Influence of Bronchial Diameter Change on the airflow dynamics Based on a Pressure-controlled Ventilation System Shuai REN1, Maolin CAI1, Yan SHI1, 2 , Weiqing Xu1 , Xiaohua Douglas Zhang3* 1 2
School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, CHINA
Beijing Engineering Research Center of Diagnosis and Treatment of Respiratory and Critical Care Medicine, Beijing Chaoyang Hospital, Beijing 100043, China 3 Faculty
of Health Sciences, University of Macau, Taipa, Macau.
SUMMARY Bronchial diameter is a key parameter that affects the respiratory treatment of mechanically ventilated patients. In this paper, to reveal the influence of bronchial diameter on the airflow dynamics of pressurecontrolled mechanically ventilated patients, a new respiratory system model is presented which combines multi-generation airways with lungs. Furthermore, experiments and simulation studies to verify the model are performed. Finally, through the simulation study, it can be determined that in airway generations 2 to 7, when the diameter is reduced to half of the original value, the maximum air pressure (maximum air pressure in lungs) decreases by nearly 16%, the maximum flow decreases by nearly 30%, and the total airway pressure loss (sum of each generation pressure drop) is more than 5 times the original value. Moreover, in airway generations 8 to 16, with increasing diameter, the maximum air pressure, maximum flow and total airway pressure loss remain almost constant. When the diameter is reduced to half of the original value, the maximum air pressure decreases by 3%, the maximum flow decreases by nearly 5%, and the total airway pressure loss increases by 200%. The study creates a foundation for improvement in respiratory disease diagnosis and treatment. KEY WORDS: respiratory system model；bronchial diameter change; airflow dynamics；simulation； experiment
Corresponding
author. Yan SHI: E-mail: [email protected]; Xiaohua Douglas Zhang: [email protected]
This article has been accepted for publication and undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process which may lead to differences between this version and the Version of Record. Please cite this article as doi: 10.1002/cnm.2929 This article is protected by copyright. All rights reserved.
1. INTRODUCTION Mechanical ventilation, which is used in the case of respiratory failure in the Intensive Care Unit (ICU), is an important life support treatment for critical patients. Mechanical ventilation provides fresh air to the lungs and ensures regular gas exchange [1]. Pressure-controlled ventilation (PCV) has been used in acute respiratory failure for several decades [2-6]. Clinical use and studies have shown that PCV could improve arterial oxygenation and decrease peak airway pressure compared with volume-controlled ventilation (VCV) [7-9]. However, when the air flows through the airways, it needs to overcome the friction caused by the airways. This friction is called airway resistance R (R=Δp/Q,Δp is the pressure drop through an airway, Q is the volumetric flow rate). Airway resistance is mainly related to the length and diameter of the airways, airflow magnitude and type of airflow [10]. To gain insight into the airway pressure-flow rate relationship of mechanically ventilated patients, the influence of bronchial diameter on the airflow dynamics is often studied based on numerical simulations. Currently, mathematical models with total respiratory resistance (R) and elastane (E) are applied in respiratory treatment research. In studies by Redlarski, Jaworski and other researchers, an electronic system is used to simulate the respiratory system where R(resistance) I(inductance) and C(capacitance) elements represent the resistances, inertances, and compliances of specific anatomical parts of the respiratory system[11-15]. Pedley derived an expression for hydraulic resistance based on a modification of fully developed Hagen–Poiseuille flow [16]. Although Redlarski and other researchers’ mathematical models do not require high computational costs because of their simple structures, resistance (R) is almost always calculated using Hagen-Poiseulle law, which is appropriate for laminar flow. It is known that flow in small airways is laminar flow because of the low flow velocity while it is turbulent flow in large airways because of the high flow velocity [17]. Bijaoui and other researchers attempted to consider airflow nonlinearities [18-23]. However, they neglected the importance of the morphological structure of the bronchial tree. Van Ertbruggen used computational fluid dynamics (CFD) to simulate flow through an anatomically based model of the airways and presented a modified expression for the hydraulic resistance of Pedley’s model [24]. Borojeni and Litwin did some researches that validate the Van Ertbruggen's model [25,26]. Compared to other CFD-studies of lung geometry, Wall used the fluid-structure interaction (FSI) approach to study airflow patterns and airway wall stress of lower airways in a real lung geometry based on CT-scans, his study has an important impact on airflow simulation in human airways [27]. Polak and Mroczka [28] built a mechanical ventilation model with complex nonlinear factors considered. The morphometry-based symmetrical structure of 23 airway generations was determined in the model. Katz derived an expression in which hydraulic resistance is expressed in terms of major and minor head losses [29]. In the normal ventilation cycle, pressure loss consists of frictional and local pressure losses, which both are the cause of respiratory resistance (R). Frictional pressure loss is caused primarily by the physical dimensions and air velocity, and local pressure loss is caused by the bifurcation of airways. Some studies mentioned above neglected the influence of local pressure loss and some studies involved the local pressure loss through calculating the coefficient of local pressure loss using a CFD method. However, as we all know, some diseases and treatments may change the bronchial diameter. For example, bronchoconstriction may occur after tracheotomy and endotracheal intubation [30-33]. Bronchial infection and obliteration may lead to bronchiectasis [34]. Furthermore, few researchers have considered the influence of bronchial diameter change on airflow dynamic characteristics and explained these experimental
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observations through mathematical models. Considering these issues, in this paper, a mathematical model that combines multi-generation airways with lungs is created. Both frictional and local pressure losses of each generation are considered in this model. The influence of bronchial diameter change on airflow dynamic characteristics is also presented in this paper. 2. MATHEMATICAL MODEL
2.1. Airway model A model is built based on the Navier-Stokes equations. This model is appropriate for airways between generation 0 and generation 16. In the modeling process, the following assumptions are made: (1) The air is incompressible. (2) One-dimensional flow is considered. (3) There is no air leakage during the working process. (4) The airways are considered to be circular rigid tubes. (5) The fully-developed flow is assumed in each time of the process. The diagram is shown in Figure 1. P1 is initial pressure of generation 0 which is equal to supply pressure. And p2 is the terminal pressure of generation 0 and it is also the initial pressure of next airway generation. Subsequent airway models are recurred from generation 0 and the terminal pressure of the final generation is equal to the pressure in lung. The airways wall is stationary with no slip. The model can be expressed as (2.1):
x
du 4 x 1 p u 2 dt D 2
(2.1)
where ρ is the density of air, x is the length of the bronchial segment, Δp is the pressure difference which is equal to p2-p1 , u is the air velocity, τ is the friction force between air and the wall, D is the diameter of the tube, 4τx/D is the component of frictional pressure loss, λρu2/2 is the component of local pressure loss, λ is the coefficient of local pressure loss and it is 1.5 when the air flows into the lungs and 1.0 inversely [35]. Further, can be described as
u2 f 2
(2.2)
According to reference [36], the Re value in pipe flow for the distinction of turbulent and laminar flow is 2000. For the air, the friction coefficient
16 f Re f 0.046 Re 0.2
f and Reynolds number can be written as Re 2000 Re 2000
Re
uD ;
(2.3)
where μ is the viscosity of air, which is 18.4×10-6pa·s (25°C, 101kpa).
