Towards dynamic PET reconstruction under flow conditions: Parameter identification in a PDE model

Louise Reips; Martin Burger; Ralf Engbers

doi:10.1515/jiip-2015-0016

Published by De Gruyter August 12, 2017

Towards dynamic PET reconstruction under flow conditions: Parameter identification in a PDE model

Louise Reips , Martin Burger and Ralf Engbers

From the journal Journal of Inverse and Ill-posed Problems

https://doi.org/10.1515/jiip-2015-0016

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Abstract

The aim of this paper is to discuss potential advances in PET kinetic models and direct reconstruction of kinetic parameters. As a prominent example we focus on a typical task in perfusion imaging and derive a system of transport-reaction-diffusion equations, which is able to include macroscopic flow properties in addition to the usual exchange between arteries, veins, and tissues. For this system we propose an inverse problem of estimating all relevant parameters from PET data. We interpret the parameter identification as a nonlinear inverse problem, for which we formulate and analyze variational regularization approaches. For the numerical solution we employ gradient-based methods and appropriate splitting methods, which are used to investigate some test cases.

Keywords: Parameter identification; dynamic PET; inverse problems; image processing; forward-backward splitting; reaction; diffusion; transport

MSC 2010: 35K57

Funding statement: This work was carried out when Louise Reips was with the Institute for Computational and Applied Mathematics, WWU Münster. Martin Burger and Ralf Engbers acknowledge partial support by the German Science Foundation (DFG) via SFB 656, Subproject B2, and Cells-in-Motion Cluster of Excellence (EXC 1003 – CiM), WWU Münster, Germany.

A Adjoint equations

The purpose of this section is the development of the parameter identification problem to allow the calculation of all the biological parameters that composes the vector p. Thus, minimizing the function below (with the regularization added) we can find the values that correspond to the desired physiological parameters

12⁢∫0T∫Ω(u-uk+12)2uk⁢𝑑x⁢𝑑t+ℛ⁢(p)+∫0T∫Ω(G⁢(p)-u)⁢q⁢𝑑x⁢𝑑t→minp

with

G⁢(p)=G⁢(p⁢(x,t))=u⁢(x,t) for all (x,t)∈Ω×[0,T].

With the associated Lagrange functional one has

ℒ⁢(u,p;q)=12⁢∫0T∫Ω(u-uk+12)2uk⁢𝑑x⁢𝑑t+ℛ⁢(p)+∫0T∫Ω(G⁢(p)-u)⁢q⁢𝑑x⁢𝑑t.

One must now calculate the optimality conditions to the problem, which means that all the partial Fréchet-derivatives must be zero. Thus, we obtain

∂⁡ℒ∂⁡u=u⁢(x,t)-uk+12⁢(x,t)uk⁢(x,t)-q⁢(x,t)=0.

The optimality conditions for k1⁢(x), k2⁢(x), k3⁢(x), V𝒯⁢(x), V𝒜⁢(x), V𝒱⁢(x), D𝒯⁢(x), D𝒜⁢(x) and D𝒱⁢(x) are

∂⁡ℒ∂⁡k1=α⁢(Λ𝒯⁢(x)⁢(k1⁢(x)-k1*)+Λ𝒜⁢(x)⁢(k1⁢(x)-k1*))-ξ⁢(Λ𝒯⁢(x)⁢Δ⁢k1⁢(x)+Λ𝒜⁢(x)⁢Δ⁢k1⁢(x))

-∫0TC𝒜⁢(x,t)⁢μ⁢(x,t)⁢𝑑t+∫0TC𝒜⁢(x,t)⁢η⁢(x,t)⁢𝑑t,

∂⁡ℒ∂⁡k2=α⁢(Λ𝒯⁢(x)⁢(k2⁢(x)-k2*)+Λ𝒱⁢(x)⁢(k2⁢(x)-k2*))-ξ⁢(Λ𝒯⁢(x)⁢Δ⁢k2⁢(x)+Λ𝒱⁢(x)⁢Δ⁢k2⁢(x))

+∫0TC𝒯⁢(x,t)⁢μ⁢(x,t)⁢𝑑t-∫0TC𝒯⁢(x,t)⁢γ⁢(x,t)⁢𝑑t,

∂⁡ℒ∂⁡k3=α⁢(Λ𝒜⁢(x)⁢(k3⁢(x)-k3*)+Λ𝒱⁢(x)⁢(k3⁢(x)-k3*))-ξ⁢(Λ𝒜⁢(x)⁢Δ⁢k3⁢(x)+Λ𝒱⁢(x)⁢Δ⁢k3⁢(x))

