Optimization of Designs for fMRI

1. Optimization of Designs for fMRI IPAM Mathematics in Brain Imaging Summer School July 18, 2008 Thomas Liu, Ph.D. UCSD Center for Functional MRI

3. Why optimize? • Scans are expensive. • Subjects can be difficult to find. • fMRI data are noisy • A badly designed experiment is unlikely to yield publishable results. • Time = Money If your result needs a statistician then you should design a better experiment. --Baron Ernest Rutherford

4. What to optimize? • Statistical Efficiency: maximize contrast of interest versus noise. • Psychological factors: is the design too boring? Minimize anticipation, habituation, boredom, etc.

5. Which is the best design? It depends on the experimental question. E

6. Possible Questions • Where is the activation? • What does the hemodynamic response function (HRF) look like?

7. Model Assumptions • Assume we know the shape of the HRF but not its amplitude. • Assume we know nothing about the HRF (neither shape nor amplitude). • Assume we know something about the HRF (e.g. it’s smooth).

8. Additive Gaussian Noise Design Matrix Nuisance Matrix Data Nuisance Parameters Parameters of Interest General Linear Model y = Xh + Sb + n

9. Stimulus Convolve w/ HRF Design matrix depends on both stimulus and HRF Parameter = amplitude of response Example 1: Assumed HRF shape

10. Design Regressor The process can be modeled by convolvingthe activity curve with a "hemodynamic response function" or HRF =  HRF Predicted neural activity Predicted response Courtesy of FSL Group and Russ Poldrack

11.  h1  h2  h3  h4 Note: Design matrix only depends on stimulus, not HRF Unknown shape and amplitude Example 2: Unknown HRF shape

12. FIR design matrix FIR estimates Courtesy of Russ Poldrack

13. Example 3: Basis Functions 4 basis functions 5 random HDRs using basis functions 5 random HDRs w/o basis functions Here if we assume basis functions, we only need to estimate 4 parameters as opposed to 20.

14. Stimulus, neural activity, field strength, vascular state Thermal noise, physiological noise, low frequency drifts, motion Test Statistic Also depends on Experimental Design!!!

15. Efficiency

16. Covariance Matrix Known Known as an A-optimal design

17. Efficiency

18. What does a matrix do? k-dimensional vector N-dimensional vector N x k matrix The matrix maps from a k-dimensional space to a N-dimensional space.

19. Xv =  u 1 1 1  u 1 1 v 1 Matrix Geometry Geometric fact: The image of the a k-dimensional unit sphere under any N x k matrix is an N-dimensional hyperellipse. k-dimensional unit sphere N-dimensional hyperellipse

20.  u  u Xv =  u 1 1 1 v 1 1 2 v 2 2 1 Singular Value Decomposition

21. Parameter Space Data Space Parameter Noise Space Xv =  u Xh 1 1 1 0  u 1 1 Good for Detection v 1 h h 0 0 Assumed HDR shape Efficiency here is optimized by amplifying the singular vector closest to the assumed HDR. This corresponds to maximizing one singular value while minimizing the others.

22. Parameter Noise Space No assumed HDR shape Parameter Space Data Space Xv =  u 1 1 1  u 1 1 v 1 Here the HDR can point in any direction, so we don’t want to preferentially amplify any one singular value. This corresponds to an equal distribution of singular values.

23. If no assumptions, then use equal singular values. Some assumptions about shape If we know the the HDR lies within a subspace spanned by a set of basis functions, we should maximize the singular values in this subspace and minimize outside of this subspace.

24. Maximize singular values associated with subspace that contains HDR Maximize singular value associated with right singular vector closest to HDR Knowledge (Assumptions) about HRF Some Total None Make singular values as equal as possible

25. Singular Values and Power Spectrum

26. Equally distributed singular values = flat power spectrum Set of dominant singular values = spread of power spectral components One dominant singular value = one dominant power spectral component f f f Knowledge (Assumptions) about HRF Some Total None Power Spectra

27. Frequency Domain Interpretation Randomized ISI single-event Boxcar Regressor Low-pass Low-pass FFT Adapted From S. Smith and R. Poldrack

28. Which is the best design? E

29. Xv =  u Xh 1 1 1 0  u 1 1 v 1 h 0 E

30. f E

31. f E

32. Detection Power When detection is the goal, we want to answer the question: is an activation present or not? When trying to detect something, one needs to specify some knowledge about the “target”. In fMRI, the target is approximated by the convolution of the stimulus with the HRF. Once we have specified our target (e.g. stimuli and assumed HRF shape), the efficiency for estimating the amplitude of that target can be considered our detection power.

33. Equal Singular values One Dominant Singular value Theoretical Trade-off HDR Estimation Efficiency Detection Power

34. Theoretical Curves Estimation Efficiency Estimation Efficiency Estimation Efficiency Estimation Efficiency

35. Efficiency vs. Power Estimation Efficiency Detection Power

36. Efficiency with Basis Functions

37. Experiments where you want to characterize in detail the shape of the HDR. Experiments where you have a good guess as to the shape (either a canonical form or measured HDR) and want to detect activation. A reasonable compromise between 1 and 2. Detect activation when you sort of know the shape. Characterize the shape when you sort of know its properties Knowledge (Assumptions) about HRF Some Total None

38. Problems with habituation and anticipation Less Predictable Question If block designs are so good for detecting activation, why bother using other types of designs?

39. is a measure of the average number of possible outcomes. Entropy Perceived randomness of an experimental design is an important factor and can be critical for circumventing experimental confounds such as habituation and anticipation. Conditional entropy is a measure of randomness in units of bits. Rth order conditional entropy (Hr) is the average number of binary (yes/no) questions required to determine the next trial type given knowledge of the r previous trial types.

40. Entropy Example A A N A A N A A A N Maximum entropy is 1 bit, since at most one needs to only ask one question to determine what the next trial is (e.g. is the next trial A?). With maximum entropy, 21 = 2 is the number of equally likely outcomes. A C B N C B A A B C N A Maximum entropy is 2 bits, since at most one would need to ask 2 questions to determine the next trial type. With maximum entropy, the number of equally likely outcomes to choose from is 4 (22).

42. Efficiency  2^(Entropy)

43. Multiple Trial Types 1 trial type + control (null) A A N A A N A A A N Extend to experiments with multiple trial types A B A B N N A N B B A N A N A B A D B A N D B C N D N B C N

44. Multiple Trial Types GLM y = Xh + Sb + n X = [X1 X2 … XQ] h = [h1Th2T … hQT]T

45. Multiple Trial Types Overview Efficiency includes individual trials and also contrasts between trials.

46. Multiple Trial Types Trade-off

47. Optimal Frequency Can also weight how much you care about individual trials or contrasts. Or all trials versus events. Optimal frequency of occurrence depends on weighting. Example: With Q = 2 trial types, if only contrasts are of interest p = 0.5. If only trials are of interest, p = 0.2929. If both trials and contrasts are of interest p = 1/3.

48. Design As the number of trial types increases, it becomes more difficult to achieve the theoretical trade-offs. Random search becomes impractical. For unknown HDR, should use an m-sequence based design when possible. Designs based on block or m-sequences are useful for obtaining intermediate trade-offs or for optimizing with basis functions or correlated noise.

49. Optimality of m-sequences

50. Clustered m-sequences