2.2 Lung model Volume Equation: According to the definition of respiratory compliance (𝐶), the compliance (𝐶) of the lung can be described as [37]
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C
dV . dpl
(2.4)
Then, the volume of the lung can be calculated by the following formula:
dV Cdpl
(2.5)
where pl is the air pressure in lungs, V is the lung volume, C is the lung compliance. mRT dlung ( can)be assumed to be an isothermal and variable volume system. In Pressure Equation: The simulated dpl V the pressure the ideal gas equation (pV =mRT), p, volume V and air mass m change with time in lung model. dt dt Therefore, the relationship between pressure dm and time dV can be calculated from the differential expression of V m the pressure [38,39]: dt dt
RT (
V2
)
1 1 dV RTq mRT 2 V V dt RTqV 2 . V CmRT
(2.6)
where R is the gas constant= 287 (J/(kg⋅K)), T is the temperature, m is the mass of air in the lungs, and q is the air mass flow.
2.3 Ventilator model In this study, the ventilation model is triggered by Bi-Level Positive Airway Pressure (BiPAP) and time. p pepap Therefore, in a normal ventilation can be given as pepap cycle, ipapthe output t , pressure oft the trventilator ;
tr pipap, tr t ti ; pv p pipap pepap (t t ), t t (t t ); r i i r ipap tr pepap , t (ti tr ).
(2.7)
where pv is the output pressure of ventilator, pipap is the inspiratory positive airway pressure, pepap is the expiratory positive airway pressure, ti is the inspiration time, and tr is the time for the pressure rise. 3. VALIDATION OF THE MATHEMATICAL MODEL
3.1 Experimental study To simplify the human respiratory system and verify the model, four-generation ventilated pipes and two lung simulators (Meinuo medical equipment Ltd) with compliance of 10 ml/cmH2O were used to simulate the bronchi and the left and right lungs.
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The experimental system, which is shown in Figure 2 and 3, consists of a ventilator (Philips BiPAP), seven Y-style joints, one differential pressure flowmeter (Envitec SpiroQuant H, accuracy: ±5 % over measuring range), one pressure sensor (All Sensors 30 inch-D-4V, accuracy: ±1.0, %span), two lung simulators, a data acquisition card (Advantech USB-4711A), a computer (Lenovo)and several pneumatic tubes with different diameters. To eliminate the disturbance caused by the high frequency of air flow, Bartwoz low-pass filter technology is used. In the experiment, symmetry of the pipes is considered, so the airflow of each generation is the same. The simulation study is conducted using Matlab/Simulink [40]. The values of ventilator settings, including inspiratory positive airway pressure (pipap), expiratory positive airway pressure (pepap), breaths per minute (BPM), inspiratory time (ti) and pressure rise time (tr) are shown in Table 1. Parameters of the airway pipes are shown in Table 2. The lung simulator compliance is 10 ml/cmH2O. The airway pressure and airflow, which are obtained by experiment and simulation respectively, are shown in Figures 4 and 5. The following results can be obtained from Figures 4 and 5: (1) The simulation and experimental results are in broad agreement, which validates the mathematical model. (2) During the inspiration process, the input air flow rises sharply to its maximum with increasing pv, followed by a sharp fall after pv reaches IPAP. Finally, the flow falls to zero when pv is equal to pl. (3) During the expiration process, along with the fall of pv, the output air flow increases rapidly to its maximum, followed by a sharp fall after pv reaches EPAP. Finally, the output air flow approaches zero when pl is equal to pv. (4) The distinction between the simulation and experimental results is caused by insufficient ventilator output pressure and air leakage during the experimental process. The lag of experimental pressure during inspiration and expiration is caused by compliance of the tube.
3.2 Simulation study The human lower respiratory tract divides into twenty-four generations from G0 to G23. The generations before G16 only play a role in gas conduction, whereas the others are responsible for gas exchange. In this study, the Weibel symmetrical description shown in Table 3[41] is used to incorporate the airway morphology. Because the gas exchange is not considered, the generations before G16 are considered in this study. Usually, regulation of airway resistance by nerves and body fluid occurs actively in small airways with inner diameter less than 2 mm. Furthermore, the diameters of G0 and G1 generations do not easily change because of their stiffness. Therefore, we divide the G0 to G16 generations into three parts: Part I: G0 and G1; Part II: G2 to G7; and Part III: G8 to G16. In the simulation process, the compliance of 25 cardiac arrest patients is used as an example [42]. Therefore, the lung simulator compliance is set 50 ml/cmH2O. The ventilation process setting is same as Table 1.
