Simulation of Pulmonary Pathophysiology During SpontaneousBreathing

Y. C. Zhao, S. E. Rees, S. Kjrgaard, S. Andreassen

Center for Model-based Medical Decision Support, Alborg University, DenmarkDepartment of Anaesthesiology, Alborg Hospital, Denmark

Abstract This paper presents a functional model of lungmechanics including a non-linear alveolar pressure volumecurve and representation of the work of respiratory musclesduring breathing. The model is used to simulate the responseto forced inspiration and expiration, and these simulationscompared to the standard results of lung function tests routinelyperformed in departments of lung medicine. The model cansimulate the characteristics of inspiratory and expiratory flowprofiles seen in normal subjects, and in patients with obstructiveor restrictive diseases.

I. INTRODUCTIONIn clinical practice linear models of respiratory mechanics

can be used to estimate resistance and elastance parametersof the respiratory system. These may provide an under-standing of the pathogenesis of lung disease, and may beused to optimize ventilator settings or adjust pharmacologicaltherapy. Whilst the parameters of these linear models canbe estimated in the clinical settings, the complexity of themodels is insufficient, meaning that these models oftenprovide a poor fit to measurements of flow and pressure, andgive parameter values which are physiologically implausible[1]. To overcome these limitations, models have been builtusing the so called functional approach [2][3][4][5][6]. Thesemodels divide the airways into three sections: upper airwaysdescribed as a flow dependent resistance; central airwaysdescribed as a transmural pressure dependent resistance; andlower airways described as a volume dependent resistance. Inthis approach, alveoli are lumped into single a compartmentwith a constant compliance. Models built using the functionalapproach have been used to simulate flows and pressuresduring mechanical ventilation. However, these models havenot been used to simulate spontaneous ventilation. In par-ticular they have not been used to simulate the flow andvolume profile that occurs during forced breathing, wherethese profiles are often used in the diagnosis of respiratorydisease.

This paper presents a modified version of the modelpreviously proposed by Barbini et al [6]. This model includesa sigmoid [7] function to represent the nonlinear properties ofboth airways and the alveoli space, and a pressure generatorto describe the work of the respiratory muscles. In doing so,it will be tested whether this model can simulate spontaneousbreathing, including forced breathing patterns in normalsubjects and patients with lung diseases.

This work was supported by the IT committee under the Danish TechnicalResearch Council.

II. MODEL OF RESPIRATORY MECHANICSMechanical and electrical analogues of respiratory me-

chanics are illustrated in figure 1. The derivation of amathematical description of these models and its detailedphysiological interpretation follows.

A. Alveoli and Thoracic CageFigure 1(a) illustrates the physiology included in the

model. Alveoli are surrounded the pleural space which isconnected with the thoracic cage. The pressure inside thealveoli and pleural space are called alveolar gas pressure (Pa)and the pleural pressure (Ppl) respectively. During breathingthe work performed by the muscles of the chest wall anddiaphragm generate pressure (Pm) in the pleural space,whilst chest wall elasticity (Ecw) exerts recoiling pressure(Pcw) against it. Ppl is therefore given by

Ppl = Pm + Pcw (1)The elastic recoil pressure of the chest wall is consideredhaving a linear relationship with the thoracic cage volume(Vtc) and describes as

Pcw = EcwVtc Pcw0 (2)where Pcw0 is a parameter representing the outward pullingpressure generated by chest wall at the relaxed state, atwhich the opposite pull of the alveoli and the chest wallreaches the equilibrium. Vtc is approximated equal to thesum of alveolar volume (Va) and central airway volume (Vc),assuming negligible pleural volume, i.e.

Vtc = Va + Vc (3)The pressure difference between Ppl and body surface

pressure (Pbs) is termed as trans-chestwall pressure (Ptc),which drives Vtc and is described as

Ptc = Ppl Pbs (4)The pressure difference between the alveoli and the pleural

space is called transpulmonary pressure (Ptp), which drivesVa change in a nonlinear relationship [8][9][10], and isdescribed as

Ptp = Pa Ppl (5)In this paper, the relationship between Ptp and Va is repre-sented by a sigmoid equation (6) proposed by Venegas et al[7]

Va = a + [b/

(1 + e(d/(Ptpc)))] (6)

Proceedings of the 2005 IEEEEngineering in Medicine and Biology 27th Annual ConferenceShanghai, China, September 1-4, 2005

