Stability analysis of Human Respiratory SystemTanmay Pal, Karan Jain and Srinivasu Maka

Department of Electrical EngineeringIndian Institute of Technology

Kharagpur, IndiaEmail: tanmaypal@iitkgp.ac.in,karanjain@ee.iitkgp.ernet.in,maka@ee.iitkgp.ernet.in

Abstract—Chemical Regulation of Human Respiratory Systemis a complex dynamical system. Primary function of this systemis to regulate gas concentrations in the Blood. It is modeled byNonlinear Delay Differential Equations. Several dynamical mod-els for respiratory regulation are available in the literature, whichare broadly classified as Comprehensive and Minimal Models.Minimal models are derived from the comprehensive modelswith some assumptions and are linearized for simplified analysis.Current literatures in these mathematical modeling approachesare directed towards describing pathological conditions, givinglittle thrust for the prediction.

Analyses of these systems are not straightforward becauseof the variable time delay. From the system’s point of view,increasing delay makes the system oscillatory. Physiologically,delay depends on the blood flow. In this model, delay in feedbackloop of alveolar oxygen and carbon dioxide partial pressuresmake the system oscillatory, causing discomfort or fatal to thesubject. Present analysis is motivated to find out the range ofdelay for comfortable operation of the system.

I. INTRODUCTION

Modeling of Respiratory system dates back to early nine-teenth century, when Haldane [1] and Bohr [2] discoveredOxygen and Carbon Dioxide transport phenomenon separately.Since then, several models have been proposed to studythis system. Respiratory models can be classified as eitherComprehensive or Minimal models. This classification comesfrom the fact that, respiratory system works with synergy withother systems. Altogether these systems maintain homeostasisof the body. Analysis of coupled system is a difficult task andoften stability analysis is done on a simpler model and therebyinfer about the whole system.

Several models on Respiratory System have been publishedin the literature. Model proposed by Grodins, Buell & Bart [3]is one of the pioneering comprehensive respiratory models,where four compartments have been considered. During thesame time, Milhorn et. al. [4], Longobardo, Cherniack &Fishman [5], Yamamoto & Hori [6] had also developed similarcomprehensive models of respiratory system. Main differencesof these models depend on the level of abstraction. In recenttimes, works by Ursino, Magosso & Avanzolini [7]-[8], Dahan,Nieuwenhuijs & Teppema [9] and Topor et. al.[10]-[11] areoriented towards the pathological conditions.

Apart from the comprehensive models, different minimalmodels have also been published in the literature. Glass &Mackey [12] and Carley & Shannon [13] formulated onedimensional model of respiratory control system. Cleave et.al. [14] studied bifurcation in two dimensional model of

respiratory system. Cooke & Turi [15] and Batzel & Tran [16]proposed two dimensional modeling of respiratory system.Here, model proposed by Khoo et. al. [17] is reduced to twodimensional model for delay dependent stability analysis.

Several control strategies have been reported in the literaturefor controlling Alveolar Minute Ventilation. Mostly two typesof control strategies have been used in the models. One ofthem is Gray type controller [18], where Oxygen and Car-bon Dioxide concentrations have summing effect (linear) onMinute Ventilation. Another type of control strategy is namedas Lloyd-Cunningham type controller [19], where multiplicityeffect (nonlinear) of these stimuli have been considered.

One of the important factors in this system is the effectof delay. Delay arises from blood flow which has an inverserelationship with cardiac output. Delay destabilizes the respi-ratory system and makes it oscillatory. It has been pointed outin the literature that congestive heart failure causes periodicbreathing [20], [21]. In this paper, a novel approach has beendeveloped to find the critical value of the delay for which thesystem remain stable. Ensuring delay not crossing the upperlimit is an important task for normal operation of the system.

This paper is organized as follows. First mathematicalmodeling of respiratory system with reduced dimension isdiscussed. Then the technique for finding poles of delaydifferential equations using Lambert W Function is considered.Finally this technique is applied for the linearized respiratorymodel for both normal and Congestive Heart Failure (CHF)cases. It is also verified from the numerical simulation byvarying the time delay. Simulation results show that these statevariables become oscillatory as the effect of increased delay,when it crosses the threshold.