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To assess the applicability of our model, a comparison of total pressure loss in air as a function of flowrate among the classic models of Otis [43], Pedley [16], Gemci [44], and Katz [29] was conducted. From Figure 6, we can see that our model is most comparable to the Katz model (ET+TB). When the flow rate is under 30 L/min, our model has good consistency with the Gemci model. Furthermore, the pressure loss in the Pedley model is lower than in the others because it does not include the local pressure loss. The Reynolds number of each generation (Q=50 L/min) is shown in Table 4, and the distribution of pressure loss is shown in Tables 5 and 6. According to Tables 5 and 6, we find that the local pressure loss accounts for over 90% of the total pressure loss. Further, the pressure loss between G2 and G7, the largest component of overall pressure loss, represents 66.5% of the total. 4. INFLUENCE OF BRONCHIAL DIAMETER CHANGE ON AIRFLOW DYNAMICS Gas exchange in the lungs is greatly influenced by the insufficient lung ventilation. Therefore, the Pmax (maximum air pressure in the human lung) and Qmax (maximum flow) are crucial to the safety and efficiency of mechanical ventilation, and the Pmax growth rate and Qmax growth rate are helpful for diagnosis [45]. When the bronchial diameter changes, the airway pressure loss changes accordingly, which affects the airway resistance. In this section, we studied the influence of bronchial diameter change on Pmax, Qmax, Pmax growth rate, Qmax growth rate and airway pressure loss. In the simulation process, the lung simulator compliance is set to 50 ml/cmH2O. The ventilation process setting is the same as in Table 1.
4.1 Influence of diameter change of G2 to G7 on airflow dynamics In the simulation process, DII = (d2, d3, d4...d7) is defined as a vector that represents the diameter of G2 to G7 generation airways. The values of diameter change of the G2 to G7 generation airways are shown in Table 7. In cases 2 and 3, the diameters are reduced by one half and one quarter, respectively. In cases 4 and 5, the diameters are one and one-half and two times the original diameter, respectively. The simulation results are presented in Figures 7 to 9. From the results, we can see the following: (1) Pmax decreases with diameter reduction, and the descent velocity rises accordingly. However, with increasing diameter, the Pmax rises slowly. When the diameter is reduced to half of the original, Pmax decreases by nearly 16%. Correspondingly, when the diameter is increased to twice the original, Pmax increases by only 2%. (2) The variation trend of Qmax is consistent with Pmax. When the diameter is reduced to half of the original, Qmax decreases by nearly 30%. Correspondingly, when the diameter is increased to twice the original, Qmax increases by only 3%. (3) The total airway pressure loss presents a reverse change with changing diameter. The total airway pressure loss is more than 5 times the original value when the diameter is reduced by half. However, when the diameter is doubled, the total airway pressure loss is nearly 40% of the original.
4.2 Influence of G8 to G16 diameter change on airflow dynamics In the simulation process, DIII = (d8, d9, d10...d16) is defined as a vector that represents the diameters of G8 to G16 generation airways. The values of G8 to G16 generation airway diameter changes are shown in Table 8. In cases 2 and 3, the diameters are decreased by one half and one quarter, respectively. In cases 4 and 5,
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the diameters are one and one-half and two times the original, respectively. The simulation results are presented in Figures 10 to 12. From the results, we can see the following: (1) Pmax declines with diameter reduction, and the descent velocity rises accordingly. However, with increasing diameter, the Pmax remains almost constant. When the diameter is reduced to half of the original, Pmax decreases by 3%. (2) The variation trend of Qmax is consistent with Pmax. When the diameter is reduced to half of the original, Qmax decreases by nearly 5%. Correspondingly, when the diameter is increased to double the original, Qmax increases by approximately 2%. (3) The trend of total airway pressure loss in G8 to G16 is consistent with that of G2 to G7. When the diameter is decreased to half of the original, the total pressure loss nearly doubles. However, when the diameter increases, the total airway pressure loss remains almost constant. 5. CONCLUSIONS In this paper, a new mathematical model combining multi-generation airways with lungs is established. The morphometry-based symmetrical structure of the 16 airway generations and local pressure loss caused by the bifurcation of airways are considered in this model. To confirm the model, a four-generation ventilation system based on two lung simulators and pneumatic pipes is established. Comparison of total pressure loss in our model and those of other researchers is also conducted, which validates our model. Then, simulation studies are conducted to analyze the influence of bronchial diameter change on Pmax, Qmax, Pmax growth rate, Qmax growth rate and total airway pressure loss. Conclusions can be summarized as follows: (1) The results of simulations and experiments are in broad agreement, which validates the mathematical model. (2) The Pmax of the lung simulator decreases with diameter reduction, and the descent velocity rises accordingly. When the diameter of G2 to G7 is decreased to half of the original, Pmax decreases by nearly 16%. Further, Pmax decreases by 3% when the diameter of G8 to G16 is reduced to half of the original. The trend of Qmax is consistent with Pmax, so when the diameter of G2 to G7 is decreased to half of the original value, Qmax decreases by nearly 30%, while Qmax decreases by 5% when the diameter of G8 to G16 is decreased to half of the original. Therefore, the upper airway diameter changes have more influence on airflow dynamics than the lower airway changes do. (3) When increasing the airway diameter, Pmax and Qmax rise accordingly. However, the range of the rise is extremely small. Therefore, reduction of the airway diameter has a greater effect than increased diameter on airflow dynamics. (4) The total airway pressure loss varies inversely with the change of diameter. Decreasing the diameter has a much greater influence on total airway pressure loss than increasing the diameter. The total airway pressure loss is more than 5 times the original value when the diameter of G2 to G7 is reduced by half and is 2 times the original value in the G8 to G16 airways. However, when the diameter of G2 to G7 doubles, the total airway pressure loss is nearly 40% of the original value, while there is no obvious change in the G8 to G16 airways. This paper presents a new model which reveals how bronchial diameter change influences airflow dynamics in a pressure-controlled ventilation system. However, the proposed mathematical model is not validated in a vivo data from real lungs. Also we have neglected the effects of airway deformation caused
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by the pressure change in ventilation process. These will be the subjects of our future investigation. In the next study, a clinical experiment based on real lungs will be conducted to verify and modify this model. In the future, medical staff may estimate the degree of bronchoconstriction and bronchiectasis inversely by calculating the Pmax, Qmax, Pmax growth rate and Qmax growth rate. ACKNOWLEDGEMENTS The research was funded by a grant (51575020) from the National Natural Science Foundation of China. None of the authors of this manuscript have any competing interests.
