*Result*: Balancing Bias and Variance in Deep Learning-Based Tumor Microstructural Parameter Mapping.
Title:
Balancing Bias and Variance in Deep Learning-Based Tumor Microstructural Parameter Mapping.
Authors:
Zou J; Department of Radiation Oncology, University of Michigan, Ann Arbor, Michigan, USA., Cao Y; Department of Radiation Oncology, University of Michigan, Ann Arbor, Michigan, USA.; Department of Radiology, University of Michigan, Ann Arbor, Michigan, USA.; Department of Biomedical Engineering, University of Michigan, Ann Arbor, Michigan, USA.
Source:
Magnetic resonance in medicine [Magn Reson Med] 2026 Mar; Vol. 95 (3), pp. 1804-1814. Date of Electronic Publication: 2025 Oct 22.
Publication Type:
Journal Article
Language:
English
Journal Info:
Publisher: Wiley Country of Publication: United States NLM ID: 8505245 Publication Model: Print-Electronic Cited Medium: Internet ISSN: 1522-2594 (Electronic) Linking ISSN: 07403194 NLM ISO Abbreviation: Magn Reson Med Subsets: MEDLINE
Imprint Name(s):
Publication: 1999- : New York, NY : Wiley
Original Publication: San Diego : Academic Press
Original Publication: San Diego : Academic Press
MeSH Terms:
References:
D. K. Jones, ed., Diffusion MRI: Theory, Methods, and Application (Oxford University Press, 2010).
O. Reynaud, “Time‐Dependent Diffusion MRI in Cancer: Tissue Modeling and Applications,” Frontiers in Physics 5 (2017): 58, https://doi.org/10.3389/fphy.2017.00058.
E. Fokkinga, J. A. Hernandez‐Tamames, A. Ianus, et al., “Advanced Diffusion‐Weighted MRI for Cancer Microstructure Assessment in Body Imaging, and Its Relationship With Histology,” Journal of Magnetic Resonance Imaging 60 (2023): 29144, https://doi.org/10.1002/jmri.29144.
L. Tang and X. J. Zhou, “Diffusion MRI of Cancer: From Low to High b‐Values,” Journal of Magnetic Resonance Imaging 49, no. 1 (2019): 23–40, https://doi.org/10.1002/jmri.26293.
Y. Cao, M. Aryal, P. Li, et al., “Diffusion MRI Correlation With p16 Status and Prediction for Tumor Progression in Locally Advanced Head and Neck Cancer,” Frontiers in Oncology 13 (2023): 998186, https://doi.org/10.3389/fonc.2023.998186.
H. J. Van Der Hulst, R. W. Jansen, C. Vens, et al., “The Prediction of Biological Features Using Magnetic Resonance Imaging in Head and Neck Squamous Cell Carcinoma: A Systematic Review and Meta‐Analysis,” Cancers 15, no. 20 (2023): 5077, https://doi.org/10.3390/cancers15205077.
H. Bollen, R. Dok, F. De Keyzer, et al., “Diffusion‐Weighted MRI and Human Papillomavirus (HPV) Status in Oropharyngeal Cancer,” Cancers 16, no. 24 (2024): 4284, https://doi.org/10.3390/cancers16244284.
O. Reynaud, K. V. Winters, D. M. Hoang, Y. Z. Wadghiri, D. S. Novikov, and S. G. Kim, “Pulsed and Oscillating Gradient MRI for Assessment of Cell Size and Extracellular Space (POMACE) in Mouse Gliomas,” NMR in Biomedicine 29, no. 10 (2016): 1350–1363, https://doi.org/10.1002/nbm.3577.
X. Jiang, H. Li, J. Xie, et al., “In Vivo Imaging of Cancer Cell Size and Cellularity Using Temporal Diffusion Spectroscopy,” Magnetic Resonance in Medicine 78, no. 1 (2017): 156–164, https://doi.org/10.1002/mrm.26356.
E. Panagiotaki, S. Walker‐Samuel, B. Siow, et al., “Noninvasive Quantification of Solid Tumor Microstructure Using VERDICT MRI,” Cancer Research 74, no. 7 (2014): 1902–1912, https://doi.org/10.1158/0008‐5472.CAN‐13‐2511.
