Non-Gaussian diffusion MRI using CTRW and FROC models for early treatment response assessment in nasopharyngeal carcinoma

Background

To evaluate the utility of non-Gaussian diffusion parameters derived from the continuous-time random walk (CTRW) and fractional order calculus (FROC) models for early treatment response assessment in nasopharyngeal carcinoma (NPC) and to compare their performance with the apparent diffusion coefficient (ADC).

Methods

Diffusion-weighted imaging (DWI) with 13 b-values was performed before and during early treatment in 61 patients with newly diagnosed NPC. Parameters derived from the CTRW and FROC models (D_CTRW, α_CTRW, β_CTRW, D_FROC, β_FROC, µ_FROC) and ADC, along with their percentage changes, were compared between residual and non-residual groups after concurrent chemoradiotherapy (CCRT) with (n = 56) or without (n = 5) induction chemotherapy (IC), and between partial responders and stable disease groups in the IC subgroup. Group comparisons were performed using the Mann–Whitney U test or t-test, and predictive performance was evaluated using receiver operating characteristic (ROC) and logistic regression analyses.

Results

For IC response, Pre-β_FROC, Pre-µ_FROC, and Pre-β_CTRW significantly differentiated partial responders from stable disease groups, with Pre-β_FROC demonstrating the highest discriminatory performance (AUC = 0.798). A combined model incorporating β_CTRW and µ_FROC further improved prediction (AUC = 0.817). For the CCRT outcome, several Pre- and Δ% parameters differed significantly between residual and non-residual groups. Δβ_CTRW% achieved the highest diagnostic accuracy (AUC = 0.822), while combining Δβ_CTRW% with ΔADC% yielded the best overall performance (AUC = 0.878).

Conclusions

Diffusion parameters derived from the CTRW and FROC models provide sensitive markers of microstructural heterogeneity in NPC and outperform conventional ADC in early treatment evaluation. Pre-β_FROC, Pre-µ_FROC, and Pre-β_CTRW are valuable for predicting IC response, whereas Δβ_CTRW% shows strong potential for identifying residual tumor after CCRT. Combined predictive models further enhance diagnostic accuracy, supporting non-Gaussian diffusion imaging as a promising biomarker for early efficacy assessment and personalized management of NPC.

Relevance statement

This study demonstrates that CTRW and FROC non-Gaussian diffusion models provide earlier and more sensitive markers of treatment response in NPC compared with conventional ADC. Their combined application significantly improves the detection of induction chemotherapy efficacy and post-CCRT residual disease, offering a promising imaging-based approach for early treatment monitoring and individualized therapeutic strategies.

Key points

Non-Gaussian diffusion models (CTRW and FROC) outperform ADC in early evaluation of treatment response in nasopharyngeal carcinoma.

Pre-β_FROC, Pre-μ_FROC, and Pre-β_CTRW effectively predict induction chemotherapy response, while Δβ_CTRW% serves as the strongest indicator of post-CCRT residual disease.

Combined Gaussian and non-Gaussian parameters significantly improve diagnostic accuracy and support personalized treatment planning.

Background

Nasopharyngeal carcinoma (NPC), originating from the epithelial lining of the nasopharynx, is highly prevalent in Southeast Asia [1]. For patients with stage II–IVA disease, concurrent chemoradiotherapy (CCRT), with or without induction chemotherapy (IC), remains the standard therapeutic approach [1, 2]. IC may reduce the tumor burden and shrink radiotherapy target volumes, thereby minimizing radiation-induced injury to adjacent critical structures [3]. However, treatment outcomes vary considerably even among patients with similar clinical stages and comparable regimens, and the benefit of IC is not uniform across individuals [4, 5]. In addition, the presence of residual tumor post-CCRT is associated with poorer local control and survival [6, 7]. Although adjuvant chemotherapy has been explored after CCRT to improve disease control in selected patients [8], identifying individuals most likely to benefit remains challenging. Collectively, these issues underscore the need for imaging biomarkers that enable early response assessment and reliable detection of residual disease to support individualized treatment strategies.

Magnetic resonance imaging (MRI) serves as the cornerstone modality for managing NPC, providing accurate tumor staging, radiotherapy target delineation, and post-treatment residual-disease evaluation. A major advantage of functional MRI is its ability to reveal microenvironmental alterations before conventional morphologic changes become apparent, offering crucial insights for early therapeutic decision-making. Diffusion-weighted imaging (DWI) has been extensively investigated for early response evaluation in NPC [9–14], and the apparent diffusion coefficient (ADC) derived from conventional DWI has been widely used in radiotherapy planning and prognostic assessment [9–15]. However, its predictive performance for early treatment response remains inconsistent across studies [11–15], potentially because the Gaussian diffusion assumption underlying ADC may not adequately capture tumor heterogeneity and complex microstructural environments [16].

