Using Simulations to Validate Models
Marco Palombo1
1Centre for Medical Image Computing (CMIC), University College London, United Kingdom

Synopsis

This lecture targets scientists and clinicians interested in learning what are the most recent developments in numerical simulations, with a particular focus on their use for the validation of commonly used diffusion MRI (dMRI) models of tissue microstructure. We will provide an overview of the aspects of dMRI techniques that can be validated with numerical phantoms, and of the range of numerical phantoms that are currently available. Examples of how to use numerical phantoms for validating dMRI techniques will be provided and future perspectives on the next-generation numerical phantoms will be discussed.

Target Audience

Scientists and clinicians interested in learning what are the most recent developments in numerical simulations, with a particular focus on their use for the validation of commonly used diffusion MRI models of tissue microstructure.

Objectives

This talk aims to provide its audience with the following:
  • Overview of the aspects of diffusion MRI techniques that can be validated with numerical phantoms.
  • Overview of the range of numerical phantoms that are currently available.
  • Examples of using numerical phantoms for validating diffusion MRI techniques.

Purpose

To provide an overview of the numerical phantoms that are currently available for validating diffusion MRI models.

Overview

Diffusion MRI (dMRI) is a key modality for imaging the microstructure of biological tissues, because of its unique sensitivity to microscopic features of cellular architecture.

Over the last decade, microstructure imaging (1) has been a successful paradigm to infer microstructural features from dMRI. Model-based microstructure imaging relies on biophysical models that relate an approximation of tissue microarchitecture to dMRI signals (2-9). In general, the approach acquires a set of images with different sensitivities and fits a specific model, chosen to represent the essential microstructural features of a tissue, to the set of signals obtained from each voxel in each image. The process yields a set of model parameters in each image voxel, which constitute quantitative maps of the modelled tissue features.

A key aspect of model-based microstructure imaging is the validation of the model assumptions and the estimated model parameters (10,11). Numerical phantoms play a unique role in providing rigorous validation that is complementary of other forms of phantoms (physical, in-vitro, ex-vivo and in-vivo). Although they in turn represent a model of the real tissue based on our current understanding, they provide a framework for validation that is both controlled (with known ground truth) and flexible.

This talk will:
  • Give an overview of classical numerical phantoms that have been widely used for validating diffusion MRI techniques.
  • Underline the limitations of classical numerical phantoms.
  • Provide examples of the most recent advancements in numerical phantoms design.

Classical numerical phantoms for validation of microstructure imaging

Numerical phantoms for validation of microstructure imaging can be divided into several types based on the models implemented (12). The most common class uses Monte Carlo (MC) simulation of diffusion (13-32), with the system implemented in the Open Source Camino software being most widely used (17). This class of phantoms has been used to validate the estimation of axon density and diameter (5,24), crossing-fibre resolution (33), exchange (20,23,34), neurite beading (35), axonal undulation (21,22), dendritic spines (36) and branching (32,37), time-dependent extra-axonal diffusion (38,39), g-ratio mapping (40) and the effect of susceptibility induced internal gradients on dMRI measurements (31).

An alternative approach is to directly solve the diffusion equations using either finite differences (27,41-43) or finite element method (44-48).

Limitations of classical numerical phantoms

A still open challenge in numerical phantoms design is their realism. In order to be a useful validation tool, numerical phantoms have to mirror the complexity of real tissue microarchitecture as close as possible. In practice, this is still an unmet need and an active area of research (12).

For instance, brain white matter (WM) has been the most targeted tissue type because of its relatively simpler microenvironment compared to gray matter (GM) or other biological tissues. However, the most used numerical phantoms for WM are still over simplifications of the reality: axons are modelled as densely packed cylinders of different radii, straight or undulating, and with or without planar dispersion. Conversely, modern microscopy techniques such as 3D electron microscopy (EM) have clearly shown that axonal morphology in WM is way more complex than packed cylinders (24).

Recent developments and future perspectives of next-generation numerical phantoms

A lot of effort has been invested by the dMRI community to design more realistic numerical phantoms. Two main strategies have been developed: one uses directly the real structure of brain tissue from 3D EM reconstructions as digital substrate to use in MC simulations (24); the other uses generative models of digital tissues based on our current knowledge derived from microscopy (29,49-51). While the first one can provide ultra-realistic digital substrates, it lacks flexibility and is unable to reproduce large volumes of tissue. On the contrary, generative models are very flexible and can be scaled to reproduce large volumes of digital tissue but they can inevitably represent only simpler architecture. Both these approaches still mainly focus on increasing the realism of WM digital phantoms, although some encouraging developments for GM digital phantoms have also been recently proposed (49).

While memory burden and computational cost still represent a bottleneck for the scalability of these more realistic numerical phantoms, the recent developments in artificial intelligence and computational hardware open exciting perspectives on the next-generation numerical phantoms for virtual experiments towards the ultimate tool for the validation of microstructure models. We will discuss the state-of-the-art numerical phantoms and future directions in this research field.

