gKFA: a geometric framework for kurtosis fractional anisotropy

Abstract

Diffusion kurtosis imaging (DKI) extends diffusion tensor imaging (DTI) by incorporating non-Gaussian diffusion effects through a fourth-order kurtosis tensor. In this work, we investigate the three-dimensional geometric structure of the kurtosis tensor and introduce a geometry-inspired anisotropy descriptor based on mapped symmetric second-order quadratic forms. This representation enables kurtosis anisotropy to be studied through lower-order tensors whose geometry is easier to analyze and whose positive definiteness can be assessed more conveniently. Based on this framework, we define geometric kurtosis fractional anisotropy (gKFA) as the geometric mean of the fractional anisotropies computed from the mapped quadratic forms. The proposed descriptor was evaluated using simulations, physical phantoms, and in vivo human and rodent brain data. In simulations and phantom experiments, gKFA showed sensitivity to directional heterogeneity and tissue-complexity changes that were only weakly reflected by conventional KFA. In the human brain dataset, additional analyses compared gKFA with conventional kurtosis fractional anisotropy (KFA), mean kurtosis (MK), axial kurtosis (AK), radial kurtosis (RK), and anisotropy measures derived from the mapped quadratic forms. These results indicate that gKFA behaves as a kurtosis-anisotropy descriptor that is closely related, but not identical, to KFA, and distinct from standard scalar kurtosis measures such as MK, AK, and RK. In both human and rodent data, gKFA showed better tissue contrast and complementary behavior relative to KFA, including sensitivity to regional differences in the rodent brain.

Introduction

Compared to diffusion tensor imaging (DTI), diffusion kurtosis imaging (DKI) incorporates non-Gaussian diffusion effects by introducing a kurtosis term into the Taylor series expansion of the diffusion-weighted signal^1–4. This non-Gaussian component captures deviations from Gaussian diffusion caused by tissue heterogeneity, cellular barriers, and complex microstructural organization. As a result, DKI provides access to additional tissue microstructure descriptors that are not available from second-order diffusion models alone. Due to this increased sensitivity, DKI has been shown to be valuable for detecting and monitoring microstructural alterations associated with a range of neurological and neurodegenerative disorders^5–10. Several scalar metrics have been derived from the kurtosis tensor to summarize non-Gaussian diffusion behavior, including mean kurtosis (MK), axial kurtosis (AK), radial kurtosis (RK), and kurtosis fractional anisotropy (KFA).

Among these, kurtosis fractional anisotropy (KFA) is commonly used to characterize the directional variability of diffusional kurtosis. While fractional anisotropy (FA) derived from DTI summarizes anisotropy of Gaussian diffusion based on second-order tensors, KFA reflects anisotropy of non-Gaussian diffusion encoded by the fourth-order kurtosis tensor. Thus, both FA and KFA summarize directional properties, but they arise from fundamentally different diffusion models and capture complementary aspects of the diffusion structure.

Despite its widespread use, FA has shown ambiguous behavior in several clinical and experimental settings. In particular, FA values can be influenced by multiple factors, including fiber crossings, partial volume effects, and proportional changes in tensor eigenvalues, which can reduce its specificity to microstructural alterations. For example, studies that examine FA changes in the substantia nigra (SN) have reported inconsistent findings between cohorts and disease stages^11–14.

Interpretation of FA becomes especially challenging in white matter regions with complex fiber configurations¹⁵. Because FA is a relative measure of eigenvalue dispersion, microstructural tissue changes, such as demyelination or axonal loss, may lead to an increase, decrease, or no measurable change in FA, depending on how the eigenvalues are affected. This limitation is less pronounced in highly coherent fiber bundles, such as the corpus callosum, the corticospinal tract, and the spinal cord, but remains problematic in regions with crossed or fanning fibers.

However, metrics derived from the kurtosis tensor have demonstrated improved sensitivity to microstructural alterations in gray matter and deep gray matter structures. For example, a rodent study¹⁶ showed that mean kurtosis was more sensitive than FA to microstructural changes in the substantia nigra and was able to detect early-stage Parkinson’s disease. Subsequent work further demonstrated that MK exhibits more pronounced differences in the major regions of the brain compared to FA¹⁷.

KFA has also been reported to exhibit elevated values in deep gray matter regions such as the thalamus and lenticular nucleus, where FA typically indicates low anisotropy¹⁸. In addition, KFA reductions have been associated with increased tissue density in tumors, aging-related changes, and myelin damage^19,20. In the context of carbon monoxide poisoning, KFA was shown to be sensitive to white matter injury and microstructural damage in both white matter and gray matter, demonstrating greater diagnostic precision and stronger correlations with neuropsychiatric scores than other kurtosis-derived metrics²¹.

Despite these advantages, the kurtosis tensor possesses a higher number of degrees of freedom than the diffusion tensor²². This increased dimensionality makes kurtosis-derived quantities harder to analyze and more sensitive to estimation instability, particularly in heterogeneous tissues and under low signal-to-noise ratio (SNR) conditions. Rather than addressing noise alone, the present work is motivated by the need to investigate the three-dimensional geometric structure of the kurtosis tensor and to derive a complementary anisotropy descriptor from lower-order geometric representations. We introduce a geometric framework that interprets the fourth-order kurtosis tensor in terms of mapped second-order quadratic forms. These quadratic forms visually approximate the dominant directional structure of the kurtosis (Orientation Distribution Function) ODF, as their principal axes tend to align with the ODF maxima observed in the visualizations, while providing a stable and interpretable geometric representation of kurtosis anisotropy. Based on this framework, we propose a descriptor, geometric kurtosis fractional anisotropy (gKFA), derived from the anisotropy of the mapped quadratic forms. We systematically evaluated the behavior of gKFA using numerical simulations, physical phantom experiments, and in vivo datasets from human and rodent brains. These analyzes included comparisons with conventional KFA and standard DKI metrics such as MK, AK, and RK. Our results indicate that gKFA is closely related, but not identical, to conventional KFA, and behaves more like a kurtosis-anisotropy descriptor than like a scalar kurtosis measure. In the human dataset, gKFA showed strong white-matter–gray-matter contrast and complementary behavior relative to KFA, while in the rodent data it showed sensitivity to regional tissue differences.

