1. Project Placenta Ethan Jewett & Megan Lewis

2. Outline Motivation Motivation Biological Background Biological Background Goal Goal Factors Factors First model (arterial dilation) First model (arterial dilation) Expanded model (trophoblast invasion) Expanded model (trophoblast invasion) Corrected arterial model Corrected arterial model Conclusion Conclusion

3. Motivation Maternal blood flow to a growing foetus affects foetal development Maternal blood flow to a growing foetus affects foetal development Too little blood flow can cause miscarriages or pre-eclampsia (a condition that causes hypertension in the mother) Too little blood flow can cause miscarriages or pre-eclampsia (a condition that causes hypertension in the mother) Question: Can something be done to alleviate these problems? Question: Can something be done to alleviate these problems?

4. Goal To model the effect of trophoblast movement, apoptosis and invasiveness on maternal arterial dilation in order to determine their effect on foetal development To model the effect of trophoblast movement, apoptosis and invasiveness on maternal arterial dilation in order to determine their effect on foetal development

5. Background: The Placenta During pregnancy, blood flows from maternal spiral arteries into the placenta, allowing nutrient/oxygen and waste exchange between the mother and the foetus During pregnancy, blood flows from maternal spiral arteries into the placenta, allowing nutrient/oxygen and waste exchange between the mother and the foetus

6. Maternal spiral arteries provide oxygen and nutrients to a growing foetus, as well as removing waste products Maternal spiral arteries provide oxygen and nutrients to a growing foetus, as well as removing waste products During the first trimester, these arteries are widened by trophoblast cells During the first trimester, these arteries are widened by trophoblast cells Trophoblast cells are produced by the foetus, and invade the uterine tissue by random motion and chemotaxis Trophoblast cells are produced by the foetus, and invade the uterine tissue by random motion and chemotaxis Once a trophoblast cell reaches an artery, it proceeds to degrade the artery wall Once a trophoblast cell reaches an artery, it proceeds to degrade the artery wall In addition, trophoblasts migrate up the spiral arteries In addition, trophoblasts migrate up the spiral arteries

7. The trophoblasts replace the smooth muscle inside the artery The trophoblasts replace the smooth muscle inside the artery The arteries can then deliver the blood required by the developing foetus The arteries can then deliver the blood required by the developing foetus

8. Factors Density of trophoblast cells in the artery Density of trophoblast cells in the artery Rate at which trophoblast cells degrade the artery Rate at which trophoblast cells degrade the artery Chemotaxis and random motion causing trophoblast cells to arrive at an artery Chemotaxis and random motion causing trophoblast cells to arrive at an artery Oxygen and temperature gradients which provide stimulus to direct trophoblast cells to an artery Oxygen and temperature gradients which provide stimulus to direct trophoblast cells to an artery Amount of muscular material trophoblasts can absorb before maturation Amount of muscular material trophoblasts can absorb before maturation

9. First model: arterial degredation by trophoblasts Artery smooth muscle Artery Ro r Artery wall Trophoblasts

10. Assumptions for first model Trophoblasts are at the artery (not worrying about invasion process) Trophoblasts are at the artery (not worrying about invasion process) Rate of change of trophoblast density is dependent on the radius of the artery Rate of change of trophoblast density is dependent on the radius of the artery When an artery reaches maximum radius, no more dilation occurs When an artery reaches maximum radius, no more dilation occurs Rate of change of the radius is dependent on density of trophoblasts Rate of change of the radius is dependent on density of trophoblasts As the radius reaches its maximum value, the rate of change of trophoblasts decreases to zero As the radius reaches its maximum value, the rate of change of trophoblasts decreases to zero

11. Artery model  R(t) – Radius of artery at time t  T(t) – Density of trophoblast cells in artery at time t  R o – Maximum radius of the artery  s – Density of trophoblasts at which the rate of increase of the radius reaches half its value (assuming Ro-R(t) fixed)  k – Parameter determining rate T(t) increases w.r.t. the radius  b – Parameter determining the mortality rate of trophoblast cells  w – Parameter affecting rate of increase of radius rate w.r.t. time Initial conditions: R(0)=r T(0)=0

12. Phase portrait Phase portrait of T(t) vs. R(t) Phase portrait of T(t) vs. R(t) (0,R o ) is a steady state (0,R o ) is a steady state Initial radius Maximum radius R o Parameters set to: Ro – 20 (length) s – 10 k – 3 (1/length) b – 4 w – 1 (1/time) r – 5 (length)

13. Arterial degredation model (non- dimensionalized)

14. Changing the value of a: Each line represents a different value of ‘a’ blue: a= 1 blue: a= 1 black: a= 2 black: a= 2 red: a= 3 red: a= 3 green: a=4 green: a=4 pink: a=5 pink: a=5 Shows as ‘a’ increases, the rate at which the radius reaches its maximum also increases and the density of trophoblasts decreases. R(t) and T(t) versus time b – Parameter determining the mortality rate of trophoblast cells w – Parameter affecting rate of increase of radius rate w.r.t. time

