A geometric function theoretic approach to MRI distortion correction using Miller Ross Poisson series

Abstract

This study focuses on the newly introduced subclass of Sakaguchi-type function associated with the Miller–Ross-type Poisson distribution series. Using the proposed operator, we derive several coefficient bound, inverse function estimates, initial logarithmic coefficient results and Feketo-Szego type inequalities. Furthermore, growth, distortion, convex combination properties, subordination outcomes, and partial sum results are derived for the defined class. Application part demonstrates how the obtained coefficient bounds can be employed to design an analytic correction operator for improving the quality of distorted Magnetic Resonance Imaging (MRI) scans. Numerical experiments on real datasets validate the effectiveness of the proposed model, showing significant improvement in terms of MSE, PSNR, and SSIM. These results highlight the interplay between geometric function theory and practical imaging problems, thereby providing a new perspective on the use of analytic function subclasses in applied sciences.

Introduction and preliminaries

Let be the class of analytic functions inside open unit disc with Montel’s normalization and is of the form

1.1

Let class is the class of all univalent function in Class it is denoted by . If for every point in class domain of the complex plane is said to be starlike with respect to a point . If the line segment joining , to any point lies in itself. It is denoted by . A function is said to be starlike if,

1.2

Miller and Ross² was introduced the function ,as a solution of fractional order initial value problem which is called the Miller-Ross function.

where is the incomplete gamma function. Using the properties of the gamma functions, the Miller–Ross function can easily be written as:

1.3

which can also be stated as:

1.4

where is the Mittag-Leffler function of two parameters. Some special values of the Miller–Ross functions can be given as follows:

1.5

1.6

1.7

The Probability mass function of the Miller–Ross-type Poisson distribution is given by:

1.8

where and is the Miller–Ross function given in 1.4^11,16. The normalized form of the Miller–Ross-type Poisson distribution is given by:

1.9

We consider the linear operator

1.10

as below:

1.11

1.12

where

1.13

and the symbol specifies the Hadamard product (convolution) of two series.

Definition We introduced a new subclass of analytic functions using operator as follows:

1.14

where

Note that maps conformally onto a symmetrical disc with respect to the real axis, which is centered at , and with a radius .

Note

(1) For in the class , we get the class , and their geometric properties were studied by Murugusundaramoorthy and et.al.

(2) For in the class , we get the class was studied by Sheza M. El-Deeb and et.al.

Supporting lemma

Lemma 1

^6,7 Let

2.1

be in the class of functions of the positive real part in , then

2.2

and for any complex number

2.3

In particular, if is a real parameter, then

2.4

Coefficient inequalities for

Theorem 1

Let be assigned to the class . Then,

3.1

3.2

Further, for a complex number

3.3

where

3.4

3.5

3.6

3.7

Proof

We consider the subordination

3.8

It can be written as :

3.9

Let

3.10

3.11

3.12

Comparing the coefficient of 3.10 and 3.11, we get

3.13

3.14

3.15

3.16

Substitute the value of and , we get

3.17

where

3.18

Applying 1.1to the above 3.6, we get the result.

Theorem 2

Let be the class . Then for a real parameter

3.19

where .

Coefficient inequalities for

With the help of the⁸ Koebe one-quarter theorem every univalent function has an inverse that satisfies the conditions and that the image of contains a disk of radius

Theorem 3

If and is the inverse function of with , the Koebe domain of the class , then

4.1

4.2

and for any complex number , we have

4.3

where .

Proof

As

4.4

is the inverse function of

4.5

From 1.1 and 4.5

4.6

From Equations 4.5 and 4.6, we can obtain

4.7

By equating the corresponding coefficients of 4.7, we have

4.8

4.9

Substitute the value of in equation4.8,

4.10

4.11

To find , from 4.9, we take in 4.8. then equation becomes,

4.12

thus

4.13

where

4.14

For any complex number ,

4.15

4.16

4.17

Hence the proof.

