Force Traction Microscopy: an inverse problem with pointwise observations

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1 Force Traction Microscopy: an inverse problem with pointwise observations Guido Vitale Dipartimento di Matematica Politecnico di Torino, Italia Cambridge, December 2011

2 Force Traction Microscopy: a biophysical problem Get information about the stress exerted by cells in their locomotion. CELL GEL BEAD Evaluate the force on the basis of the pointwise known displacement of the substratum.

3 The State of the Art Cells on a flat substrate (Dembo and Wang, 1999)

4 The State of the Art Cells on a flat substrate (Dembo and Wang, 1999) Cells embedded in a 3D enviroment (Legant, Chen et al., 2010)

5 The State of the Art Cells on a flat substrate (Dembo and Wang, 1999) Cells embedded in a 3D enviroment (Legant, Chen et al., 2010) Usual approach:

6 The State of the Art Cells on a flat substrate (Dembo and Wang, 1999) Cells embedded in a 3D enviroment (Legant, Chen et al., 2010) Usual approach: use the Green function of elasticity (exact in the flat case, approximated by finite elements in the 3D case) to determine the generated displacement u, u(x) = G(x x ) f(x ) dx.

7 The State of the Art Cells on a flat substrate (Dembo and Wang, 1999) Cells embedded in a 3D enviroment (Legant, Chen et al., 2010) Usual approach: use the Green function of elasticity (exact in the flat case, approximated by finite elements in the 3D case) to determine the generated displacement u, u(x) = G(x x ) f(x ) dx. evaluate the force field f inverting the above eqn. under Tichonov regularization

8 An alternative variational approach Borrowing techniques from Optimal Control of Partial Differential equations, we develop a variational approach, more flexible than Green functions, amenable to (Ambrosi, 2006):

9 An alternative variational approach Borrowing techniques from Optimal Control of Partial Differential equations, we develop a variational approach, more flexible than Green functions, amenable to (Ambrosi, 2006): treat arbitrary complex geometries,

10 An alternative variational approach Borrowing techniques from Optimal Control of Partial Differential equations, we develop a variational approach, more flexible than Green functions, amenable to (Ambrosi, 2006): treat arbitrary complex geometries, generalize to non homogeneous and anisotropic elasticity,

11 An alternative variational approach Borrowing techniques from Optimal Control of Partial Differential equations, we develop a variational approach, more flexible than Green functions, amenable to (Ambrosi, 2006): treat arbitrary complex geometries, generalize to non homogeneous and anisotropic elasticity, generalize to fiber reinforced elasticity,

12 An alternative variational approach Borrowing techniques from Optimal Control of Partial Differential equations, we develop a variational approach, more flexible than Green functions, amenable to (Ambrosi, 2006): treat arbitrary complex geometries, generalize to non homogeneous and anisotropic elasticity, generalize to fiber reinforced elasticity, generalize to non linear elasticity,

13 An alternative variational approach Borrowing techniques from Optimal Control of Partial Differential equations, we develop a variational approach, more flexible than Green functions, amenable to (Ambrosi, 2006): treat arbitrary complex geometries, generalize to non homogeneous and anisotropic elasticity, generalize to fiber reinforced elasticity, generalize to non linear elasticity, implement efficiently using FEM existing codes.

14 Aim of this work We aim to extend the optimal control approach in for cells migrating in a three dimensional environment:

15 Aim of this work We aim to extend the optimal control approach in for cells migrating in a three dimensional environment: derive formally the equations,

16 Aim of this work We aim to extend the optimal control approach in for cells migrating in a three dimensional environment: derive formally the equations, state the well posedness,

17 Aim of this work We aim to extend the optimal control approach in for cells migrating in a three dimensional environment: derive formally the equations, state the well posedness, approximate numerically the equations, compute and discuss the numerical results.

18 Optimal Control Approach in 3D Let f : Ω R 3 be the boundary traction and u : Ω R 3 the gel displacement. Using the Optimal Control Approach one has to solve:

19 Optimal Control Approach in 3D Let f : Ω R 3 be the boundary traction and u : Ω R 3 the gel displacement. Using the Optimal Control Approach one has to solve: where: find the minimum of J(u, f) = ε f 2 + Ou u 0 2, subjected to { C u = 0 in Ω or, for brevity : u = Sf (C u)n = f on Γ N f F adm := {g Γ N g = 0, Γ N (x o) g = 0}.

