent with the rest of the system. The parameters $c_c$ and $d_c$ must be equal with our calculations, and since $c_c$ is a normal operating pressure, it must stay within a narrow range. Likewise, $V_{c,max}$ is equivalent to dead space so not only must it also be within a narrow range but if it varies too much with respect to other parameters and states then the equations will be inconsistent. It is therefore reasonable to state that the steady-state of this model system is inherently stable for the physiological parameter ranges used here.

\nolinenumbers

redpajama_set_name: RedPajamaArXiv


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an annual member of the Cucumis genus in the family Cucurbitaceae.
The sprouting seed produces a toxic substance in its embryo.
Fruit - raw. Insipid according to one report, whilst another says that it is rather bitter. Said to have a banana-lime flavour and often sold in speciality stores in Europe and America, the fruit is not considered to be very desirable in its native area and it is only eaten in times of scarcity.
Seed - raw. Rich in oil with a nutty flavour but very fiddly to use because the seed is small and covered with a fibrous coat[K].
Used as a rootstock for C. sativus, conferring disease resistance. Since this species is less cold tolerant than C, sativus, it is probably not suitable as a rootstock in cool temperate areas.
There are no material uses listed for Cucumis metuliferus.
The seeds are vermifuge. They are ground into a fine flour, then made into an emulsion with water and eaten. It is then necessary to take a purge in order to expel the tapeworms or other parasites from the body.
Seed - sow early to mid spring in a greenhouse in a rich soil. Germination should take place within 2 weeks. Sow 2 or 3 seeds per pot and thin out to the best plant. Grow them on fast and plant out after the last expected frosts, giving them cloche or frame protection for at least their first few weeks if you are trying them outdoors.
Practical Plants is currently lacking information on propagation instructions of Cucumis metuliferus. Help us fill in the blanks! Edit this page to add your knowledge.
Requires a rich, well-drained moisture retentive soil and a very warm, sunny sheltered position. A frost-tender plant, the horned cucumber is frequently cultivated for its fruit in tropical and sub-tropical climates, there are some named varieties. It requires a hotter summer than is normally experienced in Britain in order to produce a worthwhile crop and is generally best grown in a greenhouse in this country.
There are no interactions listed for Cucumis metuliferus. Do you know of an interaction that should be listed here? edit this page to add it.
There are no polycultures listed which include Cucumis metuliferus.
This page was last modified on 4 May 2013, at 12:56.


text:
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redpajama_set_name: RedPajamaCommonCrawl