15. Changing the value of q: Each line represents a different value of ‘q’ blue: q= 1 blue: q= 1 black: q= 2 black: q= 2 red: q= 3 red: q= 3 green: q=4 green: q=4 pink: q=5 pink: q=5 Shows as ‘q’ increases, the rate at which the radius grows increases and the density of trophoblasts increases. R(t) and T(t) versus time

16. Problems Model isn’t very realistic Model isn’t very realistic Trophoblast density goes to zero as radius goes to zero... It might happen earlier! Trophoblast density goes to zero as radius goes to zero... It might happen earlier! With this model, the radius always reaches its maximum With this model, the radius always reaches its maximum Trophoblast density should be independent of the artery radius Trophoblast density should be independent of the artery radius Also would like to model flow of trophoblasts from developing foetus to an artery Also would like to model flow of trophoblasts from developing foetus to an artery

17. How to fix the problem? Make trophoblast density dependent on time, instead of on artery radius Make trophoblast density dependent on time, instead of on artery radius Develop a model which takes into account more details, specifically the flow of trophoblasts from the foetus to the maternal arteries Develop a model which takes into account more details, specifically the flow of trophoblasts from the foetus to the maternal arteries

18. Diffusion model sketch Artery smooth muscle Artery Ro r Artery wall Trophoblasts Uterine lining Trophoblast movement (diffusion) L

19. Diffusion/chemotaxis equation coupled with artery model R(t) :=Radius of artery at time t T(t) :=Density of trophoblasts in artery at time t u(x,t) :=Density of trophoblasts at position x at time t

20. Diffusion with chemotaxis plot u(x,t) vs. x at different times t values range approx. from 0 to 20

21. Fitting a curve at x=L

22. Putting estimate into ODES R(t) and T(t) versus time Each line represents a different value of ‘ψ’ blue: ψ = 1 blue: ψ = 1 black: ψ = 2 black: ψ = 2 red: ψ = 3 red: ψ = 3 green: ψ =4 green: ψ =4 pink: ψ =5 pink: ψ =5 Shows as ‘ψ’ increases, the rate at which the radius grows decreases and the rate of change of density of trophoblasts also decreases.

23. Diffusion/logistic growth equation coupled with artery model R(t) :=Radius of artery at time t T(t) :=Density of trophoblasts in artery at time t u(x,t) :=Density of trophoblasts at position x at time t

24. Diffusion with logistic growth plot u(x,t) vs. x at different times

25. Fitting a curve at x=L

26. Problems/Solutions Problem: The PDEs are too difficult to solve analytically using Maple (can be solved numerically), but since the ODES require a solution at the boundary, this is a problem Problem: The PDEs are too difficult to solve analytically using Maple (can be solved numerically), but since the ODES require a solution at the boundary, this is a problem Solution: find a function which resembles the graph of the numerical solution of the PDE at the endpoint, and use it in the original model of the artery Solution: find a function which resembles the graph of the numerical solution of the PDE at the endpoint, and use it in the original model of the artery

27. Corrected Artery model (DL) Corrected term, no longer dependent on radius Initial conditions: R(0)=r T(0)=0

28. Non-dimensionalized corrected model (DL)

29. Corrected model: changing the value of alpha Each line represents a different value of ‘α’ blue: α= 1 blue: α= 1 black: α = 2 black: α = 2 red: α = 3 red: α = 3 green: α =4 green: α =4 pink: α =5 pink: α =5 Shows as ‘α’ increases, the rate at which the radius grows increases and the rate of change of density of trophoblasts does not change. R(t) and T(t) versus time

30. Corrected model: changing the value of omega Each line represents a different value of ‘ω’ blue: ω = 1 blue: ω = 1 black: ω = 2 black: ω = 2 red: ω = 3 red: ω = 3 green: ω =4 green: ω =4 pink: ω =5 pink: ω =5 Shows as ‘ω’ increases, the rate at which the radius grows increases and the rate of change of density of trophoblasts also increases. R(t) and T(t) versus time

31. Corrected model: changing the value of psi Each line represents a different value of ‘ψ’ blue: ψ = 1 blue: ψ = 1 black: ψ = 2 black: ψ = 2 red: ψ = 3 red: ψ = 3 green: ψ =4 green: ψ =4 pink: ψ =5 pink: ψ =5 Shows as ‘ψ’ increases, the rate at which the radius grows decreases and the rate of change of density of trophoblasts also decreases. R(t) and T(t) versus time

32. Conclusion No data, making it very difficult to determine if our models are biologically correct No data, making it very difficult to determine if our models are biologically correct The models show what was expected, although with data it would be possible to determine how different parameters affect the total blood flow to the foetus The models show what was expected, although with data it would be possible to determine how different parameters affect the total blood flow to the foetus

33. Further work Model the movement down the artery Model the movement down the artery Solve the combined PDE and ODE system numerically (coding in numerics) Solve the combined PDE and ODE system numerically (coding in numerics) Find data in order to fit the results and potentially make predictions Find data in order to fit the results and potentially make predictions

34. Thank You Gerda de Vries Gerda de Vries Jim Muirhead Jim Muirhead Gustavo Carrero Gustavo Carrero And everyone we consulted And everyone we consulted