Skip to main content
Log in

Di-chromatic interpolation of magnetic resonance metabolic images

  • Research Article
  • Published:
Magnetic Resonance Materials in Physics, Biology and Medicine Aims and scope Submit manuscript



Magnetic resonance imaging with hyperpolarized contrast agents can provide unprecedented in vivo measurements of metabolism, but yields images that are lower resolution than that achieved with proton anatomical imaging. In order to spatially localize the metabolic activity, the metabolic image must be interpolated to the size of the proton image. The most common methods for choosing the unknown values rely exclusively on values of the original uninterpolated image.


In this work, we present an alternative method that uses the higher-resolution proton image to provide additional spatial structure. The interpolated image is the result of a convex optimization algorithm which is solved with the fast iterative shrinkage threshold algorithm (FISTA).


Results are shown with images of hyperpolarized pyruvate, lactate, and bicarbonate using data of the heart and brain from healthy human volunteers, a healthy porcine heart, and a human with prostate cancer.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others


  1. Ardenkjær-Larsen JH, Fridlund B, Gram A, Hansson G, Hansson L, Lerche MH, Servin R, Thaning M, Golman K (2003) Increase in signal-to-noise ratio of \(> 10,000\) times in liquid-state NMR. Proceedings of the National Academy of Sciences 100(18):10158–10163

    Article  Google Scholar 

  2. Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM journal on imaging sciences 2(1):183–202

    Article  Google Scholar 

  3. Boyd S, Boyd SP, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge

    Book  Google Scholar 

  4. Chambolle A, Pock T (2011) A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of mathematical imaging and vision 40(1):120–145

    Article  Google Scholar 

  5. Chen C, Li Y, Huang J (2013) Calibrationless parallel MRI with joint total variation regularization. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 106–114. Springer

  6. Chen HY, Larson PE, Gordon JW, Bok RA, Ferrone M, van Criekinge M, Carvajal L, Cao P, Pauly JM, Kerr AB et al (2018) Technique development of 3D dynamic CS-EPSI for hyperpolarized 13C pyruvate MR molecular imaging of human prostate cancer. Magnetic resonance in medicine 80(5):2062–2072

    Article  CAS  Google Scholar 

  7. Cui X, Mili L, Wang G, Yu H (2018) Wavelet-based joint CT-MRI reconstruction. Journal of X-ray science and technology 26(3):379–393

    Article  Google Scholar 

  8. Cunningham CH, Chen AP, Lustig M, Hargreaves BA, Lupo J, Xu D, Kurhanewicz J, Hurd RE, Pauly JM, Nelson SJ et al (2008) Pulse sequence for dynamic volumetric imaging of hyperpolarized metabolic products. Journal of magnetic resonance 193(1):139–146

    Article  CAS  Google Scholar 

  9. Cunningham CH, Lau JY, Chen AP, Geraghty BJ, Perks WJ, Roifman I, Wright GA, Connelly KA (2016) Hyperpolarized 13C metabolic MRI of the human heart: initial experience. Circulation research 119(11):1177–1182

    Article  CAS  Google Scholar 

  10. Das I (1999) On characterizing the “knee” of the pareto curve based on normal-boundary intersection. Structural optimization 18(2–3):107–115

    Article  Google Scholar 

  11. Diamond S, Boyd S (2016) CVXPY: A python-embedded modeling language for convex optimization. The Journal of Machine Learning Research 17(1):2909–2913

    Google Scholar 

  12. Dwork N, Lasry EM, Pauly JM, Balbás J (2017) Formulation of image fusion as a constrained least squares optimization problem. Journal of Medical Imaging 4(1):014003

    Article  Google Scholar 

  13. Ehman EC, Johnson GB, Villanueva-Meyer JE, Cha S, Leynes AP, Larson PEZ, Hope TA (2017) Pet/mri: where might it replace pet/ct? Journal of Magnetic Resonance Imaging 46(5):1247–1262

    Article  Google Scholar 

  14. Ehrhardt MJ, Markiewicz P, Liljeroth M, Barnes A, Kolehmainen V, Duncan JS, Pizarro L, Atkinson D, Hutton BF, Ourselin S et al (2016) PET reconstruction with an anatomical MRI prior using parallel level sets. IEEE transactions on medical imaging 35(9):2189–2199

    Article  Google Scholar 

  15. Esser E, Zhang X, Chan TF (2010) A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM Journal on Imaging Sciences 3(4):1015–1046

    Article  Google Scholar 

  16. Fessler JA (2010) Model-based image reconstruction for MRI. IEEE signal processing magazine 27(4):81–89

    Article  Google Scholar 

  17. Garzelli A (2016) A review of image fusion algorithms based on the super-resolution paradigm. Remote Sensing 8(10):797

    Article  Google Scholar 

  18. Golman K, Petersson JS, Magnusson P, Johansson E, Åkeson P, Chai CM, Hansson G, Månsson S (2008) Cardiac metabolism measured noninvasively by hyperpolarized 13C MRI. Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 59(5):1005–1013