-∫0TC𝒱⁢(x,t)⁢η⁢(x,t)⁢𝑑t+∫0TC𝒱⁢(x,t)⁢γ⁢(x,t)⁢𝑑t,

∂⁡ℒ∂⁡V𝒯=∫0TV𝒯⁢(x)⋅∇⁡μ⁢(x,t)⁢𝑑t+α⁢(V𝒯⁢(x)-V𝒯*)-ξ⁢(Λ𝒯⁢(x)⁢Δ⁢V𝒯⁢(x)),

∂⁡ℒ∂⁡V𝒜=∫0TV𝒜⁢(x)⋅∇⁡η⁢(x,t)⁢𝑑t+α⁢(V𝒜⁢(x)-V𝒜*)-ξ⁢(Λ𝒜⁢(x)⁢Δ⁢V𝒜⁢(x)),

∂⁡ℒ∂⁡V𝒱=∫0TV𝒱⁢(x)⋅∇⁡γ⁢(x,t)⁢𝑑t+α⁢(V𝒱⁢(x)-V𝒱*)-ξ⁢(Λ𝒱⁢(x)⁢Δ⁢V𝒱⁢(x)),

∂⁡ℒ∂⁡D𝒯=∫0T∇⁡C𝒯⁢(x)⋅∇⁡μ⁢(x,t)⁢𝑑t+α⁢(D𝒯⁢(x)-D𝒯*)-ξ⁢(Λ𝒯⁢(x)⁢Δ⁢D𝒯⁢(x)),

∂⁡ℒ∂⁡D𝒜=∫0T∇⁡C𝒜⁢(x)⋅∇⁡η⁢(x,t)⁢𝑑t+α⁢(D𝒜⁢(x)-D𝒜*)-ξ⁢(Λ𝒜⁢(x)⁢Δ⁢D𝒜⁢(x)),

∂⁡ℒ∂⁡D𝒱=∫0T∇⁡C𝒱⁢(x)⋅∇⁡γ⁢(x,t)⁢𝑑t+α⁢(D𝒱⁢(x)-D𝒱*)-ξ⁢(Λ𝒱⁢(x)⁢Δ⁢D𝒱⁢(x)).

We apply the Forward-Backward Splitting method for all parameters that composes the vector p to obtain

k1k+1⁢(x,y)=(1+2⁢α⁢τ-2⁢ξ⁢τ⁢Bx-2⁢ξ⁢τ⁢By)-1

×(k1k⁢(x,y)+τ⁢∫0TC𝒜⁢(x,y,t)⁢μ⁢(x,y,t)⁢𝑑t-τ⁢∫0TC𝒜⁢(x,y,t)⁢η⁢(x,y,t)⁢𝑑t+2⁢α⁢τ⁢k1*),

k2k+1⁢(x,y)=(1+2⁢α⁢τ-2⁢ξ⁢τ⁢Bx-2⁢ξ⁢τ⁢By)-1

×(k2k⁢(x,y)-τ⁢∫0TC𝒯⁢(x,y)⁢μ⁢(x,y)⁢𝑑t+τ⁢∫0TC𝒯⁢(x,y,t)⁢γ⁢(x,y,t)⁢𝑑t+2⁢α⁢τ⁢k2*),

k3k+1⁢(x,y)=(1+2⁢α⁢τ-2⁢ξ⁢τ⁢Bx-2⁢ξ⁢τ⁢By)-1

×(k3k⁢(x,y,t)+τ⁢(x,y,t)⁢∫0TC𝒱⁢(x,y,t)⁢η⁢(x,y,t)⁢𝑑t+τ⁢∫0TC𝒱⁢(x,y,t)⁢γ⁢(x,y,t)⁢𝑑t+2⁢α⁢τ⁢k3*),

V𝒯k+1⁢(x,y)=(1-α⁢τ+ξ⁢τ⁢Bx+ξ⁢τ⁢By)-1⁢(V𝒯k⁢(x,y)-τ⁢V𝒯k⁢(x,y)⋅∇⁢∫0Tμ⁢(x,y,t)⁢𝑑t+α⁢τ⁢V𝒯*),