J. Xu, X. Jiang, H. Li, et al., “Magnetic Resonance Imaging of Mean Cell Size in Human Breast Tumors,” Magnetic Resonance in Medicine 83, no. 6 (2020): 2002–2014, https://doi.org/10.1002/mrm.28056.
R. Ba, X. Wang, Z. Zhang, et al., “Diffusion‐Time Dependent Diffusion MRI: Effect of Diffusion‐Time on Microstructural Mapping and Prediction of Prognostic Features in Breast Cancer,” European Radiology 33, no. 9 (2023): 6226–6237, https://doi.org/10.1007/s00330‐023‐09623‐y.
X. Wang, R. Ba, Y. Huang, et al., “Time‐Dependent Diffusion MRI Helps Predict Molecular Subtypes and Treatment Response to Neoadjuvant Chemotherapy in Breast Cancer,” Radiology 313, no. 1 (2024): e240288, https://doi.org/10.1148/radiol.240288.
E. W. Johnston, E. Bonet‐Carne, U. Ferizi, et al., “VERDICT MRI for Prostate Cancer: Intracellular Volume Fraction Versus Apparent Diffusion Coefficient,” Radiology 291, no. 2 (2019): 391–397, https://doi.org/10.1148/radiol.2019181749.
S. Sen, S. Singh, H. Pye, et al., “ssVERDICT: Self‐Supervised VERDICT‐MRI for Enhanced Prostate Tumor Characterization,” Magnetic Resonance in Medicine 92, no. 5 (2024): 2181–2192, https://doi.org/10.1002/mrm.30186.
Z. Zhang, H. H. Wu, A. Priester, et al., “Prostate Microstructure in Prostate Cancer Using 3‐T MRI With Diffusion‐Relaxation Correlation Spectrum Imaging: Validation With Whole‐Mount Digital Histopathology,” Radiology 296, no. 2 (2020): 348–355, https://doi.org/10.1148/radiol.2020192330.
G. Lemberskiy, E. Fieremans, J. Veraart, F. M. Deng, A. B. Rosenkrantz, and D. S. Novikov, “Characterization of Prostate Microstructure Using Water Diffusion and NMR Relaxation,” Frontiers in Physics 6 (2018): 91, https://doi.org/10.3389/fphy.2018.00091.
E. Johnston, H. Pye, E. Bonet‐Carne, et al., “INNOVATE: A Prospective Cohort Study Combining Serum and Urinary Biomarkers With Novel Diffusion‐Weighted Magnetic Resonance Imaging for the Prediction and Characterization of Prostate Cancer,” BMC Cancer 16, no. 1 (2016): 816, https://doi.org/10.1186/s12885‐016‐2856‐2.
D. Wu, K. Jiang, H. Li, et al., “Time‐Dependent Diffusion MRI for Quantitative Microstructural Mapping of Prostate Cancer,” Radiology 303 (2022): 578–587, https://doi.org/10.1148/radiol.211180.
D. S. Novikov, E. Fieremans, J. H. Jensen, and J. A. Helpern, “Random Walks With Barriers,” Nature Physics 7, no. 6 (2011): 508–514, https://doi.org/10.1038/nphys1936.
Y. Cao, S. N. Davarani, D. You, et al., “In Vivo Microstructure Imaging in Oropharyngeal Squamous Cell Carcinoma Using the Random Walk With Barriers Model,” Journal of Magnetic Resonance Imaging 59, no. 3 (2024): 929–938, https://doi.org/10.1002/jmri.28831.
K. Liu, Z. Lin, T. Zheng, et al., “Improving Microstructural Estimation in Time‐Dependent Diffusion MRI With a Bayesian Method,” Journal of Magnetic Resonance Imaging 61 (2025): 724–734, https://doi.org/10.1002/jmri.29434.
E. Fieremans, G. Lemberskiy, J. Veraart, E. E. Sigmund, S. Gyftopoulos, and D. S. Novikov, “In Vivo Measurement of Membrane Permeability and Myofiber Size in Human Muscle Using Time‐Dependent Diffusion Tensor Imaging and the Random Permeable Barrier Model,” NMR in Biomedicine 30, no. 3 (2017): e3612, https://doi.org/10.1002/nbm.3612.
