TP2: Dynamic Medical Imaging Using Time-Continuous Models
Subproject leaders
Prof. Dr Martin Burger, Dr Hendrik Dirks
Project researcher
Meike Kinzel
Contact
Objectives
This work package develops models, algorithms, and implementations for processes with continuous dynamics and (nearly) continuous measurements. Continuous dynamics may arise from tracer biochemistry or from sources such as breathing, organ motion, or the continuous table movement used in some Siemens scanners. The first step is to formulate the reconstruction problem in continuous time, then represent the actual data and reconstruction using a suitable time discretisation.
An important example is list-mode PET, where temporal discretisation arises from a stochastic Poisson process. Another application is undersampled MR, where time discretisation results from the sequential acquisition of slices; the same applies to four-dimensional microscopy. Beyond this project, the results may also affect dynamic X-ray CT, where continuous dynamics are recorded through the (nearly) continuous rotation of the scanner.
A further objective is to study and separate different dynamics in images derived from indirect measurements, especially organ motion and tracer dynamics, but also external and internal motion (for example, organ and table motion) and organ motion and flow (for example, arterial motion and blood flow). In close collaboration with TP1 and TP3, we will develop variational methods and algorithms for these problems.
We expect to obtain variational models that minimise functionals of the form
$\int_0^T L_t(K_t(\rho_1,\ldots,\rho_M),f_t)\,dt+\int_0^T R_{\operatorname{Bild}}(\rho_1,\ldots,\rho_M)\,dt+\int_0^T R_{\operatorname{Bew}}(u_1,\ldots,u_M)\,dt+R_{Kin}(k_1,\ldots,k_P)$
with respect to $\rho_i,\,u_k,\,k_j$, subject to the general constraint
$\partial_t\rho_i+\nabla \cdot (\rho_iu_i)=\sum_{j=1}^M a_{ij}(k_1,\ldots,k_P)\rho_j,\quad i=1,\ldots,M.$
The unknowns are a vector of time-dependent densities $\rho_i$ (for different tracer states), their velocities $u_i$, and time-independent kinetic parameters $k_j$. Here, $L_t$ is the likelihood functional at time $t$, $K_t$ is the forward operator, and $R_{\operatorname{Bild}}$, $R_{\operatorname{Bew}}$, and $R_{Kin}$ are suitable regularisation functionals for images (total variation and variants), motion, and kinetic parameters (distance from a healthy reference value). The number $M$ of densities, the number $P$ of kinetic parameters, and the precise form of $a_{ij}$ depend on the tracer dynamics and its model. For multiple types of motion, velocities $u_i$ will be decomposed into additive components with different regularisation. We will implement these methods numerically, make them available for dynamic PET and MR in collaboration with TP4, and adapt them to specific clinical questions.
Input (from other subprojects)
Lagrangian motion-correction and registration methods (TP1)
Selection of regularisation parameters and uncertainty quantification (TP3)
Problem definitions and reconstruction software (TP4)
Output (to other subprojects)
Eulerian motion-correction methods and sparsity techniques (TP1)
Problem definitions and precise models, including regularisation parameters (TP3)
Algorithms for list-mode PET with multiple dynamics and for dynamic MR (TP4)
Previous work
H. Dirks’s dissertation, and earlier work in C. Brune’s dissertation, developed Eulerian methods for joint image reconstruction and motion estimation. Applications to biomedical data, particularly microscopy, demonstrate strong potential to improve both image quality and motion estimates. Fundamental contributions have also been made to spatially resolved estimation of kinetic PET parameters without motion correction by solving nonlinear inverse problems.
L. Reips’s dissertation took an initial step towards jointly modelling multiple dynamics using a simple perfusion model with motion and diffusion.
Modelling and variational methods
The work begins by deriving precise functionals and constraints for the different problems posed by the application partners. In close cooperation with TP3, we will study statistical models for time-continuous data, especially Poisson-distributed list-mode data, including likelihood models and suitable time discretisations of the unknowns. We will investigate different approaches to spatial and temporal regularisation of images, motion, and other kinetic parameters, and apply them to canonical cardiovascular PET (perfusion) and nephrological MR studies. The models will be analysed for mathematical properties, robustness, and other sources of uncertainty.
To obtain initial results quickly and examine models against practical questions early, we will implement test versions of numerical methods based on alternating minimisation over images, velocities, and parameters, together with primal-dual solvers for convex subproblems. These will be developed efficiently in the FlexBox environment created by H. Dirks.
Further model development and alternative approaches
The developed models will serve as a basis for further methodological advances. An initial step is selecting regularisation parameters—the weights of the individual $R$ terms relative to the likelihood $L_t$—based on results from TP3. We will also study automatic selection of multiple regularisation parameters numerically. Methods for uncertainty quantification will be adapted to the problems under study, including uncertainty due to data noise, model error, and external disturbances such as patient motion.
To improve motion estimation, we will extend the model to PET/MR data: one dataset contains both motion and dynamics (for example, PET with a suitable tracer), while another captures motion alone (for example, MR without contrast agent). Modelling the likelihood introduces a second data term that reduces underdetermination in the reconstruction and may further improve parameter and motion estimates. To separate organ motion from flows such as blood flow, we will also study a combination of Eulerian and Lagrangian methods, modelling organ motion as in TP1 with a Lagrangian approach and flow with an Eulerian approach.
For estimating kinetic parameters, we will also investigate methods based on local sparsity. Compartment models without motion are solved explicitly to obtain parameterised basis functions. With suitable parameter sampling, the solution can be represented as a sparse linear combination—ideally with only one non-zero coefficient—of these functions. We will additionally model motion, which causes further spatial mixing of the basis functions over time. Suitable variational methods using both Eulerian and Lagrangian motion models will be developed and studied numerically in collaboration with TP1.
Implementation and validation
For models that perform well in tests, we will develop and implement efficient numerical methods combining Gauss–Newton methods, convex splitting, and primal-dual formulations. Test implementations will enable continuous validation in cooperation with TP4. Final PET software implementations will be integrated into emrecon with TP4 and made directly available to Siemens Healthineers and the clinical partners.
DCE-MR implementations will be integrated with the registration software used by the clinical partners.