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Adaptive Control of Respiratory Mechanics
ADAPTIVE CONTROL OF RESPIRATORY MECHANICS
Raimo P. Hamalainen Systems Theory Laboratory, Institute of Mathematics Helsinki University of Technology Otaniemi, Finland
INTRODUCTION Hierarchy and adaptivity are typical characteristics of physiological control systems. In order to gain a deeper insight into the function of the regulators in man, the systemic approach and mathematical modelling have to be used. Large scale human systems are often divided into seemingly independent, in reality highly interconnected, blocks. When examining these blocks it is impossible to control all the relevant inputs and outputs simultaneously under experimental conditions. The only way to try to cope with the increasing number of variables in dynamical subsystems is to build a simulation model. Simulation with black box models cannot, however, explain the ability of the system to learn that is to adapt itself to new circumstances. To account for this adaptivity the model must include some kind of a performance criterion, a natural choise is a minir.ization mini~ization criterion. The system learns the new circumstances through a continuously active identification or self-optimizin~ self-o p timizin~ process correcting the operation of the system so that it would minimize imi ze the ob.iective-functional ob.i ecti ve-functional in question un~er un -.er the possible constraints on the control variables. variables .
not deviate much from a sine-curve. None of the previous models have been able to predict any changes in the individual respiratory variables of test subjects. In theRe papers the aim has merely been to prove that breathinv breathir.v is optimal witH regard to minimum effort or maximum gas exchange. Our main hypothesis is on the contrary that the system is adaptive, and so itis also supposed that it must have some criterion for decision making in adaption. When such subprocesses as breathing are concerned the criterion is probably not so clear and general as the principle of minimum energy expenditure, although the energy cost seems to be an essential component in it. It should be noted that in large scale systems partial opti~a opti~a need not necessarily give the total optimum. Further it is very difficult to define precisely all the tasks that different subsystems may perform and yet these tasks should be reflected in the corresponding performance criteria. At the first stage we have tried to find a physiologically meaningful performance criterion expressed in mechanical entities, which could succesfully explain the spontaneous breathing of test subjects. To reach this it has been necessary to extend the number of variables studied and to make the asThe mechanical breathing process is a typically adapsumptions less restrictive and more detailed. In the tive system, whose main purpose is to meet with the earlier studies for instance the differences in the ventilatory dde~and e~and ~~t~~i~ed ~~t, ~-i !')en bv the chemical state of duration of inspiration and expiration and the active the bleod. The desired ventilation ventila tion level can, however, role of muscles in the expiratory phase were neglectbe produced ma~y produce d in ma~ y ways,thus a decision is needed to ed. Moreover as far as we know this is the first time choose between the different alternative control stra- that a model is used to study the control of rib cage tegies for the r.ec ~echanical hanical system. Our work is concenand abdomen with respect to adaptivity. trated in the th e study of o f the adaptive nature of the system. Its performance principles are being cleared up A HL~ARCHICAL HI~~ARCHICAL MODEL ~~OD EL with the aid of mathe~atical o f ma t he~at i cal modelling. The models we In the ~odel ~ odel the control of respiratory mechanics is have develoDe develo~ed d will be dealt with in this paper. thought to be built up of two levels. The lower level criterion determines the the flow flow pattern pattern for for the the cycle cycle ~.10DELLHiG 'lI'E ADAPTIVITY( 0/) cri terion determines ~OD ELLING THE ADAPTI VI TY , (5) with the durations and tidal volume fixed, while the . (0 ) ( 5) Otis , ", ' The early work of Pohrer ' and O tl.S et al. were based h ·, 1 . f , "" , '~ l.gher evel crlterlon crl.terl.on el.VeS ' T 19her elVeS the overall rattern 0 on the mlnlrr.um ml.nl.ffium reSDlratorv reS~l.ratorv work rate crl crl. terl terl.on hey b th' t' f ' th l I t 'I d d on,. .