Klinik und Poliklinik für linik und Poliklinik für Anästhesiologie und nästhesiologie und Spezielle Spezielle Intensivmedizin ntensivmedizin A lumped parameter delay differential equation model of large arteries that captures reflection phenomena and integrates with modular models of the cardiovascular system. Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Sven Zenker ESCTAIC Annual Meeting Timisoara, October 2012
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Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin A lumped parameter delay differential equation model of large arteries that captures.
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KKlinik und Poliklinik für linik und Poliklinik für
• Model based data analysis may allow quantitative interpretation of monitoring data
• If the inverse problem of state and parameter estimation can be solved, and the model is mechanistic, the results may be directly interpretable in physiological terms
• Correct mechanistic models can achieve out-of-sample prediction (“extrapolation”)
KKlinik und Poliklinik für linik und Poliklinik für
• Use mechanistic model to infer determinants of these morphological features quantitatively
• Need: sufficiently simple model that – captures the phenomena of interest
– Is modular, i.e., embeds well into larger physiological models including other organs like heart, lungs, etc.
– Is formulated in the time domain, making it amenble to stepwise simulation, sequential assimilation, etc.
• Existing models:– Complicated, many (discretized) or infinitely many (distributed) parameters
(e.g., partial differential equations (PDEs), many examples in the bioengineering literature, in particular)
– Or designed to work only when fed realistic waveforms as forcing functions/input… (“t tubes”, e.g., Campbell, Burattini, Shroff: usually, assuming impedance matching)
KKlinik und Poliklinik für linik und Poliklinik für
• Pressure and volume wave propagates through elastic tubes with (in reality) non-linear elastical and viscous behaviour, many branching points, inhomogeneity in viscoelastic properties, etc., etc.
• All this can be taken into account but– yields unwieldy models which make solving the inverse
problem unrealistically hard
• Will try the simplest possible approach…
KKlinik und Poliklinik für linik und Poliklinik für
The linear system of PDEs admits a closed form d’Alembert type general solution:
As expected for a system of two PDEs, this is given in terms of two arbitrary functions ΨR and ΨL, these represent forward and backward travelling waves
KKlinik und Poliklinik für linik und Poliklinik für
• DDE reduction of Telegrapher’s Equation using suitable boundary conditions can mimic many features of real arterial waveforms
• This is achieved at the “price” of only three additional parameters (inverse problem!)
• Time domain formulation allows straightforward embedding into more complicated lumped parameter models of physiology
• In particular, the effects of the closing aortic valve can now be modeled as a simple time-varying resistance
• While the math is simple, the numerics are not: care has to be taken to correctly handle delays, discontinuities, etc. to obtain meaningful results => performance is an issue, good solver with all required properties are non-existent at this point in time
KKlinik und Poliklinik für linik und Poliklinik für