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text:
"Manchester by the Sea" explores the full depths of tragedy
PHOTO/FLICKR CREATIVE COMMONS
David Rosario, Staff Writer
Filed under Culture, Movies & Television
When someone says that they never want to see a movie again, this is usually indicative of the film's quality and suggests that there aren't enough redeeming qualities to justify investing time into it for a second viewing. "Manchester by the Sea" is an exception to this rule. Casey Affleck plays Lee Chandler, a lonely handyman who's coping with the death of his brother and tasked with looking after his 16-year-old nephew. This tragedy forces Lee to head back to his hometown and face not only those he left behind and certain inner demons that he's been trying to rid himself of for years.
Writer and director, Kenneth Lonergan will likely get Oscar consideration in the spring for his work here. There's a rawness of human emotion on display that films seldom tap into and it allows "Manchester by the Sea" to feel more like real life than a work of fiction. Every character reacts to situations in a manner that's believable and even during moments when they don't necessarily seem likeable, the "why" behind what they're doing is always crystal clear.
Casey Affleck's performance is nothing short of career defining. The heartbreak in his eyes is palpable, and you genuinely feel the pain that he's experiencing throughout the entirety of this movie. Michelle Williams also deserved recognition because despite not being in the movie all that much, her scenes with Affleck are electric and are some of the most devastating in the film
While it's difficult to imagine wanting to go through this harrowing journey with these characters again, "Manchester by the Sea" is worth seeing at least once for the power of its performances and the realness of its storytelling.
Tags: movie review
Netflix's "Murder Mystery"
Tribeca Film Festival Presents "A Day in the Life of America"
St. John's Grad "Veldor" Makes The Billboard Music Charts
Tribeca Film Festival Presents "Crown Vic"
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text:
Monday Must Read! The Long Weeping by Jessie Van Eerden
Jessie Van Eerden is the author of Glorybound (WordFarm, 2012), My Radio Radio (Vandalia Press 2016), and most recently, her collection of portrait essays,The Long Weeping, just released from Orison Books.
A West Virginia native, Jessie holds a BA in English from West Virginia University and an MFA in nonfiction writing from the University of Iowa. Her work has appeared in The Oxford American, River Teeth, Image, Bellingham Review, Willow Springs, Rock & Sling, Appalachian Heritage, Ruminate, and other publications. Her prose has been selected for inclusion in Eyes Glowing at the Edge of the Woods: Fiction and Poetry from West Virginia(Vandalia Press); Walk Till the Dogs Get Mean (Ohio University Press); Red Holler (Sarabande); Dreams and Inward Journeys: A Rhetoric and Reader for Writers, Seventh Edition (Longman); Jesus Girls: True Tales of Growing Up Female and Evangelical (Cascade Books); and Best American Spiritual Writing (Houghton Mifflin).
Jessie has taught for over fifteen years in college classrooms and in adult literacy programs. She lives in West Virginia where she directs the low-residency MFA writing program of West Virginia Wesleyan College.
Follow Jessie on Twitter: @jessievaneerden. To see what Jessie has been reading, visit her Goodreads page. To learn more about WV Wesleyan's MFA program (#wvwcmfa), visit the program website.
Visit Jessie's website: http://www.jessievaneerden.com/
Buy Jessie's beautiful books!
The Long Weeping
My Radio Radio
Glorybound
Praise for The Long Weeping
"In these startlingly honest and imaginative essays, van Eerden enlarges the world around her, giving flesh to what is all too often flattened by the outside eye, anointing places and people and the throbbing spaces between them as she searches out and sings litanies to what she calls the "larger body I belonged to and could not leave." Lush and razor-sharp, The Long Weeping shimmers with intelligence and grace. The truest essays I've read in a long time."
—Sonja Livingston, author of Ghostbread, Queen of the Fall, and Ladies Night at the Dreamland
"Van Eerden is one of the best essayists working today if judged by her craft and intellect alone, but her gifts go beyond those: she is also one of the most honest. The Long Weeping turns a visionary eye and a laser mind on subjects often simplified or even scorned by contemporary culture: white poverty; mysticism; love of family; the wisdom of modest people. Van Eerden is brave enough to say the hard things. She's strong enough to love the hard places."
—Ann Pancake, author of Given Ground, Strange as This Weather Has Been, and Me and My Daddy Listen to Bob Marley
More from Jessie Online
http://www.jessievaneerden.com/short-fiction-creative-non-fiction/
https://imagejournal.org/artist/jessie-van-eerden/
http://appalachianheritage.net/2015/09/09/interview-jessie-van-eerden/
http://cheatriverreview.com/jessie-van-eerden/
https://www.ruminatemagazine.com/blogs/ruminate-blog/works-with-soul-interview-with-jessie-van-eerden
http://memorious.org/?author=217
https://rockandsling.com/2011/03/30/an-interview-with-jessie-van-eerden/
https://web.tusculum.edu/tusculumreview/2014/05/14/jessie-van-eerden-2/
Hear Jessie Read
Happy Reading, y'all!
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Daily Prompt Catch-Up! 10 New Prompts!