To address this issue, non-Gaussian diffusion models have been proposed. The continuous-time random walk (CTRW) model characterizes tissue heterogeneity through its diffusion coefficient (D_CTRW) and temporal and spatial heterogeneity indices (α_CTRW and β_CTRW) [17–20]. The fractional-order calculus (FROC) model quantifies microstructural complexity using its derived diffusion coefficient (D_FROC), spatial fractional-order parameter (β_FROC), and spatial parameter (µ_FROC) [19–24]. Prior work has suggested that CTRW- and FROC-derived parameters are useful for monitoring chemotherapy response in rectal and esophageal squamous cell carcinoma [25, 26]. However, their value for early treatment assessment in NPC—particularly for predicting response to IC and identifying post-CCRT residual disease—remains unclear. Therefore, this study aimed to investigate whether CTRW- and FROC-derived non-Gaussian diffusion parameters can predict early response to induction chemotherapy and detect residual disease after treatment in patients with NPC.

Methods

Patient selection and treatment procedure

This retrospective study was approved by the institutional ethics committee, and the requirement for informed consent was waived because of its retrospective design. Between March 2021 and August 2025, 78 patients with biopsy-confirmed NPC were reviewed.

The inclusion criteria were as follows: (1) MRI performed before biopsy or after biopsy but before any NPC-related treatment, with no prior history of head and neck malignancy; (2) pathologically confirmed NPC staged as II–IVA according to the 8th Edition of the American Joint Committee on Cancer staging system; and (3) availability of follow-up MRI after two IC cycles or 2 weeks after the initiation of CCRT, and completion of post-CCRT MRI for residual disease assessment.

The exclusion criteria were: (1) absence of multi–b-value DWI; (2) receipt of radiotherapy or chemotherapy before baseline MRI; and (3) severe imaging artifacts or tumor volume too small during early intra-treatment to allow accurate region-of-interest (ROI) delineation.

A total of 61 patients met these criteria and were included in the final analysis (Fig. 1).

Fig. 1.

A flowchart of patient selection. Notes: CCRT, concurrent chemoradiotherapy; DWI, diffusion-weighted imaging; IC, induction chemotherapy; MRI, magnetic resonance imaging; NPC, nasopharyngeal carcinoma

All patients received intensity-modulated radiotherapy combined with concurrent chemotherapy, with or without IC. The IC regimen consisted of gemcitabine administered intravenously on days 1 and 8 and cisplatin on day 1, every 3 weeks for two to three cycles. The prescribed radiation dose for the primary tumor ranged from 66–70 Gy, 66 Gy for metastatic lymph nodes, 60 Gy for high-risk nodal regions, and 54 Gy for low-risk nodal regions, delivered in 33–35 fractions (once daily, 5 days per week). CCRT consisted of cisplatin administered weekly for four to seven cycles during radiotherapy.

Endpoints

For stage III–IVA patients (n = 56), early intra-treatment MRI was performed after two IC cycles, whereas for stage II patients (n = 5), early intra-treatment MRI was obtained 2 weeks after initiation of CCRT. All patients underwent MRI at baseline, early intra-treatment, and 1 month after CCRT completion using identical imaging protocols.

Treatment response was evaluated using RECIST 1.1 [27] criteria. After two IC cycles, the 56 patients were categorized into partial response (PR) (n = 33) and stable disease (SD) (n = 23) groups. One month after CCRT completion, all 61 patients were reassessed and classified in to the residual disease (R) (n = 19) or non-residual (NR) groups (n = 42).

MRI acquisitions

MRI examinations were performed on a 3.0-T whole-body MR system (MAGNETOM Vida, Siemens Healthcare, Germany) equipped with a 20-channel head-and-neck phased-array coil. The standardized imaging protocol included the following sequences:

Coronal T₂-weighted imaging (T₂WI)

field of view (FOV) = 28 × 28 cm, repetition time (TR) = 2828 ms, echo time (TE) = 68 ms, slice thickness = 5 mm, interslice gap = 1 mm, matrix = 288 × 192, number of excitations (NEX) = 1.

Axial T₁-weighted imaging (T₁WI)

FOV = 22 × 22 cm, TR = 790 ms, TE = 6.5 ms, slice thickness = 4 mm, interslice gap = 1 mm, matrix = 288 × 256, NEX = 2.

Axial T₂-weighted imaging (T₂WI)

FOV = 22 × 22 cm, TR = 3000 ms, TE = 72 ms, slice thickness = 4 mm, interslice gap = 1 mm, matrix = 288 × 192, NEX = 1.

DWI was performed using a readout-segmented of long variable echo-trains with 13 b-values (0, 10, 20, 50, 100, 200, 400, 600, 800, 1000, 1500, 2000, and 3000 s/mm²). Imaging parameters were as follows: FOV = 220 × 220 mm, TR = 5200 ms, TE = 70 ms, slice thickness = 4 mm, interslice gap = 20%, readout segments = 5, simultaneous multi-slice factor = 2, number of slices = 22, and diffusion mode = 3-scan trace. The total acquisition time was approximately 7 min. The 13 b-values (0–3000 s/mm²) were selected to enable robust high b-value non-Gaussian diffusion modeling (FROC and CTRW), which requires a high maximum b-value (≥ 3000 s/mm²) and sufficient sampling across low-to-high b-values for stable parameter estimation [28]. The selected number of b-values is within the range commonly reported for FROC/CTRW fitting and provides a practical trade-off between fitting robustness and acquisition time.