The remainder of the paper is organized as follows: Section “Related work” reviews related work, Section “Methodology” describes the proposed methodology, Section “Experiments” presents experimental results, Section “Discussion” discusses limitations and future directions, and Section “Conclusion” concludes the paper.

Related work

In diffusion kurtosis imaging (DKI), non-Gaussian diffusion effects are modeled by extending the logarithmic signal representation beyond the second-order (Gaussian) term. Specifically, the logarithm of the diffusion-weighted signal is approximated using a truncated Taylor series expansion in the diffusion-weighting factor b, where the second-order term describes Gaussian diffusion and the fourth-order term captures deviations from Gaussianity arising from tissue microstructural complexity:

1

where is a direction of the unit diffusion gradient, denotes the diffusion tensor, is the fully symmetric fourth-order kurtosis tensor and is the mean diffusivity. Higher-order terms are neglected under the assumption of moderate diffusion weighting. The kurtosis fractional anisotropy is defined as:

2

where , and denotes the Frobenius norm.

3

with denoting the Kronecker delta, defined as if i=j and otherwise.

Methodology

The fourth-order kurtosis form is expected to satisfy positivity conditions for physical consistency, although in practice unconstrained estimation may yield tensors that violate these conditions, especially under low-SNR conditions. Because positive definiteness is more difficult to assess directly for fourth-order tensors than for second-order matrices, we adopt the representation proposed by Song et al.²³, which maps the fourth-order kurtosis tensor to a set of symmetric second-order forms. This mapping exploits the structure of the kurtosis tensor when evaluated along diffusion gradient directions and allows sufficient positivity conditions to be assessed through lower-order tensors that are easier to interpret and analyze.

In diffusion-weighted magnetic resonance imaging (MRI), the diffusion gradient direction is a unit vector constrained to the unit sphere, i.e. . Under this condition, the fourth-order kurtosis form evaluated along can be expressed as

4

where , , and are three symmetric second-order forms derived from the components of the kurtosis tensor .

For symmetric matrices, positive definiteness is equivalent to positivity of all eigenvalues. Hence,

provides a convenient matrix-based sufficient condition for positive definiteness of the associated fourth-order tensor W.

The decomposition in Eq. (4) is also useful for analyzing positive definiteness. Define

5

Since , , and are symmetric second-order forms, their definiteness can be assessed using standard matrix criteria such as eigenvalues or Sylvester’s criterion. In particular, the following one-way implication holds.

Proposition 1

If the mapped quadratic forms , , and are all symmetric positive definite, then the fourth-order tensor W is positive definite.

Proof

Let . Since , , and are symmetric positive definite,

6

Moreover, at least one of , , and is strictly positive. Therefore, from Eq. (5),

7

which proves that W is positive definite. □

The converse does not hold in general; that is, positive definiteness of the fourth-order tensor W does not imply that each of , , and is individually positive definite.

Specifically, the matrices , , and correspond to distinct projections of the fourth-order tensor onto second-order subspaces:

8

9

10

Although the fourth-order kurtosis tensor does not admit a direct eigenvalue decomposition, its quadratic representations yield symmetric second-order forms. These second-order forms are easier to analyze because, although they are not themselves rotationally invariant, they follow the familiar transformation law of symmetric matrices, admit standard eigenvalue decomposition, and allow well-established scalar descriptors such as fractional anisotropy (FA) to be computed.

In this work, we apply the standard FA formulation to these quadratics as a simple and interpretable way to characterize anisotropy. This approach allows familiar second-order diffusion measures to be evaluated on kurtosis-derived representations and is applicable to tissues with non-Gaussian diffusion without relying on assumptions specific to white-matter fiber organization. The FA computed from the mapped quadratic forms , , and remains sensitive to angular variations in diffusion kurtosis that are suppressed when the fourth-order tensor is summarized into a single scalar.

In diffusion tensor imaging (DTI), FA is derived from the eigenvalues of the second-order diffusion tensor and provides a rotationally invariant measure of the dispersion of eigenvalues. Kurtosis fractional anisotropy (KFA) is defined analogously at the level of the fourth-order kurtosis tensor as a normalized Frobenius-norm deviation from the isotropic kurtosis tensor, and is rotationally invariant under orthogonal changes of coordinate system. Quadratic mapped forms , , and provide second-order representations that preserve the directional information contained in the kurtosis tensor.