II. MATHEMATICAL MODELING

Respiratory System is considered with three compartmentsfor modeling purpose. They are Alveolar, Brain and Tissuecompartments. Alveolar compartment works as the gatewayto external environment. Blood carries respiratory gases tothe brain & tissue compartments and they are assumed tobe constant here for simplification. There are two sensorylocations inside the body. One is Central Chemoreceptors,located near the brain and another one is Peripheral Chemore-ceptor, located near Aortic arch. Central receptors sense CO2,whereas peripheral receptors detect both O2 and CO2. Brainand Tissue compartments utilize O2 and generate CO2 fromthe metabolism. This mechanism is depicted in Figure 1.

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VACO2x1 (t) = 863QKCO2(PVCO2

− x1(t)) + VI(PICO2− x1(t))

VAO2x2 (t) = 863Q(MV PVO2

−MAx2(t) +BV −BA) + VI(PIO2− x2(t))

VI(t) = EF(GP e

−0.05x2(t−τ)(x1(t− τ)− IP ))

(1)

A =

− 863QKCO2−EFGP (IP−x1)e−0.05x2

VACO2

0

0 − 863QMA−EFGP (IP−x1)e−0.05x2

VAO2

(x1(0),x2(0))

Ad =

EFGP

(PICO2

−x1

)VACO2

e0.05x2

0.05EFGP (IP−x1)(PICO2

−x1

)VACO2

e0.05x2

EFGP

(PIO2

−x2

)VAO2

e0.05x2

0.05EFGP (IP−x1)(PIO2

−x2

)VAO2

e0.05x2

(x1(0),x2(0))

(2)

Respiratory models vary by levels of abstraction and as-sumptions. There are models in the literature with twenty sixstate variables considering additional compartments with bothO2 and CO2. Khoo et. al. [17] proposed three compartmentsand five states for their model. In the reduced order model,Batzel & Tran [16] considered only Alveolar compartmentand thereby reducing the dimension of the problem. It is hav-ing Alveolar Oxygen

(PAO2

)and Carbon Dioxide

(PACO2

)partial pressures as the state variables, neglecting centralchemoreceptor loop. Diagram of this model is shown in Figure1, inside the dotted box. The corresponding state equations ofthis model are given in Eq. 1. Symbols and their descriptionsare given in Table I.

Fig. 1. Schematic of Respiratory System

Here, Minute ventilation is a function of alveolar oxygenand carbon dioxide only. Another aspect of this model is that,minute ventilation as a function of multiplicative interaction

of alveolar oxygen and carbon dioxide. Physiologically, thisrepresents only one sensing location inside the body and thesereceptors are responsible for measuring both Alveolar O2 andCO2. There is a delay involved in this process. As the bloodpasses through the Heart, Cardiac Output determines the delay.As these receptors are nearer to the alveolar compartmentcompared to Central receptors, proper functioning of thesesensory sites determines stability of the system. This modelcan be solved for equilibrium point, using the expression ofVI(t) of Eq. 1. Equilibrium point for this model is calculatedas, {

PACO2

PAO2

}eq

=

{x1 (0)x2 (0)

}=

{41 mmHg82 mmHg

}Model described by Eq. 1 is linearized around this equi-

librium point. For Linear Delay Differential Equations of theform shown in Eq. 3, A and Ad matrices are calculated andgiven in Eq. 2. After substituting numerical values, A and Admatrices are computed below and thereby constituting a lineardelay differential model of respiratory system with two statesand single delay. Using this linearized model, delay dependentstability analysis is done.

A =

[−10.27 0

0 −1.86

]

Ad =

[−6.84 2.1613.42 −4.24

]There are several tests to determine delay-dependent stabilityof a system. One of such method is Pseudo-delay method[22]. For a system to be delay-dependent stable, eigenvaluesof (A+Ad) must be stable and (A−Ad) must be unsta-ble. Here eigenvalues of (A+Ad) becomes {−19.31;−3.89}and (A−Ad) are {−6.6; 5.6}. Hence this system is delay-dependent stable.

