UCSF Hyperpolarized MR Seminar Summer 2019, Lecture #7 "Analysis of Hyperpolarized 13C MRI" Lecturer: Peder Larson, Hsin-Yu Chen, Adam Autry Sponsored by the NIH/NIBIB-supported UCSF Hyperpolarized MRI Technology Resource Center (P41EB013598) https://radiology.ucsf.edu/research/labs/hyperpolarized-mri-tech

1. Analysis of Hyperpolarized 13C MRI
BioE 297: Hyperpolarized MR Seminar
August 16, 2019
Peder Larson, Ph.D.
Associate Professor, Department of Radiology and Biomedical Imaging, University
of California, San Francisco, CA, United States
peder.larson@ucsf.edu
https://radiology.ucsf.edu/research/labs/larson
@pezlarson

2. What do we do with HP 13C MR data?
 Parametrizations: Kinetic Modeling vs. alternatives (e.g. Area-under-curve)
 Choice of model
 Fitting algorithms
 Considerations
• Robustness of fitting
• Assumptions
• Limitations
August 16, 20192
Pyr Lac
T1 T1
kPLkTRANS
Imaging Voxel
AIF(t)
Pyr
Vessels

3. Parametrizations: Kinetic Modeling vs. alternatives
Numerous options
 Kinetic modeling (e.g. kPL)
 Unidirectional vs bidirectional
 Fixed vs free parameters
 Lactate/pyruvate
 Area-under-curve (AUCratio, Hill, et
al. PLoS One (2013).)
 Time-to-peak
 …
August 16, 2019
Daniels et al, NMR Biomed 2016, doi: 10.1002/nbm.3468
3

4.  L/P ratio (single time-point)
 AUC L/P ratio (dynamics)
 Kinetic modeling
• Assumptions
• Input functions
• Perfused voxel model
August 16, 20194

5. Ratiometric methods: Lactate/pyruvate (single time-
point)
 Simple approach – acquire single time-point lactate and pyruvate data, then
measure ratio
 Major limitation – variability with experiment timing (injection start, duration,
acquisition start time)
August 16, 20195

6. Area-under-curve methods
 Ratio of areas under curve (AUC) proportional to
kPL:
• Derived with Laplace transforms
• Assumes entire dynamics are captured and
constant-in-time flip angles
• No assumption on shape of input function (Pin)
• Assuming negligible “back-reaction”, kLP, and
consistent lactate relaxation rates, then
AUCratio is directly proportional to kPL
Hill, et al. PLoS One 8, 9 (2013), e71996.
August 16, 20196

7. AUCratio applicability
 Works best with
• Acquire data before injection OR
consistent bolus and acquisition
timing
• Constant-in-time flip angles
 Breaks down when
• Variations in lactate T1
• Acquisition starts when there is
already magnetization
• Variable flip angles and variations in
bolus timing and shape
August 16, 20197
AUCratio simulation with
metabolite-specific
(θpyruvate < θlactate),
constant-in-time flip angles

8. Is T1 consistent for HP agents?
 Largely
uninvestigated in vivo
 Extrapolated results
from in vitro (yeast)
show significantly
shorter intracellular
T1s for carboxylic
acids
August 16, 20198
Carboxylic acid T1,int T1,AC Relative T1,AC Predicted T1,int
Butyric acid 8.1±0.6 8.5 0.85 7.7
Keto‐isocaproic
acid
10.4 11 1.1 9.9
Acetic acid 9.0±1.0 10 1.0 9.0
Pyruvic acid 12.7 15 1.5 13.5
Lactic acid NA 5 0.5 4.5

9. Simulation Framework for
Evaluation of Analysis Methods
 Generate simulated data based on model
with typical parameter ranges and
experimental parameters (flip angles, TR,
start/end times)
 Measure AUCratio on ideal, noiseless data
 Repeat simulation with varying random
noise, and measure AUCratio to analyze
expected parameter accuracy
 Repeat simulation for various parameter
ranges to analyze sensitivity to
experimental and physiologic variations
August 16, 20199
Larson et al. NMR in Biomed 2018. DOI: 10.1002/nbm.3997

10. Simulation Evaluation
of AUCratio
 Constant 10º flip angle, 8x
phase encodes, TR = 5 s
 Most significant bias due to
R1L = 1/T1,lactate
 Small bias due to missing
start of acquisition
August 16, 2019
RelativeParameterAccuracy
Tarrive
Tbolus
Using gamma-
variate function
for input function,
u(t)
10

11. Simulation Evaluation
of AUCratio
 Metabolite-specific
10º(pyr)/20º(lac) flip angles,
8x phase encodes, TR = 5 s
 Bias due to R1L substantially
reduced
 Accuracy (dashed lines)
similar otherwise
 (Metabolite-specific flip angle
not helping!)
August 16, 2019
RelativeParameterAccuracy
11

12. Simulation Evaluation
of AUCratio
 Metabolite-specific variable flip
angles, 8x phase encodes, TR = 2 s
 Accuracy (dashed lines) improved,
BUT
 Large bias with differences in input
function differences and relaxation
rates
August 16, 2019
RelativeParameterAccuracy
Tarrive
Tbolus
12