The motivation of this study is to investigate the directional structure of non-Gaussian diffusion behavior in three dimensions and to represent the fourth-order kurtosis tensor in terms of well-understood second-order tensors whose geometry is more familiar and easier to analyze. In this sense, , , and should be viewed as symmetric second-order representations of the structure of directional kurtosis, rather than as quantities that uniquely reflect the properties of individual tissue such as the diameter of the axon, the content of myelin, or the exchange rates. Similarly, gKFA should be interpreted as a compact descriptor of anisotropy across these mapped quadratic representations. We define geometric kurtosis fractional anisotropy (gKFA) as the geometric mean of the FA values computed from the three mapped quadratic forms:

11

This construction provides a compact summary of anisotropy across the three quadratic representations while retaining the sensitivity to the angular heterogeneity of non-Gaussian diffusion. To assess its transformation behavior, we conducted direct numerical tensor-rotation experiments in which fourth-order tensors, represented by their 15 independent coefficients, were rotated over random three-dimensional orientations, and gKFA was recomputed after each rotation. In these experiments, KFA remained constant up to machine precision, confirming its rotational invariance, whereas gKFA varied across rotations for general tensors. gKFA is not rotationally invariant in general and is therefore better interpreted as a geometry-inspired coordinate-dependent descriptor derived from the mapped quadratic forms. Further details of the analytical proof for KFA and the direct numerical tensor-rotation experiment are provided in Supplementary Note 1. Since each FA term lies in [0, 1], gKFA is also bounded in [0, 1] whenever , , and are valid nonzero symmetric tensors for which FA is defined.

Biophysical interpretation of the mapped quadratics. The three mapped quadratic forms, , , and , are not arbitrary algebraic constructs, but direction-conditioned projections of the fourth-order kurtosis tensor evaluated on the unit sphere. Each quadratic captures how non-Gaussian diffusion varies along distinct geometric combinations of spatial directions, thereby decomposing the kurtosis tensor into complementary second-order representations. From a biophysical perspective, these quadratic forms reflect anisotropy in higher-order diffusion moments associated with orientation dispersion and compartmental heterogeneity. Although the 2nd order diffusion tensor describes the second moment of molecular displacement, the kurtosis tensor characterizes deviations from Gaussian diffusion caused by barriers, restrictions, and structural complexity. The mapped quadratics preserve this higher-order information, while enabling geometric analysis.

Fig. 10.

Scalar maps for a slice of fixed rat brain data.

Experiments

Angular resolution

We used the adaptive kernel method^24,25 to estimate ODFs for the second and fourth-order tensors that represent kurtosis. The experiment used a diffusion gradient strength b=1500 with 81 gradient directions.

Fig. 1 a) shows a configuration in which two linear fibers intersect a curved fiber, forming three (heterogeneous) intersections. In three regions, the maxima of ODFs for fourth-order tensors do not align with the underlying fiber directions. Therefore, a proper rotation is usually needed for alignment. However, Fig. 1 b) shows a DTI reconstruction. The principal directions of the corresponding second-order tensors resolve the ground truth direction in regions where only one fiber is present or where no fiber is present (background). Figs. 1 c), d) and e) represent the second-order (ellipsoid) reconstruction of the quadratics. In the crossing regions, the principal directions of the quadratics p and q follow the actual path of the underlying fibers.

Fig. 1.

Configuration with two linear fibers intersecting a curved fiber: (a) 4th-order tensor field, (b) DTI 2nd-order field, (c–e) 2nd-order reconstructions from the mapped quadratics p, q, and r.

Diffusion-weighted signals were simulated for two-fiber configurations at b = 0, 1000, 2000, and 3000 s/mm² using 33 uniformly distributed gradient directions. The ground-truth fibers were modeled with angle differences ranging from to . Appropriate eigenvalues were selected and DKI fitting was performed following a work²⁶, assuming equal volume fractions.

Angular differences are computed from the principal eigenvectors of the tensors p and q. Figure 2 shows the estimated angles (y-axis) against the ground truth fiber angles (x-axis), along with the identity line x = y. The plot indicates that the estimation errors are lower for both small and large crossing angles. These results improve on a previous study, which estimated directions by computing the ODFs maxima²⁷. That showed a reasonable estimate only for angles above and failed for the set of angles below .

Fig. 2.

Angle differences between the principal eigenvectors of the predicted tensors (p and q) and the ground-truth fiber directions for two-fiber configurations.

Sensitivity to Kurtosis Orientation

The microstructure of the tissue affects the direction dependence of diffusional kurtosis (non-Gaussian). To compare how scalar metrics from quadratics, and KFA respond to orientation changes in the kurtosis component, we used a setup with two crossing fibers. Together, these fibers are aligned with the x-axis at increasing angles as shown in Figs. 3 a) and b), the respective line plots, depicting their sensitivity. gKFA and FAs of quadratic forms are sensitive to orientation change of kurtosis, behave as a geometry-sensitive anisotropy metric, unlike conventional KFA.

Fig. 3.

Sensitivity of scalar measures to angular non-Gaussianity (kurtosis) in DKI. (a) Two-fiber configurations aligned with respect to the x-axis with crossing angles from to . (b) Corresponding response of scalar metrics: KFA is largely insensitive to angular variation, whereas quadratic-based metrics (and gKFA) vary with fiber orientation.

The quadratics visualization

We visualize the quadratic functions p, q, and r as ellipsoids alongside their corresponding fourth-order ODFs, and examine their behavior under different ground-truth configurations. We use the adaptive kernel method. Fig. 4 in the top row illustrates the configuration of the three perpendicular diffusion directions (as ellipsoids). The ellipsoids in the x-y plane (individual x and y directions) are gradually made to align with the z-direction by reducing the angle difference, and eventually all the ellipsoids point in the same direction. The rest of the rows in Fig. 4 separately show the orientations of the three quadratics p, q, and r corresponding to the configurations in the top row. Figure 5 shows similar configurations, but in 2D, considering two diffusion directions. In both cases, the first configuration is shown in blue, representing isotropic diffusion. These visual representations provide insight into the underlying relationships between the ground truth diffusion directions and the principal directions from the quadratics.