III. POLES OF DELAY DIFFERENTIAL EQUATION (DDE)Here, stability is treated through poles of the DDE and

Lambert W function is used to compute poles. Lambert W

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TABLE IDESCRIPTION OF SYMBOLS [ALL UNITS ARE IN STPD (STANDARD TEMPERATURE AND PRESSURE, DRY)]

Symbol Description UnitEF Dead space ventilationGP Peripheral Chemoreceptor gain (L/min mmHg)IP Apneic threshold for peripheral CO2 response (mmHg)KCO2 Dissociation constant for CO2 in lungs compartmentPACO2

Partial pressure of CO2 in alveolar air (mmHg)PAO2

Partial pressure of O2 in alveolar air (mmHg)PICO2

Partial pressure of CO2 in inspired air (mmHg)PIO2

Partial pressure of O2 in inspired air (mmHg)PVCO2

Partial pressure of CO2 in venous blood (mmHg)PVO2

Partial pressure of O2 in venous blood (mmHg)Q Cardiac Output (L/min)τ Transport delay from the lung compartment to the peripheral chemoreceptor (min)VACO2

Effective CO2 volume in the lung compartment (L)VAO2

Effective O2 volume in the lung compartment (L)VBCO2

Effective CO2 volume in the brain compartment (L)VI Minute Alveolar ventilation (L/min)MV , BV ,MA, BA are Constants for adjusting environmental and physiological factors

function is a complex valued function. It is defined as [23]-[24] multivalued solution of the equation α = W (α) eW (α).It is having infinite branches and the values for each branchis calculated numerically. Commercial softwares have inbuiltroutine to calculate this function for different branches1. For ageneralized system of delay differential equations, calculationof poles with a regulatory system model, as given in Eq. 3,are shown below. As a specific example, poles for respiratorysystem model are computed.

x(t) = Ax(t) +Adx(t− τ),x(t) = x0,x(t) = g(t),

t > 0t = 0t ∈ [−τ, 0)

(3)

Where A and Ad are n × n matrices and x is n × 1 vector.Characteristic Equation of this system is the determinant of,

S −A−Ade−Sτ = 0 (4)

After mathematical manipulation, Eq. 4 becomes,

τ(S −A)eSτe−Aτ = Adτe−Aτ (5)

In general, A & Ad do not commute, leading to the inequality,

τ(S −A)eSτe−Aτ 6= τ(S −A)e(S−A)τ (6)

To compensate for the inequality, an arbitrary matrix Φ isintroduced satisfying Eq. 7,

τ(S −A) eτ(S−A) = AdτΦ (7)

Eq. 7 can be written in terms of Lambert W function as,

W (AdτΦ) eW (AdτΦ) = AdτΦ (8)

Comparing Eq. 7 with Eq. 8, S can be written as,

S =1

τW (AdτΦ) +A (9)

1MATLAB provides the function value for scalar. For matrix case, eigendecomposition technique need to be used.

Here poles of the delay differential equations are eigenvaluesof the matrix S for different branches and Φ can be solvedfrom the following condition by using Eq. 9 and Eq. 5 as,

W (AdτΦ) eW (AdτΦ)+Aτ = Adτ (10)

However, finding Φ poses a problem of solving nonlinearalgebraic equations. This type of equation is solved usingnumerical optimization technique. Here, it has been solvedby using MATLAB routine fsolve. Another aspect of thisapproach is that, Lambert W function has infinite branches,making the solution of DDE as infinite series. However,stability of a delay system depends on principal branch, aseigenvalues of principal(0th) branch are the rightmost eigen-values of the system when Ad does not have repeated zeroeigenvalue. Hence, only eigenvalues of principal branch arecomputed here.

IV. RESULTS

Matrix Ad in the respiratory system considered, is rankdeficient and hence Φ matrix is needed to be calculated. Thiswas calculated using MATLAB routine fsolve. All simulationswere done using MATLAB 7.11 on 32 bit computer.