0-7803-8740-6/05/$20.00 2005 IEEE. 6128

Thoracic Cage (Vtc)

Alveoli

Va

RcRlRu

PaPmPcw

Pv

av

Pbs

Ppl

Pc Vc

Chest Wall

Diaphragm

Ptp

Ptc

Ptm

Ecwc

v

Pleural Space

u c l

(a)

Ru Rl

Va

Pc

PPl

v PaRc

+

-

Pm Pcw

Vtc Ecw

Vc

av

cv

Ptm PtpP

Pbs

1 2

+

+-

-

+

-

(b)

Fig. 1. (a) A mechanical analogue representing physiological interpretation of respiratory mechanics. Airways are lumped into three sections, i.e. upper(u), central (c), and lower airways (l). The thoracic cage is represented as a solid and dashed line, which indicates its compression due to pressure appliedby the chest wall and muscle work. (b) Respiratory mechanics is modeled and calculated in an electric analogue. Body surface pressure is referred asground in circuit analysis. The direction of inflation flow v is assumed as the positive direction.

where a corresponds to the lower asymptote of the lungvolume; b corresponds to the difference between the upperand the lower asymptote of the lung volume; c is the pressureat the inflection point between the upper and the lowerasymptote of the sigmoid curve; d indicates a pressure range,in which the most constant compliance occurs, i.e. the range(c 2d) to (c + 2d).B. Airways

The respiratory airways are divided into three sections,upper, central and lower airways, according to their func-tional importance. Upper airways are assumed to consist ofairways from mouth to extrathoracic trachea; Central airwaysare assumed from intrathoracic trachea to small bronchi(11th generation of Weibel model [11]); Lower airways areassumed to consist of bronchioles extending to the alveoli,i.e. generations (12th-23th) [11].

a) Upper Airways: Upper airways are represented asrigid pipes with considerable structural resistance to collapse,and can be modeled as flow dependent resistance (Ru)according to Rohrer [12]

Ru = k1 + k2 |v| (7)where k1 represents resistance due to laminar flow, and k2represents resistance due to turbulent flow.

b) Central Airways: Central airways can be com-pressed and correspondingly cause expiratory flow limitation(EFL) during forced expirations or during normal ventila-tion in patients with chronic obstructive pulmonary disease(COPD) [13]. Modeling this compression is therefore neces-sary to represent EFL. Compression due to high transmuralpressure (Ptm) can be represented as the difference betweenintraluminal pressure (Pc) and pleural pressure (Ppl)

Ptm = Pc Ppl (8)Golden et al [3] modeled resistance due to collapsibleairways as a compressible cylinder with a fixed lengthand variable volume Vc, whose resistance Rc is assumedproportional to 1/V 2c , and represented as

Rc = k3

(Vcmax

Vc

)2(9)

where Vcmax stands for the maximal volume of collapsibleairways when the airways are fully distended. Under fullydistended conditions, k3 represent a small viscose airwayresistance.

Previously the relationship between Vc and Ptm has beenmodeled as a nonlinear sigmoid function [6], the slope ofwhich is the airway compliance.

Ptm = a b ln

(Vcmax

Vc 1

)(10)

where a stands for the point of the maximal airway com-pliance; b is the parameter standing for a range in whichthe most constant airway compliance occurs, i.e. the range(a 2b) to (a + 2b).

c) Lower Airways: Lower airways have no rigid struc-ture and are embedded in the lung parenchyma. Due totraction arising from parenchyma on the airway walls the air-ways dilate with lung volume expansion [14], and thereforeminimize the airway resistance. The lower airway resistance(Rl) is inversely proportional to alveolar volume (Va) [15],thereby giving

Rl =k4Va

(11)where k4 is a model parameter describing different patho-physiological states of the lower airways.

