In this paper, we develop a framework for determining optimal respiratory airflow patterns for a multicompartment lung mechanics system with nonlinear resistance and compliance parameters. First, a nonlinear multicompartment lung mechanics model that accounts for nonlinearities in both the airway resistances and the lung compliances is developed. In particular, we assume that the resistive losses are characterized by a Rohrer-type model with resistive losses defined as the sum of linear and quadratic terms of the airflow. The proposed model is more realistic than those presented in the literature, since it takes into account the heterogeneity of lung anatomy and function as well as the nonlinearity of lung resistance and compliance parameters. This model can be used to provide a better understanding of pulmonary function as well as the process of mechanical ventilation. Next, using the proposed nonlinear multicompartment lung model, we develop a framework for determining optimal respiratory airflow patterns. Specifically, an optimization criterion that involves the minimization of the oxygen consumption of the lung muscles and lung volume acceleration for the inspiratory phase, and the minimization of the elastic potential energy and rapid airflow rate changes for the expiratory phase is formulated and solved. The solution to the formulated optimization problem is derived using classical calculus of variation techniques. Finally, several illustrative numerical examples are presented to illustrate the efficacy of the proposed nonlinear multicompartment lung model and the corresponding optimal airflow patterns. Comparison with experimental data shows that our nonlinear resistance model predicts the airflow patterns more accurately than linear resistance models. Moreover, the optimization criterion used in this paper also provides a more accurate prediction of the optimal airflow patterns.

References

References
1.
Campbell
,
D.
, and
Brown
,
J.
,
1963
, “
The Electrical Analogue of the Lung
,”
British J. Anaesth.
,
35
(
11
), pp.
684
692
.
2.
Wald
,
A. A.
,
Murphy
,
T. W.
, and
Mazzia
,
V. D.
,
1968
, “
A Theoretical Study of Controlled Ventilation
,”
IEEE Trans. Biomed. Eng.
,
15
(
4
), pp.
237
248
.
3.
Epstein
,
M. A. F.
, and
Epstein
,
R. A.
,
1979
, “
Airway Flow Patterns During Mechanical Ventilation of Infants: A Mathematical Model
,”
IEEE Trans. Biomed. Eng.
,
26
(
5
), pp.
299
306
.
4.
Barbini
,
P.
,
1982
, “
Non-Linear Model of the Mechanics of Breathing Applied to the Use and Design of Ventilators
,”
ASME J. Biomed. Eng.
,
4
(
4
), pp.
294
304
.
5.
Marini
,
J. J.
, and
Crooke
,
P. S.
,
1993
, “
A General Mathematical Model for Respiratory Dynamics Relevant to the Clinical Setting
,”
Am. Rev. Respir. Dis.
,
147
(
1
), pp.
14
24
.
6.
Crooke
,
P. S.
,
Hota
,
S.
,
Marini
,
J. J.
, and
Hotchkiss
,
J. R.
,
2003
, “
Mathematical Models of Passive, Pressure-Controlled Ventilation With Different Resistance Assumptions
,”
Math. Comput. Model.
,
38
(
5
), pp.
495
502
.
7.
Chellaboina
,
V.
,
Haddad
,
W. M.
,
Li
,
H.
, and
Bailey
,
J. M.
,
2010
, “
Limit Cycle Stability Analysis and Adaptive Control of a Multi-Compartment Model for a Pressure-Limited Respirator and Lung Mechanics System
,”
Int. J. Control
,
83
(
5
), pp.
940
955
.
8.
Jonson
,
B.
, and
Svantesson
,
C.
,
1999
, “
Elastic Pressure–Volume Curves: What Information Do They Convey?
Thorax
,
54
(
1
), pp.
82
87
.
9.
Crooke
,
P. S.
,
Marini
,
J. J.
, and
Hotchkiss
,
J. R.
,
2002
, “
Modeling Recruitment Maneuvers With a Variable Compliance Model for Pressure Controlled Ventilation
,”
J. Theor. Med.
,
4
(
3
), pp.
197
207
.
10.
Volyanskyy
,
K. Y.
,
Haddad
,
W. M.
, and
Bailey
,
J. M.
,
2011
, “
Pressure- and Work-Limited Neuroadaptive Control for Mechanical Ventilation of Critical Care Patients
,”
IEEE Trans. Neural Networks
,
22
(
4
), pp.
614
626
.
11.
Otis
,
A. B.
,
Fenn
,
W. O.
, and
Rahn
,
H.