text:
using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class GameFrame : MonoBehaviour {
	static GameFrame instance;
	GameFrame(){}
	void Awake(){
		if(instance == null){
			instance = this;
			DontDestroyOnLoad(gameObject);
		}else{
			Destroy(gameObject);
		}
	}

	const float GAME_DISP_WIDTH_L = -308.0f;
	public float GameDispWidthL{
		get{return GAME_DISP_WIDTH_L;}
	}
	const float GAME_DISP_WIDTH_R = 200.0f;
	public float GameDispWidthR{
		get{return GAME_DISP_WIDTH_R;}
	}
	const float GAME_DISP_HEIGHT_U = 220.0f;
	public float GameDispHeightU{
		get{return GAME_DISP_HEIGHT_U;}
	}
	const float GAME_DISP_HEIGHT_D = -200.0f;
	public float GameDispHeightD{
		get{return GAME_DISP_HEIGHT_D;}
	}

	public float width, height;
	public float Width{
		get{return width;}
	}
	public float Height{
		get{return height;}
	}
	// Use this for initialization
	void Start () {
		width = this.GetComponent<SpriteRenderer>().bounds.size.x;
		height = this.GetComponent<SpriteRenderer>().bounds.size.y;
	}

	// Update is called once per frame
}

redpajama_set_name: RedPajamaGithub


text:
https://www.bracenet.net
Unique wristbands that reduce 'ghostfishing'
Bracenet collects discarded fishing nets, sends these to Norway to have them turned into fabric and then produces unique wristbands in workshops that provide employment opportunities to disabled persons.
https://www.klattermusen.com/en/fabrics/6050_recycled-polyamide/
Klättermusen makes outdoor clothing and backpacks from recycled polyamide and polyester
Klättermusen AB
Klättermusen is a Swedish outdoor clothing company producing waterproof jackets, pants and backpacks made at least partly from recycled polyamide. The polyamide is created from post-industrial waste including packaging materials from factories, old carpets as well as discarded industrial fishing nets.
https://www.metsafibre.com/en/about-us/Bioproduct-mill/Pages/default.aspx
Metsä Group's bioproduct mill in Äänekoski, Finland, leads the way towards a carbon neutral society
Metsä Group
Metsä Group built the first next-generation bioproduct mill in Äänekoski, Finland – the largest investment of the European forest industry with the value of EUR 1.2 billion. The new mill, which began operations in the third quarter of 2017, leads the industry to a new era of resource efficiency through operating completely with no fossil fuels or fossil CO2 emissions. This makes it the most energy-efficient pulp mill in the world, utilising 100% of the production side streams for materials or renewable energy in industrial ecosystem built with partners.
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London Waste and Recycling is delivering on the circular economy with stakeholder buy-in
London Waste and Recycling Board
London is among one the world's most cosmopolitan and oldest cities, with a history spanning nearly two millennia, and one of the most cosmopolitan. As Britain's largest city and country's economic, transportation and cultural capital, over 8 million people live in London. The city is growing fast and its population is predicted to reach over 11 million by 2050. A more flexible and sustainable approach to products, housing, office space and critical infrastructure is crucial to London's ability to adapt and grow.
https://www.sfridoo.com/
Sfridoo enables manufacturers to purchase and sell industrial scraps
Sfridoo S.r.l.
Sfridoo.com is an Italian B2B publishing platform for purchasing and selling scrap materials. Using sharing economy princples to turn the circular economy into a reality, Sfridoo has already enabled more than 100 businesses to recycle and reuse industrial scraps.
Oslo takes an integrated approach to treat waste into circular bio-resources
Other (Norway )
Oslo has been developing a waste management system based on circular principles to ensure separate waste collection is maximised and transform waste into secondary raw materials. To do so it has actively engaged with citizens, farmers as well as with its city's public transportation company.
Turin spurs social inclusion with projects reducing food waste and increasing recycling
City of Turin
Facing dramatic deindustrialisation and an uncertain future, the city of Turin implemented processes that paired physical redevelopment with strategic planning to promote citywide revitalisation and economic restructuring in the 1990s. While the transformation has been profound, current challenges call for more circular strategies and an inclusive approach.
Dusseldorf is moving towards sustainable procurement with recycled office paper
City of Dusseldorf
As a densely populated and economically powerful urban area, the city of Dusseldorf recognised the challenge of climate change early on and initiated a process of low carbon and zero waste strategy development.
Amsterdam is going Circular smartly with 'learning by doing'
City of Amsterdam
In 2015 Amsterdam commissioned an in-depth study on the potential of a circular economy. The project was the first large-scale research study in the world that uses the 'city circle scan' methodology. The scan identifies the areas in which the most significant, tangible progress in realising a circular economy can be achieved. This potential impact is significant and can result in more jobs, bring added value to the city's economy, and reduce carbon dioxide emissions and material use.
Birmingham is working to reduce emissions and waste through Industrial Symbiosis
Birmingham is Britain's youngest and fastest growing city, boasting the highest quality of life of any English city outside London. The city also has the strongest economy outside the capital and is one of the first cities to adopt a proactive industrial symbiosis approach to develop a medium and long-term strategy for sustainable economic development. Often described as 'the circular economy in action', the projects born from the industrial symbiosis approach are part of Birmingham's circular economy strategy.