Finally, contrast-enhanced T₁-weighted imaging (CE-T₁WI) was acquired in axial, coronal, and sagittal planes following intravenous injection of gadodiamide (Omniscan, GE Healthcare) at 2.5 mL/s and 0.1 mmol/kg.

Imaging analysis

Diffusion metrics from conventional DWI, FROC, and CTRW models were computed using Body DiffusionLab (BoDiLab, Chengdu ZhongYing Medical Technology Co., Ltd., Chengdu, China) on an MR Station workstation. Pseudo-color parameter maps were generated with ITK-SNAP (http://www.itksnap.org/pmwiki/pmwiki.php).

For conventional DWI, the ADC was calculated using the mono-exponential model (Eq. 1):

1

where S(b) and S(0) represent signal intensities with and without diffusion weighting, respectively, and b is the diffusion-weighting factor determining diffusion sensitivity.

The FROC model was fitted according to Eq. 2:

2

where S(b) is the signal intensity without diffusion weighting, G_d is the diffusion gradient amplitude, δ is the diffusion gradient pulse width, and Δ is the gradient interval. The parameters β (dimensionless) and µ (in µm) are the intra-voxel diffusion heterogeneity and spatial diffusion parameters, respectively. The FROC model was fitted to the diffusion-weighted images using the Levenberg–Marquardt nonlinear fitting algorithm. Diffusion coefficient D was initially estimated using a mono-exponential model for lower -values (≤ 1000 s/mm²). Once D was determined, β and µ were calculated by fitting all -values.

For the CTRW model, the fitting Eq. 3:

3

where D represents the anomalous diffusion coefficient, α and β are the parameters related to temporal and spatial diffusion heterogeneity, respectively, is the Mittag-Leffler function. The diffusion coefficient D was estimated by nonlinear fitting of diffusion images with -values less than 1000 s/mm², and α and β were determined from all diffusion-weighted images (with -values ranging from 0 to 3000 s/mm²).

All diffusion parameters were independently evaluated by two observers with 10 years (J.L.) and 15 years (F.W.) of experience in head and neck oncologic imaging. Both observers were blinded to the clinical information and treatment outcomes during the evaluation. TNM staging for all patients was determined jointly by two physicians based on head and neck MRI, chest and abdominal CT, and other imaging or nuclear medicine examinations. The multi–b-value diffusion images were first imported into the Body Diffusion Lab post-processing software for image registration to ensure accurate alignment. Pixel-wise fitting was then performed using mathematical models from CTRW, FROC, and conventional DWI. By analyzing signal intensity across different b-values, the software generated multiple diffusion-related quantitative parameter maps, which were displayed as pseudo-color images. ROIs were manually drawn on axial DWI images with a b-value of 1000 s/mm² at the largest cross-sectional level of the tumor, along with the adjacent superior and inferior slices, guided by fat-saturated T2-weighted images and contrast-enhanced T1-weighted images to avoid necrotic areas, cystic components, and adjacent anatomical structures. The software then automatically propagated each manually drawn ROI onto all seven diffusion parameter maps and calculated the corresponding parameter values (Figs. 2 and 3). An independent reviewer with 15 years of MRI experience (H.M.L.) checked all ROIs delineations and resolved discrepancies between the two radiologists. For each observer, the mean value derived from the three ROIs was used as the measurement for that observer. The final quantitative value for each parameter was obtained by averaging the measurements from both observers. Inter-observer agreement was assessed using the intraclass correlation coefficient (ICC). To evaluate intra-observer reproducibility, the same observer repeated the measurements after 2 weeks, and the intra-observer ICC was calculated. Parameters measured before treatment were denoted as Pre-, and those measured during early intra-treatment were denoted as Intra-. Changes in diffusion parameters were defined as the percentage difference between intra-treatment and pre-treatment values (e.g., ΔADC%), calculated as follows: ΔADC% = [(Intra-ADC − Pre-ADC) / Pre- ADC] × 100%.

Fig. 2.

Pretreatment and intra-treatment pseudocolor maps in a responsive NPC patient. This patient showed good response to IC, with no detectable residual tumor 1 month after completion of CCRT. (A) Axial T2-weighted fat-suppressed image before treatment; (B–H) corresponding pretreatment parametric maps: Pre-ADC = 0.621 μm²/ms (B), Pre-D_FROC = 0.573 μm²/ms (C), Pre-β_FROC = 0.799 (D), Pre-µ_FROC = 3.688 μm (E), Pre-D_CTRW = 0.739 μm²/ms (F), Pre-α_CTRW = 0.731 (G), Pre-β_CTRW = 0.870 (H). (I) Axial T2-weighted fat-suppressed image after IC; (J–P) corresponding intra-treatment parametric maps: Intra-ADC = 0.857 μm²/ms (J), Intra-D_FROC = 0.803 μm²/ms (K), Intra-β_FROC = 0.890 (L), Intra-µ_FROC = 3.802 μm (M), Intra-D_CTRW = 1.058 μm²/ms (N), Intra-α_CTRW = 0.838 (O), Intra-β_CTRW = 0.950 (P). Notes: ADC, apparent diffusion coefficient; CTRW, continuous-time random walk; CCRT, concurrent chemoradiotherapy; FROC, fractional order calculus; IC, induction chemotherapy; NPC, nasopharyngeal carcinoma

Fig. 3.