N. Just, “Improving Tumour Heterogeneity MRI Assessment With Histograms,” British Journal of Cancer 111, no. 12 (2014): 2205–2213, https://doi.org/10.1038/bjc.2014.512.
J. Wu, T. Kang, X. Lan, et al., “IMPULSED Model Based Cytological Feature Estimation With U‐Net: Application to Human Brain Tumor at 3T,” Magnetic Resonance in Medicine 89, no. 1 (2023): 411–422, https://doi.org/10.1002/mrm.29429.
S. Barbieri, O. J. Gurney‐Champion, R. Klaassen, and H. C. Thoeny, “Deep Learning How to Fit an Intravoxel Incoherent Motion Model to Diffusion‐Weighted MRI,” Magnetic Resonance in Medicine 83, no. 1 (2020): 312–321, https://doi.org/10.1002/mrm.27910.
F. Grussu, M. Battiston, M. Palombo, et al., “Deep Learning Model Fitting for Diffusion‐Relaxometry: A Comparative Study,” in Computational Diffusion MRI. Mathematics and Visualization, ed. N. Gyori, J. Hutter, V. Nath, et al. (Springer International Publishing, 2021), 159–172, https://doi.org/10.1007/978‐3‐030‐73018‐5_13.
V. Golkov, A. Dosovitskiy, J. I. Sperl, et al., “Q‐Space Deep Learning: Twelve‐Fold Shorter and Model‐Free Diffusion MRI Scans,” IEEE Transactions on Medical Imaging 35, no. 5 (2016): 1344–1351, https://doi.org/10.1109/TMI.2016.2551324.
N. G. Gyori, M. Palombo, C. A. Clark, H. Zhang, and D. C. Alexander, “Training Data Distribution Significantly Impacts the Estimation of Tissue Microstructure With Machine Learning,” Magnetic Resonance in Medicine 87, no. 2 (2022): 932–947, https://doi.org/10.1002/mrm.29014.
S. C. Epstein, T. J. P. Bray, M. Hall‐Craggs, and H. Zhang, “Choice of Training Label Matters: How to Best Use Deep Learning for Quantitative MRI Parameter Estimation,” Machine Learning for Biomedical Imaging 2 (2024): 586–610, https://doi.org/10.59275/j.melba.2024‐geb5.
E. J. Fordham, P. P. Mitra, and L. L. Latour, “Effective Diffusion Times in Multiple‐Pulse PFG Diffusion Measurements in Porous Media,” Journal of Magnetic Resonance, Series A 121, no. 2 (1996): 187–192, https://doi.org/10.1006/jmra.1996.0159.
P. Virtanen, R. Gommers, T. E. Oliphant, et al., “SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,” Nature Methods 17, no. 3 (2020): 261–272, https://doi.org/10.1038/s41592‐019‐0686‐2.
D. A. Soscia, S. J. Sequeira, R. A. Schramm, et al., “Salivary Gland Cell Differentiation and Organization on Micropatterned PLGA Nanofiber Craters,” Biomaterials 34, no. 28 (2013): 6773–6784, https://doi.org/10.1016/j.biomaterials.2013.05.061.
J. Almassy, J. H. Won, T. B. Begenisich, and D. I. Yule, “Apical Ca2+−Activated Potassium Channels in Mouse Parotid Acinar Cells,” Journal of General Physiology 139, no. 2 (2012): 121–133, https://doi.org/10.1085/jgp.201110718.
M. C. Steward, Y. Seo, J. M. Rawlings, and R. M. Case, “Water Permeability of Acinar Cell Membranes in the Isolated Perfused Rabbit Mandibular Salivary Gland,” Journal of Physiology 431, no. 1 (1990): 571–583, https://doi.org/10.1113/jphysiol.1990.sp018348.
J. Zou and Y. Cao, “Accuracy and Precision of Random Walk With Barrier Model Fitting: Simulations and Applications in Head and Neck Cancers,” https://doi.org/10.48550/arXiv.2506.12228.
O. Reynaud, “Time‐Dependent Diffusion MRI in Cancer: Tissue Modeling and Applications,” Frontiers in Physics 5 (2017): 58, https://doi.org/10.3389/fphy.2017.00058.