20 Optimal Control Approach in 3D Let f : Ω R 3 be the boundary traction and u : Ω R 3 the gel displacement. Using the Optimal Control Approach one has to solve: where: find the minimum of J(u, f) = ε f 2 + Ou u 0 2, subjected to { C u = 0 in Ω or, for brevity : u = Sf (C u)n = f on Γ N f F adm := {g Γ N g = 0, Γ N (x o) g = 0}. C Hooke Elasticity Symmetric Positive Tensor, ε > 0 Tichonov Penalization, O := u (u(x 1),...,u(x N )) (δ x1,...,δ xn ) Observation Operator, u 0 = (u 1 0,...,u N 0 ) R 3N Given Displacement at (x 1,...,x N ).

21 Optimal Control Approach: the differential equations Introducing the adjoint field p, the aforementioned minimum problem is equivalent to the following coupled PDEs (see Lions 1972):

22 Optimal Control Approach: the differential equations Introducing the adjoint field p, the aforementioned minimum problem is equivalent to the following coupled PDEs (see Lions 1972): C u = 0, C p+o T Ou = O T u 0, (C p)n = 0,(C u)n = 1 p on Γ ε N u = 0, p = 0 on Γ D note: f = 1 ε p Γ N, (neglecting the constraint f F adm ).

23 Optimal Control Approach: the differential equations Introducing the adjoint field p, the aforementioned minimum problem is equivalent to the following coupled PDEs (see Lions 1972): C u = 0, C p+o T Ou = O T u 0, (C p)n = 0,(C u)n = 1 p on Γ ε N u = 0, p = 0 on Γ D note: f = 1 ε p Γ N, (neglecting the constraint f F adm ). These equations are readily implementable on a FEM code.

24 Well posedness of the 3D Mixed Problem with Boundary Control Existence, uniqueness and stability of the set of PDE system applies in the following functional framework: Ω Ω find u H0,Γ 1 D (Ω) H 2 (Ω), p W0,Γ 1,s D (Ω) H 1 2 (ΓN ) (1 s < 6 ), 5 such that q W 1,s 0,Γ D (Ω), v H0,Γ 1 D (Ω): C u v+ f v = 0, Γ N N N C p q + δ xj u δ xj q = u 0j δ xj q, j=1 f ε p+ l ie i = 0, i=1 Ω f = 0, Ω (x o) f = 0. j=1

25 Well posedness of the 3D Mixed Problem with Boundary Control Existence, uniqueness and stability of the set of PDE system applies in the following functional framework: Ω Ω find u H0,Γ 1 D (Ω) H 2 (Ω), p W0,Γ 1,s D (Ω) H 1 2 (ΓN ) (1 s < 6 ), 5 such that q W 1,s 0,Γ D (Ω), v H0,Γ 1 D (Ω): C u v+ f v = 0, Γ N N N C p q + δ xj u δ xj q = u 0j δ xj q, j=1 f ε p+ l ie i = 0, Ingredients: i=1 Ω f = 0, Ω (x o) f = 0. Existence/Uniqueness of minimum for convex coercive functionals, Existence/Uniqueness/H 2 regularity for Elasticity eqns. with H 1/2 Neumann boundary data (Ciarlet 1980), Existence/Uniqueness/regularity at Γ N for Elasticity eqns. with Borel measure as data having support in the interior of Ω (Casas 1993), Sobolev Embeddings and Trace Theorem. j=1

26 The Inverse Problem The Best Approximation Solution (BAS) for an Ill Posed Problem?f s.t. Lf = u 0 (given u 0) is: f := arg min{ g s.t. g = arg min{h Lh u 0 2 }}

27 The Inverse Problem The Best Approximation Solution (BAS) for an Ill Posed Problem?f s.t. Lf = u 0 (given u 0) is: f := arg min{ g s.t. g = arg min{h Lh u 0 2 }} For us, L = OS that maps the traction f into the displacements evaluated at the beads Ou = OSf. This operator has closed range. We can apply some well known theorems (Engl et al. 2000):

28 The Inverse Problem The Best Approximation Solution (BAS) for an Ill Posed Problem?f s.t. Lf = u 0 (given u 0) is: f := arg min{ g s.t. g = arg min{h Lh u 0 2 }} For us, L = OS that maps the traction f into the displacements evaluated at the beads Ou = OSf. This operator has closed range. We can apply some well known theorems (Engl et al. 2000): The BAS exists, it is unique and depends continuously on the data

29 The Inverse Problem The Best Approximation Solution (BAS) for an Ill Posed Problem?f s.t. Lf = u 0 (given u 0) is: f := arg min{ g s.t. g = arg min{h Lh u 0 2 }} For us, L = OS that maps the traction f into the displacements evaluated at the beads Ou = OSf. This operator has closed range. We can apply some well known theorems (Engl et al. 2000): The BAS exists, it is unique and depends continuously on the data The minimum of J, say f ε, tends strongly to the BAS of OSf = u 0 as ε 0.