https://www.letsbehonest.eu
HNST - Radically different denim (recycled denim fibres)
HNST
In September 2017, HNST (pronounce as "honest") had a collection campaign where people could drop off their old and unworn denims at more than 80 collecting points in Flanders (Belgium).
http://www.pet2pet.at/de
PET to PET Reycling:The bottle-to-bottle loop
PET to PET Reycling Österreich GmbH
In operation since 2007, PET to PET Recycling Österreich GmbH focuses on the production of food-grade recycled PET, which is used for "bottle-to-bottle recycling", a globally unique and resource-efficient recycling loop in which old PET bottles are turned into new ones.
https://www.vandesant.com/about-1
Circular comfortable furniture, made from recovered (ocean) plastics
Van de Sant Innovations BV
The company Van de Sant Innovations BV designs and manufactures comfortable sustainable furniture, made from recovered (ocean) plastics.
https://www.egger.com/shop/en_IE/timberpak-news/hermanmiller-partnership
Herman Miller are Closing the Loop on Wood Waste
Offcuts and wood waste generated during production at the Herman Miller production facility in Wiltshire is collected and delivered to Timberpak, utilising empty vehicle space and back loading to minimise their environmental footprint.
http://www.rebus.eu.com/wp-content/uploads/2017/10/REBus-case-study-Naturalmat.....
Mattresses re-designed for re-use and recycling
Portugal, Spain, United Kingdom
Production of organic, natural, luxury mattresses, toppers and bedding, using 100% biodegradable material wherever possible.
http://www.adf-global.org
Alternatives to single-use plastic bags in Morocco
Regional Activity Centre for Sustainable Consumption and Production (SCP/RAC)
Regional authority
Other (Morocco)
Cooperation and development
SCP/RAC promotes an action in Morocco to support the country in transitioning from single-use plastic bags to other green means.
https://www.switchmed.eu/en/corners/start-up
SwitchMed Green Entrepreneurship Programme
Regional Activity Center for Sustainable Consumption and Production
Other (Algeria, Egypt, Israel, Jordan, Lebanon, Morocco, Palestine, Tunisia)
The EU-funded SwitchMed Programme supports and connects stakeholders to scale-up social and eco-innovations in the Mediterranean. It promotes inclusive growth, job creation and sustainable development by supporting policy makers, eco-innovative small and medium sized enterprises, industries, start-ups and entrepreneurs in the Southern Mediterranean countries.
http://www.dgnb-system.de/en/system/version2018/
DNGB certification provides bonuses for applying circular economy principles in sustainable construction
Deutsche Gesellschaft für Nachhaltiges Bauen – DGNB e.V. (German Sustainable Building Council)
DGNB: The German Sustainable Building Council (DGNB) is a central knowledge platform for sustainable building and provides the world's most advanced sustainable building certification system which makes Circular Economy aspects measurable and assessable.
http://www.bluecity.nl
A populated circular city set up on the site of a 12.000m2 abandoned swimming pool
BlueCity
An incubator for circular entrepreneurs in and around Rotterdam, BlueCity creates a much needed tangible and inspiring example for the circular economy, closing the loops and building the city and the economy of the future. Set in a deserted swimming pool, this dynamic community of entrepreneurs, researchers, neighbours, government and education representatives serves as a breeding ground for 25+ innovative, circular companies linking their waste-streams.
http://eco.nomia.pt/pt/exemplos/762-edp
From steel production waste gases to electricity
EDP, Energias de Portugal S.A.
The integration of the steel sector into the electricity generation cycle is an example of collaboration between companies from different sectors that significantly minimizes the environmental impact of their facilities. This integration allows the steel production gases produced by the industrial processes of steel plants to be transferred to an electric company for use as additional fuel in the boilers of its plants together with the main fuel, in a hybrid generation and / or cogeneration process.
https://www.attero.nl/en/our-waste-management/your-organic-waste-is-converted-in...
Biobased substrate compost: a high quality circular ingredient for potting soil
Attero B.V.
Attero introduces a new high quality product that is used as ingredient for substrates and potting soils: biobased substrate compost. The process for achieving this quality is unique in the world of compost production.
https://www.switchmed.eu/en/corners/service-providers/actions/med-test-ii-countr...
SwitchMed - MED TEST II
Other (?)
Other (Lebanon)
Skaff Dairy Farm is a small-sized enterprise, which produces around 840 t of dairy products annually for the local market. Skaff Dairy Farm joined the MED TEST II project in order to improve quality and reduce defects, to drive out waste and continuously improve cost efficiency. The main problems the company faced was energy and finished products losses. The company was already ISO 22000:2005 certified at the project start.
https://drive.google.com/open?id=0B2MfFgUyQmgOTmhiWDlYcFdGR1U
A circular furniture system
Arcadia Design by Progetto Arcadia
Arcadia Design envisions a design that changes as its owners does.
https://www.veolia.com/en/recycling-plastic-circular-economy
Giving plastic bottles a second life through "Bottle-to-Bottle" in Rostock
Every year in Rostock, Veolia converts 1 billion PET drinks bottles into so-called recyclate from which preforms/bottles can again be manufactured.