13. 2D Dynamic Analysis
 From Phase I clinical trial
 3 good patient datasets
• Patient 1 – 10º flip
• Others –
10º(pyr)/20º(lac)
August 16, 201913
kPLAUCratio
Patient 1 Patient 3 Patient 4
Polarization Dissolution,
QC, and
delivery
Injection
Duration
Voxel
Size
Prostate Vasculature
max
Pyr
SNR
max
Pyr
time
max
Lac
SNR
max
Lac
time
max
Pyr
SNR
max
Pyr
time
max
Lac
SNR
max
Lac
time
Patient 1* 19.6% 50 s 14 s 3.5 cc 36.0 25 s 7.0 30 s 63.1 25 s 5.7 40 s
Patient 3# 18.7% 47 s 16 s 2 cc 134.9 15 s 16.7 25 s 339.9 15 s 13.6 20 s
Patient 4# 18.7% 64 s 15 s 1.2 cc 24.9 15 s 8.6 25 s 31.0 15 s 7.9 20 s
0.029 0.045 0.045

14. 3D Dynamic Prostate Cancer Analysis
 Multiband, variable flip angle
strategy
 AUCratio inconsistent compared
to kPL model
 Bottom plots show this is due to
bolus delivery variations under
variable flip angle acquisitions
August 16, 201914

15. AUCratio Summary
 Best when capturing entire dynamic curves
 Major assumption – lactate T1 is consistent
 Usable with variable flip angle strategies provided there is reproducible bolus
delivery
 Probably easier to use in preclinical studies, where there is less difference in
bolus characteristics due to rapid circulation times, and more physiologic
consistency in animal models compared to human studies
 Similarly, single time-point lactate:pyruvate ratio is fine if there is consistency in
the experiment timing (bolus delivery, vascular delivery, and acquisition time)
 Can create simulation-based calibration curves to convert to kPL
August 16, 201915

16. Kinetic modeling
 Kinetics assumed to fit to a model, which
can be expressed by a set of differential
equations:
 Fit data to model, with inclusion of solution
for differential equations
 Simple two-site model shown here
August 16, 2019
Pyr Lac
T1 T1
kPL
Imaging Voxel
u(t)
16

17. Additional considerations: RF Flip Angle compensation
 Solution: Hybrid continuous-discrete model
• Continuous evolution of z-magnetization
between RF pulses
• Discrete change in magnetization with RF
pulses
August 16, 2019
Bahrami et al. Quant Imaging Med Surg 2014. Maidens et al. IEEE-TMI 2016.
17
Continuous
Model
Discrete
Model

18. Example: Box-car input, two-site
kinetic model
 u(t) = “box-car” shape, i.e. constant for
a limited period of time
 Solve differential equations:
 Challenge: must fit or estimate
multiple parameters of input (rateinj,
tarrival, tend)
 Typical Assumptions: neglect back-
reaction, kLP = 0. Often fix T1 values
 Can be adapted for other input
shapes, such as gamma-variate
Zierhut et al, JMR 2010.
August 16, 201918
Pyr Lac
T1 T1
kPL
Imaging Voxel
u(t)

19. Example: Fitted input, two-site
kinetic model
 u(t) = Input, estimated from vascular
voxels in slice
 Estimate perfusion of pyruvate and
kPL
 Typical Assumptions: neglect back-
reaction, kLP = 0. Often fix T1 values
Maidens, Gordon, Arcak, Larson. IEEE-TMI 2016 DOI: 10.1109/TMI.2016.2574240 August 16, 201919
Pyr Lac
T1 T1
kPL
Imaging Voxel
u(t)

20. Fitting Methods: Mechanics of model fitting
 Approach: code up model and a minimization/optimization routine in your
favorite programming language
• MATLAB: lsqnonlin(), fmincon() …
 Test!
 Constrained fitting: add upper and lower bounds based on prior
results/expectations (e.g. T1,invivo < T1,solution limits on bolus duration)
 Our models typically create non-convex problems, which means the result can
depend a lot on initial guesses
• Potential solution: generate estimates from a simplified model, or based on
some other measurement source (e.g. linear fit to later data for T1 guess,
AUCratio for kPL guess, adjacent voxels fit values)
August 16, 201920

21. Robust Fitting
 Idea: separate out more complex models into sub-models
 Potentially easier to solve, less confusion and local minima than trying to fit
whole model
August 16, 201921
Perfusion
sub-model
Conversion
sub-model
Maidens, Gordon, Arcak, Larson. IEEE-TMI 2016 DOI: 10.1109/TMI.2016.2574240