C. Model DynamicsTo describe the dynamics of the respiratory system re-

quires representation of the volumes of lung compartmentsand dynamics of gas flow between these. State variables areselected describing the volume of gas in each compartment(Va1, Vc, Vtc), and differential equations formulated as:

dVadt

= va (12)dVtcdt

= v (13)dVcdt

= vc (14)va describes the net air flow into the alveoli, which can becalculated from the pressure drop over the lower airways

1where lower airway volume is lumped into the alveolar compartment

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(Pc Pa) divided by the lower airway resistance (Rl) asillustrated in figure 1(b), i.e.

va =Pc Pa

Rl(15)

v describes airflow into the respiratory system, which can becalculated from pressure gradient over the upper and centralairways (P Pc) divided by the sum of airway resistancein the upper (Ru) and central (Rc) airways as illustrated infigure 1(b), i.e.

v =P PcRu + Rc

(16)

vc describes flow bypass due to collapsible airways, whichcan be calculated from the flow entering the respiratorysystem minus the flow entering the alveolar compartment(v va) as illustrated in figure 1(b), i.e.

vc = v va (17)III. MODEL SIMULATIONS

This model includes seventeen equations with three dif-ferential equations (12-14) describing changes in state vari-ables (Va, Vtc, Vc), three equations describing flows betweencompartments (15-17) and the remaining (1-11) describingthe relationship between pressures, volumes and resistances.All equations can be solved to simulate a dynamic responseof the respiratory system to changes in ventilation, andthe resultant steady state, given values of model parame-ters {k1, k2, k3, a, b, Vcmax, k4, a, b, c, d, Ecw}, initial con-ditions of state variables (Va(0), Vtc(0), Vc(0)), and modelinputs (Pm, P , v). For spontaneous respiration inspiratoryflow (v) is not required as an input to the model, and thepressure at mouth P equals the body surface pressure Pbs.

The model is used here to simulate forced inspiration andexpiration in three cases: normal patients, obstructive lungdisease (OLD) and restrictive lung disease (RLD). OLD ischaracterized by an increase in airway resistance, because ofthickening, partial blockage and narrowing of the airways,as is commonly seen in COPD. RLD is characterized by areduction on vital capacity (VC), because of alternations inthe lung parenchyma, disease of the pleura or the chest wall,or neuromuscular apparatus.

To perform these simulations, initial values of state vari-ables are set as in Table I according to reported values offunctional residual capacity (FRC) in normal, obstructive andrestrictive lung disease [16]. Values of model parameters arefixed to represent the three conditions (Table II). OLD isassumed to be an isolated airway disease, so that modelparameters of the alveolar space are assumed to be normal.RLD is assumed to be an isolated alveolar disease, so that themodel parameters of the airways are assumed to be normal.Values of k1, k2, k3, a, b, Vcmax, k4 for airway model pa-rameters are taken from the reference [6]. Values of a, b, c, dfor normal subjects and OLD patients are set or estimatedfrom textbook values [16] using an average elastance ofalveolar space, i.e. 5 cmH2O/L [16]. For patients with RLDthese values are obtained from the work of Pereira et al [17].

Chest wall elastance (Ecw) is assumed to be normal in allcases, i.e. 5 cmH2O/L [16].

TABLE IINITIAL VALUES FOR STATE VARIABLES.

Va(0)(L) Vtc(0)(L) Vc(0)(L)Normal 3.0000 3.0748 0.0748OLD 3.5000 3.5689 0.0689RLD 1.5000 1.5735 0.0735

For each case (normal, OLD, and RLD) the model is usedto simulate flow and volume curves during forced respiration.Four simulations are performed for each case, inspiratorymuscle pressure is fixed at -25 cmH2O and expiratory musclepressure is set to 0, 30, 40, and 50 cmH2O, respectively. Toverify the model, these simulations can be compared to theresults of standard lung function tests.

Figure 2 illustrates model simulated flow volume curvesduring forced inspiration and expiration. Forced inspirationstarts at the FRC (point A), and ends at total lung capacity(TLC) (point B). Relaxed expiration returns to point A, andforced expiration ends at the RC (point C). The volumebetween point B and C is the forced vital capacity (FVC),and the forced expired volume in the first second is labeledFEV1. In clinical practice, two measurements are often usedto characterize obstructive and restrictive lung disease. Theseare a relative FEV1 (FEV1,r) calculated as FEV1 divided byFVC; and a relative FVC (FVCr) calculated as FVC dividedby the normal value of FVC. Model simulated values ofFEV1,r and FVCr are given in table III. The value of FVCin normal conditions used in calculating FVCr is assumed tobe that obtained from model simulation of forced expirationin normal subject with expiratory muscle pressure (Pm) equalto 50 cmH2O.

