1. Identification of Patient Specific Parameters for a Minimal Cardiac Model THE 26th ANNUAL INTERNATIONAL CONFERENCE OF THE IEEE ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY C. E. Hann 1, J. G. Chase 1, G. M. Shaw 2, B. W. Smith 3, 1 Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand 2 Department of Intensive Care Medicine, Christchurch Hospital, Christchurch, New Zealand 3 Centre for Model-based Medical Decision Support, Aalborg University, Aalborg, Denmark

2. Difficult task for medical staff, often trial and error – the “art of medicine” Problem compounded by lack of complete data or knowledge Goal = a minimal cardiac model to identify patient parameters and assist in diagnosis For example, increased resistance in pulmonary artery - suggests blockage common to atherosclerotic heart disease Must be done in clinical real time (3-5 minutes) indicating a need for computational simplicity This talk concentrates on mathematical and computational aspects of parameter identification from the model. Eventual goal is to determine the minimal data set for useful parameter identification Diagnosis and Treatment

3. Single Chamber Model Pressure in pulmonary vein (Ppu) Pressure in left ventricle (Plv) Pressure in aorta (Pao) Resistance of mitral valve (Rmt) Resistance of aortic valve (Rav) e.g.

4. D.E.’s and PV diagram Open on pressure, close on flow valve law Find parameters as quickly and as accurately as possible

5. Integral Method - Concept Discretised solution analogous to measured data Work backwards and find a,b,c Current method – solve D. E. numerically or analytically (simple example with analytical solution ) - Find best least squares fit of x(t) to the data - Non-linear, non-convex optimization, computationally intense integral method – reformulate in terms of integrals – linear, convex optimization, minimal computation

6. Integral Method - Concept Integrate both sides from to t ( ) Choose 10 values of t, between and seconds to form 10 equations with 3 unknowns a,b,c

7. Integral Method - Concept Linear least squares (convex problem, unique solution) MethodStarting pointCPU time (seconds)Solution Integral-0.003[-0.5002, -0.2000, 0.8003] Non-linear[-1, 1, 1]4.6[-0.52, -0.20, 0.83] Non-linear[1, 1, 1]20.8[0.75, 0.32, -0.91] Integral method is at least 1000-10,000 times faster depending on starting point Thus, very suitable for clinical application versus non-convex and non-linear methods often used

8. Integrals - Single Chamber D. E.’s are solved in MAPLE, Q1 and Q2 curves discretised. Inflow Outflow 100 mmHG 3 mmHGPressure 480000 N s 2 m 5 430000 N s 2 m 5 Inertance 81000 N s m 5 83000 N s m 5 Resistance 1.33 beats s -1 Heart rate 3.5555x10 8 m -5 10 N m -2 33000 m -3 Constant 0 m 3 DSPVR volume 0 m 3 EDPVR volume ValueSymbolDescription In practice Q1, Q2 can be obtained from echocardiography or from differentiating volume data using ultrasound Discretised curves analogous to measured data

9. Integrals – Single Chamber e(t) translated V(0)=Vmin, Q1(0)=0 (beginning of filling stage at t=0) Filling stage - Ejection stage - Choose T1, Q2(T1)=0, V(T1)=Vmax Assume are given or measured 16 values of t in filling stage 14 values of T in ejection stage 30 linear equations in 5 unknowns

10. Results – Single Chamber 0.03479868480000L2L2 0.20430876430000L1L1 0.958176881000R2R2 2.268112883000R1R1 0.02 3.55555x10 8 E es Percentage error Optimised value True value Parameter Optimised parameter values 0.060.09PV 0.060.08Q2Q2 0.17Q1Q1 Standard deviation Mean percentage error Model response error with optimised values PV curves for model with optimized values versus the model with the true values. Flows in and out for the model with optimized values versus the model with the true values. Accurate parameter identification achieved Simulation errors all less than 0.2% validating parameter identification approach

11. Conclusions Integral based optimization successfully identified patient specific parameters for a single chamber model representative of elements in larger such models. Using integrals any noise is low pass filtered Optimization is linear, convex, and has minimal computation Typically used methods are non-linear, non-convex, and require significant computation and sometimes multiple starting points Avoid problem of incorrect initial conditions increasing computational time D.E. is never required to be solved analytically or numerically Method readily extends to larger models (6+ chambers) In summary, medical staff will have rapid data on patients assisting in diagnosis and can trial and test therapies in clinical real time (3-5 minutes).

12. Acknowledgements Questions ??? Engineers and Docs Dr Geoff ChaseDr Geoff Shaw The honorary Danes AIC2, Kate, Carmen and Nick The Danes Steen Andreassen Dr Bram Smith