Optimal Experimental Design Theory

1. Optimal Experimental Design Theory

2. Motivation • To better understand the existing theory and learn from tools that exist out there in other fields • To further develop the framework • Better handle other design criteria • Improve the ability to search for the optimal design • Would like to increase the “survival rate” of the start points, threads often die due to encountering singular matrices

3. References • Texts • Atkinson, Donev, and Tobias “Optimum Experimental Designs with SAS.” • Pukelsheim. “Optimal Designs of Experiments.” • Papers and Guides • Albrecht et al. “Findings of the Joint Dark Energy Mission Figure of Merit Science Working Group.” 2008. • Coe. “Fisher Matrices and Confidence Ellipses: A Quick Start Guide and Software.” 2010. • Related Software • Fisher.py, Python– simple manipulation of Fisher matrices and plotting of ellipses • DETFast(Albrecht et al. 2006), Java – Compare expectations of cosmological constraints from different experiments with your choice of priors with a few clicks! • Fisher4Cast 4 (Bassett et al. 2009), Matlab – most sophisticated

4. Confidence Ellipses • The inverse of the Fisher matrix is the covariance matrix • Typically, we’ve only been interested in the diagonal entries of the inverse, i.e. the variance of each parameter • However, CRLB states that F-1 does bound the covariance in a matrix sense, not just the variances • We may interested in a more complete picture • Can visualize as an ellipse(oid) • The 1st standard deviation contour of the multivariate Gaussian dist.

5. Confidence Ellipses λ1 λ2

6. Fisher Matrix Operations • Knowledge about prior variances of a parameter can be incorporated by adding 1/σ2 the corresponding diagonal entry of F • e.g. removing B0 which implies we know its value (either by assumption or B0 map) • Instead, if we know that B0 is a value with some confidence, σ2, we can use that • Adding data sets, information from multiple experiments can be combined just by adding their F matrices • For example, we can consider the T1 estimate for a combined protocol using many mapping methods

7. Fisher Matrix Operations • In fact, our formulation of F = JTJ can be reduced to this • JTJ is the summation of dyads, jjT for each collected image • where j is the vector of δfi/δθ, the sensitivity of the image to changes in each tissue parameter • So, we are just adding the informations matrices associated with each image we collect

8. VOPs? • An emerging challenge in optimizing experiments for range of values is computation time • Fine sampling of say the B0 axis can cause the search to slow • This problem grow dramatically if we increase the dimensionality, find the protocol that best estimates a range of T1 and T2 and is robust to B0 and B1 effects • Multiple tissues represent many confidence ellipses, which we know can be more succinctly bounded by VOPs • However there’s a key difference from SAR optimized pulse designs, in which case SAR = bTQb, and we find b • In optimal experimental design, we know b (determined by the signal equation at the tissue we’re optimizing for), but want to find Q • But we don’t have free control over Q, i.e. F • F is restricted to the Jacobians of the pulse sequence we’re applying • We cannot use a fixed set of VOPs to summarize our Fs over a whole range of B0s because the Fs change as we are evaluating different protocols

9. VOPs on Each Image? • One workaround is to instead consider a set of potential images to collect • For example, we grid the space of flip angle, phase cycle, TR, etc. and consider what is the optimal choice of these to acquire that is robust to B0 • We can then get the VOPs for each image which gives a smaller set of matrices that bounds the information given by that image • Then we find which sum of VOPs gives the smallest confidence ellipse or CRLB • This may still be expensive with the size of the gridding and VOPs themselves require an eigenvalue decomposition for every matrix • i.e. every point in the grid in all dimensions

10. Optimal Design Theory • General Equivalence Theorem • Is a fundamental result that describes conditions that must be met if a design is optimal on some quantity of F • Provides methods for the construction and checking of designs • However, the “support points,” or the collected images are known, finding the NEX/noise variance to acquire each • Continuous and exact designs • For single response models (1 output), at most p(p+1)/2 experiments can be used to represent F of size pxp • We don’t need more than that many images • There can exist protocols with more images that are also equivalently optimal, but from a perspective of technician convenience, it’s better to choose the one with fewer series • For mcDESPOT, 7 params -> at most 28 images, we collect 27

11. Improving Stability • Previously, in order to maintain constant scan time, literally enforced an equality condition for the sum of the noise variances (“NEX”) • One practical change, which is more compatible with Newton solver, is to transform the variables • Solve for z, which can be any real value, no longer constrained, which restricted the solvers available to us and probably made the problem harder

12. Improving Stability • F can be regularized as F+εI, where ε is a small constant • Atkinson et al. note that in designs that have an objective based on a subset of the parameters, as in our case, singular F can often times appear • This way the inversion will always succeed • From before, the interpretation is that we have some tiny but constant prior information for all the variables • Therefore choosing between the best design with this prior should effectively be the same as without it

13. Criteria of Optimality • A-optimality – minimize tr(F-1), total variance • D-optimality – maximize the (log) det(F), equivalently minimize the area of ellipse • This a very popular objective, though perhaps less common in MR papers • E-optimality – minimize the largest eigenvalue of F-1 • Expect more circular ellipse, I think • Linear optimality – this is what we’ve been doing with CoV • General:

14. Compound Design • This is the term given to problems that seek to optimize over many Fs • For example the protocol that gives the best average precision in T1 for a given range