E. Fokkinga, J. A. Hernandez‐Tamames, A. Ianus, et al., “Advanced Diffusion‐Weighted MRI for Cancer Microstructure Assessment in Body Imaging, and Its Relationship With Histology,” Journal of Magnetic Resonance Imaging 60 (2023): 29144, https://doi.org/10.1002/jmri.29144.
L. Tang and X. J. Zhou, “Diffusion MRI of Cancer: From Low to High b‐Values,” Journal of Magnetic Resonance Imaging 49, no. 1 (2019): 23–40, https://doi.org/10.1002/jmri.26293.
Y. Cao, M. Aryal, P. Li, et al., “Diffusion MRI Correlation With p16 Status and Prediction for Tumor Progression in Locally Advanced Head and Neck Cancer,” Frontiers in Oncology 13 (2023): 998186, https://doi.org/10.3389/fonc.2023.998186.
H. J. Van Der Hulst, R. W. Jansen, C. Vens, et al., “The Prediction of Biological Features Using Magnetic Resonance Imaging in Head and Neck Squamous Cell Carcinoma: A Systematic Review and Meta‐Analysis,” Cancers 15, no. 20 (2023): 5077, https://doi.org/10.3390/cancers15205077.
H. Bollen, R. Dok, F. De Keyzer, et al., “Diffusion‐Weighted MRI and Human Papillomavirus (HPV) Status in Oropharyngeal Cancer,” Cancers 16, no. 24 (2024): 4284, https://doi.org/10.3390/cancers16244284.
O. Reynaud, K. V. Winters, D. M. Hoang, Y. Z. Wadghiri, D. S. Novikov, and S. G. Kim, “Pulsed and Oscillating Gradient MRI for Assessment of Cell Size and Extracellular Space (POMACE) in Mouse Gliomas,” NMR in Biomedicine 29, no. 10 (2016): 1350–1363, https://doi.org/10.1002/nbm.3577.
X. Jiang, H. Li, J. Xie, et al., “In Vivo Imaging of Cancer Cell Size and Cellularity Using Temporal Diffusion Spectroscopy,” Magnetic Resonance in Medicine 78, no. 1 (2017): 156–164, https://doi.org/10.1002/mrm.26356.
E. Panagiotaki, S. Walker‐Samuel, B. Siow, et al., “Noninvasive Quantification of Solid Tumor Microstructure Using VERDICT MRI,” Cancer Research 74, no. 7 (2014): 1902–1912, https://doi.org/10.1158/0008‐5472.CAN‐13‐2511.
J. Xu, X. Jiang, H. Li, et al., “Magnetic Resonance Imaging of Mean Cell Size in Human Breast Tumors,” Magnetic Resonance in Medicine 83, no. 6 (2020): 2002–2014, https://doi.org/10.1002/mrm.28056.
R. Ba, X. Wang, Z. Zhang, et al., “Diffusion‐Time Dependent Diffusion MRI: Effect of Diffusion‐Time on Microstructural Mapping and Prediction of Prognostic Features in Breast Cancer,” European Radiology 33, no. 9 (2023): 6226–6237, https://doi.org/10.1007/s00330‐023‐09623‐y.
X. Wang, R. Ba, Y. Huang, et al., “Time‐Dependent Diffusion MRI Helps Predict Molecular Subtypes and Treatment Response to Neoadjuvant Chemotherapy in Breast Cancer,” Radiology 313, no. 1 (2024): e240288, https://doi.org/10.1148/radiol.240288.
E. W. Johnston, E. Bonet‐Carne, U. Ferizi, et al., “VERDICT MRI for Prostate Cancer: Intracellular Volume Fraction Versus Apparent Diffusion Coefficient,” Radiology 291, no. 2 (2019): 391–397, https://doi.org/10.1148/radiol.2019181749.
S. Sen, S. Singh, H. Pye, et al., “ssVERDICT: Self‐Supervised VERDICT‐MRI for Enhanced Prostate Tumor Characterization,” Magnetic Resonance in Medicine 92, no. 5 (2024): 2181–2192, https://doi.org/10.1002/mrm.30186.
Z. Zhang, H. H. Wu, A. Priester, et al., “Prostate Microstructure in Prostate Cancer Using 3‐T MRI With Diffusion‐Relaxation Correlation Spectrum Imaging: Validation With Whole‐Mount Digital Histopathology,” Radiology 296, no. 2 (2020): 348–355, https://doi.org/10.1148/radiol.2020192330.
