Start Presentation ICSC 2005: Beijing October 25, 2005 Object-oriented Modeling in the Service of Medicine François E. Cellier, ETH Zürich Àngela Nebot, Universitat Politècnica de Catalunya

Start Presentation ICSC 2005: Beijing October 25, 2005 System Complexity and the Understandability of Models As the systems that are being analyzed by mathematical models have grown in complexity over the years, they have become increasingly difficult to interpret and maintain. Modelers need to concern themselves with the understandability and maintainability of their models. Tools need to be developed that support them in this endeavor.

Start Presentation ICSC 2005: Beijing October 25, 2005 Object-oriented Modeling The object-oriented modeling paradigm enables the modeler to encapsulate knowledge in such a way that snippets of knowledge can be translated to a language familiar to the domain expert. The complexity of the models is locally contained by encapsulation and hierarchical composition of models. Models are being made more easily understandable by exploiting the two-dimensional nature of planar graphics.

Start Presentation ICSC 2005: Beijing October 25, 2005 Graphical Modeling Models of subsystems can be encapsulated as graphical objects, called icons. The icons can be topologically interconnected to form a two-dimensional network. Sub-networks of graphical objects can be grouped together to form new objects, for which icons can be designed. In this way, systems can be hierarchically composed from sub-systems forming a tree. The leaves of the tree must be described by equations.

Start Presentation ICSC 2005: Beijing October 25, 2005 Bond Graph Modeling Bond graphs are one type of graphical object- oriented models. They describe the power flow through a physical system. Since energy and power flow are common to all types of physical systems, bond graphs are domain independent. The equation-based leaf models of bond graphs can be pre-coded for all domains.

Start Presentation ICSC 2005: Beijing October 25, 2005 The Bond Model The modeling of physical systems by means of bond graphs operates on a graphical description of energy flows. The energy flows are represented as directed harpoons. The two adjugate variables, which are responsible for the energy flow, are annotated above (intensive: potential variable, “e”) and below (extensive: flow variable, “f”) the harpoon. The hook of the harpoon always points to the left, and the term “above” refers to the side with the hook. e f P = e · f e: Effort f: Flow

Start Presentation ICSC 2005: Beijing October 25, 2005 Sources in Bond Graph Representation U 0 i v a v b U 0 + U 0 i Se I 0 I v a v b u 0 Sf u I 0   Voltage and current have opposite directions Energy is being added to the system

Start Presentation ICSC 2005: Beijing October 25, 2005 Passive Electrical Elements in Bond Graph Representation R i v a v b u C i v a v b u L i v a v b u u i R u i C u i I    Voltage and current have same directions Energy is being taken out off to the system

Start Presentation ICSC 2005: Beijing October 25, 2005 Junctions 0 e1e1 e2e2 e3e3 f1f1 f2f2 f3f3 1 e1e1 e2e2 e3e3 f1f1 f2f2 f3f3 e 1 = e 2 e 2 = e 3 f 1 – f 2 – f 3 = 0 f 1 = f 2 f 2 = f 3 e 1 – e 2 – e 3 = 0  

Start Presentation ICSC 2005: Beijing October 25, 2005 An Example I

Start Presentation ICSC 2005: Beijing October 25, 2005 An Example II v 0 = 0 P = v 0 · i 0 = 0 

Start Presentation ICSC 2005: Beijing October 25, 2005 An Example III

Start Presentation ICSC 2005: Beijing October 25, 2005 Causal Bond Graphs Every bond defines two separate variables, the effort e and the flow f. Consequently, we need two equations to compute values for these two variables. It turns out that it is always possible to compute one of the two variables at each side of the bond. A vertical bar symbolizes the side where the flow is being computed. e f

Start Presentation ICSC 2005: Beijing October 25, 2005 “Causalization” of the Sources U 0 = f(t) I 0 = f(t) U 0 i Se Sf u I 0 The source computes the effort. The flow has to be computed on the right side. The source computes the flow. The causality of the sources is fixed. 

Start Presentation ICSC 2005: Beijing October 25, 2005 “Causalization” of the Passive Elements u i R u = R · i u i R i = u / R u i C du/dt = i / C u i I di/dt = u / I The causality of resistors is free.  The causality of the storage elements is determined by the desire to use integrators instead of differentiators. 

Start Presentation ICSC 2005: Beijing October 25, 2005 “Causalization” of the Junctions 0 e1e1 e2e2 e3e3 f1f1 f2f2 f3f3 e 2 = e 1 e 3 = e 1 f 1 = f 2 + f 3  1 e1e1 e2e2 e3e3 f1f1 f2f2 f3f3 f 2 = f 1 f 3 = f 1 e 1 = e 2 + e 3  Junctions of type 0 have only one flow equation, and therefore, they must have exactly one causality bar. Junctions of type 1 have only one effort equation, and therefore, they must have exactly (n-1) causality bars.