Pretreatment and intra-treatment pseudocolor maps in a non-responsive NPC patient. This patient was insensitive to IC, with residual disease remaining 1 month after completion of CCRT. (A) Axial T2-weighted fat-suppressed image before treatment; (B–H) corresponding pretreatment parametric maps: Pre-ADC = 0.759 μm²/ms (B), Pre-D_FROC = 0.669 μm²/ms (C), Pre-β_FROC = 0.855 (D), Pre-µ_FROC = 3.022 μm (E), Pre-D_CTRW = 0.825 μm²/ms (F), Pre-α_CTRW = 0.732 (G), Pre-β_CTRW = 0.911 (H). (I) Axial T2-weighted fat-suppressed image after IC; (J–P) corresponding intra-treatment parametric maps: Intra-ADC = 0.865 μm²/ms (J), Intra-D_FROC = 0.789 μm²/ms (K), Intra-β_FROC = 0.918 (L), Intra-µ_FROC = 3.284 μm (M), Intra-D_CTRW = 1.042 μm²/ms (N), Intra-α_CTRW = 0.777 (O), Intra-β_CTRW = 0.945 (P). Notes: ADC, apparent diffusion coefficient; CTRW, continuous-time random walk; CCRT, concurrent chemoradiotherapy; FROC, fractional order calculus; IC, induction chemotherapy; NPC, nasopharyngeal carcinoma

Statistical analysis

Statistical analyses were performed using SPSS v. 23.0 (IBM Corp., Armonk, NY, USA), GraphPad Prism version 8.0 (GraphPad Software, San Diego, CA), and MedCalc v.22.01 (Calc software Ltd, Ostend, Belgium). Quantitative parameters were tested for normality. Normally distributed data were represented as mean ± standard deviation (±S), whereas those with a skewed distribution were represented as median (P25, P75). Paired-sample t-tests were used to assess differences in diffusion parameters between pre- and intra-treatment measurements. Continuous variables between the R and NR groups, as well as between the PR and SD groups, were compared using independent-sample t-tests or Mann–Whitney U-tests, as appropriate. Categorical variables were compared using the chi-square test or Fisher’s exact test. Receiver operating characteristic (ROC) curve analysis was conducted to evaluate the diagnostic performance of diffusion parameters in predicting sensitivity to induction chemotherapy and identifying residual tumor after CCRT. Variables with statistical significance (p < 0.050) were incorporated into multivariable models using binary logistic regression to construct combined diagnostic models. Differences between areas under the ROC curves (AUCs) were assessed using the DeLong test. Inter-observer and intra-observer agreement was evaluated using ICCs with 95% confidence intervals. A p-value ≤ 0.050 was considered to indicate statistical significance.

Results

Clinical characteristics and grouping of the 61 patients with NPC

In total, 61 patients with NPC were included in the final analysis. Among them, 55 had non-keratinizing carcinoma (43 undifferentiated and 12 differentiated types) and 6 had keratinizing squamous cell carcinoma. There were 38 male and 23 female patients, with a mean age of 59.59 ± 13.90 years. No significant differences in age, sex, or clinical stage were found among the groups (Table 1).

Table 1.

Clinical characteristics of patients in PR, SD, R and NR groups

Parameters	PR (n = 33)	SD (n = 23)	t/z	p	NR (n = 42)	R (n = 19)	t/z	p
Age (years)	58.24 ± 11.53	61.55 ± 10.88	-0.894	0.376	57.55 ± 13.35	64.11 ± 14.36	-1.687	0.101
Sex			0.987	0.321			1.411	0.235
Male	23	14			25	15
Female	10	9			17	4
T category			0.667	0.667			0.566	0.452
T1 + T2	7	3			12	3
T3 + T4	26	20			30	16
N category			0.010	0.919			0.218	0.641
0 + 1	18	9			22	8
2 + 3	15	14			20	11

Abbreviations: NR, non-residual group; PR, partial response group; R, residual group; SD, stable disease group

Reproducibility of diffusion parameters

The intra-observer ICCs for pre- and early intra-treatment diffusion parameters ranged from 0.806 to 0.923, whereas inter-observer ICCs ranged from 0.853 to 0.938, indicating excellent inter- and intra-observer agreement (Table 2).

Table 2.