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Table 1 Values of the main ventilation process setting Parameter
pipap (cmH2O)
Value
22
pepap (cmH2O) 4
BPM
ti (s)
tr (s)
20
1.2
0.2
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Table 2 The parameters of airway pipes
Parameter
x1 (mm)
D1 (mm)
x2 (mm)
D2 (mm)
x3 (mm)
D3 (mm)
x4 (mm)
D4 (mm)
Value
120
12
60
8
60
5
60
2.5
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Table 3 Airway parameters
Generation
Number of bronchi
d (mm)
Length (mm)
Generation
Number of bronchi
d (mm)
Length (mm)
G0 G1 G2 G3 G4 G5 G6 G7 G8
1 2 4 8 16 32 64 128 256
18 12.2 8.3 5.6 4.5 3.5 2.8 2.3 1.86
120 47.6 19 17.6 12.7 10.7 9.0 7.6 6.4
G9 G10 G11 G12 G13 G14 G15 G16
512 1024 2048 4096 8192 16384 32768 65536
1.54 1.30 1.09 0.95 0.82 0.74 0.66 0.60
5.4 4.6 3.9 3.3 2.7 2.3 2.0 1.65
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Table 4 Reynolds number of each generation (Q=50 L/min) Generation
Re number
Generation
Re number
G0 G1 G2 G3 G4 G5 G6 G7 G8
3789 2795 2054 1522 947 609 380 231 143
G9 G10 G11 G12 G13 G14 G15 G16
86 51 30 17 10 5 3 2
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Table 5 Pressure loss distribution (Q=50 L/min) Parameter
Value (pa)
Rate (%)
Total pressure loss Frictional pressure loss Local pressure loss
166 16 150
100 9.6 90.4
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Table 6 Pressure loss distribution of three parts (Q=50 L/min) Generation
Total pressure loss (pa)
Rate (%)
G0 and G1 G2 to G7 G8 to G16
44.5 110.5 11
26.8 66.5 6.7
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Table 7 Diameter change of G2 to G7 generation airways
Parameter Diameter (mm)
Case 1 DII
Case 2 0.5DII
Case 3 0.75DII
Case 4 1.5DII
Case 5 2DII
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Table 8 Diameter change of G8 to G16 generation airways
Parameter Diameter (mm)
Case 1 DIII
Case 2 0.5DIII
Case 3 0.75DIII
Case 4 1.5DIII
Case 5 2DIII
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1 u 2 2
D
p1
τ
u
p2
τ x
Figure 1. Diagram of the airway model
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1
2
3
5
4
+
A\D
6
7
PC
8
9
Figure 2. Schematic diagram of the experimental system 1. Ventilator 2. Flow sensor 3. First-generation y-style joint 4. Second-generation y-style joint 5. Third-generation y-style joint 6. Pressure sensor 7. Data acquisition card 8. Computer 9. Lung simulators
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Figure 3. Photograph of experimental apparatus
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25 simulated curve experimental curve
Pressure(cmH 2O)
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
Time(s)
Figure 4.
Airway pressure curves of experiment and simulation
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40 simulated curve experimental curve
30
Flow(L/min)
20 10 0 -10 -20 -30 -40
0
1
2
3
4
5
6
7
8
9
Time(s)
Figure 5.
Airflow curves of experiment and simulation
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350
Total pressure loss (pa)
300 250
Our model Katz model（ET+TB) Katz model (TB) Pedley model Otis model Gemci model
200 150 100 50 0 5
10
15
20
25
30
35
40
45
50
Flow(L/min) Figure 6. Comparison of total pressure loss (ET: extrathoracic region, TB: tracheobronchial tree)
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18 DII 0.5DII
16
0.75DII 1.5DII
Pressure(cmH 2O)
14
2DII 12 10 8 6 4 2
0
1
2
3
4
5
6
7
8
9
Time(s)
(a) 60 DII 0.5DII 40
0.75DII 1.5DII 2DII
Flow(L/min)
20
0
-20
-40
-60
0
1
2
3
4
5
6
7
8
9
Time(s)
(b) Figure 7. Influence of G2 to G7 diameter change on airflow dynamics. (a) Pressure of the lung simulator, (b) airflow through the airways.
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5%
P max growth rate
P max(cmH2O)
18 17 16
0% -5%
-10%
15 14 0.5
-15%
1
1.5
2
-20% 0.5
1
DII 50 45
1.5
2
0%
-10%
40
-20%
35
-30%
1
1.5
DII
Figure 8.
2
10%
Qmax growth rate
Qmax(L/min)
55
30 0.5
1.5
DII
2
-40% 0.5
1
DII
Relationship between the diameter change of G2 to G7 and Pmax, Qmax, Pmax growth rate and Qmax growth rate.
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1000 DII
Total airway pressure loss(pa)
800 600 400
0.5DII 0.75DII 1.5DII 2DII
200 0 -200 -400 -600 -800 -60
-40
-20
0
20
40
60
Flow(L/min)
Figure 9.
Relationship between the diameter change of G2 to G7 and total airway pressure loss (negative value reflects reverse flow).
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18
DIII 0.5DIII
16
0.75DIII 1.5DIII
14
Pressure(cmH 2O)
2DIII 12 10 8 6 4 2
0
1
2
3
4
5
6
7
8
9
Time(s)
(a)
60 DIII 0.5DIII 40
0.75DIII 1.5DIII 2DIII
Flow(L/min)
20
0
-20
-40
-60
0
1
2
3
4
5
6
7
8
9
Time(s)
(b) Figure 10. Influence of G8 to G16 diameter change on airflow dynamics. (a) Pressure of the lung simulator, (b) airflow through the airways.
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1%
P max growth rate
P max(cmH2O)
17.4 17.2 17 16.8 16.6 16.4 0.5
1
1.5
0% -1% -2% -3% 0.5
2
1
DIII
2
1.5
2
1%
Qmax growth rate
Qmax(L/min)
51 50 49 48 47 0.5
1.5
DIII
1
1.5
DIII
2
0% -1% -2% -3% -4% -5% 0.5
1
DIII
Figure 11. Relationship between the diameter change of G8 to G16 and Pmax, Qmax, Pmax growth rate and Qmax growth rate.
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400 DIII
Total airway pressure loss(pa)
300
200
0.5DIII 0.75DIII 1.5DIII 2DIII
100
0
-100
-200
-300 -60
-40
-20
0
20
40
60
Flow(L/min)
Figure 12.
