A damaged-informed lung model for ventilator waveforms
Deepak. K. Agrawal, Bradford J. Smith, Peter D. Sottile, David J. Albers
AA damaged-informed lung model for ventilatorwaveforms
Deepak. K. Agrawal, † Bradford J. Smith, † , ‡ Peter D. Sottile, ¶ and David J.Albers ∗ , § , k , ⊥ † Department of Bioengineering, University of Colorado Denver, Anschutz Medical Campus,Aurora, CO 80045, USA ‡ Department of Pediatrics, University of Colorado Denver, Anschutz Medical Campus,Aurora, CO 80045, USA ¶ Division of Pulmonary Sciences and Critical Care Medicine, Department ofMedicine,University of Colorado School of Medicine, Aurora, CO 80045, USA § Section of Informatics and Data Science, Department of Pediatrics, School of Medicine,University of Colorado Denver, Anschutz Medical Campus, Aurora, CO 80045, USA k Department of Bioengineering, University of Colorado Denver/Anschutz Medical Campus,Aurora, CO 80045, USA ⊥ Department of Biomedical Engineering, Columbia University, New York, NY 10032
E-mail: [email protected]
Abstract
The acute respiratory distress syndrome (ARDS) is characterized by the acute de-velopment of diffuse alveolar damage (DAD) resulting in increased vascular permeabilityand decreased alveolar gas exchange. Mechanical ventilation is a potentially lifesavingintervention to improve oxygen exchange but has the potential to cause ventilator-inducedlung injury (VILI). A general strategy to reduce VILI is to use low tidal volume and low-pressure ventilation, but optimal ventilator settings for an individual patient are difficultfor the bedside physician to determine and mortality from ARDS remains unacceptablyhigh. Motivated by the need to minimize VILI, scientists have developed models of vary-ing complexity to understand diseased pulmonary physiology. However, simple modelsoften fail to capture real-world injury while complex models tend to not be estimablewith clinical data, limiting the clinical utility of existing models. To address this gap, wepresent a physiologically anchored data-driven model to better model lung injury. Ourapproach relies on using clinically relevant features in the ventilator waveform data that a r X i v : . [ q - b i o . Q M ] O c t ontain information about pulmonary physiology, patients-ventilator interaction and ven-tilator settings. Our lung model can reproduce essential physiology and pathophysiologydynamics of differently damaged lungs for both controlled mouse model data and uncon-trolled human ICU data. The estimated parameters values that are correlated with aknown measure of lung physiology agree with the observed lung damage. In future en-deavors, this model could be used to phenotype ventilator waveforms and serve as a basisfor predicting the course of ARDS and improving patient care. Introduction
The acute respiratory distress syndrome (ARDS) is characterized by diffuse alveolar damage resultingin increased vascular permeability and decreased alveolar gas exchange.
Mechanical ventilationis an essential lifesaving therapy for ARDS that has the potential to worsen lung injury throughbarotrauma, volutrauma, and atelectrauma that are referred to collectively as ventilator induced lunginjury (VILI).
Identifying lung-protective ventilation to avoid VILI can be challenging because ofthe complex interplay between ventilator mechanics, patient-ventilator interactions, and the underlyingpulmonary physiology.
The current standard of care dictates a formulaic application of low tidalvolumes to reduce overdistension and positive end expiratory pressure to maintain patency. Thisapproach reduces VILI but does not prevent it in all cases.
One example is the ARDS Networkprotocol which can be used to guide ventilator settings to minimize VILI. While such protocols arevery helpful, but because they are not personalized, such protocols can always be improved. Thisis due partially to the heterogeneity of ARDS, both between patients and in different regions of thesame lung. In addition, management of patients with ARDS is further complicated by variable patientrespiratory effort that may lead to patient self-inflicted lung injury. Modern mechanical ventilators produce time-dependent pressure, volume, and flow waveforms thatcontain a wealth of information about respiratory mechanics, patient-ventilator interactions, and ven-tilator settings. These data can be used to trouble-shoot and optimize mechanical ventilation.
However, ventilator waveforms are typically analyzed heuristically by visual inspection and, therefore,the outcome of such an analysis is limited by individual expertise.
Therefore, our goal is to de-velop a model-inference system to quantify the characteristics of the pressure and volume waveformsof healthy and injured lungs. This type of analysis decomposes the complex characteristics of thepressure-volume waveforms into numerical values to allow tracking changes over time. One example ofthis approach that is currently used in clinical care is the driving pressure, which serves as a readout ofboth patient condition and ventilator settings. We seek to expand on that methodology to providea more comprehensive description of lung injury severity and ongoing VILI.Waveform-based analysis is a departure from traditional methods that utilize mathematical modelsto link the measured pressure and flow, such as the well-recognized single compartment model thatlumps the spatially heterogeneous lung mechanical properties into single values of resistance and com-pliance.
Due to this straightforward formulation, the single compartment model is computationallyefficient but may not be able to reproduce all of the features in measured data. On the other hand,complex multi-compartment models use many states and parameters that cannot be directly measured,such as recruitment pressure distributions, causing identifiability problems where there is no uniquesolution. As such, those model require more expansive data to estimate with any success, and requiresubstantially more computational resources. Even then, complex multi-compartment models may notproduce all the relevant features present in the pressure and volume data.
Our novel waveform-based approach offers the potential to overcome these limitations because allof the data necessary for high-fidelity analysis is contained in the pressure and volume waveforms. Webridge the gap between identifiability and fidelity by developing a systematic framework to quantifyphysiological and pathophysiological lung dynamics using mathematical models that have interpretableparameters. We anticipate that this approach will find applications in real-time clinical readouts ofventilation safety, long-term monitoring to detect changes in patient condition, and as a quantitative utcome measure for clinical trials. In addition, the relationship between components of the pressureand volume waveforms may be used to identify specific physiologic features, just as the quasi-staticcompliance is defined as the ratio of tidal volume and driving pressure.In the current study, we first identify clinically important features in typical pressure and volumewaveform data. We then separately define the pressure and volume waveforms as the sum of a set ofessential features. This approach allows independent modeling of the components of damage so thatclinical and physiologic knowledge can be used to constrain the model. The pressure and volume modelsare validated in a simulation study by demonstrating that the model has sufficient flexibility to producerelevant pressure and volume features. Model evaluation is conducted with both mouse model andhuman ICU ventilator data by comparing measurements and model predictions for pressure andvolume waveforms. We also relate changes in the model parameters to assessments of injury severityas well as qualitative features of the pressure and volume waveforms. Methods
Identifying relevant and realistic variables for the model
Our goal is to develop a lung model that can reproduce all the physiologically relevant features presentin the waveforms data such that the model could be used to understand lung pathophysiology in clinicalsettings. Therefore, it is critical to identify the appropriate complexity of the model that is necessaryto achieve the desired outcome.Mechanical ventilation is characterized using three state variables, volume, pressure and flow, anddozens of parameters that could be used to characterize a diversity of features including physiologyand ventilator settings. In a clinical setting, ventilators are initially setting pressure or a flow pattern,and as such, pressure, flow and volume are conceptualized according to this ordering. Here, for thepurposes of constructing the model, it is advantageous to begin with the less complex volume model,followed by the more complex pressure model. The flow can be derived from volume and typicallythese two variables contains much of the same information about the underlying lung mechanics incertain ventilation modes.