Fig. 4.

Visualizations for three-fiber configurations. Top row: fourth-order ODFs for varying fiber angles (perpendicular → oblate → uni-directional). Bottom rows: ellipsoid visualizations of the mapped quadratics p, q, and r.

Fig. 5.

Visualizations for two-fiber configurations. Top row: fourth-order ODFs (blue sphere indicates isotropic diffusion). Bottom rows: ellipsoid visualizations of mapped quadratics p, q, and r.

Simulations: tissue complexity

In this experiment, we examine the effect of increasing tissue complexity on the scalars. Signals in three perpendicular directions, , , and , were selected and combined following the formulation of Hansen et al.²⁰. The weights ranged from a single fiber direction (α =0) to equal contributions from all three orientations (α =1/3). The aggregate signal, , was simulated with α varying from 0 to 1/3 in increments of 0.03.

12

where VFI (volume fraction index), by construction, provides a one-dimensional index of increasing tissue complexity/orientation mixing, allowing the scalar metrics to be compared as the signal configuration varies from a coherent single-direction bundle to a maximally heterogeneous orthogonal mixture. It is obtained by computing FA on volume fractions as eigenvalues , resulting in a range of values in [0, 1]. The volume fraction indicates the configuration of three perpendicular fibers on one extreme (VFI=0, isotropic case, highest heterogeneity), and the highest value (VFI=1, anisotropic case, lowest heterogeneity) indicates a uni-directional fiber bundle.

The three signals were extracted from different data sources to show generalizability as follows:

Simulation

The diffusion-weighted signals are simulated for a voxel containing a single tensor using the Stejskal–Tanner equation²⁸, with eigenvalues . Tensors are oriented along the , , and unit directions. The combined signals are fitted using the DKI model to obtain scalar measures.

As shown in Fig. 6, FA is minimal at VFI =0, corresponding to maximal heterogeneity, reflecting orientation averaging that reduces apparent anisotropy. FA increases monotonically with VFI and reaches its maximum at VFI =1, where the configuration becomes homogeneous. In opposition, gKFA is maximized in VFI =0 and decreases with increasing homogeneity, demonstrating its sensitivity to changes in signal configuration and directional dispersion. KFA remains largely invariant across the VFI, indicating a limited sensitivity to signal heterogeneity.

Fig. 6.

Scalar measures as a function of volume fraction index (VFI) using simulated signals mixed from three perpendicular directions. The y-axis shows normalized scalar values.

Phantom

A phantom was constructed to mimic fiber-like structures equally distributed along the , , and unit directions using fresh asparagus stems placed in three perpendicular orientations within 3× 3 square blocks, with the structural map shown in Fig. 7(a). The reader is referred to^20,29 for further details. Fifteen encoding directions are used at b-values of 0, 500, 1000, 1800, 2500, and 3500 .

Fig. 7.

Phantom structure and selected regions (blue/green/red) used to extract signals in the x, y, and z directions.

Three regions of interest are selected in Fig. 7(b), highlighted in different colors (blue, green, and red), corresponding to the three dominant orientations. Individual signals are extracted from each region across six slices of the phantom. The averaged signals from the three directions are then combined using Eq. 12 to generate signals with varying volume fraction indices (VFI), representing different levels of directional heterogeneity.

Figure 8 shows that gKFA attains higher values and exhibits greater responsiveness in the maximally heterogeneous configuration (VFI =0), corresponding to equal contributions from three orthogonal directions, compared to KFA. While FA is lowest at VFI =0, it remains nonzero, which can be attributed to the processing pipeline that involves averaging diffusion-weighted signals prior to scalar estimation, followed by averaging of the resulting scalars. This averaging procedure partially preserves anisotropy despite orientation mixing. By contrast, KFA varies only weakly with VFI, indicating limited sensitivity to changes in directional heterogeneity.

Fig. 8.

Anisotropy scalars as a function of VFI using phantom-derived signals from three orthogonal bundle directions.

In vivo human brain

This data was obtained from the public dataset of Hansen and Jespersen²⁹. Human data were acquired for one healthy volunteer on a Siemens Trio 3 T system equipped with a 32-channel head coil using a double spin-echo diffusion-weighted EPI sequence. Diffusion-weighted data were recorded at b=0 and along 33 directions at b-values from 0.2 to 3.0 ms/μm in steps of 0.2 ms/μm. The encoding scheme was constructed as a combination of a 24-point spherical design and nine directions identified for rapid kurtosis estimation, and CSF suppression (inversion recovery) was employed. Imaging parameters were: TR =7200 ms, TE =116 ms, TI =2100 ms, 19 consecutive slices, isotropic spatial resolution of 2.5 mm, matrix size 96× 96, and anterior–posterior phase-encoding direction. The reported signal-to-noise ratio was approximately 39 at b=0. No additional preprocessing beyond the steps described in Section "Kurtosis metrics and correlations" was applied by us prior to the analyses reported here.

To extract approximate signals for the white matter fibers in the perpendicular unit directions (, , ), the following steps are performed: The data is fitted using the DKI model. The three sets of voxels are selected based on a threshold fractional anisotropy (FA \ge 0.6) and angle difference () between respective unit directions and the principal eigenvector.