~hey th ' ' I t' t ' used a sinusoidal ~ea lng l.ng sa lS l.S yl.ng e a,veo ar ~en l,a l.,a lon l.on eman sinusoidal ~ir flo~ shape and found t~at there ylng , t d t' th' f ~ d(4) h d (V A)· ~attern lS l.S descnbed descrl.bed ln l.n the r.odel ~odel by t ' 1 b \1 ), The overall Dattern exl.S e ~n,op l.rr.a rea, ea ~,owe inspir~tion (t ), expiration (t ) and ens ~n, op lrr:a rea , ~ng req~enc~,. req,:enc~" ~, m;e duration of inspir~tion 2 that a mlnlmur.: ml.nl.mur.: force ' am~ll.tude crlterlon crl.terl.on would glve gl.ve amDlltude + .... ,pause (t) , t h e operatln.,: operatl.n.-: eve 1 (V )( t h e c h ange 0 f + ~ -. .. v s. ( E~rther development was made by wl.ddl.wlddl- FRC 'lOb ' better resul resulvs.(E~rther FRC 1 ~ f th 'l'b 10 ) ~ t'd 1 1 b d"dl()) h ' Id'd 1 .. ' .· -eveI rom eequl.l.rl.umvaue an" an·, l l.a d"dl'O) Id~d ' . ' eequllrlumvaue a voume com ~ an ... a e , w 0 lnc l.nc u e contr~ o. dl.- (V ) and also the size of the airways (V )(see figure , ,,a o~ al.rwa~ alrwa~ dlmenSl.ons l.n the models and octal.ned ontl.mal l)T, 'f ' menSlons ln octalned. an ,'~ optlmal f eachD ee concep t 0 f preprogramrnl.ng preprogrammlng urlng d d lfOrMer ' . Th 0 cycIe d url.ng dead space vo volurr.e wit h ref~~ct re~D~ct to both criteria...•There ea space urr.e wl.th t~ both.crl.~erl.a here normal \1) normal breathing breathing is' is an an inherent inherent assumption assumption in in the the modmodhas been only on~y one attempt to find fl.nd.crl.terl.a~ el. It does not matter yet nothing is known about the criteria, whl.ch whic h air-flow corresponding neural identification could explain explal.n the shape of the th~ respiratory res~l.ratory al.r-flow existence of the corresronding curve. th minimum nunimum mean squared . .. ,.' . curve, The authors ended up Wl. with e~ c ~d ~erformance ~erformance function for since the prl.mary primary al.m aim is acceleration as a sug e~c~d networks, Sl.nce l.S first fl.r~t to ~o explain explal.n and underst~ d the the observed observed control control prl.ncl.nles. principles . humans. underst~d humans. In In aa closed closed form form their their solution solution is is aa parabolic parabolic and curve for coth inspiration and expiration and it does The system is assumed to have only one degree of
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freedom and a first order linear differential equation is used to describe the system dynamics. It is thought that the 'real' decision criterion in the respiratory control system is associated with the effectivity of oxygen transport and with the minimization of oxygen consumpt~on. The PV and p2 terms in the objective consumpt.ion. functions correspond dir~ctly dir~ctly to the oxygen cost of breathing, whereas the terms correspond to the gas exchange and to the efficiency of muscle shortening. The third and the fourth terms in the higher level objective-function account for the positive effects of resting, which may be related to the efficiency of the muscles or to the general comfortability. The weighting coefficients and B. that appear in the cost functions are cons~ant cons~ant inaividual parameters. !t ~t is clear that structural and other personal characteristic besides the system equation parameters must be taken into account in the model when we try to describe a universal biolop:ical functioning principle wi th the aid of mechanical entities. The lower level objective function to be minimized is
techiques. At first sight the upper level also looks like a dynamical optimization problem yet it is a static oroblem because of the traj2ctory traj 2ctory constraint V(t)=V*(t). Testing of the overall results of the model is being carried out at the moment. Only preliminary comDarison can be made between the model and the observed values for t " t and t , which shows good agreement 2 3 (table ,). The resemblance reseMblance o? the theoretical and individual experimental flow curves is also great ~nd ~he typical flat~ening flat~ening effect of resistiv: lo~~) lo~~) 1ng 1S demonstrated 1n the flow pattern Solut1ons . Solut10ns
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{J~I=f'(V2(t)+a'P(t)V(t))dt, {J~I=f'(V2(t)+a'P(t)V(t))dt, for
=
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0
II t,+t 2 2 (y2(t)+o2P (t))dt, for expiration ={ 2 It is assumed that the system equation
J