Therefore, in this study, we focused on two state variables, pressureand volume. Moreover, depending on the ventilator mode there can be a controlled variable, volumeor pressure, depending on whether volume-controlled or pressure-controlled ventilation is set. Thereare also hybrid ventilation modes where there is not one single controlled state variable. Generally,only the independent variables contain direct information about the respiratory mechanics of thepatient.
Here, we construct models of pressure and volume such that the models can representobserved pathophysiology present in all of these aforementioned situations.
Identifying and modeling important features in the volume and pres-sure waveform
The volume waveform has a characteristic shape that is typically independent on the ventilation modeand can be divided into two subprocesses (Fig. 1a). The first subprocess is the inspiration, denotedas A in Fig. 1a, which continues until the desired – either by the patient or according to a ventilatorsetting – tidal volume (the amount of gas delivered in that breath) is reached. The second subprocessis expiration, denoted as B in Fig. 1a.The features in the volume waveforms that we use to delineate lung damage are directly relatedto variability of these two subprocesses. Depending on the ventilator settings and lung mechanics, thegradient of the rising and falling signals can vary widely not only among patients but in the samepatient over time. Therefore, the model must be able to control each of these features independently.Accordingly, the gradients of inspiration and expiration in volume waveforms are features that mustbe variable and estimable within the volume model.The characteristic shape of the pressure waveform can vary more dramatically than the volumewaveform depending on lung mechanics and ventilation mode. In the case of pure pressure controlventilation (PCV), typically, a rectangular or trapezoidal waveform is observed.
When pressure is
Graphical representation of typical volume and pressure waveforms. (a)
Char-acteristic shape of the volume waveform is generally independent of the ventilation mode and has twodistinct subprocesses. The rising and falling of the volume signal during inspiration and expiration,respectively, are denoted as A and B. (b)
When pressure is an independent variable, there can bemultiple features in the waveform that contain useful information. The gradient of the rising signal inwhich pressure continues to increase during inspiration can have two distinct features, denoted as A1and A2. These two features define the gradient of the rising signal before and after the inflection pointsuch that there may be abrupt increases (breath 1) or decreases (breath 2) in the signal gradient. Theshape of the plateau pressure is captured using features B1 and B2 such that there may be a peak atthe beginning (B1-breath 2) and/or at the end (B2-breath 2) of plateau. Finally, the gradient of thefalling signal is captured using feature C that represents the expiration process. The baseline pressureis known as positive end-expiratory pressure (PEEP), and often used in ARDS patient to maintain anopen lung. an independent variable, such as in volume-controlled ventilation, the pressure waveform has severalimportant features that convey information about lung mechanics and ventilator-patient interaction(Fig. 1b).Based on the knowledge of physiology, clinical experience, and observation of the data, we identifiedfive features in the pressure waveform that must be captured by the model. Features one and twodetermine the gradient of the inspiration, which are denoted as A1 and A2 in Fig. 1b. The time-varying graph of inspiration can have two distinct modes where the gradient of the signal may increase(breath 1) or decrease (breath 2) during inspiration. This might correspond to nonlinear volume-dependent lung compliance (breath 1) or an increase in compliance, indicating recruitment (breath2). Features three and four are related to shape of the waveform at the start and end of the plateaupressure, which is a period of constant pressure, are denoted as B1 and B2 in Fig. 1b. There may bepeaks at the beginning (B1) and/or at the end (B2) of the plateau pressure which may correspondto inspiratory flow resistance and patient effort, respectively.
The fifth feature is related to theonset of the expiration process, and in particular, corresponds to the gradient of expiration, denotedas C in Fig. 1b. It is worth noting that we do model the constant baseline pressure is the PositiveEnd-Expiratory Pressure (PEEP) because it is a key independent variable in ARDS management, but this was not an additional feature we had to add to the model. Model validation and evaluation
In order to establish the effectiveness of our approach, we validate and evaluate the volume and pressuremodels in three steps. Model validation is necessary to show that model output have enough fidelityto capture the desired variability, which is often seen in the data. Model evaluation allows the modeloutput to represent the data via optimum parameter estimation and test whether it can be used toextract the desired outcome. Following this, first, we validate that the our volume and pressure modelshave the flexibility such that they can produce all the claimed variability in the waveform data. We dothis by showing how model parameters allow to alter the important features in the waveform data andthen how many of model parameters correspond to interpretable pathophysiology. We then evaluatethat the models are indeed able to estimate data well, or in other words the model output can represent he wide variety of waveform data accurately by estimating volume and pressure ventilation data fromindividuals – mouse models and humans – with injured and healthy, or relatively healthy in the case ofthe human ICU data, lungs. Finally, we demonstrate the model parameters capture and represent thedesired physiology and are interpretable corresponding to different lungs condition. The validation isdone with model simulations without estimating data. The evaluations are done by estimating mouseand human model data. Constructing the damage-informed lung model
Construction of the volume model : Irrespective of the state variable, the models have periodic dynamicswith a frequency defined by the respiratory rate (breaths/min). In addition to this constraint, thevolume model must have two additional features – the rate of inspiration and expiration – that mustbe changeable. We begin the volume model development by modeling the respiratory rate with asinusoidal function ( f s ): f s = sin (2 πθt − φ ) − b . (1)Here, the respiratory rate (breaths/s) is set by θ and t represents time in seconds while parameter φ allows to control the starting point in the respiratory cycle. Typical inspiration and expiration –as either volume or pressure – are not well represented by a sinusoid due to the abrupt rise from abaseline volume or pressure as shown in Fig. 1. To control the rate of inspiration or expiration whilemaintaining the periodicity, we create a periodic rectangular waveform function f b by combining thesinusoidal function with hyperbolic tangent function: f b = 12 { tanh ( a f s ) + 1 } . (2)To control the smoothness of the rectangular waveform, we added a smoothing parameter a . Theother terms (1/2, +1) are added to generate a rectangular waveform that has a zero-base value andunit amplitude. To control the duty cycle of the rectangular waveform