Reproducibility of DWI parameters pre-treatment and early intra-treatment of NPC

Parameters	Pre-treatment		Intra-treatment
	Intra-observer (95% CI)	Inter-observer (95% CI)	Intra-observer (95% CI)	Inter-observer (95% CI)
ADC (µm2/ms)	0.863 (0.771–0.918)	0.890 (0.823–0.932)	0.806 (0.677–0.884)	0.879 (0.799–0.928)
DFROC (µm2/ms)	0.923 (0.872–0.954)	0.879 (0.799–0.928)	0.872 (0.787–0.923)	0.928 (0.880–0.957)
βFROC	0.877 (0.795–0.926)	0.880 (0.799–0.928)	0.849 (0.749–0.910)	0.879 (0.798–0.927)
µFROC (µm)	0.899 (0.815–0.933)	0.920 (0.867–0.952)	0.868 (0.780–0.921)	0.853 (0.755–0.912)
DCTRW (µm2/ms)	0.916 (0.859–0.949)	0.894 (0.824–0.937)	0.827 (0.712–0.896)	0.857 (0.762–0.914)
αCTRW	0.837 (0.729–0.902)	0.905 (0.842–0.943)	0.824 (0.706–0.894)	0.896 (0.826–0.937)
βCTRW	0.918 (0.863–0.951)	0.924 (0.873–0.954)	0.879 (0.799–0.928)	0.938 (0.897–0.963)

Abbreviations: ADC, Apparent diffusion coefficient; CI, Confidence interval; CTRW, Continuous-time random walk; DWI, Diffusion-weighted imaging; FROC, Fractional-order calculus; NPC, Nasopharyngeal carcinoma

Differences in diffusion parameters and their changes between groups

Significant increases were observed in diffusion parameters after treatment in all 61 patients. Paired-sample t-tests showed statistically significant differences between pre- and early intra-treatment values for all parameters (all p < 0.001) (Table 3). Significant differences were observed between the PR and SD groups in Pre-β_FROC, Pre-µ_FROC, and Pre-β_CTRW. Comparing the R and NR groups, significant differences were identified in Pre-D_FROC, Pre-β_FROC, Pre-D_CTRW, Pre-β_CTRW, ΔADC%, ΔD_FROC%, Δβ_FROC%, ΔD_CTRW%, and Δβ_CTRW%. No statistically significant differences were found for the remaining parameters or their corresponding changes across groups (Table 4; Fig. 4).

Table 3.

Changes in diffusion parameters pre-treatment and early intra-treatment of NPC

Parameters	Pre-treatment	Intra-treatment	t	p
ADC (µm2/ms)	0.736 ± 0.096	0.894 ± 0.096	-12.079	<0.001
DFROC (µm2/ms)	0.672 ± 0.085	0.819 ± 0.106	-10.249	<0.001
βFROC	0.809 ± 0.050	0.865 ± 0.047	-7.263	<0.001
µFROC (µm)	3.281 ± 0.275	3.482 ± 0.266	-5.927	<0.001
DCTRW (µm2/ms)	0.832 ± 0.101	1.048 ± 0.102	-15.041	<0.001
αCTRW	0.792 ± 0.065	0.839 ± 0.060	-4.507	<0.001
βCTRW	0.856 ± 0.062	0.923 ± 0.044	-6.971	<0.001

Abbreviations: ADC, apparent diffusion coefficient; CTRW, Continuous-time random-walk; FROC, Fractional order calculus; NPC, Nasopharyngeal carcinoma

Table 4.

Diffusion parameters in PR, SD, R, and NR groups

Parameters	PR (n = 33)	SD (n = 23)	p	NR (n = 42)	R (n = 19)	p
Pre-ADC (µm2/ms)	0.737 ± 0.092	0.746 ± 0.088	0.58	0.719 ± 0.092	0.772 ± 0.095	0.053
Intra-ADC (µm2/ms)	0.877 (0.820,0.979)	0.892 ± 0.095	0.887	0.903 ± 0.098	0.877 ± 0.091	0.334
ΔADC%	24.341 ± 14.439	20.554 ± 13.881	0.444	26.463 ± 14.163	14.443 ± 12.134	0.002
Pre-DFROC (µm2/ms)	0.647 (0.603,0.723)	0.674 ± 0.077	0.696	0.641 (0.587,0.717)	0.707 ± 0.087	0.039
Intra-DFROC (µm2/ms)	0.795 (0.733,0.888)	0.815 ± 0.094	0.901	0.831 ± 0.116	0.796 ± 0.077	0.181
ΔDFROC%	15.502 (11.179, 35.861)	17.086 (14.570,31.852)	0.967	27.710 ± 19.353	13.589 ± 10.981	0.001
Pre-βFROC	0.826 (0.785,0.850)	0.849 (0.829,0.862)	<0.001	0.811 (0.768,0.835)	0.850 (0.811,0.859)	0.001
Intra-βFROC	0.878 (0.841,0.894)	0.882 ± 0.043	0.096	0.863 ± 0.047	0.881 (0.867,0.890)	0.414
ΔβFROC%	6.796 (2.679,11.414)	3.726 (1.904,10.423)	0.197	7.355 (4.045,13.796)	4.218 ± 6.500	0.038
Pre-µFROC (µm)	3.247 ± 0.257	3.129 ± 0.242	0.003	3.479 ± 0.278	3.487 ± 0.241	0.671
Intra-µFROC (µm)	3.469 ± 0.265	3.486 (3.136,3.605)	0.108	3.506 ± 0.289	3.514 ± 0.220	0.938
ΔµFROC%	5.600 (1.776,10.429)	8.248 (3.759,14.226)	0.154	6.140 ± 9.355	5.486 (1.796,8.938)	0.674
Pre-DCTRW (µm2/ms)	0.834 ± 0.102	0.821 ± 0.098	0.430	0.815 ± 0.100	0.871 ± 0.095	0.042
Intra-DCTRW (µm2/ms)	1.047 ± 0.104	1.058 ± 0.092	0.516	1.055 ± 0.112	1.034 ± 0.076	0.395
ΔDCTRW%	26.870 ± 15.952	26.373 (19.899,38.495)	0.221	30.770 ± 16.431	19.481 ± 10.524	0.002
Pre-αCTRW	0.793 ± 0.065	0.811 (0.758,0.868)	0.101	0.846 ± 0.060	0.823 ± 0.059	0.544
Intra-αCTRW	0.839 ± 0.062	0.837 ± 0.061	0.871	0.856 ± 0.053	0.828 ± 0.060	0.118
ΔαCTRW%	6.291 ± 10.727	3.718 ± 11.405	0.135	7.992 ± 9.881	3.699 ± 11.270	0.144
Pre-βCTRW	0.872 (0.809,0.904)	0.889 (0.858,0.912)	0.035	0.851 (0.799,0.891)	0.897 ± 0.032	<0.001
Intra-βCTRW	0.942 (0.881,0.959)	0.925 ± 0.042	0.980	0.944 (0.878,0.964)	0.916 ± 0.044	0.276
ΔβCTRW%	6.428 (2.515,12.147)	4.549 (-0.135,10.182)	0.116	8.883 (5.185,15.734)	2.322 (-2.129,4.549)	<0.001