p(t) = K'V(t) + R(VD)'V(t) does not change during the cycle. The size of the airways VD is kept at a constant value, which is determined on the higher level. The end expiration level, where the inspiration starts is V Vo and the correspondo ing boundary condition to the proolem V(0)=V(t,+t 2 )=V O ' V(t,)=VO+VT ' V(0)=V(t,)=V(t +t )=0 l 2 are also depe~dent depe~dent on the higher level minimum. Moreover there is no flow during the pause period, V~t)=O t&[t'+~2,t,:t2+t3J. In the following the V~t)=O for t&[t'+~2,t,:t2+t3J. f1rst level Solut1on Solut10n wllI be denoted hy V*(t) = =V*(t;t"t 2 ,V O'V D,V T )·
First notes on the model and some experi~ental resultf31~ be. found in ref.(9),.a more detailed report 1S stlll st1l1 under preparat10n. preparat1on. THE LOWER LEVEL 110DEL WITH ~lITH TWO DEGREES OF FREEDOIl Althoul'h Althou!"h movements of the respiratory apnaratus are generally described only with one degree of freedom, in fact a change in the lung volume is a result of the independent movements of rib cage and abdoll'en abdoll"en so the sYf1~m fr7edo~ (at sYf1~m actually actuallY has two degrees of fr7edo~ least) . The lower level performance cr1ter1on, cr1terlon, which determined the flow pattern of breathing, can also be applied with this extended systerr modelI to syst"!lT. !'lode give optimal control strategies separat"!ly separat~ly for the chest wall and the abdominal muscles. The system consists now of the two parallel components, rib cage anJ abdomen, connected in series with the lung component and it is described by a set of linear first order differential equations. P (t)-p l(t) K V (t)+R V (t) rc p rc rc rc rc Pab(t)-Ppl(t) = KabVab(t)+RabVab(t) Ppl(t) = KIVl(t)+F.IVl(t), KIVI(t)+F.IVI(t),
where P II is the change from the resting pleural pressur~ Pre pressur~ difference with respect to at~osohere, at~oso~ere, . and P b are the pressures produced by the chest wall muscl~s and by the diaphragm and the abdominal musmuscl~s cles. The elastances Krc,K b,K and the damping faclI The higher level o~;ec~~~p i~ o";ec':~'!" function if' tors R ,F ,R b,R b,R! are assume~ assume~ constant. Reference Re ference valrc a . I 1 for the re at1ve volurr.e changes V , V b' VI are ues atlve volUIT.e J (t, ,t ,t ,V ,V(t)) = 2 3 D zero at the equilibrium (FRC) level. ~~e ~olume ~olume changes of the lungs are connected to those produced by the rib cage and the abdominal move~entsby move~entsby the constraint equation V1(t)=V (t)+V (t). It is assumed that at the berinn1ng berinnlng o~rnspir~fion o~rnspir~fion ~d at the t~e end of expiration the system is in a stationary position at level V so that neither rib cage nor abdominal volume is cRanging. This gives us th, thr following where P = inspiratory muscle pressure and? = exp1ra- boundary conditions i tory muscle pressure. Constraints to this problem are Vrc(O)=Vrc(t,+t2)=KabVO/(Y.rc+Kab) the systems equation, the optimality of the airflow V(t)=V*(t) and the alveolar ventilation equation Vab(0)=Vab(t,+t2)=KrcVo/(Krc+Kab) , a.eiven . constant VA=(V T-V D)/(t,+t 2 +t ), ,.here where VA is a.eiven. Vrc(O)=Vrc(t,+t2)=Vab(0)=Vab(t,+t2)=O for each breat~. breath. Th~ total non-elast1c res1stance R(V ) Vrc(O)=Vrc(t,+t2)=Vab( 0) =Vab(t,+t2)=O D has been divided into airway resistance RA' lung tissue Further at the end of inspiration the tidal volume 'T \'T resistance R Rrr and thoracic wall resistance R " so is reached and the total flow is zero yielding the T transversality conditions that R~VD)=R;rVJ)+RLT+RTW' R~VD)=R;rVJ)+RLT+RTW' .The rela~ionship.be~we7n rela~ionship .be~we7n anatom1cal dead space and a1rway a1nlay res1stance reS1 stance 1S Qf~~ved Qf~~ ved Vrc(t,)+Vab(t,)-VO=Vr and Vrc(t,)+Vab(t,)=o Vrc(t,)+Vat(t,)= o . Vrc(t')+Vab(t,)-VO=Vr in the manner presented by Widdicombe and ::adel ~ giving The lower level objective function will in this case R (V ) = 0.082 + 0.4 0 . 082 cm H20S H20s 0 .4 cm H 0s/1 be of the form A.D VJ - 0.0825 " 2 0 . 082 5 I II t,+t 2 ... ... The model 1S presented as a two level h1erarch1cal ~lerarch1cal sysJ L(V,V,V,V L(V,V ,V ,V b'V b'V b)dt where tem in figure " where the coordination variables are o rc rc rc a a a t"t 2 ,V O'V ,Vi and V*~t). V*~t). The lowe: level ~ro~lem ~ro~lem is solved ana£yt1cally ana£ytlcally w1th wlth the class1cal var1at10nal var1atlonal