Global plastic production is estimated to be over 300Mt annually. Most conventional plastics are predominantly produced from fossil fuels and are highly resistant to biodegradation, and only a small share of about 20% of spent plastics is believed to be recycled, which is a cause for environmental concern. Biodegradable plastics would solve this concern as they are a sustainable alternative, yet these do not even cover 5% of the global plastic market. Microbial polyhydroxyalkanoates (PHAs) are a versatile group of polyesters produced by nature as prokaryotic storage materials. PHAs can be produced through sustainable bioprocess engineering and have displayed remarkable flexibility in their physical and chemical properties. PHAs are the subject of several scientific papers and numerous PHA patents have also been filed, generating significant interest in the plastic production industry. To develop overall sustainable and efficient production processes, all bioprocess steps need to be thoroughly understood and accounted for. These processes start with the selection of suitable inexpensive raw materials (microbes and enzymes), optimizing the process engineering and process regime, and conclude with the enhancement of product recovery in terms of time, energy, and material input. Microbial Biopolyester Production, Performance and Processing: Bioengineering, Characterization, and Sustainability is a compilation of eight chapters covering bacterial polyesters, green plastics and PHAs from various angles. The contents of this volume focus on sustainable practices focus on the sustainability of processes that involve the synthesis and recycling of these meterials. The volume is a useful resource for bioprocess engineers, microbiologists, biotechnologists and chemical engineers interested in the basics of biodegradable plastic production.
Process/strain optimization procedures are unavoidable in the improvement of large-scale bioproduction processes. These procedures are necessary on macroscopic (bioreactor) level and on biochemical/genetic level of producing strains. Inter alia, mathematical modeling is one of the applicable and useful methods in such procedures. Mathematical models applied for PHA biosynthesis (classified as both structured and unstructured) are denominated as formal kinetic, low-structured, dynamic, metabolicor high-structured, cybernetic, and hybrid type models, or neural networks. In the chapter at hand, each specific group of models is discussed in the light of its applicability and benefit for increased productivity, enhancing of the specific PHA biosynthesis rate, and better understanding of the intracellular metabolic regulation systems. Characteristics of production strains, particularities of mixed microbial cultures and features of industrial-scale plants cannot be described by a single type of mathematical model since it is not possible to address all the different requirements by a single model type. Therefore, it is more than necessary to fine-tune the modeling approach to each actual case in a sophisticated way. Formal-kinetic and "lowstructured" models that are relatively simple and display low computational demand, are applicable for simple cases and are beneficial for mathematical embodiment of "standard microbial cultivations and practices". Hybrid models are used by some authors to address certain deficiencies of diverse types of models. In this context, satisfying, compromised, perhaps most promising solutions can be reached if mechanistic, cybernetic, computational fluid dynamics (CFD) and neural models were combined. A hybrid modeling strategy like this generates a holistic representation of solutions for the total PHA biosynthesis process, including all advantages of the different modeling schemes. For example, application of growth media of complex composition usually entails a higher degree of model organization. For the future, really existent biotechnological systems are expected to be expressed by hybrid models of high organization.
Major attention was dedicated to the use of elementary flux modes (EFMs) and yield space analysis (YSA) to develop metabolic models of PHA biosynthesis. The implementation of these methods is reported for numerous case studies which involve modeling of metabolic networks. The chapter concludes with some case studies, where the implementation of EFMs and YSA performed as a powerful modeling tool. It includes the description of intracellular PHA generation and mobilization in the organism Cupriavidus necator, the limitation-based picture of the steady-state flux cone of the organism´s metabolic network, the detailed analysis of a multi-stage bioreactor cascade dedicated to continuous PHA production, and metabolic flux investigation in the metabolism of C. necator cultivated using glycerol.
Bioreactor cascade, Cybernetic models, Dynamic models, Elementary flux modes (EFM), Formal kinetic modelling, Hybrid models, Mathematical modelling, Metabolic models, Network modelling, Neural networks, Polyhydroxyalkanoates (PHA), Yield space analysis (YSA).
Department of Biochemical Engineering, Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia.