Relationship between the diameter change of G8 to G16 and total airway pressure loss (negative value reflects reverse flow).
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School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, CHINA
Beijing Engineering Research Center of Diagnosis and Treatment of Respiratory and Critical Care Medicine, Beijing Chaoyang Hospital, Beijing 100043, China 3 Faculty
of Health Sciences, University of Macau, Taipa, Macau.
SUMMARY Bronchial diameter is a key parameter that affects the respiratory treatment of mechanically ventilated patients. In this paper, to reveal the influence of bronchial diameter on the airflow dynamics of pressurecontrolled mechanically ventilated patients, a new respiratory system model is presented which combines multi-generation airways with lungs. Furthermore, experiments and simulation studies to verify the model are performed. Finally, through the simulation study, it can be determined that in airway generations 2 to 7, when the diameter is reduced to half of the original value, the maximum air pressure (maximum air pressure in lungs) decreases by nearly 16%, the maximum flow decreases by nearly 30%, and the total airway pressure loss (sum of each generation pressure drop) is more than 5 times the original value. Moreover, in airway generations 8 to 16, with increasing diameter, the maximum air pressure, maximum flow and total airway pressure loss remain almost constant. When the diameter is reduced to half of the original value, the maximum air pressure decreases by 3%, the maximum flow decreases by nearly 5%, and the total airway pressure loss increases by 200%. The study creates a foundation for improvement in respiratory disease diagnosis and treatment. KEY WORDS: respiratory system model；bronchial diameter change; airflow dynamics；simulation； experiment
Corresponding
author. Yan SHI: E-mail: [email protected]; Xiaohua Douglas Zhang: [email protected]
This article has been accepted for publication and undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process which may lead to differences between this version and the Version of Record. Please cite this article as doi: 10.1002/cnm.2929 This article is protected by copyright. All rights reserved.
1. INTRODUCTION Mechanical ventilation, which is used in the case of respiratory failure in the Intensive Care Unit (ICU), is an important life support treatment for critical patients. Mechanical ventilation provides fresh air to the lungs and ensures regular gas exchange [1]. Pressure-controlled ventilation (PCV) has been used in acute respiratory failure for several decades [2-6]. Clinical use and studies have shown that PCV could improve arterial oxygenation and decrease peak airway pressure compared with volume-controlled ventilation (VCV) [7-9]. However, when the air flows through the airways, it needs to overcome the friction caused by the airways. This friction is called airway resistance R (R=Δp/Q,Δp is the pressure drop through an airway, Q is the volumetric flow rate). Airway resistance is mainly related to the length and diameter of the airways, airflow magnitude and type of airflow [10]. To gain insight into the airway pressure-flow rate relationship of mechanically ventilated patients, the influence of bronchial diameter on the airflow dynamics is often studied based on numerical simulations. Currently, mathematical models with total respiratory resistance (R) and elastane (E) are applied in respiratory treatment research. In studies by Redlarski, Jaworski and other researchers, an electronic system is used to simulate the respiratory system where R(resistance) I(inductance) and C(capacitance) elements represent the resistances, inertances, and compliances of specific anatomical parts of the respiratory system[11-15]. Pedley derived an expression for hydraulic resistance based on a modification of fully developed Hagen–Poiseuille flow [16]. Although Redlarski and other researchers’ mathematical models do not require high computational costs because of their simple structures, resistance (R) is almost always calculated using Hagen-Poiseulle law, which is appropriate for laminar flow. It is known that flow in small airways is laminar flow because of the low flow velocity while it is turbulent flow in large airways because of the high flow velocity [17]. Bijaoui and other researchers attempted to consider airflow nonlinearities [18-23]. However, they neglected the importance of the morphological structure of the bronchial tree. Van Ertbruggen used computational fluid dynamics (CFD) to simulate flow through an anatomically based model of the airways and presented a modified expression for the hydraulic resistance of Pedley’s model [24]. Borojeni and Litwin did some researches that validate the Van Ertbruggen's model [25,26]. Compared to other CFD-studies of lung geometry, Wall used the fluid-structure interaction (FSI) approach to study airflow patterns and airway wall stress of lower airways in a real lung geometry based on CT-scans, his study has an important impact on airflow simulation in human airways [27]. Polak and Mroczka [28] built a mechanical ventilation model with complex nonlinear factors considered. The morphometry-based symmetrical structure of 23 airway generations was determined in the model. Katz derived an expression in which hydraulic resistance is expressed in terms of major and minor head losses [29]. In the normal ventilation cycle, pressure loss consists of frictional and local pressure losses, which both are the cause of respiratory resistance (R). Frictional pressure loss is caused primarily by the physical dimensions and air velocity, and local pressure loss is caused by the bifurcation of airways. Some studies mentioned above neglected the influence of local pressure loss and some studies involved the local pressure loss through calculating the coefficient of local pressure loss using a CFD method. However, as we all know, some diseases and treatments may change the bronchial diameter. For example, bronchoconstriction may occur after tracheotomy and endotracheal intubation [30-33]. Bronchial infection and obliteration may lead to bronchiectasis [34]. Furthermore, few researchers have considered the influence of bronchial diameter change on airflow dynamic characteristics and explained these experimental
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observations through mathematical models. Considering these issues, in this paper, a mathematical model that combines multi-generation airways with lungs is created. Both frictional and local pressure losses of each generation are considered in this model. The influence of bronchial diameter change on airflow dynamic characteristics is also presented in this paper. 2. MATHEMATICAL MODEL
2.1. Airway model A model is built based on the Navier-Stokes equations. This model is appropriate for airways between generation 0 and generation 16. In the modeling process, the following assumptions are made: (1) The air is incompressible. (2) One-dimensional flow is considered. (3) There is no air leakage during the working process. (4) The airways are considered to be circular rigid tubes. (5) The fully-developed flow is assumed in each time of the process. The diagram is shown in Figure 1. P1 is initial pressure of generation 0 which is equal to supply pressure. And p2 is the terminal pressure of generation 0 and it is also the initial pressure of next airway generation. Subsequent airway models are recurred from generation 0 and the terminal pressure of the final generation is equal to the pressure in lung. The airways wall is stationary with no slip. The model can be expressed as (2.1):
x
du 4 x 1 p u 2 dt D 2
(2.1)
where ρ is the density of air, x is the length of the bronchial segment, Δp is the pressure difference which is equal to p2-p1 , u is the air velocity, τ is the friction force between air and the wall, D is the diameter of the tube, 4τx/D is the component of frictional pressure loss, λρu2/2 is the component of local pressure loss, λ is the coefficient of local pressure loss and it is 1.5 when the air flows into the lungs and 1.0 inversely [35]. Further, can be described as
u2 f 2
(2.2)
According to reference [36], the Re value in pipe flow for the distinction of turbulent and laminar flow is 2000. For the air, the friction coefficient
16 f Re f 0.046 Re 0.2
f and Reynolds number can be written as Re 2000 Re 2000
Re
uD ;
(2.3)
where μ is the viscosity of air, which is 18.4×10-6pa·s (25°C, 101kpa).