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+p +y y22 +y y22 +r L =p +P +V )2 1 rc rc ab ab 11 rc 12 ab 13 rc ab 1= for inspiration tt[O,t,] tt[0,t ] 1 2 2 "2 "2 •••• 2 ·'2 ·'2 L2=Prc+Pab+Y21Vrc+Y22Vab+Y23(Vrc+Vab) for expirationtt[t 1 ,t 1+t ] 2 Here it has been more convenient to place the personal . . in a different position than the corweightings y .. w~ightings a0 .. . . It can be shown that the responding w~ightings formulated model has a tnique solution when not more than one of the coefficients Y1j and Y2 j is zero. Results from the model, for example in figure 2, correspond quite well to the flow and pressure curves ref.(1). t he comgiven in ref.( 1). The effect, that lowering the pliance of either part results in a smaller contributidal . volumf1~';le to that par~, par~, that is (2) tion to t~e tidal.volumf1~~e exper1ments 1S observed 1n ~n exper~ments ~s also found 1n ~n the model .
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SUMMARY Previous models of the adaptive control of respiratory mechanics have been inadequate to predict the respiratory pattern in detail, as the main variables that have been treated are respiratory frequency and dead space volume. A new hierarchical approach has been reported, which leads to significantly better understandres piratory mechanics. ing of the control principles of respiratory ins p iraThe independent control variables studied are inspiratory time, expiratory time, pause period, change of FRC-level, airway size and the flow pattern of the Hovements of the rib cage and the abdomen and cycle. Movements their contribution to the tidal volume are also treated.
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ACKNOWLEDGEMENTS The author wishes to acknowledge the support received from the Emil Aaltonen Foundation and the Foundation Automat i on, Helsinki. for Automation, REFERENCES EWSOH DAVIS,J.: (1) CAMPBELL,E.J.M.,AGOSTONI,E. and N NEWSOM Muscles Mechanics Mechanics and Neural Control The Respiratory Muscles edit .• Lloyd-Luke Ltd. ,London, 1970. 1970 . second edit.,Lloyd-Luke HAMALAI NEN,R.P. and RANTA,J. : (paper under preparapre para(2) HAMALAINEN,R.P. tion) . HAMALAINEN, R.P. and VILJANEN,A.A.:(paper VILJANEN,A.A. : (paper under pre(3) HAMALAINEN,R.P. paration) . ~~,J.: Control Control of respiratory frequency. frequen cy . J.Appl. (4) ~~,J.: 22: 3 , 1960, 1960 , pp 325 -336. Physiol. 22:3, pp.. 325-336. FENr: ,w. O. and RAHN,H.: Mechanics Mechanics of (5) OTIS,A.B., FENN,W.O. breath ing in man. J.Appl.Physiol. J. Appl.Physiol. £, £ , 1950,pp.5921950 ,pp. 592 breathing 607. ( 6 ) ROHRER,F.: Physiologie der Atembewegung. in: Hand(6) Nore.alen und un d Patologischen Patologische n P~ysiologie, Phys iologie, buch der lloIT-alen vo1.2, ed. e d . BETEE;A. 192 5 vol.2, BETP.E;A. et al., Springer Berlin, 1925 70-1 27. pp. 70-127. YAHASHIRO,S. H. and GRODINS,F.S.: GRODINS,F.S.: Optimal regula(7) Y~~SHIRO,S.~. res piratory airflow. J.Appl.Physiol. 30:5, 30 : 5 , tion of respiratory 1971, pp. 597-602. -(8) WIDDICOMBE,J.G. and NADEL, J.A.: Airway volume, airway resistance and work and force of breathing: theory . J.Appl.Physiol. 1A:5, 1R:5, 1963, ~~. P~. B63-868p theory. VILJANEN,A.A. : Coordination of neuro~;scular neuro~~scular ef(9) VILJANEN,A.A.: i n ventilation. Proceedings of the ferent systems in Respirat ory and LarInternational Symposium on Respiratory yngea~ Control Systems, London, January 1972 yngea~ (a~an g ed by the Royal College of Surgeons of (a~anged Englan d) . England)
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265
Raimo P. Hamalainen Systems Theory Laboratory, Institute of Mathematics Helsinki University of Technology Otaniemi, Finland
INTRODUCTION Hierarchy and adaptivity are typical characteristics of physiological control systems. In order to gain a deeper insight into the function of the regulators in man, the systemic approach and mathematical modelling have to be used. Large scale human systems are often divided into seemingly independent, in reality highly interconnected, blocks. When examining these blocks it is impossible to control all the relevant inputs and outputs simultaneously under experimental conditions. The only way to try to cope with the increasing number of variables in dynamical subsystems is to build a simulation model. Simulation with black box models cannot, however, explain the ability of the system to learn that is to adapt itself to new circumstances. To account for this adaptivity the model must include some kind of a performance criterion, a natural choise is a minir.ization mini~ization criterion. The system learns the new circumstances through a continuously active identification or self-optimizin~ self-o p timizin~ process correcting the operation of the system so that it would minimize imi ze the ob.iective-functional ob.i ecti ve-functional in question un~er un -.er the possible constraints on the control variables. variables .