With spring and summer on its way, more and more people are making their ways outside. Everyone wants to get outside and enjoy the beautiful weather. More motorcycles and sports cars start coming outside to play and that is when law enforcement starts seeing more accidents come across their radars. Then you add in the warm enough weather for boats, and if you wouldn't believe it, there are tons of accidents on boats too.
When you are in an accident on a boat or a motorcycle, that is considered a personal injury. Most associate car accidents with personal injuries and forget that a boat and motorcycle are considered them as well. Many of my friends and family members go to Florida on vacation from the east coast, in fact my sister is there right now! Because of Florida's gorgeous coasts and waterways, there is a huge community for recreational activity on the water, which in turn comes with personal injuries. The category is called boating, but "boating" also includes kayaks, jet skis, yachts, fishing boats, and canoes – basically anything that floats.
When you are on vacation and an accident happens, there's a big chance that the bills and case will not get resolved before you have to leave. Tons of families go on vacation to Florida and if an accident happens, you want the best help who specializes in that states rules and regulations. Personal injury attorney Tampa specialize in personal injury accidents, so if you have encountered this, you might want to consider getting an attorney.
Spring also brings out fast cars, convertibles, old showcase cars and reckless driving. Fast cars get put away for the winter and as soon as nice weather hits, they bring out the cars. No matter where you are, you know when the weather gets nice because you hear all of the motorcycles and sports cars out on the roads and highways. This is when many accidents happen due to reckless and fast driving.
What major problems occur after an accident?
The first thing to do is make sure you and your family are okay. Then it is important to start researching ways to deal with the situation at hand. Many people believe they can handle it on their own, not realizing how much insurance companies do not help during this stressful time. When this happens, people take on more stress than they need trying to figure out a way to make their personal injury accident go away and more times than not, it not only doesn't go away, but drags on for years before seeing an end result. Avoiding a common situation like this is crucial, and the way to avoid this, especially if your accident occurred in Florida, is by hiring an auto accident attorney Tampa. During an already stressful time, it is great to alleviate some of that stress by letting an attorney do the job for you.


>> resultant triples for it?
> upper graph of the RHS).
> prefer to speak about "derivations".
appropriate language visible in the specification  if it is missing.
In reply to: Michael Schneider: "RE: ACTION-178: What is ISSUE-116 (Axiomatic Triples for OWL R) about?"