Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin A lumped parameter delay differential equation model of large arteries that captures

KKlinik und Poliklinik für linik und Poliklinik für

AAnästhesiologie und nästhesiologie und

Spezielle Spezielle IIntensivmedizinntensivmedizin

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 Anästhesiologieund Spezielle Intensivmedizin

Sven Zenker

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 datawith high information content

Nonlinear interactions

Resonance phenomena

Morphology




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”)




“Prediction”

“Inference”

System states

Parameters

Quantitative

representation

of patient status

Measurement results

Diagnostic or

Therapeutic

Intervention

Mathematical model of

•Physiology

•Measurement

SingleStateVector

ProbabilityDensity functionon measurement

space

Measurement error and model stochasticity

(if present) introduce uncertainty

“Forward”

SingleMeasurement

vector

Probability densityFunction on state and

Parameter space

Measurement error, model stochasticity,

and ill-posedness introduce uncertainty

“Inverse”

“In

terp

reta

tio

n” “O

bservatio

n”

Role of forward and inverse problems

•Myocardial contractility

•Intravascular volumes

•Peripheral vasomotor tone

•….

•AP

•CVP

•HR

•….




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



Spezielle Spezielle IIntensivmedizinntensivmedizin 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…

Source: http://www.pulsecor.com

…so algorithms were developed that detect these points and compute various indices…




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

Rin(t) Rout(t)

Lossless transmission line with

•Length L

•Characteristic impedance Rc

•Propagation speed v0

•Reparametrize using delay time τ=L/v0

Pin(t) Pout(t)

x=0 x=L

P(x,t), Q(x,t)




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:

Routt) Rin(t)

x=L x=0

P(x,t), Q(x,t)

Pv

Pa

Time varying elastance ventricular model

State dependent resistance valve modelsRcap




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…



Spezielle Spezielle IIntensivmedizinntensivmedizin 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 decreasingwave 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)




…and finally…

Thank you!

Contact: [email protected]

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Klinik und Poliklinik für Anästhesiologie und Spezielle Intensivmedizin A lumped parameter delay differential equation model of large arteries that captures