not deviate much from a sine-curve. None of the previous models have been able to predict any changes in the individual respiratory variables of test subjects. In theRe papers the aim has merely been to prove that breathinv breathir.v is optimal witH regard to minimum effort or maximum gas exchange. Our main hypothesis is on the contrary that the system is adaptive, and so itis also supposed that it must have some criterion for decision making in adaption. When such subprocesses as breathing are concerned the criterion is probably not so clear and general as the principle of minimum energy expenditure, although the energy cost seems to be an essential component in it. It should be noted that in large scale systems partial opti~a opti~a need not necessarily give the total optimum. Further it is very difficult to define precisely all the tasks that different subsystems may perform and yet these tasks should be reflected in the corresponding performance criteria. At the first stage we have tried to find a physiologically meaningful performance criterion expressed in mechanical entities, which could succesfully explain the spontaneous breathing of test subjects. To reach this it has been necessary to extend the number of variables studied and to make the asThe mechanical breathing process is a typically adapsumptions less restrictive and more detailed. In the tive system, whose main purpose is to meet with the earlier studies for instance the differences in the ventilatory dde~and e~and ~~t~~i~ed ~~t, ~-i !')en bv the chemical state of duration of inspiration and expiration and the active the bleod. The desired ventilation ventila tion level can, however, role of muscles in the expiratory phase were neglectbe produced ma~y produce d in ma~ y ways,thus a decision is needed to ed. Moreover as far as we know this is the first time choose between the different alternative control stra- that a model is used to study the control of rib cage tegies for the r.ec ~echanical hanical system. Our work is concenand abdomen with respect to adaptivity. trated in the th e study of o f the adaptive nature of the system. Its performance principles are being cleared up A HL~ARCHICAL HI~~ARCHICAL MODEL ~~OD EL with the aid of mathe~atical o f ma t he~at i cal modelling. The models we In the ~odel ~ odel the control of respiratory mechanics is have develoDe develo~ed d will be dealt with in this paper. thought to be built up of two levels. The lower level criterion determines the the flow flow pattern pattern for for the the cycle cycle ~.10DELLHiG 'lI'E ADAPTIVITY( 0/) cri terion determines ~OD ELLING THE ADAPTI VI TY , (5) with the durations and tidal volume fixed, while the . (0 ) ( 5) Otis , ", ' The early work of Pohrer ' and O tl.S et al. were based h ·, 1 . f , "" , '~ l.gher evel crlterlon crl.terl.on el.VeS ' T 19her elVeS the overall rattern 0 on the mlnlrr.um ml.nl.ffium reSDlratorv reS~l.ratorv work rate crl crl. terl terl.on hey b th' t' f ' th l I t 'I d d on,. .~hey th ' ' I t' t ' used a sinusoidal ~ea lng l.ng sa lS l.S yl.ng e a,veo ar ~en l,a l.,a lon l.on eman sinusoidal ~ir flo~ shape and found t~at there ylng , t d t' th' f ~ d(4) h d (V A)· ~attern lS l.S descnbed descrl.bed ln l.n the r.odel ~odel by t ' 1 b \1 ), The overall Dattern exl.S e ~n,op l.rr.a rea, ea ~,owe inspir~tion (t ), expiration (t ) and ens ~n, op lrr:a rea , ~ng req~enc~,. req,:enc~" ~, m;e duration of inspir~tion 2 that a mlnlmur.: ml.nl.mur.: force ' am~ll.tude crlterlon crl.terl.on would glve gl.ve amDlltude + .... ,pause (t) , t h e operatln.,: operatl.n.-: eve 1 (V )( t h e c h ange 0 f + ~ -. .. v s. ( E~rther development was made by wl.ddl.wlddl- FRC 'lOb ' better resul resulvs.(E~rther FRC 1 ~ f th 'l'b 10 ) ~ t'd 1 1 b d"dl()) h ' Id'd 1 .. ' .