The averaged signals from each set of selected voxels are then composed using Eqn. 12.

Figure 9a shows the scalar measures as a function of the VFI for the selected in vivo human brain voxel configuration. gKFA attains higher values at low VFI, corresponding to maximally heterogeneous configurations where diffusion directions contribute in comparable proportions, and exhibits a non-monotonic trend with a slight initial decrease followed by an increase as VFI approaches unity. Conversely, FA increases monotonically with VFI, reflecting increasing directional coherence, while KFA shows a smoother and less pronounced dependence on VFI.

Fig. 9.

Scalar measures versus Volume Fraction Index (VFI) using signals extracted along approximately orthogonal directions from in vivo human and fixed rat brain data.

Fixed rat brain

DWI data²⁹ is acquired at b = 0 and along 33 directions at b-values from 0.2–3 in steps of 0.2 . The same procedure (as described in the previous section for the in vivo human brain) is followed to obtain the signals in the unit directions. In the data, a higher set of values is required to extract such signals.

Fig. 9b shows FA, KFA, and gKFA as functions of the Volume Fraction Index (VFI) for fixed data from the rat brain. FA increases monotonically with VFI, reflecting an increasing degree of directional coherence as the configuration becomes more homogeneous. In comparison, gKFA exhibits higher values at low VFI and a decreasing trend with increasing VFI, indicating greater sensitivity to orientation mixing. KFA varies more mildly between VFI, showing a weaker dependence on changes in directional heterogeneity.

Kurtosis metrics and correlations

Figure 11 shows FA and kurtosis-derived metric maps from in vivo human brain data. The FA map exhibits dark, curved bands (shown by a red arrow) in regions of heterogeneous white matter, which arise because FA, under the Gaussian diffusion assumption, tends to treat crossing or complex fiber configurations as more isotropic. As a result, anisotropy can be underrepresented in these regions. Both gKFA and KFA show elevated values in such areas, consistent with increased directional variability of non-Gaussian diffusion associated with fiber crossings. The gKFA map follows the overall spatial pattern of KFA but appears with higher contrast, suggesting complementary sensitivity to directional heterogeneity in such regions.

Fig. 11.

Kurtosis-derived metric maps from in vivo human brain data.

For the in vivo noise robustness experiment, we used the previously described human cfin_multib dataset. Prior to the addition of synthetic Rician noise, we applied brain masking using the DIPY median_otsu procedure (vol_idx=[0,1], median_radius=4, numpass=2, autocrop=True, dilate=1). We then restricted the analysis to the b=0, 1000, 2000, and shells and used a subsampled voxel mask for computational efficiency during fitting.

The baseline DKI fit was computed on this masked and shell-selected data set, and subsequently Rician noise was added to evaluate the SNR dependence of the correlations between the baseline and noisy estimates of KFA and gKFA. We applied no motion correction, eddy-current correction, Gibbs-ringing correction, denoising, smoothing, or additional spatial filtering before adding synthetic Rician noise. For illustration, gKFA and KFA maps at different SNR levels are shown in Fig. 12. Negative kurtosis values observed in conventional kurtosis-derived scalar maps arise primarily from noise sensitivity and the unconstrained estimation of the fourth-order kurtosis tensor.

Fig. 12.

gKFA (top row) and KFA (bottom row) maps of the human brain at SNR = 20, 40, 60, and 80 (columns).

The positivity of the fourth-order tensor W can in principle be assessed directly from its 15 independent coefficients through the quartic form , but this requires checking the positivity in all directions and is therefore less straightforward than eigenvalue-based validity checks for the mapped quadratic forms.

To analyze tissue-specific correlations, three tissue classes were defined according to FA values: gray matter (), white matter (), and intermediate tissues comprising the remaining voxels.

Figure 13 shows the Pearson correlation between the scalar maps computed from a reference in vivo human brain dataset and those obtained after adding synthetic Rician noise at increasing SNR levels. The reference data set corresponds to the original acquisition, while noisy datasets were generated by adding Rician noise with specified SNR values, where the noise scale was determined using the mean signal intensity of the volumes b=0 within the analysis mask. These results indicate that in this experimental setting, gKFA and KFA show tissue-dependent behavior, with gKFA performing more favorably in white matter.

Fig. 13.

Pearson correlation between scalar maps computed from a reference in vivo human brain dataset and those obtained after adding Rician noise at increasing SNR levels. From left to right, plots correspond to white matter, gray matter, and other tissue classes defined based on FA. The reference corresponds to the original acquisition (without additional noise), while noisy datasets are generated synthetically.

Positive-definiteness under noise: To quantify the effect of noise on the tensor positive definiteness, we performed an SNR-dependent simulation using DKI-consistent synthetic signals with repeated Rician noise realizations. For each fitted tensor, positive definiteness of the mapped quadratic forms , , and was assessed using their minimum eigenvalues, and positivity of the fourth-order tensor W was assessed numerically by evaluating the quartic form over a dense sampling of directions on the unit sphere.

Figure 15a shows that the frequency of non-positive-definite cases systematically decreased with increasing SNR. In particular, the proportion of numerically non-positive-definite tensors W decreased markedly from low to high SNR, while the proportion of cases in which at least one of , , or was non-positive-definite also decreased, although it remained consistently higher than the corresponding rate for W. The conventional KFA remained numerically defined throughout the tested SNR range.

Fig. 15.

