PDE-constrained optimization in medical image analysis

Abstract

PDE-constrained optimization problems find many applications in medical image analysis, for example, neuroimaging, cardiovascular imaging, and oncologic imaging. We review the related literature and give examples of the formulation, discretization, and numerical solution of PDE-constrained optimization problems for medical imaging. We discuss three examples. The first is image registration, the second is data assimilation for brain tumor patients, and the third is data assimilation in cardiovascular imaging. The image registration problem is a classical task in medical image analysis and seeks to find pointwise correspondences between two or more images. Data assimilation problems use a PDE-constrained formulation to link a biophysical model to patient-specific data obtained from medical images. The associated optimality systems turn out to be sets of nonlinear, multicomponent PDEs that are challenging to solve in an efficient way. The ultimate goal of our work is the design of inversion methods that integrate complementary data, and rigorously follow mathematical and physical principles, in an attempt to support clinical decision making. This requires reliable, high-fidelity algorithms with a short time-to-solution. This task is complicated by model and data uncertainties, and by the fact that PDE-constrained optimization problems are ill-posed in nature, and in general yield high-dimensional, severely ill-conditioned systems after discretization. These features make regularization, effective preconditioners, and iterative solvers that, in many cases, have to be implemented on distributed-memory architectures to be practical, a prerequisite. We showcase state-of-the-art techniques in scientific computing to tackle these challenges.

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Notes

  1. 1.

    In general, constraints can either be hard or soft. Hard constraints are a set of conditions that the variables are required to satisfy. Soft constraints are penalties for variables that appear in the objective functional; they penalize deviation of the variables from a condition.

  2. 2.

    We note that high-resolution imaging technologies have emerged; an example is CLARITY imaging (Chung and Deisseroth 2013; Tomer et al. 2014; Kutten et al. 2017); we revisit this ex-situ imaging technique briefly in Sect. 2. High-resolution imaging techniques are not yet available in a clinical setting. Resolution levels for routinely collected imaging data are between 0.5 and 5 mm in each spatial direction (depending on the imaging modality).

  3. 3.

    Similar formulations can be found in other applications, e.g., geophysical sciences Fohring et al. (2014).

  4. 4.

    An interesting direction of research is to augment these PDE constraints by more complex biophysics operators (Hogea et al. 2008a; Gooya et al. 2013; Sundar et al. 2009; Zacharaki et al. 2008, 2008, 2009). This results in additional parameters that need to be calibrated.

  5. 5.

    Applications for mass-preserving registration can be found in Burger et al. (2013), Mang and Ruthotto (2017) and Wlazło et al. (2016).

  6. 6.

    The number of unknowns is the number of discretization points in space times the number of discretization points in time for the velocity times the dimensionality of the ambient space (we invert for a time-dependent vector field).

  7. 7.

    A stationary velocity field v is a velocity field that is constant in time, as opposed to a nonstationary—i.e., time-dependent or transient—velocity field. We note that stationary velocity fields do not cover the entire space of diffeomorphisms and do not provide a Riemannian metric on this space, something that may be desirable in certain applications (Beg et al. 2005; Miller 2004; Zhang and Fletcher 2015); this requires time-dependent velocities. The work in Mang and Biros (2015) uses a Galerkin method to control the number of unknowns in time. It is shown experimentally that stationary and nonstationary velocities yield an equivalent registration quality in terms of data mismatch.

  8. 8.

    The resolution of these datasets can reach \(5 \upmu m\times 5 \upmu m\times 5 \upmu m\) resulting in \({\mathcal {O}}(4.8\hbox { TB})\) of data (if stored with half precision) Tomer et al. (2014). Overall, this results in 2.4 trillion discretization points in space.

  9. 9.

    Other distance measures, such as mutual information, normalized gradient fields, or cross correlation can be used (see Modersitzki 2004, 2009 for an overview). In our formulation, changing the distance measure will affect the first term in the objective functional and the terminal condition of the adjoint equation (8) (see Sect. 5.1.1).

  10. 10.

    More sophisticated multi-species models that, e.g., account for hypoxia, necrosis and angiogenesis can be found in Hawkins-Daarud et al. (2013), Gu et al. (2012), Rahman et al. (2017).

  11. 11.

    Models that account for the mechanical interaction of the tumor with its surroundings have been described in Chen et al. (2012), Clatz et al. (2005), Hogea et al. (2007), Mohamed and Davatzikos (2005), Weis et al. (2017), Wong et al. (2015) and Wong et al. (2017).

