Model-Based Analysis of Ventilation Inhomogeneity in Respiratory Mechanics

Individualized models of respiratory mechanics help to reduce potential harmful effects of mechanical ventilation by supporting the evaluation of patient-specific lung protective ventilation strategies. Assessing ventilation inhomogeneities might be an important aspect in optimizing ventilator settings. The aim of this study is to capture and analyze ventilation inhomogeneity by a mathematical model using clinical data. The results show that the lung physiology of mechanically ventilated patients without lung condition can be described by an inhomogeneity model revealing two alveolar compartments with median time constants of 0.4 and 3.9 s. Thus, the IHM in combination with specific ventilation maneuver might be suitable to capture lung physiology for model-based optimization of ventilator settings but requires additional image-based investigations to further support the validity of the model.


Introduction
Non-adapted ventilator settings risks are severe side effects in intensive care patients during mechanical ventilation [1].Optimized patient-specific settings can be obtained by individualizing physiological models using clinical data and parameter identification methods.Individualized models provide insight into patient's physiology that is not directly measurable.Thus, they offer significant potential to evaluate and guide personalized lung protective ventilator strategies on intensive care units [2][3][4].The concept of model-based therapy applicable at the bedside of the patient requires models that are as simple as possible, while capturing all relevant dynamics and being identifiable with limited available measurement set.
Relevant dynamics of lung mechanics are significantly affected by ventilation inhomogeneity [5].Thus, inhomogeneities in lung mechanics might provide useful information on the lung tissue response to modified ventilator settings.Currently, ventilation inhomogeneity can be captured by computed tomography (CT) [6] or by electro impedance tomography [7].An alternative approach involves the Inhomogeneity Model (IHM) of respiratory mechanics [8], which has been a strong force in the field of pulmonary physiology ever since [5].
This paper presents the assessment and analysis of the inhomogeneity model in mechanically ventilated patients to evaluate its potential for model-based therapy.

Models and Parameter Identification
First Order Model (FOM): The FOM is the simplest representation of lung mechanics and considers homogeneous ventilation.The equation of motion is given in (1) and the electrical analog is shown in Figure 1.The resistive element R (cmH 2 O•s/L) corresponds to the resistance of the central and peripheral airways and the compliant compartment C (mL/cmH 2 O) represents the elasticity of alveolar tissue and the chest wall [5] defining the respiratory time constant τ = R•C.The patient-specific parameters R and C are determined by Multiple Linear Regression using measured data samples of flow rate (V  ) as model input and airway pressure (p aw ) as model output [9].p C represents the pressure in the elastic compartment.
Inhomogeneity Model (IHM): The IHM is a twocompartment model representing two different alveolar regions by two compliances (C1, C 2 in mL/cmH 2 O) with their own local airway (R1, R2 in cmH 2 O•s/L) connected to the airway opening.This model assumes parallel ventilation inhomogeneity in the lungs described by the two time constants Thus, this model is able to simulate redistribution processes between these two compartments (Pendelluft) [5].The electrical analog is given in Figure 1, and the mathematical description is presented in state-space representation in (2) where p C1 and p C2 (cmH 2 O) are state-signals corresponding to the pressure components generated by the volumes stored in the compliant compartments C 1 and C 2 .Parameter identification is performed by minimizing the sum of squared error (SSE) between measured (p aw,meas ) and simulated airway pressure using the iterative Integral-Method (IIM) [10,11]:

Clinical Data
Measurement sets of ten mechanically ventilated patients without lung conditions were selected from a previous study [12], where Super-Syringe Maneuvers were performed.During the Super-Syringe Maneuver small volume portions (100 mL) are administered with a constant flow rate (30 L/min), followed by an airway occlusion of 3 s allowing a static pressure-volume relation.Each portion increments the alveolar pressure in the lungs.Measured airway pressure and flow were sampled at 125 Hz and are shown exemplarily in Figure 2.
Informed consent was obtained from patients or their legally authorized representative.