Positive-definiteness analysis under noise. Positive definiteness of the mapped quadratic forms was assessed using their minimum eigenvalues, whereas positivity of the fourth-order tensor W was assessed numerically by evaluating the quartic form over a dense sampling of directions on the unit sphere. The proportion of numerically non-positive-definite W tensors and the proportion of cases in which one or more mapped quadratic forms were non-positive-definite both decreased with increasing SNR.

Individually mapped quadratic forms also showed distinct sensitivity to noise (Fig. 15b). Among them, exhibited the highest non-positive-definite rate at moderate-to-high SNR, whereas the invalidity rates of and decreased more rapidly as SNR increased. These results indicate that non-positive-definite mapped quadratic forms are most common under low-SNR conditions, and support the use of explicit validity checks when computing gKFA.

Qualitative comparison with standard DKI metrics in human brain

Figure 14 compares AK, RK, MK, gKFA, , , , and KFA in the human dataset. Standard scalar kurtosis measures AK, RK, and MK display contrast patterns that differ from those of gKFA and KFA. By contrast, gKFA appears visually closer to KFA and to the mapped-form anisotropy measures, consistent with its construction as an anisotropy descriptor derived from the mapped quadratic representations of the kurtosis tensor. The maps of , , and are broadly similar, but not identical, suggesting that the three mapped forms retain partly distinct directional information.

Fig. 14.

Representative axial maps from the human dataset showing axial kurtosis (AK), radial kurtosis (RK), mean kurtosis (MK), geometric kurtosis fractional anisotropy (gKFA), the mapped-form anisotropy measures , , and , and conventional kurtosis fractional anisotropy (KFA). These maps are shown for qualitative comparison of standard scalar kurtosis metrics with the proposed mapped-quadratic anisotropy representation.

Quantitative WM–GM discrimination

To quantify tissue contrast, we compared the discrimination performance of WM-GM for standard DKI metrics AK, RK, and MK, as well as KFA, gKFA, and the mapped-form anisotropy measures , , and . Table 1 summarizes the contrast-to-noise ratio (CNR), Cohen’s d and ROC area under the curve (AUC) for this comparison. In this data set, gKFA showed the highest CNR and AUC, with KFA ranking second, and the mapped-form anisotropy measures were closely followed. By contrast, AK, RK, and MK showed substantially weaker WM–GM discrimination for this specific task, and AK performed particularly poorly. These results support the interpretation of gKFA as a kurtosis-anisotropy descriptor rather than a scalar kurtosis-magnitude measure.

Table 1.

Quantitative WM–GM discrimination performance in the human dataset. Metrics are ranked by ROC area under the curve (AUC). CNR denotes contrast-to-noise ratio and Cohen’s d denotes standardized effect size.

Metric	CNR	Cohen’s d	AUC
gKFA	1.134	1.343	0.847
KFA	0.965	1.302	0.844
FA()	0.983	1.191	0.813
FA()	0.962	1.170	0.809
FA()	0.942	1.142	0.802
MK	0.251	0.500	0.672
RK	0.427	0.952	0.642
AK	0.131	−0.212	0.514

To further assess the relationship between gKFA and other quantities derived from kurtosis, we computed voxelwise correlation matrices for the metrics evaluated. Figure 16 shows the correlation structure within the white matter. Although gKFA and KFA exhibited similar strong WM–GM discrimination at the whole-brain level, their voxelwise correlation within white matter was only modest (r=0.30), indicating that the two measures are not equivalent in anisotropic tissue. As expected from its construction, gKFA was strongly correlated with mapped-form anisotropy measures , , and . More importantly, these mapped-form anisotropy measures themselves were only moderately correlated with one another (r=0.30–0.32), suggesting that the three mapped quadratic forms retain partly distinct directional information within white matter. Additional voxelwise analyzes in whole-brain and gray-matter regions further supported the main findings of white-matter (Figs. 17 and 18). At the whole-brain level, gKFA and KFA showed a stronger overall correlation, reflecting their similar large-scale tissue contrast. Within gray matter, gKFA remained more closely associated with KFA than with the AK, RK, and MK scalar kurtosis measures. Pairwise scatter-plot analyzes (Figs. 19 and 20) further showed that gKFA is more closely related to KFA than to MK or RK, while the anisotropy measures of the mapped-form , , and were similar but not identical, indicating that the three mapped quadratic forms retain partly distinct directional information. WM–GM descriptive summary statistics (Table 2) were consistent with these observations and showed stronger tissue contrast for the gKFA, KFA and mapped-form anisotropy measures than for AK and MK in the present dataset.

Fig. 16.

Voxelwise correlation matrix within white matter for AK, RK, MK, KFA, gKFA, and the mapped-form anisotropy measures , , and . The modest correlation between gKFA and KFA within white matter indicates that the two measures are not equivalent in anisotropic tissue. As expected from its definition, gKFA is strongly correlated with the mapped-form anisotropy measures. The mapped-form anisotropy measures themselves are only moderately correlated with one another, suggesting that the three mapped quadratic forms retain partly distinct directional information in white matter.

Fig. 17.

Voxelwise correlation matrix within gray matter for AK, RK, MK, KFA, gKFA, and the mapped-form anisotropy measures , , and . Within gray matter, gKFA remains more closely associated with KFA than with the scalar kurtosis measures AK, RK, and MK.

Fig. 18.

Voxelwise correlation matrix across all masked voxels for AK, RK, MK, KFA, gKFA, and the mapped-form anisotropy measures , , and . Across all masked voxels, gKFA and KFA are strongly correlated, reflecting their similar large-scale tissue contrast.