  12. 12.

    We note that there exists a large body of literature on optimal control problems with similar PDEs as constraints in other areas. Examples can be found in Barthel et al. (2010), Croft et al. (2015), Figueiredo et al. (2011), Figueiredo and Leal (2013) and Pearson and Stoll (2013).

  13. 13.

    A naive implementation may require storage of the time history of the state and adjoint fields; one remedy is the implementation of check-pointing/domain decomposition schemes (Akcelik et al. 2002; Griewank 1992; Heinkenschloss 2005).

  14. 14.

    We note that there certainly exist parallel implementations of solvers for similar PDE-constrained optimization problems. We refer to Akcelik et al. (2002, 2006), Biros and Ghattas (1999, 2005a, b), Biegler et al. (2003, 2007) for examples.

  15. 15.

    The white matter fibre architecture can be estimated from data measured by so-called diffusion tensor imaging, a magnetic resonance imaging technique that measures the anisotropy of diffusion in the human brain. The result of this measurement is a tensor field \(\tilde{k}\) that can directly be inserted into (4).

  16. 16.

    Mutual information is a statistical distance measure that originates from information theory. As opposed to the squared \(L^2\) distance used in the present work (see Sect. 2), which is used for registering images acquired with the same modality, mutual information is used for the registration of images that are acquired with different modalities (e.g., the registration of computed tomography and magnetic resonance images). Mutual information assess the statistical dependence between two random variables (in our case image intensities; see Modersitzki 2004, 2009; Sotiras et al. 2013 for more details).

  17. 17.

    We neglect the periodic boundary conditions for simplicity.

  18. 18.

    Instead of solving an advection or continuity equation as done in the Eulerian formulation we present here, it has been suggested to solve for the state and adjoint variables using the diffeomorphism y (which involves interpolation operations instead of the solution of a transport equation) Vialard et al. (2012).

  19. 19.

    We neglect the Neumann boundary conditions for simplicity.

  20. 20.

    The data assimilation problem in Sect. 3 requires Neumann boundary conditions on \(\partial {\varOmega }_B\), with \({\varOmega }_B\subset {\varOmega }\). We use a penalty approach to approximate these boundary conditions. We apply periodic boundary conditions on \(\partial {\varOmega }\) and set the reaction and diffusion coefficients in (4a) to sufficiently small penalty parameters \(k^\epsilon \rightarrow 0\) and \(\rho ^\epsilon \rightarrow 0\) outside of \({\varOmega }_B\); see, e.g., Gholami et al. (2016), Hogea et al. (2008b), Mang (2014) and Mang et al. (2012).

  21. 21.

    Note that the \(\inf\)-\(\sup\) condition for pressure spaces arising in finite element discretizations of Stokes problems (Brezzi and Fortin 1991, p. 200ff.) is not an issue with our scheme (incompressibility constraint in diffeomorphic registration problem).

  22. 22.

    A possible remedy is to employ check-pointing or domain decomposition strategies (Akcelik et al. 2002; Griewank 1992; Heinkenschloss 2005).

  23. 23.

    By reduced space we mean that we iterate only on the reduced space of the control variable of our problem as opposed to so called full-space or all-at-once methods (see, e.g., Benzi et al. 2009; Biros and Ghattas 2005a, b; Haber and Ascher 2001; Herzog and Kunisch 2010 for more details).

  24. 24.

    The order \(\tilde{n}\) of the optimality system depends on the problem. For the diffeomorphic registration case the control variable \({\mathbf {w}}\) is given by the velocity field \({\mathbf {v}}\in {\mathbf {R}}^{dn}\), i.e., \(\tilde{n}\equiv dn\); for the tumor case \({\mathbf {w}}\) is given by \(p\in {\mathbf {R}}^{n_p}\), i.e., \(\tilde{n}\equiv n_p\).

  25. 25.

    Our implementation also features a trust region method.

  26. 26.

    Additional information on the data sets, the imaging protocol, and the preprocessing can be found in Christensen et al. (2006) and at http://www.nirep.org/.

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Acknowledgements

The authors thank the anonymous reviewers for their careful reading and their insightful comments.