Initial logarithmic coefficient bounds and Fekete-Szegö problem for

^9,10 If the function is analytic in , such that for all , then the well-known logarithmic coefficients , , of are given by

5.1

Theorem 4

Let with the logarithmic coefficients given by 5.1. Then,

5.2

5.3

and for we have

5.4

Proof

If has the form 1.14, equating the first two coefficients of the relation (5.1), we obtain

5.5

5.6

Replacing and in the above equalities 3.13 and 3.14, we obtain

5.7

5.8

5.9

Using the equation2.2,

5.10

and using the equation2.3,

5.11

We have

5.12

5.13

and in view of 2.3, we obtain the desired result.

Characterization properties

In this section we investigate the convex combination growth and distortion bounds and necessary and sufficient condition for the newly defined class

Theorem 5

Let be assigned to the class if it fulfills the inequality

6.1

equivalently, we may write

6.2

where

6.3

and as given in 1.13. Inequality 6.1is sharp.

Proof. Assume that inequality 6.1 holds, and . Then, one can put 1.14 in the form of a Schwarz function as

6.4

It must satisfy the condition

6.5

Then, we have

6.6

6.7

Hence, .

Now, for the function

6.8

such that , we have

6.9

6.10

6.11

Thus, , and the approximation 6.1is sharp.

A function defined by 1.1 and belonging to the class is said to be in the class if it also satisfies the coefficient inequality 6.1. Using the technique of proof of distortion theorems given by Silverman¹², we state the following results without proof.

Theorem 6

If a function , then

6.12

The result is sharp for the function defined as:

6.13

Theorem 7

If a function , then

6.14

The result is sharp for the extreme function defined in 6.5.

Theorem 8

Let and have the form

6.15

Then, , where

6.16

Proof

Consider . Then, we can write

6.17

Further,

6.18

6.19

Therefore,

6.20

6.21

6.22

Thus, .

Theorem 9

Let for . Then, the arithmetic mean G of is given by

6.23

and also belongs to class .

Proof

Let

6.24

6.25

6.26

To show , it is sufficient to show that

6.27

Consider

6.28

6.29

6.30

which gets .

Subordination results

Definition (Subordinating factor sequence)¹³.A sequence of complex numbers is said to be a subordinating sequence if, whenever , is regular, univalent, and convex in U, we have

7.1

Lemma 2

The sequence is a subordinating factor sequence if and only if

7.2

Theorem 10

Let and be any function in the usual class of convex functions C, then

7.3

and

7.4

where

7.5

7.6

The constant factor in equation7.3 cannot be replaced by a larger number.

Proof

Let and suppose that . Then,

7.7

Thus, the subordination result holds if

7.8

is a subordinating factor sequence, with .

7.9

By noting the fact that is an increasing function for , and in particular

7.10

then for , we have

7.11

7.12

7.13

7.14

7.15

where we have also made use of assertion 6.1 of Theorem 5. This evidently proves inequality 7.3 and hence also the subordination result 6.1 asserted by 7.3.

Inequality 7.4 follows from 7.3 by taking

7.16

Next, we consider the function

7.17

Clearly, . For this function, (52) becomes

7.18

It is easily verified that

7.19

The constant cannot be replaced by any larger one.

Partial sums

In 1997, Silverman¹⁴ investigated partial sums results for the classes of starlike and convex functions given by 1.1 and established through

8.1

8.2

Many authors have investigated partial sums for different subclasses; Recently Rosy.T and et.al were studied partial sum of starlike function based on Ruscheweyh domain¹⁵

Theorem 11

If of the form 1.1 satisfies condition 6.5, then

8.3

and

8.4

where

8.5

Proof

Now we set,

8.6

8.7

8.8

After doing some calculation we get,

8.9

8.10

By using the triangle inequalities with , we get the following inequality:

8.11

8.12

if and only if

8.13

8.14

To prove the inequality in (8.3), it sufficent to show that the left-hand side of (8.15) is bounded above by the following sum:

8.15

which is equivalent to

8.16

To prove inequality (8.4), we take

8.17

8.18

8.19

where

8.20

8.21

8.22

which completes the proof of Theorem 11.

Theorem 12

If of the form 1.1 satisfies condition 6.1, then

8.23

and

8.24

where is given by equation8.5 so we have completed the proof.

