Abstract
This chapter presents lumped models of the respiratory systems: a small number a variables of interest is selected, and the physical phenomena are modeled as differential equations on those variables. Section 2.1 details the simplest approach to model the ventilation as a linear ODE on the volume. It is based on two main parameters: resistance R and elastance E. In Section 2.2, we propose some extensions of this linear model, to include various effects, inertia, nonlinear elastic behavior, variable resistance, and some other features. Section 2.3 is dedicated to numerics: time discretization, stability, and the particular issue of approximating periodic solution. Some numerical tests are presented, in order to investigate to behavior of the different models. The last Section 2.4 is devoted to a discussion on the previous assumptions and choices, to further extensions of the presented models, and to bibliographical notes.
N.B.: The lumped models that are presented here aim at modeling the whole ventilation process with a small number of unknowns. They will also be integrated to the multi-compartment approaches detailed in Chapter 4.
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Notes
- 1.
As it is commonly used in the specialized literature, we shall adopt the “physiological” unit system. For mechanical data, it relies on the fact that pressures are measured in centimeters of water (cm H2O), and volumes in liters. SI units (i.e. in accordance with the International System of Units) can be recovered as follows: 1 cm H2O = 102 Pa. For example \( E=5\;\mathrm{cm}\;{\mathrm{H}}_2{\mathrm{OL}}^{-1}=5\times {10}^5\mathrm{Pa}\;{\mathrm{m}}^{-3},R=2\mathrm{cm}\;{\mathrm{H}}_2{\mathrm{OsL}}^{-1}=2\times {10}^5\mathrm{Pa}\;\mathrm{s}\;{\mathrm{m}}^{-3}. \)
- 2.
For example in the case of an asthma crisis, the resistance can increase drastically, inducing a drop of E/R.
- 3.
The fact that inertia in the air might affect the resistance r, which might henceforth depend on the velocity itself (represented here by V), is a different issue which will be addressed in Section 2.2.3.
- 4.
In the present context of representation of the state by a volume (and not a length), the iner- tance plays the role that the mass would play in the classical spring-mass representation of lumped systems.
- 5.
The pressure at rest within the parenchyma is known to be negative, and this seems to be neglected here. It is simply due to the fact that this intermediate pressure is not explicitly introduced in the model. But this persistent negative pressure at rest is implicitly present in the model, as will be explained in the description of the double balloon model 2.4, p. 33.
- 6.
Actually, Fig. 2.6 suggests that blow up occurs at V min and V max , with V min slightly smaller that V r , and V max slightly larger than V tlc . We shall disregard here this slight shift to alleviate notations, but it could be accounted for within the same setting.
- 7.
This energy is defined up to a constant, and the left end \( \overline{V} \) of the integration interval can be chosen arbitrarily in (V r ,V tlc ).
- 8.
This type of well-posedness is quite different from the Cauchy approach, where an initial value is given. There are many examples of ODE’s for which the Cauchy problem is well-posed, whereas, for a given forcing term, multiple periodic solutions may exist. In the context we consider here, such a situation may arise in the case where the potential energy is not a convex function of the volume (see Exercise 2.8).
- 9.
As pointed out in Chapter 7, a straight computation of the resistance of the tree based on the assumption that the flow is linear (i.e. non-inertial) leads to a value that is significantly smaller than the measured value.
- 10.
It simply means that for any b, b’ in ℝ, \( \left|{\varPhi}^{-1}(b)-{\varPhi}^{-1}\left({b}^{\prime}\right)\right|<\left|b-{b}^{\prime}\right|. \)
- 11.
Dowloadable Matlab file: PERode.m.
- 12.
In the linear setting, it corresponds to the situation where W, defined by (2.6), is not negligible compared to the tidal volume VT .
- 13.
We keep this term of lumped to qualify models based on a limited number of intrinsic parameters, i.e. which do not result from a discretization carried out to approximate the solution of a continuous (infinite dimensional) problem. Note that the distinction between lumped models and direct modeling approaches may become fuzzy as the number of parameters grows.
- 14.
Quote from [8, p. 6: “Whether or not there is any survival advantage to having smooth muscle in our lungs is still debated, but a disease such as asthma leaves little doubt that its presence can have adverse consequences”.
- 15.
This PEF appears as the maximal vertical value attained in the flow volume loop (see Fig. 2.8, p. 48).
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Maury, B. (2013). Lumped models. In: The Respiratory System in Equations. MS&A — Modeling, Simulation and Applications. Springer, Milano. https://doi.org/10.1007/978-88-470-5214-7_2
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