V𝒜k+1⁢(x,y)=(1-α⁢τ+ξ⁢τ⁢Bx+ξ⁢τ⁢By)-1⁢(V𝒜k⁢(x,y)-τ⁢V𝒜k⁢(x,y,t)⋅∇⁢∫0Tη⁢(x,y,t)⁢𝑑t+α⁢τ⁢V𝒜*),

V𝒱k+1⁢(x,y)=(1-α⁢τ+ξ⁢τ⁢Bx+ξ⁢τ⁢By)-1⁢(V𝒱k⁢(x,y)-τ⁢V𝒱k⁢(x,y)⋅∇⁢∫0Tγ⁢(x,y,t)⁢𝑑t+α⁢τ⁢V𝒱*),

D𝒯k+1⁢(x,y)=(1+α⁢τ-ξ⁢τ⁢Bx-ξ⁢τ⁢By)-1⁢(D𝒯k⁢(x,y)-τ⁢(∇⁡D𝒯k⁢(x,y)⁢∫0T∇⁡μ⁢(x,y,t)⁢𝑑t)+α⁢τ⁢D𝒯*),

D𝒜k+1⁢(x,y)=(1+α⁢τ-ξ⁢τ⁢Bx-ξ⁢τ⁢By)-1⁢(D𝒜k⁢(x,y)-τ⁢(∇⁡D𝒜k⁢(x,y)⁢∫0T∇⁡η⁢(x,y,t)⁢𝑑t)+α⁢τ⁢D𝒜*),

D𝒱k+1⁢(x,y)=(1+α⁢τ-ξ⁢τ⁢Bx-ξ⁢τ⁢By)-1⁢(D𝒱k⁢(x,y)-τ⁢(∇⁡D𝒱k⁢(x,y)⁢∫0T∇⁡γ⁢(x,y,t)⁢𝑑t)+α⁢τ⁢D𝒱*).

A good choice of τ defines a significant speedup, because the dependence on the ill-posedness of the operator K (the ill-conditioning of the matrix that represents the discretization of K) can make the iterative scheme very slow.

B Example 1: Small defects in perfusion – Regularization parameters

Table 3 shows all the regularization parameters for Example 1. The (⋅)* refers to a-priori knowledge in the regularization functional for each parameter of the problem. Whereas, for example, the velocity of the radioactive concentration in the artery has a typical value of V𝒜*, we can regularize V𝒜 by

ℛ⁢(V𝒜⁢(x))=α2⁢∫Ω(V𝒜-V𝒜*)2⁢𝑑x,

where α (values shown in the third column) denotes the regularization parameter, α∈ℝ+.

Table 3

Input regularization parameters for a first real example.

Parameter	(⋅)*	A-p. regularization (α)	Gradient regularization (ξ)
k1	0.89	0.0171	0.0008
k2	0.7	0.0158	0.0001
k3	0.85	0.0164	0.0001
Vx𝒜	0.1	0.0010	0.0001
Vy𝒜	15	1.1000	0.0001
Vx𝒯	-5	1.1220	0.0001
Vy𝒯	0.1	0.0010	0.0001
Vx𝒱	0.1	0.0010	0.0001
Vy𝒱	15	1.1000	0.0001
D𝒜	10(-3)	0.0003	0.0004
D𝒯	10(-2)	0.0003	0.0004
D𝒱	10(-3)	0.0003	0.0004

Like the a-priori regularization we apply the Gradient regularization in each parameter independently. The regularization of the gradient is designed to ensure (guarantee) smoothness in space and time, adding a bound to the spatial gradients (∇⁡k1,∇⁡k2,∇⁡k3,∇⁡V𝒜,∇⁡V𝒯,∇⁡V𝒱,∇⁡D𝒜,∇⁡D𝒯,∇⁡D𝒱). The regularization added to the terms is given by

ℛξ,Φ⁢(g)=ξ2⁢∫Φ|∇⁡g⁢(x)|2⁢𝑑x

with Φ∈Ω. Thus, the fourth column refers to the terms ξ for each biological parameter in the above equation.

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Received: 2015-2-2

Revised: 2017-4-11

Accepted: 2017-6-30

Published Online: 2017-8-12

Published in Print: 2018-4-1

Towards dynamic PET reconstruction under flow conditions: Parameter identification in a PDE model

Abstract

A Adjoint equations

B Example 1: Small defects in perfusion – Regularization parameters

References

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