Proof

The proof of Theorem 12 is similar to that of Theorem 11.

Application

Introduction

Magnetic Resonance Imaging (MRI) is the one of the most wiedely used medical imaging modalities due to its ability to provide high resolution, non invasive of soft tissue. It plays a crucial role in the diagnosis and monitoring of neurological, cardiovascular and musculoskeletal disorders. Despite these advantages, MRI images frequently suffer from intensity inhomogeneity and nonlinear signal distortions caused by nonuniform magnetic fields, coil sensitivity variations and hardware imperfections. These effects appear as gradual intensity drifts or nonlinear intensity wrapping across the image, leading to reduced contrast and loss of diagnostically important structural information.

Various proposals have been made to correct for the intensity inhomogeneity of the images obtained by MRI. They range from polynomial fitting models, bias field correction approaches, homomorphic filtering techniques, and surface coil normalisation. While the approaches have the ability of reducing the intensity inconsistencies of the images to some degree, they tend to remain rather experience-dependent. As a result, over-correction of the images as well as the instability of the images often become evident.

From the mathematical viewpoint, the intensity distortion problem of the MRI can be formulated as a nonlinear transformation of the analytical intensity distribution. It seems that the application of transformation operators with defined analytical structures and specified geometric characteristics could be useful here. Specifically, the class of analytical functions with bounded coefficients and specified geometry may serve as a basis for designing the intensity correction transformation operators.

Motivated by this viewpoint, we apply the theoretical results established in the above section, where coefficient bounds were derived for a subclass of Sakaguchi-type analytic functions associated with the Miller–Ross-type Poisson distribution series. The boundedness and subordination properties of this subclass ensure controlled growth, stability under transformation, and preservation of essential geometric features. These characteristics are especially desirable in MRI image enhancement, where uncontrolled amplification may distort anatomical boundaries.

Using Theorem 1, we construct a mathematically constrained correction operator that acts on MRI intensity values as an analytic transformation. The resulting framework provides a direct bridge between geometric function theory and practical MRI image enhancement. Unlike conventional heuristic approaches, the proposed method is supported by explicit coefficient bounds and analytic stability guarantees, offering a principled mechanism for correcting intensity inhomogeneity while preserving structural details.

^18–20Recent years have witnessed significant research efforts aimed at improving magnetic resonance imaging (MRI) quality by correcting intensity inhomogeneity and nonlinear signal distortions arising from magnetic field nonuniformity, coil sensitivity variations, and hardware imperfections. Early systematic evaluations of bias field correction techniques demonstrated that widely used methods integrated into software packages such as SPM, FSL, and FreeSurfer exhibit varying levels of performance and stability, particularly under high noise and strong inhomogeneity conditions. These studies highlighted the limitations of conventional heuristic correction models and emphasized the need for more robust and theoretically justified approaches^21–26.Subsequent developments introduced multi-scale and learning-based correction strategies to address these challenges.

Deep learning frameworks such as InhomoNet employed multi-scale feature extraction and specialized loss functions to achieve improved bias field correction compared with traditional polynomial or filtering-based techniques. More recent approaches have focused on implicit and unsupervised learning paradigms, enabling bias field estimation and correction without the requirement of ground-truth bias information, thereby improving generalizability across different MRI modalities and acquisition protocols. Clinical investigations have also demonstrated that effective post-processing bias field correction can significantly enhance quantitative MRI analysis, including segmentation accuracy and biomarker reliability in applications such as breast and brain MRI.

Parallel to these advances in medical image processing, mathematically rigorous frameworks have increasingly been explored for image enhancement and signal correction. In particular, recent studies have demonstrated that coefficient-based constructions from geometric function theory, such as those involving^26–28Hankel determinants and subclasses of starlike and convex functions, can be effectively employed to design image enhancement algorithms with improved stability and structural preservation. These analytic approaches offer inherent mathematical constraints that regulate intensity transformation and contrast amplification, thereby reducing overcorrection and artifact introduction.