· -eveI rom eequl.l.rl.umvaue an" an·, l l.a d"dl'O) Id~d ' . ' eequllrlumvaue a voume com ~ an ... a e , w 0 lnc l.nc u e contr~ o. dl.- (V ) and also the size of the airways (V )(see figure , ,,a o~ al.rwa~ alrwa~ dlmenSl.ons l.n the models and octal.ned ontl.mal l)T, 'f ' menSlons ln octalned. an ,'~ optlmal f eachD ee concep t 0 f preprogramrnl.ng preprogrammlng urlng d d lfOrMer ' . Th 0 cycIe d url.ng dead space vo volurr.e wit h ref~~ct re~D~ct to both criteria...•There ea space urr.e wl.th t~ both.crl.~erl.a here normal \1) normal breathing breathing is' is an an inherent inherent assumption assumption in in the the modmodhas been only on~y one attempt to find fl.nd.crl.terl.a~ el. It does not matter yet nothing is known about the criteria, whl.ch whic h air-flow corresponding neural identification could explain explal.n the shape of the th~ respiratory res~l.ratory al.r-flow existence of the corresronding curve. th minimum nunimum mean squared . .. ,.' . curve, The authors ended up Wl. with e~ c ~d ~erformance ~erformance function for since the prl.mary primary al.m aim is acceleration as a sug e~c~d networks, Sl.nce l.S first fl.r~t to ~o explain explal.n and underst~ d the the observed observed control control prl.ncl.nles. principles . humans. underst~d humans. In In aa closed closed form form their their solution solution is is aa parabolic parabolic and curve for coth inspiration and expiration and it does The system is assumed to have only one degree of
,,11 0
0
o
262
freedom and a first order linear differential equation is used to describe the system dynamics. It is thought that the 'real' decision criterion in the respiratory control system is associated with the effectivity of oxygen transport and with the minimization of oxygen consumpt~on. The PV and p2 terms in the objective consumpt.ion. functions correspond dir~ctly dir~ctly to the oxygen cost of breathing, whereas the terms correspond to the gas exchange and to the efficiency of muscle shortening. The third and the fourth terms in the higher level objective-function account for the positive effects of resting, which may be related to the efficiency of the muscles or to the general comfortability. The weighting coefficients and B. that appear in the cost functions are cons~ant cons~ant inaividual parameters. !t ~t is clear that structural and other personal characteristic besides the system equation parameters must be taken into account in the model when we try to describe a universal biolop:ical functioning principle wi th the aid of mechanical entities. The lower level objective function to be minimized is
techiques. At first sight the upper level also looks like a dynamical optimization problem yet it is a static oroblem because of the traj2ctory traj 2ctory constraint V(t)=V*(t). Testing of the overall results of the model is being carried out at the moment. Only preliminary comDarison can be made between the model and the observed values for t " t and t , which shows good agreement 2 3 (table ,). The resemblance reseMblance o? the theoretical and individual experimental flow curves is also great ~nd ~he typical flat~ening flat~ening effect of resistiv: lo~~) lo~~) 1ng 1S demonstrated 1n the flow pattern Solut1ons . Solut10ns
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{J~I=f'(V2(t)+a'P(t)V(t))dt, {J~I=f'(V2(t)+a'P(t)V(t))dt, for
=
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0
II t,+t 2 2 (y2(t)+o2P (t))dt, for expiration ={ 2 It is assumed that the system equation
J