Start Presentation ICSC 2005: Beijing October 25, 2005 “Causalization” of the Bond Graph U0.e U0.f L1.fL1.e R1.eR1.f R2.f R2.e C1.fC1.e U0.e R1.f U 0.e = f(t) U 0.f = L 1.f + R 1.f dL 1.f /dt = U 0.e / L 1 dC 1.e /dt = C 1.f / C 1 C 1.f = R 1.f – R 2.f R 2.f = C 1.e / R 2 R 1.e = U 0.e – C 1.e R 1.f = R 1.e / R 1 e

Start Presentation ICSC 2005: Beijing October 25, 2005 The Four Base Variables of the Bond Graph Methodology Beside from the two adjugate variables e and f, there are two additional physical quantities that play an important role in the bond graph methodology: p =   e · dt Generalized Momentum: Generalized Position: q =   f · dt

Start Presentation ICSC 2005: Beijing October 25, 2005 Relations Between the Base Variables e f qp  R C I Resistor: Capacity: Inductivity: e = R( f ) q = C( e ) p = I( f )  Arbitrarily non-linear functions in 1 st and 3 rd quadrants  There cannot exist other storage elements besides C and I.

Start Presentation ICSC 2005: Beijing October 25, 2005 EffortFlow Generalized Momentum Generalized Position efpq Electrical Circuits Voltage u (V) Current i (A) Magnetic Flux  (V·sec) Charge q (A·sec) Translational Systems Force F (N) Velocity v (m / sec) Momentum M (N·sec) Position x (m) Rotational Systems Torque T (N·m) Angular Velocity  (rad / sec) Torsion T (N·m·sec) Angle  (rad) Hydraulic Systems Pressure p (N / m 2 ) Volume Flow q (m 3 / sec) Pressure Momentum Γ (N·sec / m 2 ) Volume V (m 3 ) Chemical Systems Chem. Potential  (J / mol) Molar Flow (mol/sec) -Number of Moles n (mol) Thermodynamic Systems Temperature T (K) Entropy Flow S’ (W / K) -Entropy S (J / K )

Start Presentation ICSC 2005: Beijing October 25, 2005 Hemodynamics The hemodynamics describe the flow of blood through the heart and the blood vessels, i.e., the flow of blood through the cardiovascular system. The hemodynamics of the human body can be interpreted as a hydromechanical system. Blood is similar to water, blood vessels can be inerpreted as pipes, and the heart chambers act as hydraulic pumps. Some of the chambers and vessels contain valves that act like check valves, preventing a backflow.

Start Presentation ICSC 2005: Beijing October 25, 2005 Transporter Model I Diagram Window Icon Window

Start Presentation ICSC 2005: Beijing October 25, 2005 Transporter Model II Equation window Documentation window

Start Presentation ICSC 2005: Beijing October 25, 2005 Container Model Diagram Window Icon Window

Start Presentation ICSC 2005: Beijing October 25, 2005 Valve Model (Transporter) Diagram Window

Start Presentation ICSC 2005: Beijing October 25, 2005 Heart Chamber (Container) Diagram Window Icon Window

Start Presentation ICSC 2005: Beijing October 25, 2005 Left Ventricle (Heart Chamber) Diagram Window

Start Presentation ICSC 2005: Beijing October 25, 2005 The Heart Diagram Window Icon Window

Start Presentation ICSC 2005: Beijing October 25, 2005 The Thorax Diagram Window

Start Presentation ICSC 2005: Beijing October 25, 2005 The Cardiovascular System Icon Window

Start Presentation ICSC 2005: Beijing October 25, 2005 The Cardiovascular System Heart Rate Controller Myocardiac Contractility Controller Peripheric Resistance Controller Venous Tone Controller Coronary Resistance Controller Central Nervous System Control (Qualitative Model) Regenerate Heart Circulatory Flow Dynamics Carotid Sinus Blood Pressure Recode Hemodynamical System (Quantitative Model) TH B2 Q4 D2 Q6 PAC Pressure of the arteries in the brain.

Start Presentation ICSC 2005: Beijing October 25, 2005 Simulation Results (Valsalva Maneuver)

Start Presentation ICSC 2005: Beijing October 25, 2005 Summary Object-oriented graphical modeling has helped us translate a hydro-mechanical model of the cardiovascular system into a representation that medical personnel can interpret and deal with. The knowledge at each layer was suitably encapsulated for limiting the local complexity to a level that can be represented on a single screen. No manual translation from the high-level representation to executable simulation code is needed. The graphical model at each level contains all of the model equations at that level and underneath it. Hence the model can be compiled in a fully automated fashion and simulated thereafter.