2.2 Lung model Volume Equation: According to the definition of respiratory compliance (𝐶), the compliance (𝐶) of the lung can be described as [37]
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C
dV . dpl
(2.4)
Then, the volume of the lung can be calculated by the following formula:
dV Cdpl
(2.5)
where pl is the air pressure in lungs, V is the lung volume, C is the lung compliance. mRT dlung ( can)be assumed to be an isothermal and variable volume system. In Pressure Equation: The simulated dpl V the pressure the ideal gas equation (pV =mRT), p, volume V and air mass m change with time in lung model. dt dt Therefore, the relationship between pressure dm and time dV can be calculated from the differential expression of V m the pressure [38,39]: dt dt
RT (
V2
)
1 1 dV RTq mRT 2 V V dt RTqV 2 . V CmRT
(2.6)
where R is the gas constant= 287 (J/(kg⋅K)), T is the temperature, m is the mass of air in the lungs, and q is the air mass flow.
2.3 Ventilator model In this study, the ventilation model is triggered by Bi-Level Positive Airway Pressure (BiPAP) and time. p pepap Therefore, in a normal ventilation can be given as pepap cycle, ipapthe output t , pressure oft the trventilator ;
tr pipap, tr t ti ; pv p pipap pepap (t t ), t t (t t ); r i i r ipap tr pepap , t (ti tr ).
(2.7)
where pv is the output pressure of ventilator, pipap is the inspiratory positive airway pressure, pepap is the expiratory positive airway pressure, ti is the inspiration time, and tr is the time for the pressure rise. 3. VALIDATION OF THE MATHEMATICAL MODEL
3.1 Experimental study To simplify the human respiratory system and verify the model, four-generation ventilated pipes and two lung simulators (Meinuo medical equipment Ltd) with compliance of 10 ml/cmH2O were used to simulate the bronchi and the left and right lungs.
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The experimental system, which is shown in Figure 2 and 3, consists of a ventilator (Philips BiPAP), seven Y-style joints, one differential pressure flowmeter (Envitec SpiroQuant H, accuracy: ±5 % over measuring range), one pressure sensor (All Sensors 30 inch-D-4V, accuracy: ±1.0, %span), two lung simulators, a data acquisition card (Advantech USB-4711A), a computer (Lenovo)and several pneumatic tubes with different diameters. To eliminate the disturbance caused by the high frequency of air flow, Bartwoz low-pass filter technology is used. In the experiment, symmetry of the pipes is considered, so the airflow of each generation is the same. The simulation study is conducted using Matlab/Simulink [40]. The values of ventilator settings, including inspiratory positive airway pressure (pipap), expiratory positive airway pressure (pepap), breaths per minute (BPM), inspiratory time (ti) and pressure rise time (tr) are shown in Table 1. Parameters of the airway pipes are shown in Table 2. The lung simulator compliance is 10 ml/cmH2O. The airway pressure and airflow, which are obtained by experiment and simulation respectively, are shown in Figures 4 and 5. The following results can be obtained from Figures 4 and 5: (1) The simulation and experimental results are in broad agreement, which validates the mathematical model. (2) During the inspiration process, the input air flow rises sharply to its maximum with increasing pv, followed by a sharp fall after pv reaches IPAP. Finally, the flow falls to zero when pv is equal to pl. (3) During the expiration process, along with the fall of pv, the output air flow increases rapidly to its maximum, followed by a sharp fall after pv reaches EPAP. Finally, the output air flow approaches zero when pl is equal to pv. (4) The distinction between the simulation and experimental results is caused by insufficient ventilator output pressure and air leakage during the experimental process. The lag of experimental pressure during inspiration and expiration is caused by compliance of the tube.
3.2 Simulation study The human lower respiratory tract divides into twenty-four generations from G0 to G23. The generations before G16 only play a role in gas conduction, whereas the others are responsible for gas exchange. In this study, the Weibel symmetrical description shown in Table 3[41] is used to incorporate the airway morphology. Because the gas exchange is not considered, the generations before G16 are considered in this study. Usually, regulation of airway resistance by nerves and body fluid occurs actively in small airways with inner diameter less than 2 mm. Furthermore, the diameters of G0 and G1 generations do not easily change because of their stiffness. Therefore, we divide the G0 to G16 generations into three parts: Part I: G0 and G1; Part II: G2 to G7; and Part III: G8 to G16. In the simulation process, the compliance of 25 cardiac arrest patients is used as an example [42]. Therefore, the lung simulator compliance is set 50 ml/cmH2O. The ventilation process setting is same as Table 1.