G. Lemberskiy, E. Fieremans, J. Veraart, F. M. Deng, A. B. Rosenkrantz, and D. S. Novikov, “Characterization of Prostate Microstructure Using Water Diffusion and NMR Relaxation,” Frontiers in Physics 6 (2018): 91, https://doi.org/10.3389/fphy.2018.00091.
E. Johnston, H. Pye, E. Bonet‐Carne, et al., “INNOVATE: A Prospective Cohort Study Combining Serum and Urinary Biomarkers With Novel Diffusion‐Weighted Magnetic Resonance Imaging for the Prediction and Characterization of Prostate Cancer,” BMC Cancer 16, no. 1 (2016): 816, https://doi.org/10.1186/s12885‐016‐2856‐2.
D. Wu, K. Jiang, H. Li, et al., “Time‐Dependent Diffusion MRI for Quantitative Microstructural Mapping of Prostate Cancer,” Radiology 303 (2022): 578–587, https://doi.org/10.1148/radiol.211180.
D. S. Novikov, E. Fieremans, J. H. Jensen, and J. A. Helpern, “Random Walks With Barriers,” Nature Physics 7, no. 6 (2011): 508–514, https://doi.org/10.1038/nphys1936.
Y. Cao, S. N. Davarani, D. You, et al., “In Vivo Microstructure Imaging in Oropharyngeal Squamous Cell Carcinoma Using the Random Walk With Barriers Model,” Journal of Magnetic Resonance Imaging 59, no. 3 (2024): 929–938, https://doi.org/10.1002/jmri.28831.
K. Liu, Z. Lin, T. Zheng, et al., “Improving Microstructural Estimation in Time‐Dependent Diffusion MRI With a Bayesian Method,” Journal of Magnetic Resonance Imaging 61 (2025): 724–734, https://doi.org/10.1002/jmri.29434.
E. Fieremans, G. Lemberskiy, J. Veraart, E. E. Sigmund, S. Gyftopoulos, and D. S. Novikov, “In Vivo Measurement of Membrane Permeability and Myofiber Size in Human Muscle Using Time‐Dependent Diffusion Tensor Imaging and the Random Permeable Barrier Model,” NMR in Biomedicine 30, no. 3 (2017): e3612, https://doi.org/10.1002/nbm.3612.
N. Just, “Improving Tumour Heterogeneity MRI Assessment With Histograms,” British Journal of Cancer 111, no. 12 (2014): 2205–2213, https://doi.org/10.1038/bjc.2014.512.
J. Wu, T. Kang, X. Lan, et al., “IMPULSED Model Based Cytological Feature Estimation With U‐Net: Application to Human Brain Tumor at 3T,” Magnetic Resonance in Medicine 89, no. 1 (2023): 411–422, https://doi.org/10.1002/mrm.29429.
S. Barbieri, O. J. Gurney‐Champion, R. Klaassen, and H. C. Thoeny, “Deep Learning How to Fit an Intravoxel Incoherent Motion Model to Diffusion‐Weighted MRI,” Magnetic Resonance in Medicine 83, no. 1 (2020): 312–321, https://doi.org/10.1002/mrm.27910.
F. Grussu, M. Battiston, M. Palombo, et al., “Deep Learning Model Fitting for Diffusion‐Relaxometry: A Comparative Study,” in Computational Diffusion MRI. Mathematics and Visualization, ed. N. Gyori, J. Hutter, V. Nath, et al. (Springer International Publishing, 2021), 159–172, https://doi.org/10.1007/978‐3‐030‐73018‐5_13.
V. Golkov, A. Dosovitskiy, J. I. Sperl, et al., “Q‐Space Deep Learning: Twelve‐Fold Shorter and Model‐Free Diffusion MRI Scans,” IEEE Transactions on Medical Imaging 35, no. 5 (2016): 1344–1351, https://doi.org/10.1109/TMI.2016.2551324.
N. G. Gyori, M. Palombo, C. A. Clark, H. Zhang, and D. C. Alexander, “Training Data Distribution Significantly Impacts the Estimation of Tissue Microstructure With Machine Learning,” Magnetic Resonance in Medicine 87, no. 2 (2022): 932–947, https://doi.org/10.1002/mrm.29014.