22. “Input-less” Fitting as a Robust Model
 Actual pyruvate signal as input, change in lactate
as output
 No assumptions or fitting of pyruvate signal or
input function u(t)
 Pros: Reduced number of parameters to fit,
insensitive to fitting errors in pyruvate (e.g.
incorrect bolus shape), works with any sampling
strategy
 Cons: No estimate of perfusion, using input
shape could help constrain fit results, errors in
pyruvate can propogate noise
August 16, 2019
Inpsired by: Khegai, et al. NMR Biomed 2014, Bahrami, et al. Quant Imaging Med Surg 2014.
22
Only fit change in Lactate:
MZ,L[n]
MZ,L[n+1]

23. Simulation: Constant flip angles
 Bolus characteristics and T1 values
are fixed to nominal values (center
values in these plots)
 Input-less fitting slightly better than
AUCratio
 Changes in input (Tarrival, Tbolus)
cause major problems for Fitting with
Input
August 16, 201923

24. Simulation: Metabolite-specific
flip angles
 Bolus characteristics and T1 values
are fixed to nominal values (center
values in these plots)
 Input-less fitting slightly better than
AUCratio
 Changes in input (Tarrival, Tbolus)
cause major problems for Fitting with
Input
 Slight performance improvement
over constant flip angles, except for
B1 errors (larger flips used here)
August 16, 201924

25. Simulation: Metabolite-
specific, variable flip angles
 Bolus characteristics and T1 values
are fixed to nominal values (center
values in these plots)
 Changes in input (Tarrival, Tbolus)
cause major problems for Fitting with
Input and AUCratio
 AUCratio performance improved if
bolus is known
 Input-less now the most robust,
except to B1 errors
 Overall slight performance
improvement over other flip angle
schedules August 16, 201925

26. Simulation: Fitting bolus
characteristics
 T1 values are fixed to nominal
values
 Allow for fitting on bolus
characteristics (Tarrival, Tbolus)
 Improves robustness of Fitting with
Input at expense of slightly
decreased accuracy
August 16, 201926

27. Simulation:
Fitting T1
 Bolus
characteristics
(Tarrival, Tbolus)
are fixed to
nominal values
 Allow for fitting
on T1
 Large increases
in expected
variance of fits
August 16, 201927

28. Assuming unidirectional conversion
 Bidirectional model is very
poorly conditioned (Swisher et
al. MRM 2014.)
• Hard to separate decay and
metabolic conversion
 Lac-to-pyr in HP experiment
measured to be typically order
of magnitude of more less than
pyr-to-lac
August 16, 2019
Pyruvate Lactate
KPyrLac
T1,Pyr T1,Lac
kLacPyr
28

29. Alternative Tissue Models
 Problem: Potentially large fraction of pyruvate signal comes from
vascular and/or extracellular compartments, which are a potential
confounder for metabolism conversion
 Alternative models
a. Two-site kinetic model with input
b. “Perfused model” – separate extravascular and intravascular
compartments
c. Full model – separate Extravascular/extracellular,
intracellular, and intravascular compartments
 Key for evaluation: Adding more model parameters will always fit
data better, so use Akaike Information Criteria (AIC) which
balances fit quality with number of model parameters
 Assumptions
• Neglect kLP, lactate transport
• Vascular input function (VIF) estimated from heart voxels
• Fixed T1P = 45s, T1L = 25s
Bankson et al. Cancer Research 2016. doi: 10.1158/0008-5472.CAN-15-0171 August 16, 201929
Models b and c are comparable by AIC
(lower AIC is better)

30. Additional Considerations
 Multiple conversion pathways (e.g. alanine, bicarbonate from pyruvate)
• AUCratio still applies (assuming uni-directional conversion)
• Kinetic modeling easily adapted
 Fitting Constraints
• Spatial constraints similar to compressed sensing also possible
 Magnitude vs complex data
• Noise statistics changed by magnitude operation
• Should use appropriate maximum likelihood estimator – example in fit_kPL()
August 16, 201930

31. Hyperpolarized-MRI-Toolbox
https://github.com/LarsonLab/hyperpolarized-mri-toolbox
 Kinetic Modeling
• fit_pyr_kinetics() –input-less model, for pyruvate to lactate
(optionally bicarbonate and alanine)
• fit_kPL() – old version of input-less model
• fit_kPL_withinput – box-car input, two-site model
• fit_kPL_withgammainput – gamma-variate shape input, two-site
model
• ompute_* - AUCratio, time-to-peak, mean-time metrics
• test_fit_* functions for examples
 Feeling adventurous?
• Perfused model now in “perfused_model” branch
• Make sure to ‘git pull’ regularly as this is a work in progress
August 16, 201932

32. Live Demo
 https://mybinder.org/v2/gh/LarsonLab/hyperpolarized-mri-toolbox/notebooks
August 16, 201933

33. Recommendations
 Input-less kPL is just as robust as AUC ratios, and provides results in units of
[1/s] that can then be compared more generally
 Ratiometric methods require reproducible Experimental setup (good flip angle
calibration, consistent injection, consistent cardiovascular physiology)
 Perfused voxel model is promising as a more accurate model, but does require
additional estimates or measurements of vascular input function
 Metabolite-specific and Variable flip angles may improve SNR and expected
accuracy, but can also create additional instability in model
August 16, 201934