    Article  CAS  Google Scholar 

  19. Golman K, Thaning M et al (2006) Real-time metabolic imaging. Proceedings of the National Academy of Sciences 103(30):11270–11275

    Article  CAS  Google Scholar 

  20. Gordon JW, Autry AW, Tang S, Graham JY, Bok RA, Zhu X, Villanueva-Meyer JE, Li Y, Ohilger MA, Abraham MR et al (2020) A variable resolution approach for improved acquisition of hyperpolarized 13C metabolic MRI. Magnetic Resonance in Medicine

  21. Gordon JW, Vigneron DB, Larson PE (2017) Development of a symmetric echo planar imaging framework for clinical translation of rapid dynamic hyperpolarized 13C imaging. Magnetic resonance in medicine 77(2):826–832

    Article  CAS  Google Scholar 

  22. Grant M, Boyd S (2008) Graph implementations for nonsmooth convex programs. In: Blondel V, Boyd S, Kimura H (eds.), Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences, pp. 95–110. Springer-Verlag Limited. Accessed 6 Feb 2019

  23. Grant M, Boyd S (2014) CVX: Matlab software for disciplined convex programming, version 2.1. Accessed 6 Feb 2019

  24. Grist JT, McLean MA, Riemer F, Schulte RF, Deen SS, Zaccagna F, Woitek R, Daniels CJ, Kaggie JD, Matys T et al (2019) Quantifying normal human brain metabolism using hyperpolarized [1-13C] pyruvate and magnetic resonance imaging. NeuroImage 189:171–179

    Article  CAS  Google Scholar 

  25. Knoll F, Holler M, Koesters T, Otazo R, Bredies K, Sodickson DK (2016) Joint MR-PET reconstruction using a multi-channel image regularizer. IEEE transactions on medical imaging 36(1):1–16

    Article  Google Scholar 

  26. Kurhanewicz J, Vigneron DB, Ardenkjaer-Larsen JH, Bankson JA, Brindle K, Cunningham CH, Gallagher FA, Keshari KR, Kjaer A, Laustsen C et al (2019) Hyperpolarized 13C MRI: path to clinical translation in oncology. Neoplasia 21(1):1–16

    Article  Google Scholar 

  27. Larson PE, Chen HY, Gordon JW, Korn N, Maidens J, Arcak M, Tang S, Criekinge M, Carvajal L, Mammoli D et al (2018) Investigation of analysis methods for hyperpolarized 13C-pyruvate metabolic MRI in prostate cancer patients. NMR in Biomedicine 31(11):e3997

    Article  Google Scholar 

  28. Larson PE, Hu S, Lustig M, Kerr AB, Nelson SJ, Kurhanewicz J, Pauly JM, Vigneron DB (2011) Fast dynamic 3D MR spectroscopic imaging with compressed sensing and multiband excitation pulses for hyperpolarized 13c studies. Magnetic resonance in medicine 65(3):610–619

    Article  CAS  Google Scholar 

  29. Lau AZ, Chen AP, Hurd RE, Cunningham CH (2011) Spectral-spatial excitation for rapid imaging of DNP compounds. NMR in Biomedicine 24(8):988–996

    Article  CAS  Google Scholar 

  30. Li Z, Leung H (2009) Fusion of multispectral and panchromatic images using a restoration-based method. IEEE transactions on geoscience and remote sensing 47(5):1482–1491

    Article  Google Scholar 

  31. Liu J, Liu T, de Rochefort L, Ledoux J, Khalidov I, Chen W, Tsiouris AJ, Wisnieff C, Spincemaille P, Prince MR et al (2012) Morphology enabled dipole inversion for quantitative susceptibility mapping using structural consistency between the magnitude image and the susceptibility map. Neuroimage 59(3):2560–2568

    Article  Google Scholar 

  32. Lukin A, Kubasov D (2004) High-quality algorithm for bayer pattern interpolation. Programming and Computer Software 30(6):347–358

    Article  Google Scholar 

  33. Mehranian A, Belzunce MA, Prieto C, Hammers A, Reader AJ (2017) Synergistic PET and SENSE MR image reconstruction using joint sparsity regularization. IEEE transactions on medical imaging 37(1):20–34

    Article  Google Scholar 

  34. Mugler JP III, Driehuys B, Brookeman JR, Cates GD, Berr SS, Bryant RG, Daniel TM, De Lange EE, Downs JH, Erickson CJ et al (1997) MR imaging and spectroscopy using hyperpolarized 129 Xe gas: preliminary human results. Magnetic resonance in medicine 37(6):809–815

    Article  Google Scholar 

  35. Nelson SJ, Kurhanewicz J, Vigneron DB, Larson PE, Harzstark AL, Ferrone M, van Criekinge M, Chang JW, Bok R, Park I et al (2013) Metabolic imaging of patients with prostate cancer using hyperpolarized [1-13C] pyruvate. Science translational medicine 5(198):198ra108

