1. Team 5 – Dr. Ross Chuan Xue David Tello Skyler Speakman Rina Santos Shilpa Das Gupta

2. Mathematical Models of the Delipidation Cascade of Low Density Lipoproteins (LDL)

3. PURPOSE: Why? Small, dense LDL has been linked to coronary artery disease (CAD). We’d like to evaluate what measure to use: LDL itself, Apo-B/Apo-A1 Ratio, MR, C-Reactive Protein…and how prescribed drugs play a part in the process. What? Devise a mathematical model of LDL size distribution that is characterized by its components and processes.

4. List of Chemical Players for the Model LDL Components Cholesterol Ester (CE) Triglyceride (TG) Protein – ApoB-100 CETP (Cholesterol Ester Transfer Protein) Note: Component list is not all-inclusive HDL Components Cholesterol Ester (CE) Triglyceride (TG) Protein – ApoA-I HL (Hepatic Lipase) Liver and artery wall (Source and Sink)

5. Components of LDL ApoB-100CE TG

6. Components of HDL ApoA-ICE TG

7. Team 5’s Initial Model involving ODE’s Assume we have reaction: Which can be described by the following system:

8. Team 5’s Initial Model involving ODE’s Steady states of the system

9. Team 5 – 2 nd Model of LDL Delipidation Packard’s Diagram Chemical reactions incorporated: 1.CETP facilitates the exchange of CE for TG from VLDL to LDL 2. CETP also facilitates the process taking CE from HDL to LDL. 3.HL removes TG from LDL transforming LDL/IDL to small, dense LDL These three reactions are directly responsible for LDL delipidation.

10. Team 5 – 2 nd Model of LDL Delipidation i : # of CE wrapped in LDL particle j : # of TG wrapped in LDL particle denotes LDL particle Blue reactions push LDL distribution to small dense direction Green reaction pushes LDL distribution back to large buoyant direction. Elevated VLDL concentration in blood enhance blue reaction and have a impact making LDL smaller denser.

11. Team 5 – 2 nd Model of LDL Delipidation

12. FINAL MODEL

13. The Final Model Previous models required 3 objects to interact simultaneously. However, the probability for 3 particles to interact is much smaller than 2 particles. This lead us to disband the “swapping” ability of CETP for a more realistic series of reactions.

14. Pool of CETP We introduced a pool (finite number) of CETP. CETP exists in 3 states:  Free  Carrying CE  Carrying TG CETP serves as a transport for CE and TG between the lipoproteins (both HDL and LDL). There is no direct interaction between lipoproteins.

15. Actions of CETP The aforementioned series of reactions can be broken down into 4 sub-reactions:  CETP may pick up a CE or TG  CETP may drop off a CE or TG

16. Hepatic Lipase CETP accounts for all of the interactions involving CE and TG except Hepatic Lipase. Hepatic Lipase is a enzyme that takes TG from LDL. This process contributes to the formation of small dense LDL.

17. Chemical Reactions:

18. System of Differential Equations:

20. Numerical Approximations using MATLAB An example of a source distribution for LDL The final distribution of LDL corresponding to the above source distribution

21. Increased CETP This allows for more exchanges between LDL. This is represented by the smoother graph on the right.

22. Increased Hepatic Lipase This removes TG from LDL. The distribution has shifted towards less TG in LDL.

23. Increased HDL source HDL removes CE from the system. The distribution has shifted towards fewer CE and an abundance of TG in the LDL. Note the decrease in sdLDL and increase in VLDL.

24. Summary Through studying biological papers and speaking with a cardiologist, Dr. Duprez, we developed a model that resembled the processes involved in the LDL delipidation cascade. We formed a system of ODE’s to describe our model and simulated the system in Matlab. We explored the numerical model and interpreted the results.

25. Future Work Incorporate real source terms and reaction rates to improve our ODE model, and add in drugs to test the model. Continue steady-state analysis of our ODE model. Try to interpret the macroscopic flow rate in the 1 st order hyperbolic PDE model we get in terms of microscopic chemical reaction rates. Continue adding features to the Matlab code.

26. Acknowledgements IMA Dr. Ross Professor Braun Professor Santosa Dr. Duprez M.D. Valjean Eleander “John” from Rice University

27. Thank you!

29. CETP deposits CE on HDL instead of on LDL. HDL then leaves the system. Resulting in higher TG and lower CE on VLDL / IDL

30. CETP CETP (P) CE CETP with CE (C) TG CETP with TG (T)

31. HDL CE TG ApoA-I CE i TG j ApoA-I with i cholesterol ester(CE) and j triglyceride (TG)

32. LDL CE TG ApoB-100 CE i TG j ApoB-100 with i cholesterol (CE) and j triglyceride (TG)

33. Size, Data and Composition Table from Camilla Anderson’s PhD Thesis, 2003