that sets inspiratory:expiratoryratio, we used parameter b such that zero value of b corresponds to 1:1 I:E ratio.In Fig. 1a shows additional model features: the rate of inspiration and expiration. To representthese two rates independently we created two separate submodels that define the volume ( V ) usingthe rectangular waveform as a base waveform: V = A v ( f v + f v ) , (3)where f v = [ n X i =1 { β f b ( i ) + (1 − β ) f v ( i − } ] f b max ( f v ) , (4)and f v = [ n X i =1 { β f b ( i ) + (1 − β ) f v ( i − } ] (1 − f b ) max ( f v ) . (5)Here, β and β control the gradient of the inspiration and expiration, respectively, while A v controlsthe amplitude of the volume waveform. Construction of the pressure model : We begin building the pressure model as we did the volumemodel, by modeling the respiratory rate and the I:E ratio. The pressure model has five features thatmust be changeable, the gradient of the rising signal during inspiration at low (1) and high (2) volume,the shape of the peaks at the beginning (3) and end (4) of the plateau pressure, and the rate of changeof the pressure during expiration (5). While volume and pressure are coupled in several ways, the mostfoundational coupling is via their period. We enforce this constraint by requiring that both models ave the same respiratory frequency ( θ ) in their base periodic sinusoid: f s = sin (2 πθt − φ ) − b . (6)Because the pressure may lag or lead the volume, we include a phase shift term, φ in the sinusoid.Additionally, to account for variations in the I:E ratio we added the parameter b . We then create arectangular waveform submodel f b as we did for the volume model using the hyperbolic tangent, or: f b = 12 { tanh ( a f s ) + 1 } . (7)The five key features in pressure are represented with three submodels: (i) f p defines the rates ofpressure change during inspiration and expiration, (ii) f p determines the peaks at the beginningand end of the pressure plateau, and (iii) f p specifies the gradient of the initial rising signal duringinspiration, leaving us with the full the pressure model ( P ): P = f p + f p + f p + A p . (8)The constant parameter A p corresponds to the baseline pressure value (PEEP). The rates of pressurechange during inspiration and expiration (see A2, and C in Fig. 1b, respectively) are: f p = A p ( f p + f p ) , (9)where f p = n X i =1 { β f b ( i ) + (1 − β ) f p ( i − } f b max ( f p ) , (10)and f p = n X i =1 { β f b ( i ) + (1 − β ) f p ( i − } (1 − f b ) max ( f p ) . (11)Here, β and β control the gradient during inspiration and expiration, respectively. The next setof features, the peaks at the beginning and end of plateau pressure (see B1, and B2 in Fig. 1b), aremodeled by: f p = A p | ( f p ) | max ( | ( f p ) | ) , (12)where f p = 1 β n X i =1 [ f p ( i −
1) + { f b ( i ) − f b ( i − } , (13) f p = f p f b , (14)and f p = 1 β n X i =1 [ f p ( i −
1) + { f p ( i ) − f p ( i − } ] . (15)The parameters β and β control the shape of both the peaks. Finally, the gradient of the initial rateof inspiration, (A1 in Fig. 1b), is modeled by: f p = A p f p { − ( f p + f p ) } max [ f p { − ( f p + f p ) } ] , (16)where f p ( t ) = sin (2 πθt − φ ) − b , (17) nd f p = 12 { tanh ( a f p ) + 1 } . (18)The position, shape and gradient of the rising signal, produced by f p submodel are controlled usingthe parameters φ , b and a , respectively. Mouse Mechanical Ventilation Experiments
A nine week old female BALB/c mouse (Jackson Laboratories, Bar Harbor, ME, USA) was studied un-der a University of Colorado Anschutz Medica Campus Institutional Animal Care and used Committee(IACUC)-approved protocol ( µ L IP 5% dextrose lactated Ringer’s solution. Respiratoryefforts were suppressed with 0.8 mg/kg pancuronium bromide administered at 90 min intervals. Heartrate was monitored via electrocardiogram.Baseline ventilation, consisting of a tidal volume (Vt) = 6 ml/kg, PEEP = 3 cmH O, and respira-tory rate (RR) = 250 BPM, was applied for 10 mins with recruitment maneuvers at 3 min intervals.Pressure and volume were recorded with a custom flowmeter based on our previously published de-sign (REF SAMER PAPER). Three types of ventilation were recorded for analysis: LowVT-PEEP0,consisting the baseline ventilation with PEEP = 0 cmH O, LowVT-PEEP12 that was the baselineventilation with PEEP = 12 cmH O, and HighPressure that consisted of (Pplat) = 35 cmH O atPEEP = 0 cmH O with RR = 60 BPM. Lung injury was induced with a 0.15 ml lavage with warmsaline. This fluid was pushed into the lung with an additional 0.3 ml air, and suction was applied to thetracheal cannula with an approximate return of 0.05 ml. The mouse was ventilated for 10 mins with aplateau pressure (Pplat) = 35 cmH O, PEEP = 0 cmH O, and respiratory rate (RR) = 60 BPM andthe LowVT-PEEP0, LowVT-PEEP12, and HighPressure ventilation was recorded again.
Human Data Collection
Between June 2014 and January 2017, adult patients admitted to the University of Colorado Hospitalmedical intensive care unit (MICU) at risk for or with ARDS and requiring mechanical ventilationwere enrolled within 12 hours of intubation. At risk patients were defined as intubated patients withhypoxemia and a mechanism of lung injury known to cause ARDS, who had not yet met chest x-rayor oxygenation criteria for ARDS. To facilitate the capture of continuous ventilator data, only patientsventilated with a Hamilton G5 ventilator were included. Patients requiring mechanical ventilation onlyfor asthma, COPD, heart failure, or airway protection were excluded. Additionally, patients less than18 years of age, pregnant, or imprisoned were excluded. The University of Colorado Hospital utilizes aventilator protocol that incorporated the ARDS network low tidal volume protocol with the low PEEPtitration table. The Colorado Multiple Institutional Review Board approved this study and waivedthe need for informed consent.Baseline patient information including age, gender, height, and initial P/F ratio were collected.Continuous ventilator data were collected using a laptop connected to the ventilator and using Hamil-ton DataLogger software (Hamilton, v5.0, 2011) to obtain pressure, flow, and volume measurements.Additionally, the DataLogger software allowed collection of ventilator mode and ventilator settingsbased on mode (i.e.: set tidal, respiratory rate, positive end-expiratory pressure (PEEP), and fractioninspired oxygen (FiO )). Data were collected until extubation or for up to seven days per patient. Parameter estimation methodology
Estimating model parameters is relatively straightforward when the model is identifiable given data,or, the model is constructed such that every state and parameter is uniquely estimable and there areenough data to uniquely estimate every state and parameter uniquely.
In practice, most models re not identifiable even with ideal data. Moreover, in clinical settings – where we eventually want touse this model – the data are often noisy and difficult to use. Given this reality, we must use careto set up the inference task such that we can ensure robust results with quantifiable uncertainty. This forces three issues, how to choose and limit model features estimated, how to choose an inferencemethodology, and how to manage uncertainty quantification.First issue of limiting model features estimated is important to minimize identifiability failurewhere there is no unique solution in terms of best parameters values for a given data. We employ twoapproaches for managing identifiability failure.