Notes: ADC, apparent diffusion coefficient; CTRW, Continuous time random walk; FROC, Fractional order calculus; NR, non-residual group; PR, partial response group; R, residual group; SD, stable disease group

Fig. 4.

Diffusion parameter differences in PR, SD, NR, and R groups. Notes: CTRW, continuous-time random walk; FROC, fractional order calculus; NR, non-residual group; PR, partial response group; R, residual group; SD, stable disease group

Diagnostic performance of diffusion parameters

Distinguishing PR vs. SD groups

Univariable logistic regression identified Pre-β_FROC, Pre-µ_FROC, and Pre-β_CTRW as significant predictors (p = 0.003, 0.006, and 0.037). ROC analysis showed that Pre-β_FROC achieved the highest diagnostic performance, with an AUC of 0.798, sensitivity of 75.76%, and specificity of 78.26%. DeLong testing showed a significant difference between Pre-β_FROC and Pre-β_CTRW (p = 0.018). Multivariable logistic regression identified Pre-µ_FROC and Pre-β_CTRW as independent predictors for distinguishing PR from SD (p = 0.003 and 0.009). The combined model (Pre-µ_FROC + Pre-β_CTRW) achieved an AUC of 0.817, with a sensitivity of 90.94% and specificity of 60.87%. DeLong testing showed that the combined model differed significantly from Pre-β_CTRW alone (p = 0.033), while no differences was observed compared to the Pre-β_FROC and Pre-β_CTRW (Table 5; Fig. 5A).

Table 5.

Diagnostic performance of diffusion parameters for treatment response and residual disease

Parameters	AUC (95% CI)	Sensitivity (%)	Specificity (%)	Cut off value
aPre-βFROC	0.798 (0.677–0.920)	75.76	78.26	0.827
aPre-µFROC (µm)	0.717 (0.580–0.854)	72.73	65.22	3.183
aPre-βCTRW	0.667 (0.519–0.814)	60.61	78.26	0.858
aPre-µFROC  + Pre-βCTRW	0.817 (0.702–0.932)	90.94	60.87	0.293
bΔADC%	0.743 (0.617–0.869)	57.14	89.47	21.763
bPre-DFROC (µm2/ms)	0.666 (0.520–0.812)	69.05	63.16	0.681
bΔDFROC%	0.726 (0.597–0.854)	61.90	84.21	18.324
bPre-βFROC	0.776 (0.639–0.912)	85.71	68.42	0.842
bΔβFROC%	0.667 (0.522–0.811)	76.19	52.63	3.726
bPre-DCTRW (µm2/ms)	0.649 (0.503–0.796)	35.71	89.47	0.770
bΔDCTRW%	0.706 (0.576–0.835)	59.52	84.21	26.373
bPre-βCTRW	0.784 (0.670–0.899)	57.14	94.74	0.856
bΔβCTRW%	0.822 (0.709–0.935)	80.95	78.95	4.549
bΔADC% + ΔβCTRW%	0.878 (0.788–0.969)	85.71	78.95	0.339

Notes: ADC, apparent diffusion coefficient; CCRT, concurrent chemoradiotherapy; CTRW, Continuous time random walk; CI, Confidence interval; FROC, Fractional order calculus; IC, induction chemotherapy; ^apredicting stable disease from partial response groups; ^bpredicting residual from non-residual groups

Fig. 5.