p(t) = K'V(t) + R(VD)'V(t) does not change during the cycle. The size of the airways VD is kept at a constant value, which is determined on the higher level. The end expiration level, where the inspiration starts is V Vo and the correspondo ing boundary condition to the proolem V(0)=V(t,+t 2 )=V O ' V(t,)=VO+VT ' V(0)=V(t,)=V(t +t )=0 l 2 are also depe~dent depe~dent on the higher level minimum. Moreover there is no flow during the pause period, V~t)=O t&[t'+~2,t,:t2+t3J. In the following the V~t)=O for t&[t'+~2,t,:t2+t3J. f1rst level Solut1on Solut10n wllI be denoted hy V*(t) = =V*(t;t"t 2 ,V O'V D,V T )·
First notes on the model and some experi~ental resultf31~ be. found in ref.(9),.a more detailed report 1S stlll st1l1 under preparat10n. preparat1on. THE LOWER LEVEL 110DEL WITH ~lITH TWO DEGREES OF FREEDOIl Althoul'h Althou!"h movements of the respiratory apnaratus are generally described only with one degree of freedom, in fact a change in the lung volume is a result of the independent movements of rib cage and abdoll'en abdoll"en so the sYf1~m fr7edo~ (at sYf1~m actually actuallY has two degrees of fr7edo~ least) . The lower level performance cr1ter1on, cr1terlon, which determined the flow pattern of breathing, can also be applied with this extended systerr modelI to syst"!lT. !'lode give optimal control strategies separat"!ly separat~ly for the chest wall and the abdominal muscles. The system consists now of the two parallel components, rib cage anJ abdomen, connected in series with the lung component and it is described by a set of linear first order differential equations. P (t)-p l(t) K V (t)+R V (t) rc p rc rc rc rc Pab(t)-Ppl(t) = KabVab(t)+RabVab(t) Ppl(t) = KIVl(t)+F.IVl(t), KIVI(t)+F.IVI(t),
where P II is the change from the resting pleural pressur~ Pre pressur~ difference with respect to at~osohere, at~oso~ere, . and P b are the pressures produced by the chest wall muscl~s and by the diaphragm and the abdominal musmuscl~s cles. The elastances Krc,K b,K and the damping faclI The higher level o~;ec~~~p i~ o";ec':~'!" function if' tors R ,F ,R b,R b,R! are assume~ assume~ constant. Reference Re ference valrc a . I 1 for the re at1ve volurr.e changes V , V b' VI are ues atlve volUIT.e J (t, ,t ,t ,V ,V(t)) = 2 3 D zero at the equilibrium (FRC) level. ~~e ~olume ~olume changes of the lungs are connected to those produced by the rib cage and the abdominal move~entsby move~entsby the constraint equation V1(t)=V (t)+V (t). It is assumed that at the berinn1ng berinnlng o~rnspir~fion o~rnspir~fion ~d at the t~e end of expiration the system is in a stationary position at level V so that neither rib cage nor abdominal volume is cRanging. This gives us th, thr following where P = inspiratory muscle pressure and? = exp1ra- boundary conditions i tory muscle pressure. Constraints to this problem are Vrc(O)=Vrc(t,+t2)=KabVO/(Y.rc+Kab) the systems equation, the optimality of the airflow V(t)=V*(t) and the alveolar ventilation equation Vab(0)=Vab(t,+t2)=KrcVo/(Krc+Kab) , a.eiven . constant VA=(V T-V D)/(t,+t 2 +t ), ,.here where VA is a.eiven. Vrc(O)=Vrc(t,+t2)=Vab(0)=Vab(t,+t2)=O for each breat~. breath. Th~ total non-elast1c res1stance R(V ) Vrc(O)=Vrc(t,+t2)=Vab( 0) =Vab(t,+t2)=O D has been divided into airway resistance RA' lung tissue Further at the end of inspiration the tidal volume 'T \'T resistance R Rrr and thoracic wall resistance R " so is reached and the total flow is zero yielding the T transversality conditions that R~VD)=R;rVJ)+RLT+RTW' R~VD)=R;rVJ)+RLT+RTW' .The rela~ionship.be~we7n rela~ionship .be~we7n anatom1cal dead space and a1rway a1nlay res1stance reS1 stance 1S Qf~~ved Qf~~ ved Vrc(t,)+Vab(t,)-VO=Vr and Vrc(t,)+Vab(t,)=o Vrc(t,)+Vat(t,)= o . Vrc(t')+Vab(t,)-VO=Vr in the manner presented by Widdicombe and ::adel ~ giving The lower level objective function will in this case R (V ) = 0.082 + 0.4 0 . 082 cm H20S H20s 0 .4 cm H 0s/1 be of the form A.D VJ - 0.0825 " 2 0 . 082 5 I II t,+t 2 ... ... The model 1S presented as a two level h1erarch1cal ~lerarch1cal sysJ L(V,V,V,V L(V,V ,V ,V b'V b'V b)dt where tem in figure " where the coordination variables are o rc rc rc a a a t"t 2 ,V O'V ,Vi and V*~t). V*~t). The lowe: level ~ro~lem ~ro~lem is solved ana£yt1cally ana£ytlcally w1th wlth the class1cal var1at10nal var1atlonal
