Continuous System Modeling

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Object-oriented Modeling in the Service of
Medicine
François E. Cellier, ETH Zürich
Àngela Nebot, Universitat Politècnica de Catalunya
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System Complexity and theUnderstandability 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.

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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.

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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.

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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.

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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.

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Sources in Bond Graph Representation
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Passive Electrical Elements in Bond Graph
Representation
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?
?
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Junctions
?
?
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An Example I
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An Example II
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An Example III
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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.

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Causalization of the Sources
U0 f(t)
I0 f(t)
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Causalization of the Passive Elements
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Causalization of the Junctions
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.
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Causalization of the Bond Graph
e
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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

Generalized Momentum
q ? f dt
Generalized Position
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Relations Between the Base Variables
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Effort Flow Generalized Momentum Generalized Position
e f p q
Electrical Circuits Voltage u (V) Current i (A) Magnetic Flux ? (Vsec) Charge q (Asec)
Translational Systems Force F (N) Velocity v (m / sec) Momentum M (Nsec) Position x (m)
Rotational Systems Torque T (Nm) Angular Velocity ? (rad / sec) Torsion T (Nmsec) Angle ? (rad)
Hydraulic Systems Pressure p (N / m2) Volume Flow q (m3 / sec) Pressure Momentum G (Nsec / m2) Volume V (m3)
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 )
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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.

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Transporter Model I
Icon Window
Diagram Window
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Transporter Model II
Equation window
Documentation window
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Container Model
Icon Window
Diagram Window
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Valve Model (Transporter)
Diagram Window
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Heart Chamber (Container)
Icon Window
Diagram Window
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Left Ventricle (Heart Chamber)
Diagram Window
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The Heart
Icon Window
Diagram Window
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The Thorax
Diagram Window
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The Cardiovascular System
Icon Window
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The Cardiovascular System
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Simulation Results (Valsalva Maneuver)
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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.