Funding

This material is based upon work supported by AFOSR grants FA9550-17-1-0190; by NSF Grant CCF-1337393; by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Numbers DE-SC0010518 and DE-SC0009286; by NIH Grant 10042242; by DARPA Grant W911NF-115-2-0121; and by the Technische Universität München—Institute for Advanced Study, funded by the German Excellence Initiative (and the European Union Seventh Framework Programme under grant agreement 291763). Any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the AFOSR, DOE, NIH, DARPA, and NSF. Computing time on the Texas Advanced Computing Center’s (TACC) systems was provided by an allocation from TACC and the NSF. Computing time on the High-Performance Computing Center’s (HLRS) Hazel Hen system (Stuttgart, Germany) was provided by an allocation of the federal project application ACID-44104. This work was completed in part with resources provided by the University of Houston Center for Advanced Computing and Data Science (CACDS)

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Correspondence to Andreas Mang.

Appendices

Appendix 1: Computing \(\det \nabla y\)

We do not differentiate the deformation map y, but transport \(\psi \mathrel {\mathop :}=\det \nabla \phi\) where \(y^{-1}(x) = \phi (x,t=1)\). That is, we solve

$$\begin{aligned} \partial _t \psi + v\cdot \nabla \psi&= \psi \nabla \cdot v&\quad \text {in}\;\;{\varOmega }\times (0,1] \end{aligned}$$
(13a)
$$\begin{aligned} \psi&= 1&\quad \text {in}\;\;{\varOmega }\times \{0\} \end{aligned}$$
(13b)

with periodic boundary conditions on \(\partial {\varOmega }\).

Remark 6

If we start from an identity map \({\text {id}}\) (zero velocity field) the determinant of the deformation gradient will have a value of one. Hence, we require that the determinant of the deformation gradient remains strictly positive for all \(x \in {\varOmega }\) for the computed deformation map to be locally diffeomorphic. Strictly speaking, we also require that the map y is smooth, and has a smooth inverse for y to be a diffeomorphism. If we use (slightly more than) \(H^2\)-regularity for v we meet these requirements from a theoretical perspective (assuming that the images are smooth) (Beg et al. 2005; Vialard et al. 2012). However, we note that ensuring that the discrete deformation map y (which does not appear in our formulation) is a diffeomorphism, requires sophisticated discretization schemes (see, e.g., Burger et al. 2013). In our formulation, we transport the data. Since the deformation map y does not appear explicitly, we decided to compute the determinant of the deformation gradient by computing the solution of the transport problem in Appendix 1. We found that this is more stable than computing \(\det \nabla y\) from y using our numerical scheme.

Appendix 2: Gauss–Newton Krylov algorithm

Here, we showcase the individual steps necessary for solving (1) in Algorithm 3 (Newton iterations; outer iterations) and Algorithm 4 (solution of the system in (12), inner iterations). We refer to Mang and Biros (2015, 2017), Gholami et al. (2016) for details on the particular implementation of our nonlinear solver. We rely on PETSc’s TAO package (Balay et al. 2016; Munson et al. 2017) for our distributed-memory solver (Mang et al. 2016; Gholami et al. 2017) (see Sect. 5.5 for details). The steps are the same as outlined in Algorithms 3 and 4, respectively.

figurec
figured

Appendix 3: Newton step

Here, we describe the Newton step for the two problems described in Sects. 2.2 and 3.2. The Newton step is obtained by computing the second variation of (6). We will see that the structure of the PDE operators that appear in the reduced space Hessian is very similar to the first-order optimality systems in Sect. 5.1.

Diffeomorphic registration

The nonlinearity, ill-posedness, and the dimensionality of the search space makes the solution of this variational problem computational challenging. Most available implementations use first-order iterative schemes to compute a minimizer to the variational optimization problem (see, e.g., Avants et al. 2011; Beg et al. 2005; Chen and Lorenz 2011; Ha et al. 2010; Hart et al. 2009; Vialard et al. 2012). To increase the rate of convergence, they typically do not use \(g_v\) in their iterative scheme, but the gradient in the Sobolev space induced by the regularization operator \({\mathcal {A}}\), i.e.,

$$\begin{aligned} \tilde{g}_v = v + (\beta {\mathcal {A}})^{-1}\int _0^1\lambda \nabla m{\mathrm {d}}t. \end{aligned}$$