    Article  Google Scholar 

  36. Pock T, Cremers D, Bischof H, Chambolle A (2009) An algorithm for minimizing the mumford-shah functional. In: 2009 IEEE 12th International Conference on Computer Vision, pp. 1133–1140. IEEE

  37. Rudin LI, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Physica D: nonlinear phenomena 60(1–4):259–268

    Article  Google Scholar 

  38. Scheinberg K, Goldfarb D, Bai X (2014) Fast first-order methods for composite convex optimization with backtracking. Foundations of Computational Mathematics 14(3):389–417

    Article  Google Scholar 

  39. Schramm G, Holler M, Rezaei A, Vunckx K, Knoll F, Bredies K, Boada F, Nuyts J (2017) Evaluation of parallel level sets and bowsher’s method as segmentation-free anatomical priors for time-of-flight PET reconstruction. IEEE transactions on medical imaging 37(2):590–603

    Article  Google Scholar 

  40. Schroeder MA, Clarke K, Neubauer S, Tyler DJ (2011) Hyperpolarized magnetic resonance: a novel technique for the in vivo assessment of cardiovascular disease. Circulation 124(14):1580–1594

    Article  Google Scholar 

  41. Tang S, Milshteyn E, Reed G, Gordon J, Bok R, Zhu X, Zhu Z, Vigneron DB, Larson PE (2019) A regional bolus tracking and real-time B1 calibration method for hyperpolarized 13C MRI. Magnetic resonance in medicine 81(2):839–851

    Article  CAS  Google Scholar 

  42. Wang Y, Cao N, Liu Z, Zhang Y (2017) Real-time dynamic MRI using parallel dictionary learning and dynamic total variation. Neurocomputing 238:410–419

    Article  Google Scholar 

  43. Wang ZJ, Ohliger MA, Larson PE, Gordon JW, Bok RA, Slater J, Villanueva-Meyer JE, Hess CP, Kurhanewicz J, Vigneron DB (2019) Hyperpolarized 13c mri: State of the art and future directions. Radiology 291(2):273–284

    Article  Google Scholar 

  44. Wu H, Zhao S, Zhang J, Lu C (2019) Remote sensing image sharpening by integrating multispectral image super-resolution and convolutional sparse representation fusion. IEEE Access 7:46562–46574

    Article  Google Scholar 

  45. Xing Y, Reed GD, Pauly JM, Kerr AB, Larson PE (2013) Optimal variable flip angle schemes for dynamic acquisition of exchanging hyperpolarized substrates. Journal of magnetic resonance 234:75–81

    Article  CAS  Google Scholar 

Download references


The authors would like to thank Roselle Abraham, Rahul Aggarwal, Robert Bok, Hsin-Yu Chen, John Kurhanewicz, James Slater, and Daniel Vigneron for their assistance in the imaging of human subjects. The authors would like to thank Gennifer T. Smith for her helpful suggestions regarding the editing of this document. ND would like to thank the Quantitative Biosciences Institute at UCSF and the American Heart Association as funding sources for this work.


ND has received post-doctoral training funding from the American Heart Association (Grant number 20POST35200152). ND has received funding from the Quantitative Biosciences Institute at UCSF (no grant number). JG has received funding from the National Institute of Health/National Institute of Biomedical Imaging and Bioengineering (Grant number U01EB026412). PL has received funding from the National Institute of Health (Grant number NIH R01 HL136965).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Nicholas Dwork.

Ethics declarations

Conflict of interest

No conflicts of interest, financial or otherwise, are declared by the authors.

Human and animal participants

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards. MR data of humans were gathered with institutional review board (IRB) approval and Health Insurance Portability and Accountability Act (HIPAA) compliance. Informed consent was obtained from all individual participants included in the study. All applicable international, national, and/or institutional guidelines for the care and use of animals were followed. Animal experiments were done in accordance with relevant laws and ethics under permission from The Animal Experiments Inspectorate of Denmark.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ND would like to thank the Quantitative Biosciences Institute at UCSF and the American Heart Association as funding sources for this work.

Appendix: Fast iterative shrinkage threshold algorithm

Appendix: Fast iterative shrinkage threshold algorithm

The fast iterative shrinkage threshold algorithm (FISTA) solves problems of the form:

$$\begin{aligned} \underset{x\in {\mathbb {R}}^N}{\text {minimize}} \quad F(x) + G(x), \end{aligned}$$

where F is differentiable and G has a simple proximal operator [2, 38]. The FISTA algorithm with line search is described in Algorithm 2. Note that \(\langle \cdot ,\cdot \rangle\) represents an inner product. To initialize the algorithm, set \(v^{(0)} = x^{(0)}\), where \(x^{(0)}\) is the initial guess and can be any value. Select a \(t_0>0\), and select a maximum number of iterations K. Select a backtracking line search parameter \(r\in (0,1)\) (a common choice of r is 0.9) and select a step size scaling parameter \(s>1\) (a common choice of s is 1.25).

figure b

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dwork, N., Gordon, J.W., Tang, S. et al. Di-chromatic interpolation of magnetic resonance metabolic images. Magn Reson Mater Phy 34, 57–72 (2021).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: