Share：
Share this content in WeChat
X
[Chinese][PDF]291
• Review •
Research progress on fractional order calculus models in the diagnosis and treatment response prediction of malignant tumors
LI Wenxin  WANG Xiaochun 

Cite this article as: LI W X, WANG X C. Research progress on fractional order calculus models in the diagnosis and treatment response prediction of malignant tumors[J]. Chin J Magn Reson Imaging, 2025, 16(6): 228-234. DOI:10.12015/issn.1674-8034.2025.06.035.


[Abstract] Malignant tumors are one of the leading causes of mortality worldwide. The heterogeneity of tumors in genetics and histology significantly impacts their diagnostic and therapeutic outcomes. Traditional imaging techniques have made significant progress in the evaluation of malignancies and are widely used in clinical practice. The fractional order calculus (FROC) diffusion model, developed based on fractional-order calculus theory, provides a novel approach for non-invasively assessing intra-tumoral heterogeneity by quantifying water molecule diffusion characteristics and tissue homogeneity through multiparametric analysis. This model complements conventional imaging techniques and has been extended to applications in tumors of the central nervous, digestive, urinary, and female reproductive systems for diagnosis and treatment response prediction. However, clinical translation still faces challenges including the absence of technical standardization and insufficient multi-center compatibility. This review examines the application and value of FROC models in tumor diagnosis and therapeutic response prediction across various organ systems, summarizes current research limitations, and outlines future directions for advancing tumor heterogeneity assessment.
[Keywords] malignant tumors；magnetic resonance imaging；fractional order calculus diffusion model；diagnosis；treatment response prediction

LI Wenxin1   WANG Xiaochun2*  

1 College of Medical Imaging, Shanxi Medical University, Taiyuan 030001, China

2 Department of Radiology, First Hospital of Shanxi Medical University, Taiyuan 030001, China

Corresponding author: WANG X C, E-mail: 2010xiaochun@163.com

Conflicts of interest   None.

Received  2025-03-26
Accepted  2025-06-10
DOI: 10.12015/issn.1674-8034.2025.06.035
Cite this article as: LI W X, WANG X C. Research progress on fractional order calculus models in the diagnosis and treatment response prediction of malignant tumors[J]. Chin J Magn Reson Imaging, 2025, 16(6): 228-234. DOI:10.12015/issn.1674-8034.2025.06.035.

[1]
BRAY F, LAVERSANNE M, SUNG H, et al. Global cancer statistics 2022: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries[J]. CA Cancer J Clin, 2024, 74(3): 229-263. DOI: 10.3322/caac.21834.
[2]
YODA R A, CIMINO P J. Classification and grading of central nervous system tumors according to the World Health Organization 5th edition[J]. Semin Neurol, 2023, 43(6): 833-844. DOI: 10.1055/s-0043-1776793.
[3]
SYGA S, JAIN H P, KRELLNER M, et al. Evolution of phenotypic plasticity leads to tumor heterogeneity with implications for therapy[J/OL]. PLoS Comput Biol, 2024, 20(8): e1012003 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/39121170/. DOI: 10.1371/journal.pcbi.1012003.
[4]
MOFFET J J D, FATUNLA O E, FREYTAG L, et al. Spatial architecture of high-grade glioma reveals tumor heterogeneity within distinct domains[J/OL]. Neurooncol Adv, 2023, 5(1): vdad142 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/38077210/. DOI: 10.1093/noajnl/vdad142.
[5]
WU F Y, LIU Y K, JIANG C P, et al. Progress in diagnosis and treatment of whole-body magnetic resonance imaging in hematologic malignancies[J]. Chin J Magn Reson Imag, 2025, 16(1): 228-234. DOI: 10.12015/issn.1674-8034.2025.01.037.
[6]
INGENERF M, SCHMID-TANNWALD C. Diffusion-weighted imaging in Crohn's disease[J]. Die Radiol, 2023, 63(2): 27-33. DOI: 10.1007/s00117-023-01191-y.
[7]
MAO C, HU L, JIANG W, et al. Discrimination between human epidermal growth factor receptor 2 (HER2)-low-expressing and HER2-overexpressing breast cancers: a comparative study of four MRI diffusion models[J]. Eur Radiol, 2024, 34(4): 2546-2559. DOI: 10.1007/s00330-023-10198-x.
[8]
TRAMONTANO L, CAVALIERE C, SALVATORE M, et al. The role of non-Gaussian models of diffusion weighted MRI in hepatocellular carcinoma: a systematic review[J/OL]. J Clin Med, 2021, 10(12): 2641 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/34203995/. DOI: 10.3390/jcm10122641.
[9]
LI W E, CHAI R M. Research progresses of non-Gaussian of diffusion weighted imaging models in hepatocellular carcinoma[J]. Chin J Magn Reson Imag, 2024, 15(9): 194-200. DOI: 10.12015/issn.1674-8034.2024.09.034.
[10]
SUI Y, WANG H, LIU G Z, et al. Differentiation of low- and high-grade pediatric brain tumors with high b-value diffusion-weighted MR imaging and a fractional order Calculus model[J]. Radiology, 2015, 277(2): 489-496. DOI: 10.1148/radiol.2015142156.
[11]
FENG C, WANG Y C, DAN G Y, et al. Evaluation of a fractional-order Calculus diffusion model and bi-parametric VI-RADS for staging and grading bladder urothelial carcinoma[J]. Eur Radiol, 2022, 32(2): 890-900. DOI: 10.1007/s00330-021-08203-2.
[12]
GUO J T, WANG X C. Research progress in multimodal function magnetic resonance imaging in staging and grading of bladder cancer[J]. Chin J Magn Reson Imag, 2024, 15(2): 229-234. DOI: 10.12015/issn.1674-8034.2024.02.038.
[13]
ZHOU X J, GAO Q, ABDULLAH O, et al. Studies of anomalous diffusion in the human brain using fractional order Calculus[J]. Magn Reson Med, 2010, 63(3): 562-569. DOI: 10.1002/mrm.22285.
[14]
PADHANI A R, LIU G Y, KOH D M, et al. Diffusion-weighted magnetic resonance imaging as a cancer biomarker: consensus and recommendations[J]. Neoplasia, 2009, 11(2): 102-125. DOI: 10.1593/neo.81328.
[15]
GAGLIARDI T, ADEJOLU M, DESOUZA N M. Diffusion-weighted magnetic resonance imaging in ovarian cancer: exploiting strengths and understanding limitations[J/OL]. J Clin Med, 2022, 11(6): 1524 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/35329850/. DOI: 10.3390/jcm11061524.
[16]
SUI Y, XIONG Y, JIANG J, et al. Differentiation of low- and high-grade gliomas using high b-value diffusion imaging with a non-Gaussian diffusion model[J]. AJNR Am J Neuroradiol, 2016, 37(9): 1643-1649. DOI: 10.3174/ajnr.A4836.
[17]
TANG L, SUI Y, ZHONG Z, et al. Non-Gaussian diffusion imaging with a fractional order Calculus model to predict response of gastrointestinal stromal tumor to second-line sunitinib therapy[J]. Magn Reson Med, 2018, 79(3): 1399-1406. DOI: 10.1002/mrm.26798.
[18]
LUO Y, MENG N, HUANG Z, et al. The value of monoexponentia, fractional order Calculus models and 18F-FDG PET imaging in evaluating the proliferation status of lung adenocarcinoma[J]. Chin J Magn Reson Imag, 2022, 13(10): 121-126. DOI: 10.12015/issn.1674-8034.2022.10.018.
[19]
LUO Y, JIANG H, MENG N, et al. A comparison study of monoexponential and fractional order Calculus diffusion models and 18F-FDG PET in differentiating benign and malignant solitary pulmonary lesions and their pathological types[J/OL]. Front Oncol, 2022, 12: 907860 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/35936757/. DOI: 10.3389/fonc.2022.907860.
[20]
SHENG R F, ZHANG Y F, SUN W, et al. Staging chronic hepatitis B related liver fibrosis with a fractional order Calculus diffusion model[J]. Acad Radiol, 2022, 29(7): 951-963. DOI: 10.1016/j.acra.2021.07.005.
[21]
WEN Q Q, TONG H Y, YANG H Y. Impacts of diffusion time on DWI parameters of stretched-exponential model and fractional order Calculus model in mice brains in vitro[J]. Chin J Med Imag Technol, 2018, 34(12): 1761-1766. DOI: 10.13929/j.1003-3289.201804068.
[22]
THORBINSON C, KILDAY J P. Childhood malignant brain tumors: balancing the bench and bedside[J/OL]. Cancers (Basel), 2021, 13(23): 6099 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/34885207/. DOI: 10.3390/cancers13236099.
[23]
OZTEK M A, NODA S M, ROMBERG E K, et al. Changes to pediatric brain tumors in 2021 World Health Organization classification of tumors of the central nervous system[J]. Pediatr Radiol, 2023, 53(3): 523-543. DOI: 10.1007/s00247-022-05546-w.
[24]
SEO M, CHOI Y, LEE Y SOO, et al. Glioma grading using multiparametric MRI: head-to-head comparison among dynamic susceptibility contrast, dynamic contrast-enhancement, diffusion-weighted images, and MR spectroscopy[J/OL]. Eur J Radiol, 2023, 165: 110888 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/37257338/. DOI: 10.1016/j.ejrad.2023.110888.
[25]
XU J Q, REN Y, ZHAO X Y, et al. Incorporating multiple magnetic resonance diffusion models to differentiate low- and high-grade adult gliomas: a machine learning approach[J]. Quant Imaging Med Surg, 2022, 12(11): 5171-5183. DOI: 10.21037/qims-22-145.
[26]
FILHO A M, LAVERSANNE M, FERLAY J, et al. The GLOBOCAN 2022 cancer estimates: Data sources, methods, and a snapshot of the cancer burden worldwide[J]. Int J Cancer, 2025, 156(7): 1336-1346. DOI: 10.1002/ijc.35278.
[27]
LEE S, KANG T W, SONG K D, et al. Effect of microvascular invasion risk on early recurrence of hepatocellular carcinoma after surgery and radiofrequency ablation[J]. Ann Surg, 2021, 273(3): 564-571. DOI: 10.1097/SLA.0000000000003268.
[28]
CHEN J J, GUO Y X, GUO Y L, et al. Preoperative assessment of microvascular invasion of hepatocellular carcinoma using non-Gaussian diffusion-weighted imaging with a fractional order Calculus model: A pilot study[J/OL]. Magn Reson Imaging, 2023, 95: 110-117 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/34506910/. DOI: 10.1016/j.mri.2021.09.003.
[29]
XIE J H, LONG L L, LI C H, et al. Non-Gaussian diffusion-weighted imaging: the value of stretch index modeland fractional Calculus model in predicting microvascular invasion of hepatocellular carcinoma before operation[J]. J Clin Radiol, 2022, 41(12): 2250-2256. DOI: 10.13437/j.cnki.jcr.2022.12.021.
[30]
XIE J H, LI C H, CHEN Y D, et al. Potential value of the stretched exponential and fractional order Calculus model in discriminating between hepatocellular carcinoma and intrahepatic cholangiocarcinoma: an animal experiment of orthotopic xenograft nude mice[J/OL]. Curr Med Imaging, 2023 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/36946482/. DOI: 10.2174/1573405619666230322123117.
[31]
GUO Y X, CHEN J J, ZHANG Y F, et al. Differentiating Cytokeratin 19 expression of hepatocellular carcinoma by using multi-b-value diffusion-weighted MR imaging with mono-exponential, stretched exponential, intravoxel incoherent motion, diffusion kurtosis imaging and fractional order Calculus models[J/OL]. Eur J Radiol, 2022, 150: 110237 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/35278979/. DOI: 10.1016/j.ejrad.2022.110237.
[32]
MA L, GUO L L, ZHU X Y, et al. Diffusion-weighted MRI of advanced gastric cancer: correlations of the apparent diffusion coefficient with Borrmann classification, proliferation and aggressiveness[J/OL]. Abdom Radiol (NY), 2025 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/39815027/. DOI: 10.1007/s00261-024-04718-6.
[33]
HOU Y J, SANG Z T, LI Q, et al. Advanced multiparametric MRI strategies for tumor restaging after neoadjuvant therapy in locally advanced gastric cancer[J]. Ann Surg Oncol, 2025, 32(5): 3382-3391. DOI: 10.1245/s10434-025-16972-z.
[34]
KARAMAN M M, TANG L, LI Z Y, et al. In vivo assessment of Lauren classification for gastric adenocarcinoma using diffusion MRI with a fractional order Calculus model[J]. Eur Radiol, 2021, 31(8): 5659-5668. DOI: 10.1007/s00330-021-07694-3.
[35]
LI J, ZHANG H K, BEI T X, et al. Advanced diffusion-weighted MRI models for preoperative prediction of lymph node metastasis in resectable gastric cancer[J]. Abdom Radiol (NY), 2025, 50(3): 1057-1068. DOI: 10.1007/s00261-024-04559-3.
[36]
SIEGEL R L, KRATZER T B, GIAQUINTO A N, et al. Cancer statistics, 2025[J]. CA A Cancer J Clinicians, 2025, 75(1): 10-45. DOI: 10.3322/caac.21871.
[37]
ZHOU M, HUANG H Y, BAO D Y, et al. Assessment of prognostic indicators and KRAS mutations in rectal cancer using a fractional-order Calculus MR diffusion model: whole tumor histogram analysis[J]. Abdom Radiol (NY), 2025, 50(2): 569-578. DOI: 10.1007/s00261-024-04523-1.
[38]
ZHOU M, BAO D Y, HUANG H Y, et al. Utilization of diffusion-weighted derived mathematical models to predict prognostic factors of resectable rectal cancer[J]. Abdom Radiol (NY), 2024, 49(9): 3282-3293. DOI: 10.1007/s00261-024-04239-2.
[39]
GUO R, LU F, LIN J, et al. Multi-b-value DWI to evaluate the synergistic antiproliferation and anti-heterogeneity effects of bufalin plus sorafenib in an orthotopic HCC model[J/OL]. Eur Radiol Exp, 2024, 8(1): 43 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/38467904/. DOI: 10.1186/s41747-024-00448-y.
[40]
ZHOU M, CHEN M Y, CHEN M N, et al. Predictive value of mono-exponential and multiple mathematical models in locally advanced rectal cancer response to neoadjuvant chemoradiotherapy[J]. Abdom Radiol (NY), 2025, 50(3): 1105-1116. DOI: 10.1007/s00261-024-04588-y.
[41]
ZHOU M, HUANG H Y, BAO D Y, et al. Fractional order Calculus model-derived histogram metrics for assessing pathological complete response to neoadjuvant chemotherapy in locally advanced rectal cancer[J/OL]. Clin Imag, 2024, 116: 110327 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/39454478/. DOI: 10.1016/j.clinimag.2024.110327.
[42]
BAI B M, CUI L, CHU F N, et al. Multiple diffusion models for predicting pathologic response of esophageal squamous cell carcinoma to neoadjuvant chemotherapy[J]. Abdom Radiol, 2024, 49(12): 4216-4226. DOI: 10.1007/s00261-024-04474-7.
[43]
