Data generation

The synthetic AAA model is shown in figure 6.1. Its geometry incorporates an asymmet-ric distribution of intraluminal thrombus (ILT) and a thin layer of 1 mm of aneurysmatic arterial wall. Its diameter in the descending aorta is 22 mm from where it enlarges to mimic a rather large aneurysm with a maximum diameter of 72 mm. For the compu-tational model, symmetry in longitudinal and lateral direction is used and, despite not being completely physiological, Dirichlet conditions according to (2.39) are prescribed at the proximal inlet of the aneurysm. For later reference, the luminal (inner) model surface is referred to asΓ_I and the outer model surface is referred to as Γ_O.

Figure 6.1.: Discretization of the synthetic AAA model which utilizes symmetry in lateral and longitudinal direction. The model incorporates an asymmetric distribution of ILT and a patch-wise distribution of the growth parametercϑ (cϑ,P ∈ {0.05,1.0,2.0}) in the arterial wall.

The ILT is modeled as a coupled compressible neo-Hookean material [99] whose strain energy function is given in terms of the first invariant I_1,e of the elastic component C_e of right Cauchy-Green tensor according to

ΨILT =c1(I1,e−3) + c1

β₁(Je−2β1−1) (6.1) with c₁ = 18.0 kPa and β₁ = 4.5 taken from Maier et al. [148]. The arterial wall is modeled as the nearly-incompressible neo-Hookean type material given in (5.12) and (5.13).

For the simulation of growth according to (2.65), a patch-wise spatial distributionc_ϑ,P of the growth parameter c_ϑ is prescribed as shown in figure 6.1. Growth is simulated over a generic time period¯t. Prior to the growth simulation, the model is prestressed by the application of an orthonormal pressure on the inner surfaceΓ_I of the synthetic AAA in the initial configuration. By applying a pressure ofp = 1.0666e−2 MPa≈80 mmHg, diastolic conditions are sought. Thereby, the MULF approach is applied to project the pressurized state to the initial configuration [77]. The overall nonlinear solution is controlled by the load-control scheme given in figure 5.2 whereby the prediction-phase is not of interest for this example. Except for the simplification in geometry, this setup very closely resembles the approach pursued in state-of-the-art patient-specific computational experiments.

6.1. Data generation A synthetic measurement is given by the solution of the forward problemA(θ =c_ϑ,P) as

D_ΓO :=C(A(c_ϑ,P)) +ζ, (6.2)

whereby ζ is a vector of white Gaussian noise (σ = 0.1, which corresponds to a signal-to-noise ratio (SNR) ≈ 30%) and C is the restriction to the degrees of freedom onΓO. The associated push-forward of the model surfaceΓ_O is given by

Γ^A(c_O ^ϑ,P):=ϕ^A(cϑ,P)+ζ(Γ_O), (6.3) i.e., the outer model surface deformed via the displacements D_ΓO. The current repre-sentation of this surface is given according to (3.50).

6.1.1. Data registration via surface matching

To introduce a systematic error into the data generation, a registration step is applied.

Thereby, the practical situation where the measurements D_ΓO are extracted from longi-tudinal image data is emulated. Since the synthetic model is not based on image data, this registration step cannot be performed in the sense of image registration as introduced in chapter 3.2. Instead, a registration algorithm is applied that operates on surface data directly. To this end, the surface matching algorithm introduced in Vaillant and Glaunès [221] is used here. It applies the LDDMM framework [see e.g. 57]. Since the specific properties of such an algorithm are not of interest for the effect to be shown, the reader is referred to appendix E for an introduction of the surface matching framework. In sum-mary, for two surfacesT1⊂Ω^I₀ andT2 ⊂Ω^I_tembedded in the ambient spaceΩ = Ωˆ ^I₀∪Ω^I_t, the surface matching tries to find a diffeomorphism ϕ_R(ˆx) withϕ_R(ˆx) = ˆx+V_R such that an energy functional based on the surface current distance

kT₂−ϕ_R(T₁)kW^∗ (6.4)

is minimized, see (E.3) for details. Consequently, when applied to the model surfacesΓ_O and the deformed model surface Γ^A(c_O ^ϑ,P), the matching results in an optimal mapping Γ_O 7→Γ^A(c_O ^ϑ,P)with respect to the normk·k_W^∗. The quality of this mapping is depicted in figure 6.2. It can be seen that the bulk of point-wise distances betweenΓ^A(c_O ^ϑ,P) and ϕ_R(Γ_O) is < 0.1 mm. This indicates that the surface matching accurately performs the mapping Γ_O 7→ Γ^A(c_O ^ϑ,P). However, the in-plane accuracy of the registration, which cannot be detected by the surface current normk·k_W^∗, might be severely corrupted. This setup closely resembles the application of image registration as outlined in chapter 3.2 where uniqueness of deformations is enforced by modeling assumptions not necessarily in line with the underlying physical processes.

The deformation resulting from the application of the surface matching algorithm allows for two different definitions of measurements. On the one hand, the point-wise measurement

V_R,ΓO :=C_R,ΓO(V_R) (6.5)

is generated by restricting the registered deformation to the associated n^Γ_dof degrees of freedom of the boundaryΓO. This defines a likelihood in terms of the similarity measure

0.0 0.5 1.0 1.5 2.0 2.5 distance d [mm]

0 20 40 60 80 100

%ofverticeswithdistance<d

Γ_O→Γ^A(c_O ^ϑ,P) ϕR(ΓO)→Γ^A(c_O ^ϑ,P)

Figure 6.2.: Result of the surface matching algorithm in terms of the projected distance (3.44).

The initial distances betweenΓO andΓ^A(c_O ^ϑ,P) is shown in blue. The result of the registration in terms of the distance between the targetΓ^A(c_O ^ϑ,P)and the registered surfaceϕR(ΓO)is shown in green.

(3.38) via

−logp(θ|Z) =−logp_D(c_ϑ|V_R,ΓO) = 1

2σ_D² kC(A(c_ϑ))−V_R,ΓOk²

R^nΓdof. (6.6) On the other hand, the registered surface

Γ^R_O:=ϕ_R(Γ_O) (6.7)

can be used as a measurement in the space of currents. According to (3.69), a negative log-likelihood is given by

−logp(θ|Z) =−logp_W(c_ϑ|Γ^R₀) = 1

2σ_N² kϕ^A(cϑ)(Γ_O)−Γ^R_Ok²_W^∗. (6.8) Both likelihood functions could have been equivalently defined if the measurements V_R,ΓO had been obtained from image registration. This highlights the use of the sur-face matching algorithm as an emulation of systematic errors introduced through the application of image registration.

6.1.2. Inverse problem setup

The result of the registration step is the creation of a registered displacement measure-ment VR,ΓO that closely resembles the result of a measurement process in applications involving image registration. Furthermore, by just changing the definition of the space of measurements from point-wise displacement measurements to surface measurements, the same measurement is obtained as a measured surface Γ^R_O. Since both measurements are subject to the exact same systematic error, the resulting parameter estimates can be compared. A prerequisite for this comparison is the consistent determination of the noise parametersσD andσN. Obviously,σD is best chosen to be the standard deviation of the