Motivated by these developments, the present work integrates geometric function theory with MRI image enhancement by exploiting coefficient bounds derived for a subclass of Sakaguchi-type functions associated with the Miller–Ross-type Poisson distribution series. Unlike purely data-driven or heuristic correction methods, the proposed approach provides a mathematically constrained intensity transformation framework, ensuring analytic stability while enhancing MRI image quality. This establishes a meaningful bridge between abstract analytic function theory and practical medical imaging applications.

Experiment setup

To validate the applicability of the proposed framework, a controlled simulation study was performed. An MRI slice was chosen as the reference image and normalised to the intensity range [0,1]. To simulate realistic distortions, a polynomial-based intensity warping was applied, defined as:

9.1

where I is the normalized reference image intensity and are distortion parameters. In this study, the chosen values were , which produce noticeable nonlinear intensity shifts while preserving the global image structure. This emulates the type of distortions encountered in clinical MRI due to field inhomogeneity and hardware nonlinearity.

The correction was then applied using an analytic transformation derived from Theorem 1:

9.2

where and are correction coefficients constrained by the coefficient bounds of the class :

9.3

Here, and are functions of the Miller–Ross parameters, while R, S, t are the Janowski domain parameters. This ensures that the correction transformation remains analytic and univalent in the unit disc, preventing overcorrection and ensuring stability.

The corrected image was then compared with both the distorted and the original reference images for qualitative and quantitative evaluation.

Algorithm part

1. Take the input image.

2. If the image is RGB, convert it to grayscale. Normalise its intensity range to [0,1].

3. Stimulate distortion by adding nonlinear intensity components by using the equation 9.1.

4. Calculate the coefficient bounds of and .

5. Apply the fractional polynomial correction to the image by using the equation 9.2.

6. Evaluate the quality metrics such as MSE, PSNR, SSIM, NIQE and BRIQUE.

7. Display the values.

The proposed MRI image enchancement framework follows a structured and analytically guided image processing pipeline, as illustrated in Figure1. The pipeline begins with the acquisition of an input MRI image, If the input image is in RGB format, it is first converted into a grayscale image to ensure intensity based processing. The grayscale image is then normalized to the range[0,1], which improves numerical stability and enables consistent application of nonlinear transformation.

Fig. 1.

flowchart of the proposed method.

To modelise realistic MRI intensity inhomogeneity and nonlinear signal distortion, an artifical distortion model is applied to the normalized image using the nonlinear intensity transformation given in equation9.1. Subsequently, the analytically derived coefficient bounds and obtained from the theorem 1 for the considered subclass of Sakaguchi-type function associated the Miller Ross type Poisson distribution series are computed. These coefficient ensure that the correction operator satisfies theoretical stability and boundedness conditions.

Using the computed coefficients, an analytic correction operator defined in equation9.2 is applied to the distorted image.

Dataset

In this study, we utilise imaging data from The Cancer Genome Atlas Lung Adenocarcinoma (TCGA-LUAD) collection, a publicly accessible dataset hosted by The Cancer Imaging Archive (TCIA). The TCGA-LUAD dataset forms a key component of The Cancer Genome Atlas (TCGA) initiative, which systematically catalogues and analyses genomic alterations responsible for cancer development, thereby facilitating advancements in diagnosis and therapy. Although the complete data set includes thousands of radiological scans in multiple modalities, for this study, we restricted our analysis to a curated subset of 100 MRI images. Figure2 illustrates sample images from the selected dataset.

Fig. 2.

Sample images of the data set.

The TCGA-LUAD collection primarily targets lung adenocarcinoma (LUAD), the most common subtype of non-small cell lung cancer (NSCLC). It provides a comprehensive set of radiological images alongside clinical outcomes and genomic annotations, enabling integrative research in radiogenomics and precision oncology. Although magnetic resonance imaging is not the dominant modality within this collection compared to CT, our selected subset of MRI images offers a suitable basis for investigating signal correction and intensity normalisation using the proposed analytic model. By combining structural imaging information with theoretical correction methods, this dataset supports experimental validation while maintaining clinical relevance.

Evaluation metrics

The performance of the proposed correction method was assessed using three widely recognised metrics in image processing and medical imaging: Mean Squared Error (MSE), Peak Signal-to-Noise Ratio (PSNR), and Structural Similarity Index (SSIM).

1) Mean Squared Error (MSE)

MSE quantifies the average squared difference between pixel intensities of the reference and processed images. It is defined as:

9.4

where I is the original reference image, is the processed image, and denotes the image dimensions. Lower values of MSE indicate higher accuracy in pixel-level restoration.

2) Peak Signal-to-Noise Ratio (PSNR)

PSNR provides a logarithmic measure of image fidelity relative to the maximum possible signal intensity L. It is defined as:

9.5

Higher PSNR values correspond to better image quality. For medical images, values above 25–30 dB are generally considered acceptable, while lower values suggest visible degradation.

3) Structural Similarity Index (SSIM)

SSIM evaluates perceptual image quality by comparing luminance, contrast, and structural similarity between two images. It is defined as:

9.6

where and represent local means, and denote standard deviations, and is the covariance of the two images. SSIM values range from 0 to 1, with values closer to 1 indicating stronger structural similarity and better perceptual quality.

4) Naturalness Image Quality Evaluator (NIQE)

NIQE is a no-reference image quality metric that evaluates perceptual naturalness based on statistical regularities of natural images. Unlike MSE, PSNR, and SSIM, NIQE does not require a reference image. Lower NIQE values indicate better perceptual quality and natural appearance, while higher values suggest the presence of distortions or unnatural patterns.

5) Blind/Referenceless Image Spatial Quality Evaluator (BRISQUE)

BRISQUE is another no-reference image quality assessment metric that measures distortions based on locally normalised luminance coefficients. Lower BRISQUE scores indicate higher perceptual quality and fewer visible artifacts, making it useful when reference images are not available.

Together, these five metrics provide a comprehensive assessment of the proposed correction framework: MSE reflects pixel-level accuracy, PSNR measures overall fidelity, SSIM evaluates structural similarity, while NIQE and BRISQUE assess perceptual naturalness without requiring reference images.

Discussion

The proposed analytic correction framework was rigorously evaluated using a combination of reference-based and no-reference image quality metrics, including Mean Squared Error (MSE), Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), Naturalness Image Quality Evaluator (NIQE), and Blind/Referenceless Image Spatial Quality Evaluator (BRISQUE). Together, these metrics provide a comprehensive assessment of the framework’s effectiveness in restoring distorted MRI images. Table1 presents a comparative quantitative performance evaluation of distorted and corrected MRI images using standard image quality metrics, namely MSE, PSNR, SSIM, NIQE, and BRISQUE. The comparison between the distorted and corrected images reveals clear improvements across all measures, confirming the robustness of the method.

Table 1.

Comparative metrics of Distorted image and Corrected Image.

Metrics	Distored image	Corrected Image
MSE	0.003509	0.001617
PSNR	24.8197	28.1758
SSIM	0.9693	0.9865
NIQE	6.9541	6.9767
BRIQUE	48.6485	48.6375

The MSE provides a direct measure of pixel-level differences between the reference and the processed images. In this study, the distorted image exhibited an MSE of 0.003509, which was reduced to 0.0061617 after applying the proposed correction operator. Such a significant decrease highlights the capability of the correction method to effectively compensate for nonlinear distortions introduced during MRI acquisition. From a clinical perspective, reducing MSE ensures that pixel intensities better represent true tissue properties, particularly in homogeneous regions such as soft tissues and organs where intensity inhomogeneity can mask pathologies. The lower error ensures that diagnostic details are preserved, making the corrected image more reliable for subsequent interpretation and analysis.

The PSNR is one of the most widely used measures for image quality assessment, reflecting the ratio between the maximum possible signal and the noise introduced by distortions. The distorted MRI image achieved a PSNR of 24.8187 dB, a value that suggests noticeable degradation in image fidelity. After correction, the PSNR increased significantly to 28.1758 dB, yielding an improvement of approximately 3.4 dB. In medical imaging, even a modest increase of 1–2 dB is regarded as meaningful, as it indicates improved visibility of structural features and boundaries. Therefore, the observed gain of 3.4 dB is highly significant. Higher PSNR directly translates to improved diagnostic capability, as it reduces noise-like distortions that may obscure critical lesions or tissue abnormalities. The improvement demonstrates that the analytic correction model successfully enhances global image fidelity while maintaining consistency across intensity ranges.

