Solution of an inverse problem to estimate airway resistance in mechanically ventilated patients from 3-D electrical impedance tomography data

Emily Heavner; Jennifer L. Mueller; Tzu-Jen Kao; Nilton Barbosa da Rosa Jr.; Patrick J. Offner; Ellen Burnham

doi:10.3934/ammc.2025004

Article Contents

March 2025, Volume 3, 90-116. Doi: 10.3934/ammc.2025004

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Solution of an inverse problem to estimate airway resistance in mechanically ventilated patients from 3-D electrical impedance tomography data

1.
Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
2.
School of Biomedical Engineering, Colorado State University, Fort Collins, CO 80523, USA
3.
GE Healthcare, Niskayuna, NY 12309, USA
4.
Dept. of Medicine, University of Colorado Anschutz Medical Campus, Aurora, CO 80217, USA

^*Corresponding author: Jennifer L. Mueller
^*Corresponding author: Jennifer L. Mueller

Received: September 2024

Revised: February 2025

Early access: March 2025

Published: March 2025

Research reported in this publication was supported by the National Institute Of Biomedical Imaging And Bioengineering of the National Institutes of Health under Award Number R01EB026710-S1. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

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Abstract

We investigate the use of electrical impedance tomography, a non-invasive, non-ionizing imaging modality that provides real-time images of the ventilation distribution throughout the lung at the bedside, to image patients with acute hypoxic respiratory failure at the bedside and as data to inform an algorithm to determine the airway resistance throughout the bronchial tree. In this work, 3-D Electrical Impedance Tomography (EIT) difference image reconstructions and ventilator data are used in a multi-compartment lung model to solve an inverse problem to estimate the airway resistance along the bronchial tree. The method is demonstrated on five hospitalized patients who received mechanical ventilation as part of their hospital care. Comparisons of the EIT and airway resistance reconstructions are shown to be in good agreement with computed tomography (CT) scans and/or chest x-rays taken as part of the patients' care, and the results are consistent with their clinical condition. Patients with acute hypoxic respiratory failure often have heterogeneous distribution of ventilation, which makes the optimization of ventilator settings more difficult and increases the risk of ventilator-induced lung injury. Knowledge of the airway distribution along the bronchial tree could aid in determining personalized ventilator settings in such patients.

Keywords:

Mathematics Subject Classification: Primary: 35A01, 65L10; Secondary: 65L12, 65L20, 65L70.

Citation:

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Figure 1. The wearable electrode applicator textile (WEAT) belt on a healthy volunteer

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Figure 2. Screenshots from the ventilator

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Figure 3. Left: Definition of the chest quadrants in DICOM orientation. UR = upper right, UL = upper left, LR = lower right, LL = lower left. Right: A depiction of a 32 compartment, 5 generation lung with blue airways defined as upper right and left lung (URL and ULL, respectively) and red airways defined as right and left lower lung (LRL and LLL, respectively)

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Figure 4. Computed resistance vectors and lung volumes by subject

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Figure 5. CXR of Subject A

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Figure 6. Snapshot of EIT reconstructions at full inspiration of Subject A, displayed in DICOM orientation

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Figure 7. Resistance values for each airway for Subject A. Results are in DICOM orientation

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Figure 8. Left: CXR of Subject B. Right: Slice from the abdominal/pelvic CTA of Subject B

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Figure 9. Snapshot of EIT reconstructions at full inspiration of Subject B, displayed in DICOM orientation

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Figure 10. Resistance values for each airway for Subject B. Values in the right lower lung are not provided since the EIT reconstructions indicated this region was not being ventilated. Results are in DICOM orientation

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Figure 11. CXRs of Subject C taken 5 days before EIT data collection (left) and 6 days after EIT data collection (right)

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Figure 12. Slice from the abdominal/pelvic CT scan (without contrast) of Subject C

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Figure 13. Snapshot of EIT reconstructions at full inspiration of Subject C, displayed in DICOM orientation

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Figure 14. Resistance values for each airway for Subject C. For this subject, the volume of the right lower lung was less compared to the other quadrants and this was accounted for this by defining several branches to have zero airflow. Results are in DICOM orientation

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Figure 15. CXR of Subject D taken the same day as the EIT data collection

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Figure 16. Snapshot of EIT reconstructions at full inspiration of Subject D, displayed in DICOM orientation

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Figure 17. Resistance values for each airway for Subject D. For Subject D, the volume of the left lung is zero since the EIT reconstructions indicated this region was not being ventilated. Results are in DICOM orientation

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Figure 18. CT scan slices of subject 122. Left: At the level of the upper row of electrodes. Right: At the level of the lower row of electrodes

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Figure 19. Snapshot of EIT reconstructions at full inspiration of Subject E displayed in DICOM orientation

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Figure 20. Resistance values for each airway for Subject E. Results are not in DICOM orientation

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Figure 21. Resistance vector outputs from Algorithm 1 for all five subjects in this study

