1. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Lecture 12: Multiscale Bio-Modeling and Visualization Organ Models II: Heart, Cardiovascular Circulation and Reactive Fluid Transport Chandrajit Bajaj http://www.cs.utexas.edu/~bajaj

2. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Blood Circulation

3. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

4. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

5. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

6. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

7. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

8. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Heart Organ System

9. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Active Transport

10. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Transport of Reactive Substances through Fluids

11. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Transport of Reactive Substances through Fluids To extend the model of fluid hydrodynamics with chemical kinetics to handle flow of reactive substances through fluids. To establish a particle-mesh simulation technique for reactive flow transport.

12. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Basic Fluid Dynamics Equations in [Stam99] The incompressible Navier-Stokes equations for inviscid fluids For the velocity u = (u, v, w), –Conservation of mass –Conservation of momentum advection diffusion pressure external force

13. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Fluids (contd) Helmholtz-Hodge decomposition –“Any vector field is the sum of a mass conserving field and a gradient field.” Projection operator P The combined Navier-Stokes equations –Using the fact that and, the following equation is obtained:

14. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Updating the Velocity Field The general procedure 1.The add force step: f  Update the velocity field for the effect of external forces.  Implementation: –Simple.

15. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Advect Step 2.The advect step: Use the method of characteristics for the effect of advection: a semi- Lagrangian scheme –Implementation: Build a particle tracer and linear (or cubic) interpolator.

16. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Diffuse step 3.The diffuse step: –Use an implicit method for the effect of viscosity. –Implementation: Use the linear solver POIS3D from FISHPAK after discretization.

17. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Project step 4.The project step: P(w 3 ) Apply the projection operator to make the velocity field divergent-free. –Implementation: Use the linear solver POIS3D form FISHPAK after discretization.

18. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Moving Substances through the Fluid A non-reactive substance is advected by the fluid while diffusing at the same time. The following equation can be used to evolve density, temperature, etc. –Dissipation term

19. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Introduction to Chemical Kinetics What is chemical kinetics? A branch of kinetics that studies the rates and mechanisms of chemical reactions. Stoichiometric equation –A, B, E, F : chemical species (reactants & products) –a, b, e, f : stoichiometric coefficients

20. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Reaction Reaction rate (a.k.a. rate law) –Describes the rate r of change of the concentrations, denoted by [*], of reactants and products.

21. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Reaction Rate How to decide the reaction rate r : –r : a function of the concentrations of species present at time t, –For a large class of chemical reactions, it is proportional to the concentration of each reactant/product raised to some power. When, for example, only a forward reaction occurs, –Once the rate is determined, [A], [B], [C] and [D] are updated by integrating the rate law over time interval.

22. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Rate Coefficient Dependence Rate coefficient k –Is a function of both temperature and pressure. –Usually, the pressure dependence is ignored. –For many homogeneous reactions, Arrhenius equation A = const. Ea = activation energy R = universal gas constant 8.314x10^-3 kJ/(mol. K)

23. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Extension to Reactive Fluids Update of velocity field Evolution of density and temperature Application of chemical reaction Update of reaction- related parameters [Step1][Step3][Step2][Step4] The simulation technique by [Sta99] and [FSJ01] comprises [Step1] and [Step2] and [IKC04] for [Step3] and [Step 4]

24. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Grid values used in this method discretized grid Velocity Molar concentration Pressure Temperature Reaction rate Several values are defined at the center of the grid cell grid celldefined values

25. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Added control factors Control parameterDescription the chemical reaction type the stoichiometric coefficients the molar masses the reaction rate law the reaction rate constant the heat source term the vorticity coefficient the divergence control factor

26. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Computation Flow Computation of the fluid’s velocity field Evolution of the density & temperatureApplication of chemical reaction Update of reaction-related parameters

27. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin [Step1] Update of velocity field Uses a modified mass conservation equation, as in [FOA03], to control the expansion/contraction of reactive gases: The divergence constraint  is determined for each cell according to the reaction process that occurs in the region. –Determined in [Step4] after the application of chemical kinetics. The pressure is computed through the modified Poisson equation:

28. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin [Step2] Evolution of density and temperature Density field –Similarly as in [Sta99] and [FSJ01] except that multiple substances in the gas mixture are handled:

29. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Reactive Fluids –Each substance is evolved separately. Molar concentrations and densities are related by molar masses. Evolve.

30. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Temperature Fields Temperature field –Similarly as in [Sta99] and [FSJ01] except that a heat source term is added. –The heat source term is updated for each cell in [Step4] to reflect the occurring chemical reaction in the region.

31. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin [Step3] Application of chemical reaction The reaction process is applied for each cell in the reaction system. ① Determine the reaction rate ② Then, the new concentration vector c is updated by integrating the differential equations over  t:

32. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin [Step4] Update of reaction-related parameters The updated density d, temperature T, and reaction rate r influence the velocity through the heat source term external force f and the  value. – The temperature update is completed by taking care of the heat source term defined by –The buoyancy force, as proposed in [FSJ01], is updated:

33. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Velocity confinement ③ The vorticity confinement force, as proposed in [FSJ01], is updated according to or ④ The resulting external force is applied to the momentum conservation equation in each time frame. ⑤ The  value, determined by or,is applied to the modified mass conservation equation in the next time frame.

34. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Vorticity confinement f conf : vorticity confinement force –Use a vorticity confinement method by Steinhoff and Underhill. –Inject the energy lost due to numerical dissipation back into the fluid using a forcing term. –Reduce the numerical dissipation inherent in semi-Lagrangian schemes. –Implementation: straightforward

35. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Computation Flow Computation of the fluid’s velocity field Evolution of the density & temperatureApplication of chemical reaction Update of reaction-related parameters

36. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Animation Results – Reactive substance in a gaseous flow

37. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Additional Reading 1.J. Stam “Stable Fluids”, SIGGRAPH 1999, 121-128. 2.N. Foster, D. Metaxas, “Modeling the motion of a hot turbulent gas”, SIGGRAPH 1997, 181-188 3.G. Yngve, J. O’Brien, J. Hodgins. Animating explosions. SIGGRAPH 2000. 29-36 4.R. Fedkiw, J. Stam, H. Jensen. “Visual simulation of smoke”. SIGGRAPH 2001, 23-30. 5.W. Gates “Animation of Reactive Fluids”, Ph.D. Thesis, UBC, 2002 6.B. Feldman, J. O’Brien, O. Arikan. Animating suspended particle explosions. TOG, 22(3):23-40. 2003. 7.I. Ihm, B. Kang, D. Cha “Animation of Reactive Gaseous Fluids through Chemical Kinetics”, ACM/Siggraph Symp. on Computer Animation (2004)

38. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Heart Organ System I

39. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Heart Disorders I

40. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Heart Disorder II

41. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Heart Disorder III

42. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Heart Disorder IV

43. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Summary of [Stam99] Based on the full Navier-Stokes equations Based on an ‘unconditionally’ stable computational model –Semi-Lagrangian integration scheme Easy to implement Appropriate for gas and smoke Suffers from ‘numerical dissipation’ –The flow tends to dampen rapidly. –[Fedkiw01] attempts to solve this problem.

44. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Basic Equations in [Fedkiw01] The incompressible Euler equations “Gases are modeled as inviscid, incompressible, constant density fluids.” The equations for the evolution of the temperature T and the smoke’s density 

45. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Updating the Velocity Field 1.The add force step: f –Update the velocity field for the effect of forces. f user : user-defined force (for any purpose) f buoy : gravity and buoyancy forces

46. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Advection 2.The advect step: - (u  ) u –Use the method of characteristics for the effect of advection: a semi-Lagrangian scheme –Implementation: Build a particle tracer and linear interpolator. Same as [Stam99]

47. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Project step 3.The project step: P(w 3 ) Apply the projection operator to make the velocity field divergent- free. Same as [Stam99] –Implementation: Impose free Neumann boundary conditions at the occupied voxels. Use the conjugate gradient method with an incomplete Choleski pre- conditioner. : Poisson equation

48. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Moving Substances through the Fluid Use the semi-Lagrangian scheme.