For a system of linear DDE, number of eigenvalues becomeinfinite, but here the eigenvalues for principal branch havebeen calculated. Using this technique, poles for the linearizedmodel is determined, and it is tabulated in Table II. As thedelay increases, both the poles move towards the imaginaryaxis. At the delay of 0.29 min, at least one pole enters intothe right half plane. Trajectory of the poles with increasingdelay is shown in Fig. 2. At this point, the oscillation beginsand as the delay increases, more poles continue to cross theimaginary axis.

It was postulated that congestive heart failure (CHF) willincrease the delay in blood flow and thereby make the systemoscillatory. Literature shows, maximum allowable delay foran adult person is 60 s [25]. Evidently, allowable delay

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TABLE IIPOLES OF PRINCIPAL BRANCH FOR LINEARIZED MODEL

delay(min) 0.1 0.15 0.2 0.28

poles

{−4.87 + i16.92

−4.23− i1.70× 10−10

} {−1.91 + i12.69

−4.38 + i1.81× 10−10

} {−0.75 + i10.15

−4.51 + i1.01× 10−09

} {0.02 + i7.69

−4.69 + 7.28× 10−10

}

Fig. 2. Movement of Poles in complex plane

must come below this range. Here, the limit for stability iscalculated as 0.28 min (16.8 s). From a theoretical pointof view, it can be interpreted that, as the delay crosses thethreshold limit, sustained oscillations would be producedin Alveolar Oxygen, Carbon Dioxide partial pressure andminute ventilation. However, in a physiological system,these quantities can never become unstable, and variationsof those parameters must be within a comfortable region.This region can be calculated as 0.27 min(16.2 s), at whichcondition, at least one pole is near to the imaginary axis.When this pole crosses the imaginary axis, the systembecomes oscillatory. These oscillations are manifested bycheyne-stokes respiration, which is an oscillatory pattern ofbreathing.

Simulation of the system using DDE23 is given in Figure- 3. Figure - 3(a) shows simulation without oscillation fornominal delay, and Figure - 3(b) shows oscillations due toincreased delay.

V. CONCLUSION

Linearized minimal model with single time delay for res-piratory system is analyzed and simulated using an iterativemethod. These results demonstrate stability of the human res-piratory system on the complex plane in a simplified manner.It also shows, effect of cardiovascular system on the stabilityproperty of respiratory system. As the delay increases, it willmake respiratory system oscillatory. However, there is anothersensory location, for which, critical delay calculation is also

(a) Stable Solution(τ=0.15 min)

(b) Oscillatory Solution(τ=0.28 min)

Fig. 3. Solution of Minimal Model

necessary, and this method is not suitable for system with morethan one delay. Nevertheless, this method is used to analyzeand verify a simplified respiratory model and thereby, effect ofcardiovascular system on respiratory system is demonstrated.

REFERENCES

[1] J. S. Haldane and J. G. Priestley, “The regulation of the lung-ventilation,”The Journal of Physiology, vol. 32, no. 3-4, pp. 225–266, 1905.

[2] C. Bohr, K. Hasselbalch, and A. Krogh, “Ueber einen in biologischerbeziehung wichtigen einfluss, den die kohlensaurespannung des blutesauf dessen sauerstoffbindung ubt1,” Skandinavisches Archiv Fur Physi-ologie, vol. 16, no. 2, pp. 402–412, 1904.

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[3] F. S. Grodins, J. Buell, and A. J. Bart, “Mathematical analysis anddigital simulation of the respiratory control system,” Journal of AppliedPhysiology, vol. 22, no. 2, pp. 260–276, 1967.

[4] H. T. Milhorn Jr., R. Benton, R. Ross, and A. C. Guyton, “A math-ematical model of the human respiratory control system,” BiophysicalJournal, vol. 5, no. 1, pp. 27–46, 1965.

[5] G. S. Longobardo, N. S. Cherniack, and A. P. Fishman, “Cheyne-stokes breathing produced by a model of the human respiratory system,”Journal of Applied Physiology, vol. 21, no. 6, pp. 1839–1846, 1966.

[6] W. Yamamoto and T. Hori, “Phasic air movement model of respira-tory regulation of carbon dioxide balance,” Computers and BiomedicalResearch, vol. 3, no. 6, pp. 699–717, 1970.