Acknowledgements

This work was supported by EPSRC grants EP/N018702/1.

References

1. Alexander DC, Dyrby TB, Nilsson M, & Zhang H (2019) Imaging brain microstructure with diffusion MRI: practicality and applications. Nmr Biomed 32(4):e3841.

2. Stanisz GJ, Szafer A, Wright GA, & Henkelman RM (1997) An analytical model of restricted diffusion in bovine optic nerve. Magn Reson Med 37(1):103-111.

3. Assaf Y & Basser PJ (2005) Composite hindered and restricted model of diffusion (CHARMED) MR imaging of the human brain. Neuroimage 27(1):48-58.

4. Jespersen SN, Kroenke CD, Ostergaard L, Ackerman JJ, & Yablonskiy DA (2007) Modeling dendrite density from magnetic resonance diffusion measurements. Neuroimage 34(4):1473-1486.

5. Alexander DC, et al. (2010) Orientationally invariant indices of axon diameter and density from diffusion MRI. Neuroimage 52(4):1374-1389.

6. Fieremans E, Jensen JH, & Helpern JA (2011) White matter characterization with diffusional kurtosis imaging. Neuroimage 58(1):177-188.

7. Zhang H, Schneider T, Wheeler-Kingshott CA, & Alexander DC (2012) NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain. Neuroimage 61(4):1000-1016.

8. Scherrer B, et al. (2016) Characterizing brain tissue by assessment of the distribution of anisotropic microstructural environments in diffusion-compartment imaging (DIAMOND). Magn Reson Med76(3):963-977.

9. Palombo M, et al. (2019) SANDI: a compartment-based model for non-invasive apparent soma and neurite imaging by diffusion MRI. arXiv preprint.

10. Novikov DS, Kiselev VG, & Jespersen SN (2018) On modeling. Magn Reson Med 79(6):3172-3193.

11. Novikov DS, Fieremans E, Jespersen SN, & Kiselev VG (2019) Quantifying brain microstructure with diffusion MRI: Theory and parameter estimation. Nmr Biomed 32(4):e3998.

12. Fieremans E & Lee HH (2018) Physical and numerical phantoms for the validation of brain microstructural MRI: A cookbook. Neuroimage 182:39-61.

13. Szafer A, Zhong J, & Gore JC (1995) Theoretical model for water diffusion in tissues. Magn Reson Med 33(5):697-712.

14. Ford JC & Hackney DB (1997) Numerical model for calculation of apparent diffusion coefficients (ADC) in permeable cylinders--comparison with measured ADC in spinal cord white matter. Magn Reson Med37(3):387-394.

15. Peled S (2007) New perspectives on the sources of white matter DTI signal. IEEE Trans Med Imaging26(11):1448-1455.

16. Balls GT & Frank LR (2009) A simulation environment for diffusion weighted MR experiments in complex media. Magn Reson Med 62(3):771-778.

17. Cook PA, et al. (2006) Camino: open-source diffusion-MRI reconstruction and processing. In 14th scientific meeting of the international society for magnetic resonance in medicine (Vol. 2759, p. 2759). Seattle WA, USA.

18. Hall MG & Alexander DC (2009) Convergence and parameter choice for Monte-Carlo simulations of diffusion MRI. IEEE Trans Med Imaging 28(9):1354-1364.

19. Nilsson M, et al. (2009) On the effects of a varied diffusion time in vivo: is the diffusion in white matter restricted? Magn Reson Imaging 27(2):176-187.

20. Nilsson M, et al. (2010) Evaluating the accuracy and precision of a two-compartment Karger model using Monte Carlo simulations. J Magn Reson 206(1):59-67.

21. Nilsson M, Latt J, Stahlberg F, van Westen D, & Hagslatt H (2012) The importance of axonal undulation in diffusion MR measurements: a Monte Carlo simulation study. Nmr Biomed 25(5):795-805.

22. Brabec J, Lasic S, & Nilsson M (2020) Time-dependent diffusion in undulating thin fibers: Impact on axon diameter estimation. Nmr Biomed 33(3):e4187.

23. Fieremans E, Novikov DS, Jensen JH, & Helpern JA (2010) Monte Carlo study of a two-compartment exchange model of diffusion. Nmr Biomed 23(7):711-724.

24. Lee HH, et al. (2019) Along-axon diameter variation and axonal orientation dispersion revealed with 3D electron microscopy: implications for quantifying brain white matter microstructure with histology and diffusion MRI. Brain Struct Funct 224(4):1469-1488.

25. Landman BA, et al. (2010) Complex geometric models of diffusion and relaxation in healthy and damaged white matter. Nmr Biomed 23(2):152-162.

26. Yeh CH, et al. (2013) Diffusion microscopist simulator: a general Monte Carlo simulation system for diffusion magnetic resonance imaging. PLoS One 8(10):e76626.

27. Li JR, Calhoun D, Poupon C, & Le Bihan D (2014) Numerical simulation of diffusion MRI signals using an adaptive time-stepping method. Phys Med Biol 59(2):441-454.