30 The Inverse Problem The Best Approximation Solution (BAS) for an Ill Posed Problem?f s.t. Lf = u 0 (given u 0) is: f := arg min{ g s.t. g = arg min{h Lh u 0 2 }} For us, L = OS that maps the traction f into the displacements evaluated at the beads Ou = OSf. This operator has closed range. We can apply some well known theorems (Engl et al. 2000): The BAS exists, it is unique and depends continuously on the data The minimum of J, say f ε, tends strongly to the BAS of OSf = u 0 as ε 0. The same holds with noisy data u 0, provided the noise level tends to zero too.

31 Numerical results: a Validation Test Set f given and evaluate the displacement solving numerically u given = Sf given (a linear elliptic PDE, the linear elasticity problem).

32 Numerical results: a Validation Test Set f given and evaluate the displacement solving numerically u given = Sf given (a linear elliptic PDE, the linear elasticity problem). Observe the displacement u 0 = Ou given (possibly perturbed with artificial noise).

33 Numerical results: a Validation Test Set f given and evaluate the displacement solving numerically u given = Sf given (a linear elliptic PDE, the linear elasticity problem). Observe the displacement u 0 = Ou given (possibly perturbed with artificial noise). Solve the Optimal Control Problem (for a range of ε) given u 0, thus obtaining u and f.

34 Numerical results: a Validation Test Set f given and evaluate the displacement solving numerically u given = Sf given (a linear elliptic PDE, the linear elasticity problem). Observe the displacement u 0 = Ou given (possibly perturbed with artificial noise). Solve the Optimal Control Problem (for a range of ε) given u 0, thus obtaining u and f. Evaluate the errors f f given, Ou u 0, etc...

35 Numerical Setup A µm 3 ellipsoid in a µm 3 cube. 300 beads are located with a mean distance from the origin of 17.88µm (fig. on the right) Mesh: tethraedron aspect ratio ranging between 0.6µm (near the ellipsoid) and 10µm near the external border, 7392 d.o.f. (fig. on the left).

Numerical Setup A 10 10 20µm 3 ellipsoid in a 100 3 µm 3 cube. 300 beads are located with a mean distance from the origin of 17.88µm (fig.

36 Numerical Setup A µm 3 ellipsoid in a µm 3 cube. 300 beads are located with a mean distance from the origin of 17.88µm (fig. on the right) Mesh: tethraedron aspect ratio ranging between 0.6µm (near the ellipsoid) and 10µm near the external border, 7392 d.o.f. (fig. on the left). Computational cost of the algorithm: 2 elliptic PDE solved with FEM and an iterative sparse solver N o dof vs Green function approach (N o dof) 3/2 (Legant, Chen et al. 2010)

37 Numerical Setup: given traction and noise Fig.: Given Force (left) and Displacement (right), u given = Sf given

the data u 0 = Ou given = OSf given as (w any unitary vector): u i 0 w = u given(x i) w+ν

38 Numerical Setup: given traction and noise Fig.: Given Force (left) and Displacement (right), u given = Sf given We artificially perturb the data u 0 = Ou given = OSf given as (w any unitary vector): u i 0 w = u given(x i) w+ν Unf( 1 2, 1 2 ), Unf(a, b) is the a uniform probability distribution on ]a, b[, ν = 0.4µm the noise level (cfr. Legant, Chen et al. 2010).

39 Numerical Results (I) e 2(Ou) e 2(f) e 09 1e 08 1e 07 1e 06 1e 05 ε e 09 1e 08 1e 07 1e 06 1e 05 ε Fig.: Ou u0 vs ε (Left) f f given L 2 (ΓN ) vs ε (Right). u 0 f given L 2 (ΓN ) 1e+05 2e+05 5e+05 1e+06 f Ou u 0 Fig.: L curve (Left). Curvature of L curve (Right) k 1e 09 1e 08 1e 07 1e 06 1e 05 ε Noisy data (empty circles), non noisy data (filled one).

40 Numerical Results (II) Fig.: The reconstructed force field f (Left). Error f f given(right). Here: ε = ε opt = (corner of the L curve).

41 Sensitivity Analysis: observation points Plots of f f given L 2 (ΓN ) f given L 2 (ΓN ) vs ε e 2 (f) Number of Beads: Fig.:150 beads (filled circles), 300 beads (Reference Setup, plus) and 450 beads (empty circles) e 09 1e 08 1e 07 1e 06 1e 05 ε

42 Sensitivity Analysis: observation points Plots of f f given L 2 (ΓN ) f given L 2 (ΓN ) vs ε e 2 (f) Number of Beads: Fig.:150 beads (filled circles), 300 beads (Reference Setup, plus) and 450 beads (empty circles) e 09 1e 08 1e 07 1e 06 1e 05 ε e 2 (f) e 09 1e 08 1e 07 1e 06 1e 05 ε Beads Distance: Fig.: The mean distance of the observation points from the origin is 17.88µm (Reference Setup, squares), 19.63µm (plus signs), 20.52µm (filled circles), and 21.41µm (empty circles).

43 Sensitivity Analysis: mesh and boundary Plots of f f given L 2 (ΓN ) f given L 2 (ΓN ) vs ε. Irregular border Γ N : Fig.: Γ N smooth (Reference Setup, filled circles), Γ N perturbed (empty circles) as below. 0.7 e 2 (f) e 09 1e 08 1e 07 1e 06 1e 05 ε

44 Sensitivity Analysis: mesh and boundary Plots of f f given L 2 (ΓN ) f given L 2 (ΓN ) vs ε. Irregular border Γ N : Fig.: Γ N smooth (Reference Setup, filled circles), Γ N perturbed (empty circles) as below. 0.7 e 2 (f) e 09 1e 08 1e 07 1e 06 1e 05 ε e 2 (f) Finer Grids: Fig.: Filled Circles: dof (Finest grid). Empty Circles: dof. Plus signs: 7392 dof (Reference Setup) 0.3 1e 09 1e 08 1e 07 1e 06 1e 05 ε

45 Remarks The numerical simulations yield:

46 Remarks The numerical simulations yield: Reconstruction Error 30 40%.

47 Remarks The numerical simulations yield: Reconstruction Error 30 40%. Good agreement between given and reconstructed pattern of f.

48 Remarks The numerical simulations yield: Reconstruction Error 30 40%. Good agreement between given and reconstructed pattern of f. Well behaviour of the L curve method.

49 Remarks The numerical simulations yield: Reconstruction Error 30 40%. Good agreement between given and reconstructed pattern of f. Well behaviour of the L curve method. Strong dependence on beads distance.

50 Remarks The numerical simulations yield: Reconstruction Error 30 40%. Good agreement between given and reconstructed pattern of f. Well behaviour of the L curve method. Strong dependence on beads distance. Weak dependence on beads number (after a minimum number is registered).

51 Conclusions and Future Goals We have:

52 Conclusions and Future Goals We have: proved well posedness of the Boundary Control with Pointwise Observation Problem,

53 Conclusions and Future Goals We have: proved well posedness of the Boundary Control with Pointwise Observation Problem, write down a FEM code to solve the latter,

54 Conclusions and Future Goals We have: proved well posedness of the Boundary Control with Pointwise Observation Problem, write down a FEM code to solve the latter, done some numerical tests.

55 Conclusions and Future Goals We have: proved well posedness of the Boundary Control with Pointwise Observation Problem, write down a FEM code to solve the latter, done some numerical tests. We hope to:

56 Conclusions and Future Goals We have: proved well posedness of the Boundary Control with Pointwise Observation Problem, write down a FEM code to solve the latter, done some numerical tests. We hope to: mesh up the actual cell shape,

57 Conclusions and Future Goals We have: proved well posedness of the Boundary Control with Pointwise Observation Problem, write down a FEM code to solve the latter, done some numerical tests. We hope to: mesh up the actual cell shape, use our tool with real biological data,

58 Conclusions and Future Goals We have: proved well posedness of the Boundary Control with Pointwise Observation Problem, write down a FEM code to solve the latter, done some numerical tests. We hope to: mesh up the actual cell shape, use our tool with real biological data, extend the approach to more complex material behavior.

59 Credits Newton Institute, Cambridge Organizers of Inverse problem in Science and Engineering Luigi Preziosi and Paolo Tilli (Torino) Davide Ambrosi, Marco Verani and Simone Pezzuto (Milano)

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