ESCTAIC Annual Meeting 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

Joint work with Jonathan Rubin Dept. of Mathematics University of Pittsburgh Gilles Clermont Dept. of Critical Care Medicine University of Pittsburgh Medical Center

Increasing amounts of data with high information content Resonance phenomena Morphology Nonlinear interactions

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-of-sample prediction (“extrapolation”)

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

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

Arterial blood pressure and flow waveform is dependent on location… Source: http://ai.stanford.edu/~mitul/me382/web/spring_reportp7.htm

…has many features that reflect physiological state… Source: Liang et al., Clin Sci (1995) 669-679

…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

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)

What goes on physiologically… 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…

Starting point: Telegrapher’s Equation Hyperbolic system of 2 coupled linear PDEs describing dynamic relationship of pressure and flow in a lossless elastic tube: With RC characteristic impedance, v0 characteristic velocity Q(x,t) flow, P(x,t) pressure Can be thought of as composed of infenitesimally small inductances and shunt capacitances…

Simplifying assumptions Losslessness (negligible resistance terms, i.e., only capacitance and inductance (“inertia”) matter Homogeneity Linearity Primitive “topology” (for this component)

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

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

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

Boundary conditions: schematic x=0 x=L P(x,t), Q(x,t) Pin(t) Rin(t) Rout(t) Pout(t) Lossless transmission line with Length L Characteristic impedance Rc Propagation speed v0 Reparametrize using delay time τ=L/v0

After some algebra we find… …an expression for ΨR in terms of known things ΨR analogous thanks to symmetry…

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.

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 Pa x=L x=0 P(x,t), Q(x,t) Routt) Rin(t)

Implementation issues DDE system, need adequate solver State dependent resistances introduce discontinuities, need careful event handling and detection

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…

Results: transients, valve behaviour

Rough explorations of what this can do… Changing outflow resistance (“arteriolar resistance”) Increasing arteriolar resistance, everything else unchanged

Rough explorations of what this can do… Changing tube impedance Increasing aortic stiffness, everything else unchanged

Rough explorations of what this can do… Changing delay time Increasing delay time (=increasing length of tube or decreasing wave velocity, everything else unchanged

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

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…)

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)

Contact: zenker@uni-bonn.de …and finally… Thank you! Contact: zenker@uni-bonn.de