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Figure 22. Left: True volume for Subject C. The plot is in time (seconds) versus volume (liters). The overall volume difference is 1 L from start of inhale to end of inhale. Right: Subject C volume results for original EIT volume but doubled pressure value

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Table 1. Summary of patient information

Patient ID	Age	Sex	Height (cm)	Weight (kg)	BMI
A	47	Male	182.9	67.6	20.2
B	73	Male	182.9	71.7	21.4
C	52	Male	177.8	70.6	22.3
D	68	Male	160	65.2	25.5
E	62	Male	180.3	68.9	21.2

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Table 2. Clinical reasons for intubation for each patient

Patient ID	Reason for intubation
A	Upper gastrointestinal bleed with hematemesis and concern for aspiration
B	Acute hypoxic respiratory failure due to bacterial pneumonia, cardiogenic pulmonary edema, and shock
C	Acute hypoxic respiratory failure due to viral pneumonia, complicated by acute respiratory distress syndrome
D	Acute hypoxic respiratory failure due to pneumonia (possibly fungal)
E	Acute hypoxic respiratory failure due to bacterial pneumonia, with encephalopathy due to meningitis

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Table 3. Summary of airways defined to be in the upper versus the lower lung. The trachea or generation 0 is defined to be in neither the lower or upper lung as EIT data cannot collect information on the trachea since it is above the electrode placement

Generation	Airways in Upper Lung	Airways in Lower Lung	Airways in Generation
0	0	0	$ 2^0=1 $
1	2	0	$ 2^1=2 $
2	2	2	$ 2^2=4 $
3	2	6	$ 2^3=8 $
4	4	12	$ 2^4=16 $
5	8	24	$ 2^5=32 $
6	16	48	$ 2^6=64 $
7	32	96	$ 2^7=128 $
8	64	192	$ 2^8=256 $
9	128	284	$ 2^9=512 $
$\vdots $	$\vdots $	$\vdots $	$\vdots $
23	$ 2^{22}=4,194,304 $	$ 2^{24}-2^{22}=12,582,912 $	$ 2^{23}=16,777,216 $

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Table 4. Table of patient input parameters for the RCCB lung model taken from the ventilator. All values represent the total lung

Patient ID	Min/Max Pressure (cm H$ _2 $0)	Ventilator compliance (L/cm H$ _2 $0) on inhalation	Ventilator compliance (L/cm H$ _2 $0) on exhalation
A	5/17	0.07	0.11
B	5/16	0.035625	0.04417
C	8/19	0.0552	0.06846
D	5/14	0.19	0.335
E	5.2/14	0.0405	0.06

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Table 5. Table of outputs. The second column is the total resistance value (cm H$ _2 $0/(L/s)) reported from ventilator screenshots. The third column is the resistance vector output from Algorithm 1. The total computed resistance value (cm H$ _2 $0/(L/s)), in the fourth column, is the computed total resistance using equation (9)

Patient ID	Ventilator Resistance	Estimated Resistance Vector (cm H$ _2 $0/(L/s))	Computed Total Resistance
A	10	$ [5.78;6.34;4.22;0.31;0.001] $	10.045
B	16	$ [8.88;9.5;6.13;1.75;0.3758] $	15.4
C	10	$ [5.79;6.35;4.22;0.2872;0.001] $	10.06
D	2	$ [0.69;1.1;1.097;0.34;0.0716] $	1.5651
E	11	$ [6.03;7.07;5.39;1.88;0.214] $	11.1578

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Table 6. Table of inverse problem relative errors between the lung volume curve from EIT reconstructions and the RCCB lung volume output using the resistance vector from Table 5

Patient ID	2-norm Error (%)	SSIM Error
A	38.29%	0.8197
B	32.17%	0.8797
C	27.93%	0.9313
D	58.06%	0.6275
E	31.3%	0.8633

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Table 7. Table of 2-norm relative error of resistance vector results between iterations of Algorithm 1 for all patients

	2-norm relative error (%) between iterations
Patient ID	A	B	C	D	E
Iteration 1 to 2	38.42	19.15	2.87	23.02	$ 5.0109 \text{e}{-6} $
Iteration 2 to 3	$ 6.4115 \text{e}{-4} $	$ 1.1911 \text{e}{-4} $	2.87	13.61	$ 1.5711 \text{e}{-6} $
Iteration 3 to 4	$ 5.8801 \text{e}{-4} $	$ 2.0410 \text{e}{-5} $	0.66	3.4	$ 8.8284 \text{e}{-7} $
Iteration 4 to 5	$ 5.7435 \text{e}{-4} $	$ 2.6716 \text{e}{-6} $	0.59	3.37	$ 2.2823 \text{e}{-7} $
Iteration 5 to 6	$ 5.6130 \text{e}{-4} $	$ 9.6670 \text{e}{-7} $	0.53	0.01	$ 6.0198 \text{e}{-8} $

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