Fig. 19.

Pairwise voxelwise scatter plots illustrating the relationships of gKFA with KFA, MK, and RK in the human dataset. The plots show that gKFA is more closely related to KFA than to the scalar kurtosis measures MK and RK.

Fig. 20.

Pairwise voxelwise scatter plots among the mapped-form anisotropy measures , , and . The mapped-form anisotropy measures are similar but not identical, indicating that the three mapped quadratic forms retain partly distinct directional information.

Table 2.

WM and GM descriptive statistics for the evaluated metrics in the human cfin_multib dataset.

Metric	WM mean	WM median	GM mean	GM median
AK	0.528	0.579	0.569	0.576
RK	0.965	1.093	0.522	0.531
MK	0.668	0.822	0.535	0.547
KFA	0.715	0.735	0.438	0.407
gKFA	0.864	0.886	0.562	0.538
FA()	0.866	0.913	0.578	0.552
FA()	0.874	0.926	0.573	0.544
FA()	0.874	0.931	0.577	0.548

Sensitivity to regional tissue differences

As an illustrative example of its behavior, we evaluated the sensitivity of gKFA to regional differences in brain tissue using a data set of control rodents previously reported in¹⁶. Such analyses may help assess whether gKFA captures spatial variations in tissue microstructure that could be relevant for studies of neurodegenerative changes.

For rodent regional analysis, the regions of interest (ROIs) were defined manually in native diffusion space following the procedure used in our previous study³⁰. ROIs were selected in anatomically relevant regions involved in Parkinsonian pathology, namely the thalamus, substantia nigra, hippocampus, striatum and sensorimotor cortex. ROIs were delineated on the images b=0 with the aid of FA maps using ImageJ software, according to the Paxinos and Franklin mouse brain atlas³¹. No separate atlas registration step was applied; instead, ROIs were drawn directly on the diffusion data with reference to the anatomical atlas and the corresponding diffusion contrast. An atlas-guided illustration of the rodent ROI delineation procedure is provided in Supplementary Figure S1.

The sensitivity to metrics was examined in five anatomically and functionally relevant regions: thalamus (th), substantia nigra (sn), hippocampus (hc), striatum (str), and dorsal motor nucleus of the vagus (DMV). For each region, the mean scalar values were calculated and compared using one-way analysis of variance (ANOVA).

ANOVA revealed a marginal overall effect for conventional KFA (F = 2.21, p = 0.085), while gKFA demonstrated a statistically significant regional effect (F = 3.44, p = 0.016), indicating a greater sensitivity to inter-regional differences in the present dataset. The Post-hoc Tukey HSD analysis identified significant pairwise differences for gKFA between DMV and substantia nigra, as well as between substantia nigra and thalamus (p < 0.05). KFA did not produce significant pairwise differences. These results suggest that gKFA provides an improved discrimination of regional differences compared to conventional KFA in this data set. The corresponding boxplots and statistical comparisons are shown in Fig. 21 and Table 3, respectively.

Fig. 21.

Distribution of metric values across five mouse brain regions: KFA (left) and gKFA (right).

Table 3.

Multiple comparisons of regional mean gKFA values using Tukey HSD.

Group 1	Group 2	p-adj	Lower	Upper	Reject
dmv	hc	0.5299	−0.1883	0.0552	False
dmv	sn	0.0373	−0.2486	−0.0052	True
dmv	str	0.9918	−0.1405	0.1029	False
dmv	th	0.9998	−0.1148	0.1286	False
hc	sn	0.6210	−0.1821	0.0613	False
hc	str	0.7950	−0.0740	0.1694	False
hc	th	0.4318	−0.0483	0.1951	False
sn	str	0.1027	−0.0136	0.2298	False
sn	th	0.0249	0.0121	0.2555	True
str	th	0.9738	−0.0960	0.1474	False

Consistent with previous diffusion kurtosis imaging studies in transgenic mouse models, the thalamus emerged as a particularly sensitive region. Following three months of chronic exposure to rotenone, gKFA showed a significant alteration in the thalamus relative to controls (p < 0.039), while KFA did not reach statistical significance (p = 0.067). Given the central role of the thalamus in the motor circuitry of the basal ganglia, this finding is anatomically and functionally plausible and suggests that gKFA is sensitive to regional differences in the structure of directional kurtosis in a region implicated in neurodegeneration.

In general, this regional analysis suggests that gKFA may provide complementary sensitivity to regional differences compared to conventional KFA in biologically relevant brain structures.

Discussion

Conventional kurtosis fractional anisotropy (KFA) summarizes the anisotropy of the fourth-order kurtosis tensor into a scalar, analogous to FA in DTI. As such, it provides a compact measure of kurtosis anisotropy but necessarily collapses the full three-dimensional directional structure of the kurtosis tensor. This dimensional reduction may limit sensitivity to directional heterogeneity in tissues with a complex diffusion organization. By comparison, gKFA is derived from a geometric decomposition of the kurtosis tensor into multiple symmetric second-order quadratic forms. Each mapped quadratic form encodes directional combinations of kurtosis-tensor components in a lower-order representation whose geometry is easier to analyze than that of the full fourth-order tensor. The motivation of the present work is therefore to study non-Gaussian diffusion behavior in 3D through these reduced-order geometric representations. By retaining coordinate dependence, gKFA captures aspects of directional kurtosis structure that are averaged out in rotationally invariant summaries. In this sense, gKFA should be interpreted as a geometry-inspired descriptor of anisotropy across the mapped quadratic forms rather than as a replacement for conventional rotationally invariant kurtosis measures. The proposed framework may also be useful for constrained DKI simulation, since mapped quadratic forms provide lower-order geometric objects with explicit matrix-based validity checks. This could support synthetic-data generation for training and evaluating deep-learning methods for kurtosis-parameter estimation. However, this should be understood as a future direction.