,
1950
, “
Mechanics of Breathing in Man
,”
J. Appl. Physiol.
,
2
(
11
), pp.
592
607
.
12.
Wood
,
L.
,
Engel
,
L.
,
Griffin
,
P.
,
Despas
,
P.
, and
Macklem
,
P.
,
1976
, “
Effect of Gas Physical Properties and Flow on Lower Pulmonary Resistance
,”
J. Appl. Physiol.
,
41
(
2
), pp.
234
244
.
13.
Peslin
,
R.
,
Ying
,
Y.
,
Gallina
,
C.
, and
Duvivier
,
C.
,
1992
, “
Within-Breath Variations of Forced Oscillation Resistance in Healthy Subjects
,”
Eur. Respir. J.
,
5
(
1
), pp.
86
92
.
14.
Tomalak
,
W.
,
Peslin
,
R.
, and
Duvivier
,
C.
,
1998
, “
Variations in Airways Impedance During Respiratory Cycle Derived From Combined Measurements of Input and Transfer Impedances
,”
Eur. Respir. J.
,
12
(
6
), pp.
1436
1441
.
15.
Younes
,
M.
,
Kun
,
J.
,
Masiowski
,
B.
,
Webster
,
K.
, and
Roberts
,
D.
,
2001
, “
A Method for Noninvasive Determination of Inspiratory Resistance During Proportional Assist Ventilation
,”
Am. J. Respir. Crit. Care Med.
,
163
(
4
), pp.
829
839
.
16.
Avanzolini
,
G.
,
Barbini
,
P.
,
Bernardi
,
F.
,
Cevenini
,
G.
, and
Gnudi
,
G.
,
2001
, “
Role of the Mechanical Properties of Tracheobronchial Airways in Determining the Respiratory Resistance Time Course
,”
Ann. Biomed. Eng.
,
29
(
7
), pp.
575
586
.
17.
Crooke
,
P.
,
Kongkul
,
K.
,
Lenbury
,
Y.
,
Adams
,
A.
,
Carter
,
C.
,
Marini
,
J.
, and
Hotchkiss
,
J.
,
2005
, “
Mathematical Models for Pressure Controlled Ventilation of Oleic Acid-Injured Pigs
,”
Math. Med. Biol.
,
22
(
1
), pp.
99
112
.
18.
Younes
,
M.
,
Puddy
,
A.
,
Roberts
,
D.
,
Light
,
R.
,
Quesada
,
A.
,
Taylor
,
K.
,
Oppenheimer
,
L.
, and
Cramp
,
H.
,
1992
, “
Proportional Assist Ventilation: Results of an Initial Clinical Trial
,”
Am. Rev. Respir. Dis.
,
145
(
1
), pp.
121
129
.
19.
Laubscher
,
T.
,
Heinrichs
,
W.
,
Weiler
,
N.
,
Hartmann
,
G.
, and
Brunner
,
J.
,
1994
, “
An Adaptive Lung Ventilation Controller
,”
IEEE Trans. Biomed. Eng.
,
41
(
1
), pp.
51
59
.
20.
Dojat
,
M.
,
Brochard
,
L.
,
Lemaire
,
F.
, and
Harf
,
A.
,
1992
, “
A Knowledge-Based System for Assisted Ventilation of Patients in Intensive Care Units
,”
Int. J. Clin. Monit. Comput.
,
9
(
4
), pp.
239
250
.
21.
Sinderby
,
C.
,
Navalesi
,
P.
,
Beck
,
J.
,
Skrobik
,
Y.
,
Comtois
,
N.
,
Friberg
,
S.
,
Gottfried
,
S.
, and
Lindstrom
,
L.
,
1999
, “
Neural Control of Mechanical Ventilation in Respiratory Failure
,”
Nat. Med.
,
5
(
12
), pp.
1433
1436
.
22.
Li
,
H.
, and
Haddad
,
W. M.
,
2013
, “
Model Predictive Control for a Multicompartment Respiratory System
,”
IEEE Trans. Control Syst. Technol.
,
21
(
5
), pp.
1988
1995
.
23.
Hou
,
S. P.
,
Meskin
,
N.
, and
Haddad
,
W. M.
,
2014
, “
Output Feedback Sliding Mode Control for a Linear Multicompartment Lung Mechanics System
,”
Int. J. Control
,
87
(
10
), pp.
2044
2055
.
24.
Mead
,
J.
,
1960
, “
Control of Respiratory Frequency
,”
J. Appl. Physiol.
,
15
(
3
), pp.