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To assess the applicability of our model, a comparison of total pressure loss in air as a function of flowrate among the classic models of Otis [43], Pedley [16], Gemci [44], and Katz [29] was conducted. From Figure 6, we can see that our model is most comparable to the Katz model (ET+TB). When the flow rate is under 30 L/min, our model has good consistency with the Gemci model. Furthermore, the pressure loss in the Pedley model is lower than in the others because it does not include the local pressure loss. The Reynolds number of each generation (Q=50 L/min) is shown in Table 4, and the distribution of pressure loss is shown in Tables 5 and 6. According to Tables 5 and 6, we find that the local pressure loss accounts for over 90% of the total pressure loss. Further, the pressure loss between G2 and G7, the largest component of overall pressure loss, represents 66.5% of the total. 4. INFLUENCE OF BRONCHIAL DIAMETER CHANGE ON AIRFLOW DYNAMICS Gas exchange in the lungs is greatly influenced by the insufficient lung ventilation. Therefore, the Pmax (maximum air pressure in the human lung) and Qmax (maximum flow) are crucial to the safety and efficiency of mechanical ventilation, and the Pmax growth rate and Qmax growth rate are helpful for diagnosis [45]. When the bronchial diameter changes, the airway pressure loss changes accordingly, which affects the airway resistance. In this section, we studied the influence of bronchial diameter change on Pmax, Qmax, Pmax growth rate, Qmax growth rate and airway pressure loss. In the simulation process, the lung simulator compliance is set to 50 ml/cmH2O. The ventilation process setting is the same as in Table 1.
4.1 Influence of diameter change of G2 to G7 on airflow dynamics In the simulation process, DII = (d2, d3, d4...d7) is defined as a vector that represents the diameter of G2 to G7 generation airways. The values of diameter change of the G2 to G7 generation airways are shown in Table 7. In cases 2 and 3, the diameters are reduced by one half and one quarter, respectively. In cases 4 and 5, the diameters are one and one-half and two times the original diameter, respectively. The simulation results are presented in Figures 7 to 9. From the results, we can see the following: (1) Pmax decreases with diameter reduction, and the descent velocity rises accordingly. However, with increasing diameter, the Pmax rises slowly. When the diameter is reduced to half of the original, Pmax decreases by nearly 16%. Correspondingly, when the diameter is increased to twice the original, Pmax increases by only 2%. (2) The variation trend of Qmax is consistent with Pmax. When the diameter is reduced to half of the original, Qmax decreases by nearly 30%. Correspondingly, when the diameter is increased to twice the original, Qmax increases by only 3%. (3) The total airway pressure loss presents a reverse change with changing diameter. The total airway pressure loss is more than 5 times the original value when the diameter is reduced by half. However, when the diameter is doubled, the total airway pressure loss is nearly 40% of the original.
4.2 Influence of G8 to G16 diameter change on airflow dynamics In the simulation process, DIII = (d8, d9, d10...d16) is defined as a vector that represents the diameters of G8 to G16 generation airways. The values of G8 to G16 generation airway diameter changes are shown in Table 8. In cases 2 and 3, the diameters are decreased by one half and one quarter, respectively. In cases 4 and 5,
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the diameters are one and one-half and two times the original, respectively. The simulation results are presented in Figures 10 to 12. From the results, we can see the following: (1) Pmax declines with diameter reduction, and the descent velocity rises accordingly. However, with increasing diameter, the Pmax remains almost constant. When the diameter is reduced to half of the original, Pmax decreases by 3%. (2) The variation trend of Qmax is consistent with Pmax. When the diameter is reduced to half of the original, Qmax decreases by nearly 5%. Correspondingly, when the diameter is increased to double the original, Qmax increases by approximately 2%. (3) The trend of total airway pressure loss in G8 to G16 is consistent with that of G2 to G7. When the diameter is decreased to half of the original, the total pressure loss nearly doubles. However, when the diameter increases, the total airway pressure loss remains almost constant. 5. CONCLUSIONS In this paper, a new mathematical model combining multi-generation airways with lungs is established. The morphometry-based symmetrical structure of the 16 airway generations and local pressure loss caused by the bifurcation of airways are considered in this model. To confirm the model, a four-generation ventilation system based on two lung simulators and pneumatic pipes is established. Comparison of total pressure loss in our model and those of other researchers is also conducted, which validates our model. Then, simulation studies are conducted to analyze the influence of bronchial diameter change on Pmax, Qmax, Pmax growth rate, Qmax growth rate and total airway pressure loss. Conclusions can be summarized as follows: (1) The results of simulations and experiments are in broad agreement, which validates the mathematical model. (2) The Pmax of the lung simulator decreases with diameter reduction, and the descent velocity rises accordingly. When the diameter of G2 to G7 is decreased to half of the original, Pmax decreases by nearly 16%. Further, Pmax decreases by 3% when the diameter of G8 to G16 is reduced to half of the original. The trend of Qmax is consistent with Pmax, so when the diameter of G2 to G7 is decreased to half of the original value, Qmax decreases by nearly 30%, while Qmax decreases by 5% when the diameter of G8 to G16 is decreased to half of the original. Therefore, the upper airway diameter changes have more influence on airflow dynamics than the lower airway changes do. (3) When increasing the airway diameter, Pmax and Qmax rise accordingly. However, the range of the rise is extremely small. Therefore, reduction of the airway diameter has a greater effect than increased diameter on airflow dynamics. (4) The total airway pressure loss varies inversely with the change of diameter. Decreasing the diameter has a much greater influence on total airway pressure loss than increasing the diameter. The total airway pressure loss is more than 5 times the original value when the diameter of G2 to G7 is reduced by half and is 2 times the original value in the G8 to G16 airways. However, when the diameter of G2 to G7 doubles, the total airway pressure loss is nearly 40% of the original value, while there is no obvious change in the G8 to G16 airways. This paper presents a new model which reveals how bronchial diameter change influences airflow dynamics in a pressure-controlled ventilation system. However, the proposed mathematical model is not validated in a vivo data from real lungs. Also we have neglected the effects of airway deformation caused
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by the pressure change in ventilation process. These will be the subjects of our future investigation. In the next study, a clinical experiment based on real lungs will be conducted to verify and modify this model. In the future, medical staff may estimate the degree of bronchoconstriction and bronchiectasis inversely by calculating the Pmax, Qmax, Pmax growth rate and Qmax growth rate. ACKNOWLEDGEMENTS The research was funded by a grant (51575020) from the National Natural Science Foundation of China. None of the authors of this manuscript have any competing interests.