In the first approach, we estimate all parametersbut constrain their ranges to lie within physically possible values while in the second approach, we fixmany low-impact, low-sensitivity parameters, and estimate a limited number of parameters that arechosen based on features present in the waveform data. For example, in the mouse-model data, shownin Fig. 6 and Supplementary Fig. S6, the peaks at the plateau pressure did not appear, and because ofthis, we did not estimate parameters that control those peaks ( β , β and A p ). Similarly, for the thehuman data, shown in Fig. 7, the characteristic shape of the volume and pressure waveforms remain thesame at different time points except for significant variations in the peak amplitudes. Therefore, for thefirst breath we estimated all the parameters but kept certain parameters ( β , β , a , b , β ) constant inthe second breath to maintain the characteristic shape of the volume and pressure waveforms betweenthe two breaths.Second and third issues are choosing an inference methodology that would allow to estimate statesand parameters of the model effectively, and the respective uncertainties in the estimated parame-ters. While stochastic methods, e.g., Markov Chain Monte Carlo (MCMC), might guarantee tofind global minima and quantifying uncertainty in the estimated parameters values, they are generallyquite slow. On the other hand, deterministic methods, e.g., Nelder-Mead optimization, are substan-tially faster and by choosing many initial conditions, a robust solution may be obtained. Therefore,here we focused on a smoothing or optimization task that employ deterministic inference scheme.In this study, we used MATLAB FMINCON function, which is a gradient-based minimizationalgorithm for nonlinear functions. To ensure a robust solution and to quantify uncertainty we ad-ditionally used MATLAB MULTISTART function that performs optimization starting from multiplestart points. MULTISTART effectively boostraps the optimization, uniformly sampling optimizationinitial conditions across a provided interval. We determine realistic lower and upper bound values (con-straints) for each case using an iterative method and these bounds define the constraints employed bythe parameter estimation problem in the optimization scheme. A full description for the computationaland mathematical aspects and implementation of parameter estimation methodology can be found inref. 56. This approach not only allows to determine the best fit parameter values but the respectiveuncertainties as well while trying to find global or multiple minima depending on the solution surfacefor each parameter.
Results
We validate and evaluate the lung models using numerical simulations and measured data, respec-tively. In the validation step, we demonstrate that the models have the flexibility to the desiredvariability through simulations and identify the parameters that correspond to interpretable patho-physiology by analyzing simulated pressure-volume waveforms. In the evaluation step, we demonstratethat the model parameters capture and represent the desired physiology and are interpretable byestimating volume and pressure ventilation data.
Validation of volume and pressure models
Validation of volume model : Figure 2 shows the volume model and the three submodels it is constructedfrom, detailed in Eqns. 1-5. The volume model is a sum of the inspiration and expiration submodels,and is shown as the top plot of Fig. 2. The effective variability in rates of inspiration and expiration,specified by β and β respectively, is shown in Fig. 3a and 3b. The respective variation in the submodelsis shown in Supplementary Fig. S1a and b, respectively. Additionally, the peak amplitude value of Time (sec)
Func t i on s f b1 f v1 f v2 V Figure 2:
Simulated response of various submodels that make up the injury-inclusivevolume model ( V ). A periodic rectangular waveform submodel f b is used to create two more sub-models ( f v and f v ) through which the gradient of the rising and falling signals in the volume waveformare controlled, respectively. Equations 1-5 were used to simulate the response of each submodel withparameter values θ = 0.3, a = 200, b = 0.7, φ = 0, β = 30, β = 10, A v = 1. Time (sec) V o l u m e Time (sec) V o l u m e Time (sec) V o l u m e Time (sec) V o l u m e β = 10 β = 20 β = 40 β = 2.5 β = 10 β = 20b = 0.7b = 0.35 𝜃 = 0.3 𝜃 = 0.45 a bc d Figure 3:
Demonstrating the volume model flexibility by varying specific parameters thatallow altering the gradient of the rising and falling signals, respiratory rate and I:E ratio.
The gradient of the rising and falling signals can be altered using the (a) β and (b) β parameters,respectively. Increased values of these parameters increase the transient time for the signal to reachthe same volume level. (c) Changes in the respiratory frequency ( θ ) change the period of the breathwhile (d) the I:E ratio (inspiratory to expiratory time ratio) can be modified using the b parameter.The output of the model ( V ) was calculated using Eqns. (1)-(5) while considering θ = 0.3, a = 200, b = 0.7, φ = 0, β = 10, β = 10, A v = 1. The respective variation in the submodels that make thevolume model is shown in Fig. S1 for each case. Additional control on these features is shown in theSupplementary Fig. S2the volume waveform can be changed by altering A v ; this variability is shown in the SupplementaryFig. S2a. Variations in respiratory rate are controlled by the respiratory frequency ( θ ) and is shown n Fig. 3c and in the Supplementary Fig. S1c. The I:E ratio is represented through the parameter b that changes the duty cycle of the rectangular base waveform, and is shown in Fig. 3d and inthe Supplementary Fig. S1d. Finally, the starting point of the breath in the breathing cycle andthe smoothness of the volume waveform are set by φ and a respectively, and are shown in theSupplementary Figs. S2b and c, respectively. Validation of pressure model : Figure 1b demonstrates the features of the pressure waveform thatwe deem important for understanding lung function and lung damage. Each of these features in Fig. 1bis controlled by a specific submodel with associated parameters that dictate the shape of that featurewhile its contribution is controlled via the respective amplitude term. Figure 4 shows the pressuremodel and the ten submodels it is constructed from, detailed in Eqns. 6-18.