A key limitation of the present study is that no direct biophysical validation of , , , or gKFA is provided. Although metrics are derived systematically from the kurtosis tensor and appear sensitive to tissue complexity and directional heterogeneity, the present results do not establish a one-to-one correspondence with specific microstructural features such as axonal diameter, myelin content, fiber crossing angle, or water exchange. In addition, confounding factors such as T2 relaxation, partial-volume effects, and model mismatch can influence the kurtosis tensor and therefore the mapped quantities derived from it. Consequently, gKFA should currently be interpreted as a descriptor of the structure of directional kurtosis in 3D. A dedicated Monte Carlo study varying controlled microstructural parameters would be required to establish such a specificity.

The regional behavior of gKFA, for example in the thalamus, is interpreted more appropriately as reflecting differences in the structure of directional kurtosis than a uniquely identified underlying tissue mechanism. Thus, gKFA may provide complementary information in tissues where conventional measures such as FA or KFA offer a less differentiated description of non-Gaussian diffusion behavior. More generally, many pathological processes, including neurodegeneration, demyelination, inflammation, tumor infiltration, and synaptic loss, can alter tissue complexity in ways that are not captured solely by directional coherence. Geometry-based descriptors derived from the kurtosis tensor may therefore be useful to probe such changes, although their biological specificity remains to be established^32,33.

Several methodological aspects also need to be considered. We expanded the set of benchmarks to include the standard DKI metrics mean kurtosis (MK), axial kurtosis (AK), and radial kurtosis (RK). These measures quantify important but different aspects of the kurtosis signal, which reinforces the interpretation of gKFA as a kurtosis-anisotropy measure rather than a scalar kurtosis-magnitude measure. KFA remains the most direct comparator to gKFA, since both are intended to summarize kurtosis anisotropy. By contrast, MK, AK and RK are not direct analogues of the proposed metric, but rather complementary measures of kurtosis magnitude and directional averaging. At the same time, the current set of benchmarks remains incomplete. In particular, the study has not yet included a comparison with noise-corrected KFA or with generalized fractional anisotropy (GFA). Accordingly, gKFA is best viewed as a geometrically motivated kurtosis-anisotropy descriptor that is closely related, but not identical, to KFA, while its relationship to other established measures remains an important topic for future work.

In the current work, we also rely on region-of-interest definitions based on FA thresholds and angular criteria, which may introduce selection bias. In addition, the physical phantoms with idealized orthogonal configurations cannot fully capture biological complexity such as fiber dispersion, partial-volume contamination, or inter-compartmental exchange. Future work should therefore evaluate gKFA under more realistic experimental conditions, examine how acquisition design, including b-value sampling and gradient-direction schemes, may affect its precision and sensitivity, and test the metric in larger datasets and disease cohorts. Broader benchmarking against noise-corrected KFA, GFA, and microstructure-informed simulations will be necessary to establish the robustness and biological interpretability of gKFA. In the present study, a direct comparison with the noise-correction scheme of Glenn et al. was not possible because the required repeated independent acquisitions were not available^18,22.

Conclusion

We presented a geometric framework for diffusion kurtosis that maps the fourth-order kurtosis tensor to a set of symmetric quadratic forms, enabling a geometric interpretation of the kurtosis anisotropy. Based on this formulation, we defined geometric kurtosis fractional anisotropy (gKFA) as a scalar descriptor of orientation-dependent non-Gaussian diffusion.

The results of simulations, physical phantoms, and in vivo human and rodent data indicate that gKFA is sensitive to directional heterogeneity of non-Gaussian diffusion in both white matter and deep gray matter regions such as the thalamus. These findings suggest that gKFA captures aspects of kurtosis anisotropy that are not fully reflected by conventional scalar summaries of the kurtosis tensor. The gKFA is intended as an additional descriptor that complements existing diffusion and kurtosis measures by providing a geometric view of non-Gaussian diffusion anisotropy.

Author contributions

S.K. conceptualized the study, developed the methodology, implemented the software, performed validation and formal analysis, conducted the investigation, curated the data, prepared the original draft, reviewing and editing the manuscript. A.B. contributed to the review and editing and created the visualizations. J.K. contributed to reviewing and editing the manuscript, project administration, funding acquisition M.H. contributed to reviewing and editing. A.K. contributed to reviewing and editing and data resources

Funding

This article has been produced with the financial support of the European Union under the LERCO project number CZ.10.03.01/00/22_003/0000003 via the Operational Programme Just Transition. The work and contributions were supported by the project SP2026/032 ‘Biomedical Engineering systems XXI’; and the Research Council of Norway under FRIPRO Researcher Project 302624.

Data availability

The datasets analyzed in Section "Sensitivity to regional tissue differences" originate from a previously published study in the Journal of Neurochemistry (doi:10.1111/jnc.15449). These data are available from the corresponding author (Amit Khairnar) of that study on reasonable request. Additional data used in this work were previously reported in Hansen and Jespersen, “Kurtosis fractional anisotropy, its contrast, and estimation by proxy” (Scientific Reports, 6 (1), 23999, 2016). No new raw datasets were generated for the current study.

Declarations

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-53456-w.