325
336
.
25.
Yamashiro
,
S.
, and
Grodins
,
F.
,
1971
, “
Optimal Regulation of Respiratory Airflow
,”
J. Appl. Physiol.
,
30
(
5
), pp.
597
602
.
26.
Hämäläinen
,
R. P.
, and
Viljanen
,
A. A.
,
1978
, “
A Hierarchical Goal-Seeking Model of the Control of Breathing I–II
,”
Biological Cybernetics
,
29
(
3
), pp.
151
166
.
27.
Hämäläinen
,
R. P.
, and
Viljanen
,
A. A.
,
1978
, “
Modeling the Respiratory Airflow Pattern by Optimization Criteria
,”
Biol. Cybern.
,
29
(
3
), pp.
143
149
.
28.
Li
,
H.
, and
Haddad
,
W. M.
,
2012
, “
Optimal Determination of Respiratory Airflow Patterns Using a Nonlinear Multicompartment Model for a Lung Mechanics System
,”
Comput. Math. Methods Med.
,
2012
, pp.
1
11
.
29.
Proctor
,
D. F.
,
Hardy
,
J. B.
, and
McLean
,
R.
,
1949
, “
Studies of Respiratory Air Flow; Significance of the Normal Pneumotachogram
,”
Bull. Johns Hopkins Hosp.
,
85
(
4
), pp.
253
280
.
30.
Hämäläinen
,
R. P.
, and
Sipilä
,
A.
,
1984
, “
Optimal Control of Inspiratory Airflow in Breathing
,”
Optim. Control Appl. Methods
,
5
(
2
), pp.
177
191
.
31.
Weibel
,
E. R.
,
1963
,
Morphometry of the Human Lung
,
Academic
,
New York
.
32.
Hou
,
S. P.
,
Meskin
,
N.
, and
Haddad
,
W. M.
,
2014
, “
A General Multicompartment Lung Mechanics Model With Nonlinear Resistance and Compliance Respiratory Parameters
,”
American Control Conference
(
ACC
), Portland, OR, June 4–6, pp.
566
571
.
33.
Svantesson
,
C.
,
Sigurdsson
,
S.
,
Larsson
,
A.
, and
Jonson
,
B.
,
1998
, “
Effects of Recruitment of Collapsed Lung Units on the Elastic Pressure–Volume Relationship in Anaesthetised Healthy Adults
,”
Acta Anaesthesiol. Scandinavica
,
42
(
10
), pp.
1149
1156
.
34.
Bitzén
,
U.
,
Niklason
,
L.
,
Göransson
,
I.
, and
Jonson
,
B.
,
2010
, “
Measurement and Mathematical Modelling of Elastic and Resistive Lung Mechanical Properties Studied at Sinusoidal Expiratory Flow
,”
Clin. Physiol. Funct. Imaging
,
30
(
6
), pp.
439
446
.
35.
Dombi
,
J.
, and
Gera
,
Z.
,
2005
, “
The Approximation of Piecewise Linear Membership Functions and Łukasiewicz Operators
,”
Fuzzy Sets Syst.
,
154
(
2
), pp.
275
286
.
36.
Campbell
,
E. J. M.
,
Agostoni
,
E.
, and
Davis
,
J. N.
,
1970
,
The Respiratory Muscles: Mechanics and Neural Control
,
Lloyd-Luke
,
London
.
37.
McGregor
,
M.
, and
Becklake
,
M. R.
,
1961
, “
The Relationship of Oxygen Cost of Breathing to Respiratory Mechanical Work and Respiratory Force
,”
J. Clin. Investig.
,
40
(
6
), pp.
971
980
.
38.
Georgopoulos
,
D.
, and
Roussos
,
C.
,
1996
, “
Control of Breathing in Mechanically Ventilated Patients
,”
Eur. Respir. J.
,
9
(
10
), pp.
2151
2160
.
39.
Bonmarchand
,
G.
,
Chevron
,
V.
,
Jusserand
,
D.
,
Girault
,
C.
,
Moritz
,
F.
,
Leroy
,
J.
,
Pasquis
,
P.
, and
Chopin
,
C.
,
1996
, “
Increased Initial Flow Rate Reduces Inspiratory Work of Breathing During Pressure Support Ventilation in Patients With Exacerbation of Chronic Obstructive Pulmonary Disease
,”
Intensive Care Med.
,
22
(
11
), pp.
1147
1154
.
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