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## Optimal Experimental Design Theory

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Presentation Transcript

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

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

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.

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

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

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

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

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

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

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

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:

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

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