TABLE IIIRELATIVE FEV1 AND FVC AS FRACTION OF NORMAL

Pm = 30cmH2O Pm = 40cmH2O Pm = 50cmH2OFEV1,r FVCr FEV1,r FVCr FEV1,r FVCr

Normal 0.82 0.98 0.83 0.99 0.84 1.00OLD 0.35 0.79 0.35 0.80 0.35 0.81RLD 0.96 0.48 0.96 0.48 0.97 0.48

Figure 2(a) illustrates a flow-volume curve of forcedventilation in a normal subject. The maximal expiratory flow(MEF) is about 8 L/s, and is only achieved during forcedexpiration. The peak expiratory flow obtained in each forcedexpiration varies only slightly when increasing expiratoryeffort, i.e. when increasing Pm from 30 to 50 cmH2O, astandard result of lung function tests on normal subjects.The values of FEV1,r (Table III), 0.82-0.84, for the normalsubject are consistent with reported values [16]. Figure 2(b)illustrates forced ventilation in a patient with OLD. The sim-ulated value of MEF is about 1.8 L/s, substantially lower thannormal. MEF is almost achieved during passive expiration(Pm = 0) simulating flow limitation in relaxed expiration,a simulation which is consistent with clinical findings insevere COPD patients [13]. The calculated value of FEV1,r

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TABLE IIMODEL PARAMETER VALUES FOR NORMAL, OLD AND RLD.

k1 k2 k3 a b Vcmax k4 a b c d Ecw

cmH2O s/L cmH2O s2/L2 cmH2O s/L cmH2O cmH2O L cmH2O s L L cmH2O cmH2O cmH2O/L

Normal 0.5 0.2 0.2 0.35 2.0 0.1 9.5 1.3 5.2 8.0 4.0 5.0OLD 0.5 0.2 0.2 1.05 6.0 0.1 47.5 1.3 5.2 8.0 4.0 5.0RLD 0.5 0.2 0.2 0.35 2.0 0.1 9.5 1.0 2.6 13.3 5.84 5.0

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 66

4

2

0

2

4

6

8

Flow

(L/s)

Lung volume (L)

AC B

Pm=50

Pm=40

Pm=30

Pm=0

MEF

Pm=25

Normal

(a)

2.5 3 3.5 4 4.5 5 5.5 6 6.52

1.5

1

0.5

0

0.5

1

1.5

2

Lung volume (L)

Pm=50Pm=40

Pm=30

MEF

Pm=0A B C

Pm=25

OLD

(b)

1 1.5 2 2.5 34

3

2

1

0

1

2

3

4

5

6

Lung volume (L)

MEFPm=50

Pm=40

Pm=30

Pm=0A BC

Pm=25

RLD

(c)Fig. 2. Model simulated flow-volume profiles during forced inspiration and expiration. Inspiratory and expiratory flow are represented using negative andpositive values respectively. FEV1 for an expiratory muscle pressure of 40 cmH20 is indicated by a circle. Solid lines represent forced inspiration at Pm= -25 cmH20 and forced expiration at Pm = 50 cmH2O. Dotted lines represent forced expiration at Pm = 40 cmH2O. Dashed lines represent forcedexpiration at Pm= 30 cmH2O. Dashed-dotted lines represent passive expiration. Subplots illustrate flow-volume curves for (a) a normal subject; (b) anOLD patient; and (c) a RLD patient.

and FVCr (Table III) are around 0.35 and 0.80 respectively,typical of OLD patients [16]. Figure 2(c) illustrates forcedventilation in a patient with RLD. MEF is close to normal,FEV1,r is higher than normal and FVCr is reduced to 0.48(Table III), typical of OLD patients [16].

IV. CONCLUSIONThis paper has described a model of respiratory mechanics

modified from previous models built using the functionalapproach. By including representation of nonlinear pressure-volume curve of alveolar space and muscle pressure, themodel can be used to simulate forced respiration. The resultsof these simulations are consistent with the usual results oflung function test in normal subjects and in patients withOLD and RLD, illustrating that quite complex pathologiescan be simulated with relatively simple models. Furtherwork is required to determine whether unique values canbe obtained for all model parameters in the clinical setting,enabling tuning of the model to the individual patient.

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[15] A. Lumb, Nunns Applied Respiratory Physiology. U.K: Butterworth& Co., 1997.

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[17] C. Pereira, B. Julien, S. Rosselli, E. Combourieu, C. Pommier,J. Perdrix, J. Richard, M. Badet, S. Gaillard, F. Philit, and C. Guerin,Sigmoidal equation for lung and chest wall volume-pressure curves inacute respiratory failure, J Appl Physiol, vol. 95, no. 5, pp. 206471,2003.

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