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Table 1 Values of the main ventilation process setting Parameter
pipap (cmH2O)
Value
22
pepap (cmH2O) 4
BPM
ti (s)
tr (s)
20
1.2
0.2
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Table 2 The parameters of airway pipes
Parameter
x1 (mm)
D1 (mm)
x2 (mm)
D2 (mm)
x3 (mm)
D3 (mm)
x4 (mm)
D4 (mm)
Value
120
12
60
8
60
5
60
2.5
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Table 3 Airway parameters
Generation
Number of bronchi
d (mm)
Length (mm)
Generation
Number of bronchi
d (mm)
Length (mm)
G0 G1 G2 G3 G4 G5 G6 G7 G8
1 2 4 8 16 32 64 128 256
18 12.2 8.3 5.6 4.5 3.5 2.8 2.3 1.86
120 47.6 19 17.6 12.7 10.7 9.0 7.6 6.4
G9 G10 G11 G12 G13 G14 G15 G16
512 1024 2048 4096 8192 16384 32768 65536
1.54 1.30 1.09 0.95 0.82 0.74 0.66 0.60
5.4 4.6 3.9 3.3 2.7 2.3 2.0 1.65
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Table 4 Reynolds number of each generation (Q=50 L/min) Generation
Re number
Generation
Re number
G0 G1 G2 G3 G4 G5 G6 G7 G8
3789 2795 2054 1522 947 609 380 231 143
G9 G10 G11 G12 G13 G14 G15 G16
86 51 30 17 10 5 3 2
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Table 5 Pressure loss distribution (Q=50 L/min) Parameter
Value (pa)
Rate (%)
Total pressure loss Frictional pressure loss Local pressure loss
166 16 150
100 9.6 90.4
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Table 6 Pressure loss distribution of three parts (Q=50 L/min) Generation
Total pressure loss (pa)
Rate (%)
G0 and G1 G2 to G7 G8 to G16
44.5 110.5 11
26.8 66.5 6.7
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Table 7 Diameter change of G2 to G7 generation airways
Parameter Diameter (mm)
Case 1 DII
Case 2 0.5DII
Case 3 0.75DII
Case 4 1.5DII
Case 5 2DII
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Table 8 Diameter change of G8 to G16 generation airways
Parameter Diameter (mm)
Case 1 DIII
Case 2 0.5DIII
Case 3 0.75DIII
Case 4 1.5DIII
Case 5 2DIII
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1 u 2 2
D
p1
τ
u
p2
τ x
Figure 1. Diagram of the airway model
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1
2
3
5
4
+
A\D
6
7
PC
8
9
Figure 2. Schematic diagram of the experimental system 1. Ventilator 2. Flow sensor 3. First-generation y-style joint 4. Second-generation y-style joint 5. Third-generation y-style joint 6. Pressure sensor 7. Data acquisition card 8. Computer 9. Lung simulators
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Figure 3. Photograph of experimental apparatus
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25 simulated curve experimental curve
Pressure(cmH 2O)
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
Time(s)
Figure 4.
Airway pressure curves of experiment and simulation
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40 simulated curve experimental curve
30
Flow(L/min)
20 10 0 -10 -20 -30 -40
0
1
2
3
4
5
6
7
8
9
Time(s)
Figure 5.
Airflow curves of experiment and simulation
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350
Total pressure loss (pa)
300 250
Our model Katz model（ET+TB) Katz model (TB) Pedley model Otis model Gemci model
200 150 100 50 0 5
10
15
20
25
30
35
40
45
50
Flow(L/min) Figure 6. Comparison of total pressure loss (ET: extrathoracic region, TB: tracheobronchial tree)
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18 DII 0.5DII
16
0.75DII 1.5DII
Pressure(cmH 2O)
14
2DII 12 10 8 6 4 2
0
1
2
3
4
5
6
7
8
9
Time(s)
(a) 60 DII 0.5DII 40
0.75DII 1.5DII 2DII
Flow(L/min)
20
0
-20
-40
-60
0
1
2
3
4
5
6
7
8
9
Time(s)
(b) Figure 7. Influence of G2 to G7 diameter change on airflow dynamics. (a) Pressure of the lung simulator, (b) airflow through the airways.
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5%
P max growth rate
P max(cmH2O)
18 17 16
0% -5%
-10%
15 14 0.5
-15%
1
1.5
2
-20% 0.5
1
DII 50 45
1.5
2
0%
-10%
40
-20%
35
-30%
1
1.5
DII
Figure 8.
2
10%
Qmax growth rate
Qmax(L/min)
55
30 0.5
1.5
DII
2
-40% 0.5
1
DII
Relationship between the diameter change of G2 to G7 and Pmax, Qmax, Pmax growth rate and Qmax growth rate.
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1000 DII
Total airway pressure loss(pa)
800 600 400
0.5DII 0.75DII 1.5DII 2DII
200 0 -200 -400 -600 -800 -60
-40
-20
0
20
40
60
Flow(L/min)
Figure 9.
Relationship between the diameter change of G2 to G7 and total airway pressure loss (negative value reflects reverse flow).
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18
DIII 0.5DIII
16
0.75DIII 1.5DIII
14
Pressure(cmH 2O)
2DIII 12 10 8 6 4 2
0
1
2
3
4
5
6
7
8
9
Time(s)
(a)
60 DIII 0.5DIII 40
0.75DIII 1.5DIII 2DIII
Flow(L/min)
20
0
-20
-40
-60
0
1
2
3
4
5
6
7
8
9
Time(s)
(b) Figure 10. Influence of G8 to G16 diameter change on airflow dynamics. (a) Pressure of the lung simulator, (b) airflow through the airways.
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1%
P max growth rate
P max(cmH2O)
17.4 17.2 17 16.8 16.6 16.4 0.5
1
1.5
0% -1% -2% -3% 0.5
2
1
DIII
2
1.5
2
1%
Qmax growth rate
Qmax(L/min)
51 50 49 48 47 0.5
1.5
DIII
1
1.5
DIII
2
0% -1% -2% -3% -4% -5% 0.5
1
DIII
Figure 11. Relationship between the diameter change of G8 to G16 and Pmax, Qmax, Pmax growth rate and Qmax growth rate.
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400 DIII
Total airway pressure loss(pa)
300
200
0.5DIII 0.75DIII 1.5DIII 2DIII
100
0
-100
-200
-300 -60
-40
-20
0
20
40
60
Flow(L/min)
Figure 12.
Relationship between the diameter change of G8 to G16 and total airway pressure loss (negative value reflects reverse flow).
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