Time (sec)
Func t i on s f b2 f p11 f p12 f p13 f p21 f p22 f p23 f p24 f p32 f p33 P Figure 4:
Simulated timing response of various submodels that make up the injury-inclusive pressure model ( P ). A periodic rectangular waveform ( f b ) serves as a basis to create othersubmodels that contribute to the pressure model. The overall shape of the pressure waveform, whichdefines gradient of the inspiration and expiration signals are formed using f p submodel comprised ofthe rising signal of f p (A2) and falling signal of f p (C). The shape of the plateau pressure is definedby f p , where the output of f b is processed via f p , f p and f p to produce peaks at the beginning(B1) and end (B2) of the plateau pressure. The shape of the rising signal at low volume (A1) is definedby f p , where a short pulse is produced via f p and reshaped via f p . Note that the amplitude terms A p , A p and A p control the amplitude of f p , f p and f p submodels, respectively. Equations 6-18were used to simulate the response of each submodel with parameter values θ = 0.3, a = 200, b =0.7, φ = 0, a = 10, b = 0.9, φ = -0.6, β = β = 5, β = 1.001, β = 1.1111, A p = 1, A p = 0.5, A p = 0.5, A p = 0.The validation of the model flexibility is shown in Fig. 5 and Supplementary Fig. S2d-i. We carryout this validation by varying five features of the pressure waveform; variation in the rate of change ofthe pressure before (A1 in Fig. 1b) and after (A2 in Fig. 1b) the inflection point during inspiration; theshape of the peaks at the beginning (B1 in Fig. 1b) and end (B2 in Fig. 1b) of the plateau pressure;and variation in the rate of change of the pressure during expiration (C in Fig. 1b). In brief, thesefeatures are controlled by the following parameters. Time (sec) P r e ss u r e β = 5 β = 10 β = 20 Time (sec) P r e ss u r e β = 5 β = 10 β = 20 Time (sec) P r e ss u r e a = 2.5 a = 10 a = 40 Time (sec) P r e ss u r e β = 1.1 β = 1.05 β = 1.001 a bc d Figure 5:
Demonstrating the pressure model flexibility by altering physiologically relevantfeatures. (a)
The initial gradient of the pressure signal during inspiration at low volume (A1) iscontrolled by the a parameter (b) The gradient of the rising signal after the inflection point (A2),is controlled by the β parameter. (c) The shapes of the peaks at the beginning (B1) and at theend (B2) of the plateau are regulated by the β parameter when A p = 0.5. (d) The gradient of thefalling signal (C) during expiration can be modified by the β parameter. Equations 6-18 were usedto simulate the response of the pressure model while considering θ = 0.3, a = 200, b = 0.7, φ = 0, a = 10, b = 0.9, φ = -0.6, β = β = 5, β = 1.001, β = 1.1111, A p = 1, A p = 0, A p = 0.5, A p = 0. A zoomed-in view of each plot is shown inside the respective plot to highlight the changes inthe waveform. The respective variations in the submodels that make the pressure model is shown inFig. S3 for each case. Additional control on these features is shown in the Supplementary Fig. S2The initial gradient of the pressure during inspiration (A1) is controlled by the a parameter suchthat higher values of a result in a slower rising signal as seen in Fig. 5a and in the SupplementaryFig. S3a. The shape of the initial gradient signal before inflection point can be altered using the b parameter as shown in the Supplementary Fig. S2d. And the amplitude of the initial gradientalteration is controlled by the A p parameter as shown in the Supplementary Fig. S2e. The gradientof pressure dynamics at inspiration after the inflection point (A2) is specified by β such that highervalues of β result in a slower rising signal as seen in Fig. 5b and in the Supplementary Fig. S3b.The shapes of the peaks at the beginning (B1) and end (B2) of the plateau pressure are controlled byseveral parameters. The overall shape of the peaks is controlled by the β parameter for a given β as can be observed in Fig. 5c and in the Supplementary Fig. S3c. The sharpness of these peaks canbe altered further by the β parameter for a given shape of the peaks as shown in the SupplementaryFig. S2f. The amplitude of the peaks is controlled by the A p parameter whose effect can be seen in theSupplementary Fig. S2g. Additionally, we can control individual peaks by the parameter β as shownin the Supplementary Fig. S2h and i. And finally, variation in the gradient of pressure dynamics atexpiration (C) is specified by β such that higher values of β result in a slower falling signal as seen inFig. 5d and in the Supplementary Fig. S3d. Along with these, the I:E ratio is characterized by the b parameter in the same way that parameter b controls the I:E ratio in the volume model, cf Fig. 3d. inking model parameters to lung function The next step of our model validation is to demonstrate how the model parameters can be related tophysiology. In the volume model, we focus on three parameters that have physiological meaning: β , β and A v . The rate of inspiration is controlled by the β parameter, which is shown as feature A inFig. 1a. Higher values of β result in a lower inspiratory flow rate (Supplementary Fig. S4). Duringpressure control ventilation (PCV), inspiratory flow rate can change due to reduction in lung complianceand/or increase in lung resistance. Alternatively, during volume-controlled ventilation (VCV), thisfeature corresponds to the set inspiratory flow rate. The gradient of expiration is controlled by the β parameter and is captured as feature B in Fig. 1a. Higher values of β result in a longer expiration(Supplementary Fig. S4). This parameter is directly proportional to the expiratory time constant whichis the product of resistance and compliance. Finally, the tidal volume in VCV is represented by theamplitude parameter, A v . In PCV, higher values of A v for the same pressure waveform would suggestan increase in the overall compliance (Supplementary Fig. S4). There are several other parameters inthe volume model that represent settings controlled by the ventilator such as respiratory frequency,I:E ratio etc. A short description of how these, and other, model parameters contribute to the modelis provided in the Supplementary Table S1.In the pressure model, we identified five parameters that are associated with aspects of lung com-pliance during VCV: a , b , β , A p and A p . During PCV, these (and other) parameters may bedirectly controlled via ventilator. The gradient of the initial rising pressure signal (A1) is controlled bythe a parameter and higher values of a result in slower pressure rise at low volume while maintainingthe shape of the gradient as shown in the Supplementary Fig. S5. We can therefore directly relate thisparameter to the low volume compliance during VCV such that higher values of a would suggest anincrease in the low volume compliance and vice versa.The shape of the initial rising pressure signal at the onset of inspiration (A1) is also controlledby the b parameter such that higher values of b result in slower pressure rise at low volume whilechanging the shape of the gradient as shown in the Supplementary Fig. S5. Note that parameters a and b control the same feature in the pressure waveform (A1) but different aspects of it which mightbe relevant to distinguish the cases where alveoli recruitment varies substantially at low volume.The gradient of the rising signal above the inspiratory inflection point (A2) is controlled by the β parameter, and higher values of β result in slower pressure rising signal as shown in the SupplementaryFig. S5. We relate this parameter directly to the high volume compliance during VCV such that highervalues of β would suggest an increase in the high volume compliance and vice versa.The pressure value at the plateau is defined using the A p parameter, and higher values of A p result in higher values of the plateau pressure as shown in the Supplementary Fig. S5. This parameteris inversely related to the overall lung compliance such that increasing values of A p would suggest areduction in the compliance and vice versa, given the tidal volume does not change.Finally, change in the upper inflection point (UIP) can be directly related to the A p parametersuch that higher values of A p increase the value of UIP in the waveform while maintaining the shapeof the pressure waveform as shown in the Supplementary Fig. S5.It is important to note that these interpretations are valid only when a change is observed in one ofthe variables (volume or pressure) while having the other features of the waveforms fixed. There may becases where both volume and pressure waveforms change simultaneously and, in those cases, additionalinterpretation is needed to establish the relationships between pressure and volume parameters. Forexample, when there is a change in the amplitude of volume and pressure simultaneously, A v / A p ratioshould be considered to determine the over change in the lung compliance. Model evaluation with animal and human data
In the previous sections, we validated that the model can simulate the diversity of observable volumeand pressure features we had previously identified as important. The validation is carried out withoutdata, and therefore without an inference task. Here, we begin the data-driven model evaluation byshowing that the model is indeed flexible enough to estimate the pathophysiology we designed it toestimate.