First-order methods are usually inefficient; they require a lot of iterations to converge to a specific tolerance; we have analyzed this in Mang and Biros (2015). An alternative is to apply variants of Newton’s method to the KKT system. To derive the operators for applying Newton’s method we have to compute the second variations of (6). We arrive at the incremental state and adjoint equation

$$\begin{aligned} \partial _t \tilde{m} + \nabla \tilde{m} \cdot v + \nabla m \cdot \tilde{v}&= 0&{\mathrm{in}}\;\; {\varOmega } \times (0,1], \end{aligned}$$
(14a)
$$\begin{aligned} \tilde{m}&= 0&\quad {\mathrm{in}}\;\; {\varOmega } \times \{0\}, \end{aligned}$$
(14b)
$$\begin{aligned} -\partial _t \tilde{\lambda } - \nabla \cdot (v\tilde{\lambda } + \tilde{v}\lambda )&= 0&\quad \text {in}\;\; {\varOmega } \times [0,1), \end{aligned}$$
(14c)
$$\begin{aligned} \tilde{\lambda }&= - \tilde{m}&\quad {\mathrm{in}}\;\; {\varOmega } \times \{1\}, \end{aligned}$$
(14d)

and the expression for the reduced space Hessian matvec

$$\begin{aligned} {\mathcal {H}}(v)\tilde{v} \mathrel {\mathop :}=\beta {\mathcal {A}}[\tilde{v}] + {\mathcal {K}}[\int _0^1\tilde{\lambda }\nabla m + \lambda \nabla \tilde{m}{\mathrm {d}}t\;]&\quad\text {in}\,\, {\varOmega }, \end{aligned}$$
(14e)

with periodic boundary conditions on \(\partial {\varOmega }\). The incremental control variable \(\tilde{v}\) corresponds to the search direction of our scheme. The transport equations for the incremental state and adjoint fields \(\tilde{m}\) and \(\tilde{\lambda }\) are hidden in the integro-differential operator in (14e); we have to solve (14a) and (14c) every time we apply \({\mathcal {H}}\) to a new vector \(\tilde{v}\). We use a Gauss–Newton approximation to the true Hessian to ensure that the operator is positive definite far away from the optimum. This corresponds to neglecting all terms that involve \(\lambda\) in (14c) and (14e).

Data assimilation

The expressions for evaluating the Hessian operator are derived by computing the second variations for (9). The Hessian matvec is given by \({\mathcal {H}}\tilde{p} \mathrel {\mathop :}=\beta \tilde{p} - {\varPhi }^\mathsf {T}\tilde{\lambda }_0\), \({\mathcal {H}} : {\mathbf {R}}^{n_p} \rightarrow {\mathbf {R}}^{n_p}\). To be able to evaluate this expression we have to solve the incremental state and adjoint equations given by

$$\begin{aligned} \partial _t \tilde{m} - \nabla \cdot k \nabla \tilde{m} - \rho (1-2m)\tilde{m}&= 0&\quad {\mathrm{in}}\;{\varOmega }_B\times (0,1], \end{aligned}$$
(15a)
$$\begin{aligned} \tilde{m}&={\varPhi }\tilde{p}&\quad {\mathrm{in}}\;{\varOmega }_B\times \{0\}, \end{aligned}$$
(15b)
$$\begin{aligned} -\partial _t \tilde{\lambda } - \nabla \cdot k \nabla \tilde{\lambda } - \rho (1-2m - 2\lambda )\tilde{\lambda }&= 0&\quad {\mathrm{in}}\;{\varOmega }_B\times [0,1), \end{aligned}$$
(15c)
$$\begin{aligned} \tilde{\lambda }&= -{\mathcal {Q}}^\mathsf {T}{\mathcal {Q}} \tilde{m}&\quad {\mathrm{in}}\;{\varOmega }_B\times \{1\}, \end{aligned}$$
(15d)

with Neumann boundary conditions on \(\partial {\varOmega }_B\). For the Gauss–Newton approximation to the true Hessian the terms that involve \(\lambda\) in (15c) need to be dropped.

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Mang, A., Gholami, A., Davatzikos, C. et al. PDE-constrained optimization in medical image analysis. Optim Eng 19, 765–812 (2018). https://doi.org/10.1007/s11081-018-9390-9

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Keywords

  • Optimal control
  • Medical imaging
  • PDE-constrained optimization
  • Memory-distributed algorithms

Mathematics Subject Classification

  • 49K20
  • 65Y05
  • 65M32
  • 65K10
  • 76D55
  • 68U10
  • 35M10