Analysis
Each inspiration cycle was selected and referenced to its plateau pressure (p Plat ), reached at the end of occlusion.Patient-specific FOM parameters (R, C) and IHM parameters (R 1 , C 1 , R 2 , C 2 ) were identified for each cycle.The identified parameters of the cohort were plotted in boxplots illustrating the median and interquartile range (IQR) to present general trends with respect to plateau pressure.

Model Individualization
Overall 381 breathing cycles were available to identify model parameters.FOM identification leads to physiological plausible values in every case tested, whereas IHM identification revealed partly negative and un-phy-  siological parameter values in 39 cases.These erroneous identifications occurred mainly in breathing cycles in low pressure regions.Figure 3 shows comparisons of measured and simulated model responses of the FOM and IHM in low and high pressure regions.Obviously, the pressure relaxation during the occlusion is more pronounced in higher pressure regions.Fitting the IHM to the data assigns these relaxation effects redistribution processes.Pressure responses at low plateau pressure show no relaxation effects and thus impair IHM identification.
The identified parameters resulting from the successfully fitted cycles are summarized in a statistical analysis and presented as overall cohort medians and IQR in Table 1 and in terms of plateau pressure in Figure 4. Generally, the individualized IHM parameters indicate two heterogeneous compartments with significant different time constants of τ 1 = 3.9 s (IQR: 2.1 -7.7) and τ 2 = 0.4 s (IQR: 0.2 -0.5).In addition the global median time constant of the FOM equals 0.9 s (IQR: 0.6 -1.1).
The pressure dependency of the FOM parameters show a trend in terms of resistance, and a parabolic trend for the compliance with a maximum value at p Plat = 20 cmH 2 O.The IHM parameter of compartment 1 reveal an increase in R 1 and a parabolic trend of C 1 similar to C. R 2 and C 2 tend to remain constant, with R 2 being in the same orders of magnitude as R.

Discussion
The presented analysis shows the pressure dependency of the identified parameters of the FOM and IHM in patients without lung condition.
FOM identification shows a parabolic trend in compliance C increasing by more than factor 2 with increasing pressure.The maximal compliance was reached at 20 cmH 2 O.
IHM identification reveals inhomogeneous ventilation represented by two compartments with significant different time constants.The compartment with the larger compliance shows similar behavior as the global compliance trend of the FOM.Simultaneously, the compliance of the second compartment is smaller by factor 3. Similar findings of inhomogeneity of two different compartments with various time constants were obtained with EIT, where inhomogeneity was related to regional dynamics differences in ventral and dorsal areas in patients under general anesthesia without lung condition [13].Thus, these model-based results may indicate the IHM as an alternative approach to obtain measures of dynamic changes of inhomogeneous lung aeration.Still, the interpretation of model parameters, in particular, the validity of the identified compartments are only valid if the model assumption is correct.
However, the same model prediction quality could be obtained by the viscoelastic model (VEM), which describes the observed by the same equation but different coefficients [5,14].In this case, measured data of flow rate and airway pressure lead to the problem of undistinguishable models.It is unclear whether the observed dynamics can be assigned to viscoelastic or inhomogeneity characteristics.Thus, further investigations combined with imaging methods are necessary to analyze both dynamics separately to further validate the model assumptions of the IHM.
Once the IHM is fully validated, it might offer a new possibility to easily assess ventilation inhomogeneities to evaluate and guide personalized lung protective ventilator strategies on intensive care units.

Figure 2 .
Figure 2. Airway pressure and flow rate during a Super-Syringe Maneuver, initiated after baseline ventilation.

Figure 1 .
Figure 1.Measured and simulated pressure of First Order Model (FOM) and Inhomogeneity Model (IHM) at various plateau pressures of 4 and 36.5 cm H 2 O.

Figure 4 .
Figure 4. Statistical analysis of identified model parameters in terms of plateau pressure (p Plat ).Top line: FOM parameters, Bottom lines: IHM parameters.

Table 1 . Medians and IQR from FOM and IHM identified parameters
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