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+p +y y22 +y y22 +r L =p +P +V )2 1 rc rc ab ab 11 rc 12 ab 13 rc ab 1= for inspiration tt[O,t,] tt[0,t ] 1 2 2 "2 "2 •••• 2 ·'2 ·'2 L2=Prc+Pab+Y21Vrc+Y22Vab+Y23(Vrc+Vab) for expirationtt[t 1 ,t 1+t ] 2 Here it has been more convenient to place the personal . . in a different position than the corweightings y .. w~ightings a0 .. . . It can be shown that the responding w~ightings formulated model has a tnique solution when not more than one of the coefficients Y1j and Y2 j is zero. Results from the model, for example in figure 2, correspond quite well to the flow and pressure curves ref.(1). t he comgiven in ref.( 1). The effect, that lowering the pliance of either part results in a smaller contributidal . volumf1~';le to that par~, par~, that is (2) tion to t~e tidal.volumf1~~e exper1ments 1S observed 1n ~n exper~ments ~s also found 1n ~n the model .
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SUMMARY Previous models of the adaptive control of respiratory mechanics have been inadequate to predict the respiratory pattern in detail, as the main variables that have been treated are respiratory frequency and dead space volume. A new hierarchical approach has been reported, which leads to significantly better understandres piratory mechanics. ing of the control principles of respiratory ins p iraThe independent control variables studied are inspiratory time, expiratory time, pause period, change of FRC-level, airway size and the flow pattern of the Hovements of the rib cage and the abdomen and cycle. Movements their contribution to the tidal volume are also treated.
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ACKNOWLEDGEMENTS The author wishes to acknowledge the support received from the Emil Aaltonen Foundation and the Foundation Automat i on, Helsinki. for Automation, REFERENCES EWSOH DAVIS,J.: (1) CAMPBELL,E.J.M.,AGOSTONI,E. and N NEWSOM Muscles Mechanics Mechanics and Neural Control The Respiratory Muscles edit .• Lloyd-Luke Ltd. ,London, 1970. 1970 . second edit.,Lloyd-Luke HAMALAI NEN,R.P. and RANTA,J. : (paper under preparapre para(2) HAMALAINEN,R.P. tion) . HAMALAINEN, R.P. and VILJANEN,A.A.:(paper VILJANEN,A.A. : (paper under pre(3) HAMALAINEN,R.P. paration) . ~~,J.: Control Control of respiratory frequency. frequen cy . J.Appl. (4) ~~,J.: 22: 3 , 1960, 1960 , pp 325 -336. Physiol. 22:3, pp.. 325-336. FENr: ,w. O. and RAHN,H.: Mechanics Mechanics of (5) OTIS,A.B., FENN,W.O. breath ing in man. J.Appl.Physiol. J. Appl.Physiol. £, £ , 1950,pp.5921950 ,pp. 592 breathing 607. ( 6 ) ROHRER,F.: Physiologie der Atembewegung. in: Hand(6) Nore.alen und un d Patologischen Patologische n P~ysiologie, Phys iologie, buch der lloIT-alen vo1.2, ed. e d . BETEE;A. 192 5 vol.2, BETP.E;A. et al., Springer Berlin, 1925 70-1 27. pp. 70-127. YAHASHIRO,S. H. and GRODINS,F.S.: GRODINS,F.S.: Optimal regula(7) Y~~SHIRO,S.~. res piratory airflow. J.Appl.Physiol. 30:5, 30 : 5 , tion of respiratory 1971, pp. 597-602. -(8) WIDDICOMBE,J.G. and NADEL, J.A.: Airway volume, airway resistance and work and force of breathing: theory . J.Appl.Physiol. 1A:5, 1R:5, 1963, ~~. P~. B63-868p theory. VILJANEN,A.A. : Coordination of neuro~;scular neuro~~scular ef(9) VILJANEN,A.A.: i n ventilation. Proceedings of the ferent systems in Respirat ory and LarInternational Symposium on Respiratory yngea~ Control Systems, London, January 1972 yngea~ (a~an g ed by the Royal College of Surgeons of (a~anged Englan d) . England)
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265