28. Ginsburger K, et al. (2018) Improving the Realism of White Matter Numerical Phantoms: A Step toward a Better Understanding of the Influence of Structural Disorders in Diffusion MRI. Front Phys 6.

29. Ginsburger K, et al. (2019) MEDUSA: A GPU-based tool to create realistic phantoms of the brain microstructure using tiny spheres. Neuroimage 193:10-24.

30. Palombo M, Gabrielli A, Servedio VD, Ruocco G, & Capuani S (2013) Structural disorder and anomalous diffusion in random packing of spheres. Sci Rep 3:2631.

31. Palombo M, Gentili S, Bozzali M, Macaluso E, & Capuani S (2015) New insight into the contrast in diffusional kurtosis images: does it depend on magnetic susceptibility? Magn Reson Med 73(5):2015-2024.

32. Palombo M, et al. (2016) New paradigm to assess brain cell morphology by diffusion-weighted MR spectroscopy in vivo. Proc Natl Acad Sci U S A 113(24):6671-6676.

33. Ramirez-Manzanares A, Cook PA, Hall M, Ashtari M, & Gee JC (2011) Resolving axon fiber crossings at clinical b-values: an evaluation study. Med Phys 38(9):5239-5253.

34. Brusini L, Menegaz G, & Nilsson M (2019) Monte Carlo Simulations of Water Exchange Through Myelin Wraps: Implications for Diffusion MRI. IEEE Trans Med Imaging 38(6):1438-1445.

35. Budde MD & Frank JA (2010) Neurite beading is sufficient to decrease the apparent diffusion coefficient after ischemic stroke. Proc Natl Acad Sci U S A 107(32):14472-14477.

36. Palombo M, Ligneul C, Hernandez-Garzon E, & Valette J (2017) Can we detect the effect of spines and leaflets on the diffusion of brain intracellular metabolites? Neuroimage 182:283-293.

37. Vincent M, Palombo M, & Valette J (2020) Revisiting double diffusion encoding MRS in the mouse brain at 11.7T: Which microstructural features are we sensitive to? Neuroimage 207:116399.

38. Burcaw LM, Fieremans E, & Novikov DS (2015) Mesoscopic structure of neuronal tracts from time-dependent diffusion. Neuroimage 114:18-37.

39. Lam WW, Jbabdi S, & Miller KL (2015) A model for extra-axonal diffusion spectra with frequency-dependent restriction. Magn Reson Med 73(6):2306-2320.

40. Stikov N, et al. (2011) Bound pool fractions complement diffusion measures to describe white matter micro and macrostructure. Neuroimage 54(2):1112-1121.

41. Chin CL, Wehrli FW, Hwang SN, Takahashi M, & Hackney DB (2002) Biexponential diffusion attenuation in the rat spinal cord: computer simulations based on anatomic images of axonal architecture. Magn Reson Med 47(3):455-460.

42. Hwang SN, Chin CL, Wehrli FW, & Hackney DB (2003) An image-based finite difference model for simulating restricted diffusion. Magn Reson Med 50(2):373-382.

43. Russell G, Harkins KD, Secomb TW, Galons JP, & Trouard TP (2012) A finite difference method with periodic boundary conditions for simulations of diffusion-weighted magnetic resonance experiments in tissue. Phys Med Biol 57(4):N35-46.

44. Hagslatt H, Jonsson B, Nyden M, & Soderman O (2003) Predictions of pulsed field gradient NMR echo-decays for molecules diffusing in various restrictive geometries. Simulations of diffusion propagators based on a finite element method. J Magn Reson 161(2):138-147.

45. Moroney BF, Stait-Gardner T, Ghadirian B, Yadav NN, & Price WS (2013) Numerical analysis of NMR diffusion measurements in the short gradient pulse limit. J Magn Reson 234:165-175.

46. Nguyen DV, Li JR, Grebenkov DS, & Le Bihan D (2014) A finite element method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging. J. Comp. Phys. 263:283-302

47. Li JR, et al. (2019) SpinDoctor: A MATLAB toolbox for diffusion MRI simulation. Neuroimage202:116120.

48. Beltrachini L, Taylor ZA, & Frangi AF (2015) A parametric finite element solution of the generalised Bloch-Torrey equation for arbitrary domains. J Magn Reson 259:126-134.

49. Palombo M, Alexander DC, & Zhang H (2019) A generative model of realistic brain cells with application to numerical simulation of the diffusion-weighted MR signal. Neuroimage 188:391-402.

50. Callaghan R, Alexander DC, Zhang H, & Palombo M (2019) Contextual Fibre Growth to Generate Realistic Axonal Packing for Diffusion MRI Simulation. Information Processing in Medical Imaging, Ipmi 2019 11492:429-440.

51. Reuter JA, et al. (2019) FAConstructor: an interactive tool for geometric modeling of nerve fiber architectures in the brain. Int J Comput Assist Radiol Surg 14(11):1881-1889.

Proc. Intl. Soc. Mag. Reson. Med. 28 (2020)