Volume and pressure models responses closely agree with the experimental datafrom a representative mouse in healthy and injured condition.
In the first row, the measuredresponse is shown in solid lines while the model inferred response is shown in dashed lines. Changesin the volume and pressure submodels are shown in the second and third rows, respectively (in solidlines). The volume and pressure models shown in Eqns. 1-5 and 6-18 were used to generate the best-fitmodel response using estimated mean parameter values shown in Table 1, respectively. The respectiveuncertainties in the parameter values are shown in Table 1 estimations for each breath.
Parameter selection and estimation : As mentioned in the Methods section, we do not alwaysestimate every parameter. In particular, we did not infer parameters that control features that were notobserved in the data to reduce confounding problems. In more detail, for the mouse model experimentsshown in Fig. 6, we estimated a , b , φ , β , β A v , a , b , φ , a , b , φ , β , β , A p , A p , A p , and heldthe parameters that control the pressure plateau peaks, β , β , A p , constant. For the retrospectivehuman data-based evaluation, shown in Fig. 7, we estimated a , b , φ , A v a , b , φ , φ , β , β , β , A p , A p , A p , A p . We then held the variables that control the shape and the gradients at inspirationand expiration – constant, including β , β , a , b β . In order to maintain the coupling betweenvolume and pressure models, respiratory rate, θ , was kept constant for each dataset. Data selection : Each data set contained thousands of breaths. In an effort to perform a morecontrolled evaluation, we isolated a single breath in each case that is representative of the breathsin that data set and performed the parameter estimation and evaluation on those data. The best-fitparameter values for Fig. 6 and 7 are shown in Table 1 and for Supplementary Fig. S6 and S7 areshown in Supplementary Table S2 with 95% confidence intervals with respect to mean.
Broad model evaluation : Figure 6 shows two breaths measured in the same mouse when healthy
Damaged informed-lung model can accurately follow two different breaths ofan ICU patient with ARDS.
In the first row, the measured response is shown in solid lines whilethe model inferred response is shown in dashed lines. Changes in the volume and pressure submodelsare shown in the second and third rows, respectively (in solid lines). The volume and pressure modelsshown in Eqns. 1-5 and 6-18 were used to generate the best-fit model response using estimated meanparameter values shown in Table 1, respectively. The respective uncertainties in the parameter valuesare shown in Table 1 estimations for each breath.(green) and after lung injury (orange) during ventilation with Pplat = 35 cmH O and PEEP =0 cmH O. The model estimates are shown in dashed lines and the submodels of the volume andpressure waves are shown in the 2nd and 3rd rows, respectively. Low tidal volume ventilation measure-ments for these two time points are shown in the supplement for PEEP = 0 cmH O (see SupplementaryFig. S6a) and PEEP = 12 cmH O (see Supplementary Fig. S6b). The model states and parameterswere also estimated using data from two patients with ARDS, shown in Fig. 7 and in the Supplemen-tary Fig. S7. As can be seen in the figures (Fig. 6, 7, and Supplementary Fig. S6, S7), the models areable to accurately estimate all data and their observed pathophysiology.
Estimated model parameters correspond to interpretable pathophysi-ology
Our final evaluation step is to show that the values of the estimated parameters for data sets corre-sponding to different phenotypes – injured/damaged versus healthy – have physiological meaning. Inother words, that differences in the estimated parameter values reflect different phenotypic states ofthe subject in a manner that is consistent with the pathophysiology.
Mouse model, PCV : Figure 6 shows two different breaths of mouse model data, one healthy breathat the beginning of the experiment, and one injured breath at the end of the experiment with ≈ to particular parameters we are focusing on for this evaluation. Model parameter interpretations aredetailed in the Supplementary Table S1.In the volume model, we observed the injured lung showed slower estimated inspiration, quantifiedby an increase in β , and a faster expiration, quantified by a reduction in both β and A v comparedto the healthy lung model estimates. This leads directly to an interpretation of a reduction in lungcompliance.In the pressure model, we observed a decrease in b , which could be inferred as a reduction inthe compliance in the injured versus healthy lung data. However, we also observed two parametersrelated to lung compliance, a and β , indicate increased overall lung compliance as the lungs becomemore damaged. These results seem to be contradictory with each other and with the volume model.The data shown in Fig. 6 corresponds to PCV, where volume was an independent variable whilethe pressure signal was the ventilator controller variable. Therefore, any changes in the pressurewaveform correspond to the ventilator settings and not the respiratory mechancs. These results, makean important point: it is essential to see the relative change in the parameters that control thesefeatures and to synthesize the model-based inference in a holistic fashion, instead of focusing on anyone parameter or feature in isolation given lung mechanics depends on both pressure and volume signalmutually.Ideally, we would expect pressure signal to be the same over time in PCV, but our mouse modelventilator is not a perfect controller since it uses a piston pump. However, larger changes in the volumesignal would be expected, considering a significant change in respiratory mechanics over time. In thisspecific case, we observed a much greater change in β compared to a and β , and hence changes in he volume waveform are dominating over changes in the pressure waveform. Moreover, A v / A p ratiois reduced in the injured case (Supplementary Table S3). By considering the model-based parameterestimates and ventilator mode in total, the conclusion is that the injured lung is estimated to havesubstantially lower compliance than the healthy lung. Mouse model, VCV : The second mouse model evaluation, which includes variations in PEEP duringVCV, has PV loops indicating a reduced compliance in the injured case compared to the healthy casefor both PEEPs. The full PEEP-varied results are shown in the Supplementary Fig. S6.In the volume model, the healthy lung with PEEP has a slower rate of inspiration leadingto an interpretation of mildly worse compliance in comparison to the injured lung, as quantified bythe β parameter value (Supplementary Fig. S6, Table S1, S2). In contrast, the pressure modelindicates a reduction in compliance in the injured lung as quantified by lower values of parameters a , b and β , and elevated estimates of in A p , cf Supplementary Table S1, S2. In contrast tothe results shown in Fig. 6, here, changes in parameter estimates in the pressure model were muchlarger in comparison to the observed differences in the volume model. This is expected since thetidal volumes were approximately equal during VCV, and the reduction in compliance is reflected inincreased pressure. This effect can be inferred by analyzing A v / A p ratio where we observed reductionthis ratio in the injured cases at both the PEEPs (Supplementary Table S3). Human ICU data-driven evaluation : Fig. 7 shows two different breaths of an ICU patient withARDS that were taken near extubation when ARDS has nearly resolved (see Methods section, Table 1).The ventilator mode was human-triggered, a mode that is not possible in our mouse ventilators and is acommonly used ventilator mode in the ICU. Of the thousands of breaths available, we selected breathswithout dyssynchrony. PV loops for these cases suggest that lung compliance is increased at the latertime point. We found that the model-estimated parameters suggest the same interpretation. Betweenthe early and later breath data respectively, we observed an increase in A v / A p ratio indicating increasein compliance, cf Table 1, Supplementary Table S1-S3. The set reduction in PEEP was reflected in areduction in A p .The cases where some patient effort is present, additional model parameters might be used tounderstand the interaction between the ventilator and the respiratory mechanics. Such a case is shownin the Supplementary Fig. S7, where PV loops for these cases suggest that lung compliance is increasedat the later time point. The model estimated parameters show the ratio of tidal volume to the plateauwas increasing, quantified by the A v / A p ratio increasing (Supplementary Table S3), suggesting anincrease in the compliance from when the patient had acute ARDS to the point of extubation. Inthe pressure waveform, inspiration is happening at a slower rate in the later breath as suggested byan increase in the b and β parameters, also indicating higher compliance. But, while tidal volumeappeared to be the same in the two breaths, a significant increase in the β parameter indicatinga reduction in the compliance. This effect is likely to be a result of patient effort to overcome theventilator. This can be inferred from an increased value of A P /A P in the later breath, suggestingan increase in the inspiratory flow resistance and patient effort.To further validate our finding, we used single-compartment model, which was developed by ourresearch group earlier, to extract the relevant respiratory parameters and found a qualitativeagreement between the outcomes of the two models (see Supplementary Table S3) and thereby, furthervalidating our parameter estimation and interpretation scheme.Overall, these results suggest that our model can not only reproduce a wide variety of waveformdata but also capable of extracting clinically relevant information from the waveforms that might allowto understand injured lung dynamics systematically. Discussion
We developed a damage-informed lung model that represents pressure and volume time-series data byreconstructing the waveforms from a modular set of subcomponents. We demonstrate the efficacy ofthe model using a combination of simulations for validation along with mouse and human data forevaluation. The model was able to simulate desired physiology and pathophysiology, accurately esti-mate volume and pressure waveforms, and distinguish healthy from injured lungs based on parameter stimation. The model is novel because of the flexibility afforded by the waveform-based approach.Furthermore, we directly incorporate clinical and physiologic knowledge and hypotheses regarding im-portant and observable lung pathophysiology into the model. The model is also limited using priorknowledge so as to not have the capability to estimate every possible variation in PV waveforms,but rather is constrained to estimate the features of the ventilator data that are the most clinicallyimpactful.Our approach of developing a model that incorporates clinical insights and limits the model torepresent meaningful physiology and pathophysiology appears capable of reproducing a wide range ofventilator waveform including pressure- and volume-controlled ventilation in healthy and lung-injuredmice and humans (Fig. 6, 7 and Supplementary Fig. S6, S7). This approach lives between a machinelearning approach, were the model is flexible enough to estimate every feature and must then discernwhich features are important through regularization to prevent overfitting, and the fully mechanisticlung modeling approach where the observed physiology must emerge from the proposed lung mechanics.It is possible that taking this middle path will help advance all approaches.The most direct application of our modeling approach is to quantify the qualitative physiologicalinterpretation of pressure and volume data. An experienced clinician or physiology can infer the sta-tus of a patient, the safety of ongoing ventilation, the presence of ventilator dyssynchrony, and otherimportant details from visual inspection. However, we currently do not yet have methods to quantita-tively identify all of these characteristics in ventilator data. The entire waveform may be utilized andthis provides a rich repository of data that is challenging and time consuming to use for diagnosis andtreatment. Alternative, these data are, for example, by summarizing in scalar values for resistance andcompliance and this may cast aside important details. Our approach offers a methodology for con-densing the pressure-volume data to assess ongoing VILI, track changes in injury severity over time,and estimate injury phenotypes (Fig. 6, 7, Table 1 and Supplementary Fig. S6, S7, Table S2). Thesephenotypes could be used for to categorize and understand lung injury, serve as outcome measuresfor interventions, and may describe the impacts of VILI and dyssynchrony, and VILI.. This isreminiscent of current interest in the driving pressure, which is derived from the pressure waveformand has been linked to ventilation safety and ARDS outcomes.
Lung injury diagnosis and decision-making are based in part on interpretation of the pressure,volume, and flow waveforms, such as the aforementioned driving pressure. However, different patho-physiologic mechanisms can lead to the same observed waveform features. For example, increaseddriving pressure could be a result of derecruitment (alveolar collapse) or alveolar flooding.
Inother words, the human-based inference using single waveform data can be ill-posed. Our modelingapproach suggests that the ill-posed nature of the inference problem can be addressed in two ways.First, we can quantify the potential observed impact of different pathophysiologic-driven features inthe waveforms using experimental data. Second, by estimating over many similar but varied breaths, itmay be possible to better triangulate the most probable pathophysiologic drivers because the primarydriver of damage will likely be present and significant despite breadth variations while more extraneousdetails will not be consistently expressed in every breadth.Then in future studies we can look at the relationship between parameters. The model we presentdoes not fully couple pressure and volume. We have taken this approach in the current study topreserve flexibility so that we can accurately recapitulate a wide variety of clinically and experimentallyobserved features in the pressure and volume signals, including the effects of ventilator dyssynchrony.This fidelity and flexibility is not always possible with rigid coupling between pressure and volume datalike, for example, in a single compartment model where pressure is defined as the sum of linear resistiveand elastic contributions. This is not to say that pressure and volume are totally independent in ourmodel because we utilize the same respiratory rate for both. In future studies we will link specificcomponents of the pressure and volume waveforms through physiologically-relevant parameters suchas nonlinear lung elastance or inspiratory and expiratory flow resistance.As secondary application of our modeling approach, and a method to incorporate the physiologiccoupling between pressure and volume data, is to utilize the outputs from the model presented here asinputs for compartment models. Currently, most compartment models are fit to measured data usingregression. In a model with few parameters (e.g. only resistance and compliance) this is feasible for real- ime analysis. However, as model complexity increases to include representations of nonlinear tissueelastance, recruitment dynamics, and other factors it is no longer possible to perform the regressionsin a clinically-applicable timescale. If our waveform-based model is used to process the data prior toanalysis using a compartment model then it is possible to formulate the problem entirely of ordinarydifferential equations and this opens up a range of more efficient inference machinery. Finally, our work here has several notable limitations.
First , our evaluations were performed withsingle, but typical, breaths of mouse and human ventilator data. We took this approach becauseeach breath is, in some sense, a single controlled experiment and our goal was to demonstrate thefunctionality of the model under varied conditions.
Second , our evaluation was conducted using healthyand severely lung-injured mice as well as a single human data set. This is sufficient for proof in principlethat the model can capture physiologic differences. However, establishing that the model can accuratelydifferentiate more specifically defined phenotypes will require evaluation on much larger populations.
Third , we relied on the expert knowledge of a single critical care physician to determine the clinicallyimportant characteristics of the pressure and volume waveforms and it is likely that differing opinionswill exist among intensivists. Collecting and synthesizing such information will require a differentqualitative study. Moreover, that there may be differing opinions regarding what should and shouldnot be included in the model. This does not negate our methodology or our model. Instead, it suggestsfuture work is necessary to better understand and verify clinically important features. Alternatively,we may instead seek to link model features to patient outcomes, thus establishing the importantcharacteristics of the model by linking those parameters to outcomes.In summary, we developed a physiologically anchored and data-driven lung model that can repro-duce the important features pressures and volumes during mechanical ventilation. The performance ofthe model was verified with experimental and clinical data in healthy and injured lungs to demonstratemodel efficacy in robustly estimating interpretable parameters. This methodology represents a depar-ture from many lung modeling efforts, and suggests future directions of work that can provide anotherpathway for better understanding lung function during mechanical ventilation and can potentially forma bridge between experimental physiology and clinical practice.
Grants
This work was supported by National Institutes of Health R01 “Mechanistic machine learning,” LM012734and LM006910 “Discovering and applying knowledge in clinical databases,” along with R00 HL128944,and K24 HL069223.
Author Contribution
D.K.A., and D.J.A. conception and design of research; D.K.A., B.J.S., and P.D.S. performed exper-iments; D.K.A., B.J.S., and D.J.A. analyzed data; D.K.A., B.J.S., P.D.S., and D.J.A. interpretedresults of experiments; D.K.A. prepared figures; D.K.A. drafted manuscript; D.K.A., B.J.S., P.D.S.,and D.J.A. edited and revised manuscript; D.K.A., B.J.S., P.D.S., and D.J.A. approved final versionof manuscript.
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E-mail: [email protected] a r X i v : . [ q - b i o . Q M ] O c t Time (sec) F un c t i on s Time (sec) F un c t i on s Time (sec) F un c t i on s Time (sec) F un c t i on s β = 10 β = 20 β = 40 β = 2.5 β = 10 β = 20 a bc d f b1 f v1 f v2 V f b1 f v1 f v2 V f b1 f v1 f v2 Vf b1 f v1 f v2 V b = 0.7b = 0.35 𝜃 = 0.3 𝜃 = 0.45 Figure S1: Showing the effect of parameter variations onto the submodels that form the volumemodel for the cases shown in Fig. 3. Changing (a) β and (b) β changes the gradient ofthe rising and falling signals, respectively. Increased value of these parameters increases thetransient time for the signal to reach the same volume level. (c) Change in the respiratoryfrequency ( θ ) changes the periodicity of the breath while (d) parameter b changes the I:E ratio(inspiratory to expiratory time ratio). Here, V is the output of the model, which was calculatedusing Eqns. (1)-(5) while considering θ = 0.3, a = 200, b = 0.7, φ = 0, β = 10, β = 10, A v = 1. Y-axis was normalized to represent all the submodels in a sequential manner.2 Time (sec) V o l u m e A v = 1 A v = 0.5 A v = 0.25 Time (sec) V o l u m e Φ = 0 Φ = -0.8 Φ = 0.8 Time (sec) V o l u m e a = 200 a = 10 a = 5 Time (sec) P r e ss u r e A p3 = 0 A p3 = 0.25 A p3 = 0.5 Time (sec) P r e ss u r e β = 1.25 β = 1.1 β = 1.05 Time (sec) P r e ss u r e b = 0.8 b = 0.9 b = 1 Time (sec) P r e ss u r e A p2 = 0 A p2 = 0.25 A p2 = 0.5 Time (sec) P r e ss u r e A p2 = 0 A p2 = 0.1 A p2 = 0.2 Time (sec) P r e ss u r e A p2 = 0 A p2 = 0.2 A p2 = 0.3 b ca e fdg h i Volume WaveformPressure Waveform
Figure S2: (a-c) Further variations in the volume waveform can be achieved by changing the(a) A v , (b) φ and (c) a parameters. (d-i) Further variations in the pressure waveform canbe achieved by changing the (d) b , and (e) A p parameters; (f) β parameter when β = 2.5and A p = 0.5; (g-i) A p parameter when (g) β = 5, (h) β = 2.5 (i) β = 10; To simulatethe response of the volume and pressure models, Eqns (1)-(5) and Eqns (6)-(18) were used,respectively, at the parameter values θ = 0.3, a = 200, b = 0.7, φ = 0, β = 10, β = 10, A v = 1, a = 200, b = 0.7, φ = 0, a = 10, b = 0.9, φ = -0.6, β = β = 5, β = 1.001, β =1.1111, A p = 1, A p = 0, A p = 0.5, A p = 0. A zoomed-in view of each plot is shown insidethe respective plot to highlight the changes in the waveform.3 Time (sec) F un c t i on s f b2 f p11 f p12 f p13 f p21 f p22 f p23 f p24 f p32 f p33 Pβ = 5 β = 10 β = 20 Time (sec) F un c t i on s f b2 f p11 f p12 f p13 f p21 f p22 f p23 f p24 f p32 f p33 Pβ = 5 β = 10 β = 20 Time (sec) F un c t i on s f b2 f p11 f p12 f p13 f p21 f p22 f p23 f p24 f p32 f p33 Pa = 2.5 a = 10 a = 40 Time (sec) F un c t i on s f b2 f p11 f p12 f p13 f p21 f p22 f p23 f p24 f p32 f p33 Pβ = 1.1 β = 1.05 β = 1.001 a bc d Figure S3 Showing the effect of parameter variations onto the submodels that make up thepressure model for the cases shown in Fig. 5. (a) Feature A1, which corresponds to the shapeof the rising signal gradient at the beginning of inspiration, is produced via f p submodel wherechanges in parameter a alters the gradient of the signal before inflection point. (b) FeatureA2 is controlled via f p function, that is a part of f p submodel and changing β changesthe shape of feature A2. (c) Features B1 and B2, which correspond to peaks at the beginningand end of plateau respectively, are incorporated via f p submodel and their shapes can bealtered by changing β parameter ( A p = 0.5) (d) Feature C, which corresponds to the shapeof falling signal gradient, is controlled via f p function where changing β changes the shapeof this feature. Equations (6)-(18) were used to simulate the response of the pressure modelwhile considering θ = 0.3, a = 200, b = 0.7, φ = 0, a = 10, b = 0.9, φ = -0.6, β = β =5, β = 1.001, β = 1.1111, A p = 1, A p = 0, A p = 0.5, A p = 0. Y-axis was normalized torepresent all the submodels in a sequential manner.4 time V Variation in
10 20 30 40 P V Variation in
10 20 30 40 P V Variation in
10 20 30 40 P V Variation in A v
10 20 30 40 P V Variation in a
10 20 30 40 P V Variation in b
10 20 30 40 P V Variation in
10 20 30 40 50 P V Variation in A p1
10 20 30 40 P V Variation in A p3