S. C. Epstein, T. J. P. Bray, M. Hall‐Craggs, and H. Zhang, “Choice of Training Label Matters: How to Best Use Deep Learning for Quantitative MRI Parameter Estimation,” Machine Learning for Biomedical Imaging 2 (2024): 586–610, https://doi.org/10.59275/j.melba.2024‐geb5.
E. J. Fordham, P. P. Mitra, and L. L. Latour, “Effective Diffusion Times in Multiple‐Pulse PFG Diffusion Measurements in Porous Media,” Journal of Magnetic Resonance, Series A 121, no. 2 (1996): 187–192, https://doi.org/10.1006/jmra.1996.0159.
P. Virtanen, R. Gommers, T. E. Oliphant, et al., “SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,” Nature Methods 17, no. 3 (2020): 261–272, https://doi.org/10.1038/s41592‐019‐0686‐2.
D. A. Soscia, S. J. Sequeira, R. A. Schramm, et al., “Salivary Gland Cell Differentiation and Organization on Micropatterned PLGA Nanofiber Craters,” Biomaterials 34, no. 28 (2013): 6773–6784, https://doi.org/10.1016/j.biomaterials.2013.05.061.
J. Almassy, J. H. Won, T. B. Begenisich, and D. I. Yule, “Apical Ca2+−Activated Potassium Channels in Mouse Parotid Acinar Cells,” Journal of General Physiology 139, no. 2 (2012): 121–133, https://doi.org/10.1085/jgp.201110718.
M. C. Steward, Y. Seo, J. M. Rawlings, and R. M. Case, “Water Permeability of Acinar Cell Membranes in the Isolated Perfused Rabbit Mandibular Salivary Gland,” Journal of Physiology 431, no. 1 (1990): 571–583, https://doi.org/10.1113/jphysiol.1990.sp018348.
J. Zou and Y. Cao, “Accuracy and Precision of Random Walk With Barrier Model Fitting: Simulations and Applications in Head and Neck Cancers,” https://doi.org/10.48550/arXiv.2506.12228.
Grant Information:
EB016079 United States NH NIH HHS
Contributed Indexing:
Keywords: deep learning; diffusion MRI; head and neck cancers; model fitting; tissue microstructure
Entry Date(s):
Date Created: 20251023 Date Completed: 20251229 Latest Revision: 20251229
Update Code:
20260130
DOI:
10.1002/mrm.70154
PMID:
41126537
Database:
MEDLINE
*Further Information*
*Purpose: Time-dependent diffusion MRI enables quantification of tumor microstructural parameters useful for diagnosis and prognosis. Nevertheless, current model fitting approaches exhibit suboptimal bias-variance trade-offs; specifically, nonlinear least squares fitting (NLLS) demonstrated low bias but high variance, whereas supervised deep learning methods trained with mean squared error loss (MSE-Net) yielded low variance but elevated bias. This study investigates these bias-variance characteristics and proposes a method to control fitting bias and variance.
Methods: Random walk with barrier model was used as a representative biophysical model. NLLS and MSE-Net were reformulated within the Bayesian framework to elucidate their bias-variance behaviors. We introduced B2V-Net, a supervised learning approach using a loss function with adjustable bias-variance weighting, to control bias-variance trade-off. B2V-Net was evaluated and compared against NLLS and MSE-Net numerically across a wide range of parameters and noise levels, as well as in vivo in patients with head and neck cancer.
Results: Flat posterior distributions that were not centered at ground truth parameters explained the bias-variance behaviors of NLLS and MSE-Net. B2V-Net controlled the bias-variance trade-off, achieving a 56% reduction in standard deviation relative to NLLS and an 18% reduction in bias compared to MSE-Net. In vivo parameter maps from B2V-Net demonstrated a balance between smoothness and accuracy.
Conclusion: We demonstrated and explained the low bias-high variance of NLLS and the low variance-high bias of MSE-Net. The proposed B2V-Net can balance bias and variance. Our work provided insights and methods to guide the design of customized loss functions tailored to specific clinical imaging needs.
(© 2025 International Society for Magnetic Resonance in Medicine.)*