TEOH J Y, KAMAT A M, BLACK P C, et al. Recurrence mechanisms of non-muscle-invasive bladder cancer: a clinical perspective[J]. Nat Rev Urol, 2022, 19(5): 280-294. DOI: 10.1038/s41585-022-00578-1.
[44]
WANG H J, CAI Q, HUANG Y P, et al. Amide proton transfer-weighted MRI in predicting histologic grade of bladder cancer[J]. Radiology, 2022, 305(1): 127-134. DOI: 10.1148/radiol.211804.
[45]
FAN Z C, GUO J T, ZHANG X Y, et al. Non-Gaussian diffusion metrics with whole-tumor histogram analysis for bladder cancer diagnosis: muscle invasion and histological grade[J/OL]. Insights Imaging, 2024, 15(1): 138 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/38853200/. DOI: 10.1186/s13244-024-01701-z.
[46]
LI Z H, DAN G Y, TAMMANA V, et al. Predicting the aggressiveness of peripheral zone prostate cancer using a fractional order Calculus diffusion model[J/OL]. Eur J Radiol, 2021, 143: 109913 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/34464907/. DOI: 10.1016/j.ejrad.2021.109913.
[47]
HE Y S, QI X, ZHOU M X, et al. Improved differentiation of prostate cancer using advanced diffusion models: a comparative study of mono-exponential, fractional-order-Calculus, and multi-compartment models[J/OL]. Abdom Radiol (NY), 2025 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/39964371/. DOI: 10.1007/s00261-024-04684-z.
[48]
LIU G Q, LU Y, DAI Y M, et al. Comparison of mono-exponential, bi-exponential, kurtosis, and fractional-order Calculus models of diffusion-weighted imaging in characterizing prostate lesions in transition zone[J]. Abdom Radiol (NY), 2021, 46(6): 2740-2750. DOI: 10.1007/s00261-020-02903-x.
[49]
ZHANG A N, HU Q M, SONG J C, et al. Value of non-Gaussian diffusion imaging with a fractional order Calculus model combined with conventional MRI for differentiating histological types of cervical cancer[J/OL]. Magn Reson Imag, 2022, 93: 181-188 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/35988835/. DOI: 10.1016/j.mri.2022.08.014.
[50]
SHAO X, AN L, LIU H, et al. Cervical carcinoma: evaluation using diffusion MRI with a fractional order Calculus model and its correlation with histopathologic findings[J/OL]. Front Oncol, 2022, 12: 851677 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/35480091/. DOI: 10.3389/fonc.2022.851677.
[51]
ZHANG J C, SUN Y N, YANG Q, et al. Fractional order Calculus model diffusion weighted imaging for evaluating pathological classification and differentiation degree of cervical cancer[J]. Chin J Med Imag Technol, 2024, 40(11): 1730-1734. DOI: 10.13929/j.issn.1003-3289.2024.11.020.
[52]
WANG C H, WANG G Y, ZHANG Y F, et al. Differentiation of benign and malignant breast lesions using diffusion-weighted imaging with a fractional-order Calculus model[J/OL]. Eur J Radiol, 2023, 159: 110646 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/36577184/. DOI: 10.1016/j.ejrad.2022.110646.
[53]
WANG F, SUN Y N, ZHANG B T, et al. Differentiation of benign and malignant breast lesions using DWI with a fractional-order Calculus model based on SMS technique[J]. Chin J Magn Reson Imag, 2024, 15(1): 48-54. DOI: /doi.org/10.12015/issn.1674-8034.2024.01.008.
[54]
TANG C L, LI F, HE L T, et al. Comparison of continuous-time random walk and fractional order Calculus models in characterizing breast lesions using histogram analysis[J/OL]. Magn Reson Imaging, 2024, 108: 47-58 [2025-03-25]. https://pubmed.ncbi.nlm.nih.gov/38307375/. DOI: 10.1016/j.mri.2024.01.012.

PREV Progress of multimodal MRI in evaluating biomarkers related to treatment and prognosis of cervical cancer
NEXT Correction to: Predictive value of ADCmean combined with PSAD in clinically significant prostate cancer with PI-RADS score ≥ 3
  


LINK


Tel & Fax: +8610-67113815    E-mail: editor@cjmri.cn