Klinik und Poliklinik fr Ansthesiologie und Spezielle Intensivmedizin

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin A lumped parameter delay differential equation model of large arteries that captures reflection phenomena and integrates with modular models of the cardiovascular system. Sven Zenker Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin ESCTAIC Annual Meeting Timisoara, October 2012

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Joint work with Jonathan Rubin Gilles Clermont Dept. of Mathematics Dept. of Critical Care Medicine University of Pittsburgh Medical Center

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Increasing amounts of data with high information content Resonance phenomena Morphology Nonlinear interactions

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Motivation • 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-ofsample prediction (“extrapolation”)

Role of forward and inverse problems Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin “Forward” “Interpretation” (if present) introduce uncertainty • AP “Prediction” Quantitative representation • Peripheral of patient status vasomotor tone • …. Probability Density function on measurement space System states Parameters Mathematical model of • Physiology • Measurement • CVP Diagnostic or Therapeutic Intervention • HR Measurement • …. results “Inference” Probability density Function on state and Parameter space Measurement error, model stochasticity, and ill-posedness introduce uncertainty “Inverse” Single Measurement vector “Observation” • Myocardial contractility • Intravascular volumes Measurement error and model stochasticity Single State Vector

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Motivation • Information contained in features of the data that the mechanistic model cannot reproduce cannot be extracted in this way… • Ubiquitously measure pressures invasively in the ICU and (less ubiquitously) the OR • Typically 125 – 250 Hz data with many interesting features • This potential source of information about the patient’s physiological sate is mostly ignored in clinical practice

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Arterial blood pressure and flow waveform is dependent on location… Source: http: //ai. stanford. edu/~mitul/me 382/web/spring_reportp 7. htm

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin …has many features that reflect physiological state… Source: Liang et al. , Clin Sci (1995) 669 -679

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin …with commercially available attempts to exploit them for monitoring purposes in a more or less ad hoc way… …so algorithms were developed that detect these points and compute various indices… Source: http: //www. pulsecor. com

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Goal • 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)

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin What goes on physiologically… • Pressure and volume wave propagates through elastic tubes with (in reality) nonlinear elastical and viscous behaviour, many branching points, inhomogeneity in viscoelastic properties, 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…

Klinik und Poliklinik für Anästhesiologie und Starting point: Telegrapher’s Equation Spezielle Intensivmedizin Hyperbolic system of 2 coupled linear PDEs describing dynamic relationship of pressure and flow in a lossless elastic tube: With • RC characteristic impedance, v 0 characteristic velocity • Q(x, t) flow, P(x, t) pressure Can be thought of as composed of infenitesimally small inductances and shunt capacitances…

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Simplifying assumptions • Losslessness (negligible resistance terms, i. e. , only capacitance and inductance (“inertia”) matter • Homogeneity • Linearity • Primitive “topology” (for this component)

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Idea: Reduce this to delay differential equation (DDE) Following the idea pursued by Shroff, Burattini, and others, we want to reduce this PDEs to DDEs, but without assuming matching terminal impedances

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin First step: general solution 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

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Boundary conditions • To achieve the desired specific solution, we need to define boundary conditions • We choose these to permit generic embedding into larger physiological models

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Boundary conditions: schematic x=L x=0 P(x, t), Q(x, t) Pin(t) Rout(t) Lossless transmission line with • Length L • Characteristic impedance Rc • Propagation speed v 0 • Reparametrize using delay time τ=L/v 0 Pout(t)

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin After some algebra we find… …an expression for ΨR in terms of known things ΨR analogous thanks to symmetry…

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Approximate… So now we can actually compute everything we need to set up a DDE system with fixed delays, which are integer multiples of τ.

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin To test, … We embedded this new component into a simple model of the cardiovascular system: Time varying elastance ventricular model Pv State dependent resistance valve models Rcap x=0 x=L Pa P(x, t), Q(x, t) Routt) Rin(t)

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Implementation issues • DDE system, need adequate solver • State dependent resistances introduce discontinuities, need careful event handling and detection

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Disclaimer before we look at results No serious attempts were made to meaningfully parametrize this other than plugging in values from the literature to the extent available…

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Results: transients, valve behaviour

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Rough explorations of what this can do… Changing outflow resistance (“arteriolar resistance”) Increasing arteriolar resistance, everything else unchanged

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Rough explorations of what this can do… Changing tube impedance Increasing aortic stiffness, everything else unchanged

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Rough explorations of what this can do… Changing delay time Increasing delay time (=increasing length of tube or decreasing wave velocity, everything else unchanged

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Summary • 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

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Future Work • Parameter and state estimation from real data: computational challenges attributable to DDE model • Query interpretability of results • If unsatisfactory: – Smoother valve model (“Spikyness”) – Non-lossless, possibly tapered tube – More complex vessel “topologies” (branching…)

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin Acknowledgement This work was, in part, supported by • The German Israeli Foundation (GIF, Young Investigator Grant No. 2249) • the NIH • the DFG (Sachbeihilfe ZE 904/2)

Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin …and finally… Thank you! Contact: zenker@uni-bonn. de