ROC curves for predicting treatment response and residual disease in NPC. (A) Prediction of IC treatment response. (B) Identification of residual disease one month after CCRT. Notes: ADC, apparent diffusion coefficient; CTRW, continuous-time random walk; CCRT, concurrent chemoradiotherapy; FROC, fractional order calculus; IC, induction chemotherapy; NPC, nasopharyngeal carcinoma

Distinguishing NR vs. R groups after CCRT

Univariable logistic regression identified Pre-D_FROC, Pre-β_FROC, Pre-D_CTRW, Pre-β_CTRW, ΔADC%, ΔD_FROC%, Δβ_FROC%, ΔD_CTRW%, and Δβ_CTRW% as significant predictors (p = 0.039, 0.005, 0.048, 0.002, 0.005, 0.010, 0.036, 0.013, and 0.002, respectively). Among them, Δβ_CTRW% showed the highest diagnostic efficiency, achieving an AUC of 0.822 with a sensitivity of 85.71% and specificity of 78.95%. Multivariable logistic regression identified ΔADC% and Δβ_CTRW% as independent predictors (p = 0.008 and 0.002). The combined model (ΔADC% + Δβ_CTRW%) achieved the best diagnostic performance, with an AUC of 0.878, sensitivity of 85.71%, and specificity of 78.95%. DeLong tests showed that this model performed significantly better than Pre-D_FROC, Pre-D_CTRW, ΔADC%, ΔD_FROC%, ΔD_CTRW%, and Δβ_FROC% (p = 0.003, 0.003, 0.028, 0.021, 0.019, and 0.002, respectively) (Table 5; Fig. 5B).

Discussion

This study is among the few investigations exploring the clinical value of quantitative parameters derived from non-Gaussian diffusion models—namely CTRW and FROC—in the early evaluation of treatment response in primary NPC. Beyond assessing the response to IC, we further examined the ability of these parameters to identify early post-CCRT residual disease. Our findings show that indices reflecting microstructural heterogeneity from both CTRW and FROC models offer promising biomarkers for predicting IC response and detecting early residual tumor.

D_CTRW and D_FROC describe diffusion behavior in a manner analogous to ADC, yet capture non-Gaussian characteristics of tissue diffusion that better reflect the complex tumor microenvironment [17, 18, 21, 22, 24]. β_FROC characterizes intra-voxel heterogeneity, while µ_FROC inversely correlates with the effective diffusion path length of water molecules [21, 22, 24]. α_CTRW reflects the temporal aspect of diffusion, and β_CTRW reflects spatial heterogeneity; increasing structural complexity typically leads to larger deviations of β_CTRW and α_CTRW from unity [17–19]. Across all patients, ADC, D_FROC, and D_CTRW significantly increased after treatment, consistent with biological changes induced by chemoradiotherapy, including cell death, reduced cellularity, and expansion of the extracellular space. These observations align with previous reports describing increased diffusion after effective treatment [12]. Treatment-induced necrosis and cystic alterations may also partially homogenize tissue composition [12], contributing to elevated β_FROC, α_CTRW, β_CTRW, and µ_FROC. Similar intra-treatment alterations in non-Gaussian parameters have been reported in studies of rectal cancer undergoing neoadjuvant therapy [26].

A key finding is that pre-β_FROC, pre-µ_FROC, and pre-β_CTRW were significant predictors of IC response, whereas ADC did not provide meaningful discrimination. These results highlight the limited sensitivity of Gaussian diffusion metrics in heterogeneous head and neck tumors and suggest that non-Gaussian parameters may help identify patients unlikely to benefit from IC. Early stratification of responders and non-responders may assist clinicians in optimizing treatment duration, avoiding unnecessary toxicity, and individualizing therapeutic decisions. Tumors in the responder group showed lower pre-β_FROC and pre-β_CTRW, suggesting denser cellular architecture and richer vascularity [20, 29]—conditions that may facilitate chemotherapeutic delivery and improve treatment response [30, 31]. Higher β_FROC, in contrast, may reflect necrotic or cystic regions associated with hypoxia and acidosis, features known to promote resistance to chemoradiotherapy [32, 33]. In our study, neither ADC nor diffusion-associated parameters predicted IC response, consistent with some reports [13, 14] but differing from others [11, 12], possibly due to variation in acquisition protocols and evaluation timepoints. High-quality multi–b-value acquisition is essential for accurate non-Gaussian model fitting. In this study, a readout-segmented echo-planar imaging DWI sequence was employed to enhance image clarity and mitigate geometric distortion in the anatomically complex head and neck region [34–36]. Building on previous evidence stating that simultaneous multi-slice acceleration with a factor of 2 can preserve image quality while improving spatial resolution and lesion conspicuity in readout-segmented echo-planar imaging—such as in studies of the NPC and parotid gland tumors [35, 36]—simultaneous multi-slice was incorporated in the present protocol to achieve three aims: shortening overall scan time, increasing patient throughput, and obtaining more stable quantitative diffusion parameters.