[7] M. Ursino, E. Magosso, and G. Avanzolini, “An integrated model of thehuman ventilatory control system: the response to hypoxia,” ClinicalPhysiology, vol. 21, no. 4, pp. 465–477, 2001.

[8] ——, “An integrated model of the human ventilatory control system:the response to hypercapnia,” Clinical Physiology, vol. 21, no. 4, pp.447–464, 2001.

[9] A. Dahan, D. Nieuwenhuijs, and L. Teppema, “Plasticity of centralchemoreceptors: Effect of bilateral carotid body resection on centralco2 sensitivity,” PLoS Med, vol. 4, no. 7, pp. 1195–1204, 2007.

[10] Z. Topor, M. Pawlicki, and J. Remmers, “A computational modelof the human respiratory control system: Responses to hypoxia andhypercapnia,” Annals of Biomedical Engineering, vol. 32, no. 11, pp.1530–1545, 2004.

[11] Z. L. Topor, K. Vasilakos, M. Younes, and J. E. Remmers, “Modelbased analysis of sleep disordered breathing in congestive heart failure,”Respiratory Physiology & Neurobiology, vol. 155, no. 1, pp. 82–92,2007.

[12] M. Mackey and L. Glass, “Oscillation and chaos in physiological controlsystems,” Science, vol. 197, no. 4300, pp. 287–289, 1977.

[13] D. W. Carley and D. C. Shannon, “A minimal mathematical model ofhuman periodic breathing,” Journal of Applied Physiology, vol. 65, no. 3,pp. 1400–1409, 1988.

[14] J. Cleave, M. Levine, P. Fleming, and A. Long, “Hopf bifurcations andthe stability of the respiratory control system,” Journal of TheoreticalBiology, vol. 119, no. 3, pp. 299 – 318, 1986.

[15] K. L. Cooke and J. Turi, “Stability, instability in delay equationsmodeling human respiration,” Journal of Mathematical Biology, vol. 32,pp. 535–543, 1994.

[16] J. Batzel and H. Tran, “Stability of the human respiratory control systemi. analysis of a two-dimensional delay state-space model,” Journal ofMathematical Biology, vol. 41, pp. 45–79, 2000.

[17] M. C. Khoo, R. E. Kronauer, K. P. Strohl, and A. S. Slutsky, “Factorsinducing periodic breathing in humans: a general model,” Journal ofApplied Physiology, vol. 53, no. 3, pp. 644–659, 1982.

[18] J. S. Gray, “The multiple factor theory of the control of respiratoryventilation,” Science, vol. 103, no. 2687, pp. 739–744, 1946.

[19] D. J. C. Cunningham and B. B. Lloyd, “A quantitative approach tothe regulation of human respiration,” in The Regulation of HumanRespiration, D. J. C. Cunningham and B. B. Lloyd, Eds. Oxford,UK: Blackwell Scientific Publications, 1963, pp. 331–349.

[20] U. Corra, M. Pistono, A. Mezzani, A. Braghiroli, A. Giordano, P. Lan-franchi, E. Bosimini, M. Gnemmi, and P. Giannuzzi, “Sleep and exer-tional periodic breathing in chronic heart failure: Prognostic importanceand interdependence,” Circulation, vol. 113, no. 1, pp. 44–50, 2006.

[21] T. P. Olson and B. D. Johnson, “Quantifying oscillatory ventilationduring exercise in patients with heart failure,” Respiratory Physiology& Neurobiology, vol. 190, no. 0, pp. 25 – 32, 2014.

[22] K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems(Control Engineering). Birkhauser Boston, 2003.

[23] R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth, “On thelambertw function,” Advances in Computational Mathematics, vol. 5,pp. 329–359, 1996.

[24] S. Yi, P. Nelson, and A. Ulsoy, Time-Delay Systems: Analysis andControl Using the Lambert W Function. World Scientific, 2010.

[25] T. Young, M. Palta, J. Dempsey, J. Skatrud, S. Weber, and S. Badr, “Theoccurrence of sleep-disordered breathing among middle-aged adults,”New England Journal of Medicine, vol. 328, no. 17, pp. 1230–1235,1993.

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