MRI serves as the primary imaging tool for evaluating residual disease in NPC, as biopsy is often limited by anatomical constraints [37, 38]. While some studies have suggested ADC differences between R and NR groups [11, 12], our results did not confirm this pattern, consistent with the findings of Ai QYH et al. [14]. Notably, lower pre-D_FROC and Pre-D_CTRW values were associated with complete tumor clearance after treatment, possibly reflecting the sensitivity of densely structured and highly vascularized tumors to chemoradiotherapy. The magnitude of change in these parameters also corresponded with treatment response. Another clinically relevant observation is the strong performance of Δβ_CTRW% in detecting residual disease one month after CCRT. This parameter showed higher diagnostic accuracy than ΔADC%, suggesting that CTRW-derived spatial heterogeneity metrics are more sensitive to subtle treatment-induced alterations. Combining Δβ_CTRW% with ΔADC% further improved diagnostic performance, demonstrating the complementary value of Gaussian and non-Gaussian diffusion metrics for treatment monitoring. At the same time, a ΔADC% of approximately 21.76% was indicative of residual disease, closely matching previously reported thresholds for head and neck squamous cell carcinoma [39, 40]. Compared with absolute parameter values, percentage changes are less susceptible to scanner- and protocol-related variability and therefore serve as more robust biomarkers. Additionally, ΔD_FROC% and ΔD_CTRW% provided similar predictive value. Interestingly, α_CTRW and its percentage change did not differ significantly between groups. The dissociation between α_CTRW and β_CTRW suggest that microstructural alterations in NPC predominantly affect spatial diffusion heterogeneity rather than temporal characteristics. Similar selective alterations in α_CTRW or β_CTRW have been reported in other tumor entities [24]. This phenomenon indicates that the biological underpinnings of non-Gaussian diffusion features are highly complex, and their underlying mechanisms remain to be further elucidated through future research.

This study has some limitations. First, early therapeutic response does not necessarily translate into long-term clinical outcomes; therefore, prospective survival analyses are needed to determine the prognostic relevance of CTRW and FROC parameters. Second, the sample size was relatively small and derived from a single institution, necessitating multicenter validation. Third, the selected timepoints for evaluating the response to IC and for assessing residual disease after CCRT may not represent the optimal window for early response imaging. Although researchers differ in the assessment schedules they adopt, the ideal timepoint for such evaluations is yet to be clearly established [10–14]. Moreover, factors such as pre-treatment tumor volume [41], inflammatory markers [42], and EBV DNA levels [43] were not incorporated. Finally, although this study provides preliminary evidence supporting the utility of non-Gaussian diffusion parameters in NPC, deeper exploration of the biological mechanisms linking microenvironmental alterations and diffusion behavior is warranted.

Conclusion

Non-Gaussian diffusion parameters derived from CTRW and FROC models offer clinically useful information for early assessment of treatment response in NPC. These metrics improve the prediction of IC effectiveness and enhance the detection of post-CCRT residual disease, supporting their potential integration into routine MRI protocols for individualized treatment planning.

Abbreviations

ADC

Apparent diffusion coefficient

AUC

Area under the curve

CTRW

Continuous-time random walk

CCRT

Concurrent chemoradiotherapy

DWI

Diffusion-weighted imaging

FROC

Fractional order calculus

IC

Induction chemotherapy

ICC

Intraclass correlation coefficient

MRI

Magnetic resonance imaging

NPC

Nasopharyngeal carcinoma

NR

Non-residual group

PR

Partial response group

SD

Stable disease group

R

Residual group

ROI

Region of interest

ROC

Receiver operating characteristic

Author contributions

J.L., J.Z., and Y.F.Q. devised the experiment. F.W. and H.M.L. designed the tables and figures. J.L. and H.F.S. performed the data analysis. Y.F.Q. and J.Z. revised the manuscript. L.J. wrote the original draft. F.W., H.M.L., J.L. and M.C. were responsible for data collection, and M.X.L. handled manuscript editing. Q.Y., J.Z., and Y.F.Q. are co-corresponding authors. All authors have read and approved the final manuscript.

Funding

This study was supported by the Key Projects of Natural Science Research in Universities of Anhui Province (No. 2024AH050749) and Youth Science Fund Projects of Anhui Medical University (No. 2021xjk114).

Data availability

The data that support the findings of this study are available from Y.F.Q. but restrictions apply to the availability of these data, which were used under license for the current study, and therefore are not publicly available. Data are, however, available from the authors upon reasonable request and with permission from Y.F.Q.

Declarations

Ethics approval and consent to participate

This retrospective study was approved by the Institutional Ethics Committee of Anhui Medical University (Ethics No. 83244611; effective date: April 18, 2024). The requirement for informed consent was waived owing to the retrospective nature of the study. All procedures were performed in accordance with the ethical standards of the institutional research committee and the principles outlined in the Declaration of Helsinki.

Consent for publication

Not applicable.

Competing interests

One of the authors (Mengxiao Liu) is an employee of Siemens Healthcare. His contribution to this manuscript was limited to language editing and manuscript revision, and he was not involved in the study design, data acquisition, analysis, or interpretation of the results. The other authors of this manuscript declare no relationships with any companies whose products or services may be related to the subject matter of the article.