The SSIM metric evaluates image quality from a perceptual perspective, considering luminance, contrast, and structural similarity between the reference and processed images. The distorted image recorded an SSIM of 0.9693, which, while relatively high, still indicated subtle structural deviations due to nonlinear distortion. After correction, the SSIM improved to 0.9865, approaching near-perfect similarity with the reference image. This improvement is particularly significant in MRI imaging, where structural preservation is critical for accurate diagnosis. Unlike natural images, where perceptual similarity may tolerate minor distortions, medical images demand very high structural fidelity. The correction method’s ability to increase SSIM demonstrates that it not only reduces error but also preserves diagnostic structures such as lesion boundaries, tissue interfaces, and subtle anatomical features. This ensures that the corrected images remain clinically interpretable and suitable for downstream tasks such as segmentation and classification.

NIQE is a no-reference metric designed to measure the perceptual naturalness of an image based on statistical features. For the distorted image, the NIQE score was 6.9541, while the corrected image achieved a slightly improved score of 6.9467. Although the numerical difference may appear small, it nevertheless indicates that the corrected images exhibit marginally improved perceptual naturalness. This result is consistent with the observation that NIQE is less sensitive to small corrections compared to reference-based metrics such as MSE or SSIM. However, the improvement, though minor, is aligned with the general enhancement trend, confirming that the correction method not only restores pixel accuracy but also improves the perceptual realism of the MRI images. From a clinical perspective, a lower NIQE score implies reduced artificial distortions and more natural intensity distributions, enhancing radiologist confidence in diagnostic interpretation.

The effectiveness of the suggested analytic correction model is strongly supported by the comparative assessment across every measure. In MSE, PSNR, and SSIM, where error correction, fidelity enhancement, and structural preservation were obviously apparent, the most noticeable changes were seen. These measures precisely indicate how well the correcting operator can recover MRI intensities and preserve diagnostic structures. The NIQE and BRISQUE criteria, meanwhile, indicate even minor changes but offer more proof from a no-reference viewpoint that the adjustment helps to improve perceptual quality. Visually reinforced in figure3, this trend is seen as the corrected picture surpasses the distorted picture over five measurements in the comparative graph. The clear contrast of the two story lines draws attention to the dependability and reliability of the suggested correction model. Notably, the improvements were consistent throughout all five measures, therefore highlighting the robustness of the suggested technique in simulated MRI distortions.

Fig. 3.

Comparative graph between the distorted images and corrected images.

The comparative evaluation across all metrics strongly validates the effectiveness of the proposed analytic correction model. The most pronounced improvements were observed in MSE, PSNR, and SSIM, where error reduction, fidelity enhancement, and structural preservation were clearly evident. These metrics directly reflect the ability of the correction operator to restore MRI intensities and preserve diagnostic structures. Meanwhile, the NIQE and BRISQUE metrics, although showing smaller improvements, provide additional evidence that the correction contributes positively to perceptual quality from a no-reference perspective. Importantly, the improvements were consistent across all five metrics, demonstrating the robustness of the proposed method in simulated MRI distortions. The combination of significant gains in reference-based metrics and modest yet consistent enhancements in no-reference metrics indicates that the method can improve both numerical accuracy and perceptual quality in a controlled setting. While these results are promising, further studies and clinical validation are required before the method can be applied in real-world medical settings.

Conclusion

In this work, we investigated geometric properties of a subclass of Sakaguchi-type functions defined via the Miller–Ross-type Poisson distribution operator. Several analytic results were established, including sharp coefficient inequalities, inverse function estimates, and Fekete–Szegö bounds. The study also explored distortion, growth, convexity, subordination, and partial sums, thereby giving a comprehensive characterization of the newly defined class. Importantly, we bridged the gap between theoretical function theory and practical imaging by applying the results to MRI correction. The proposed analytic correction operator successfully reduced nonlinear distortions in MRI scans while preserving structural fidelity, as confirmed by quantitative metrics. This dual contribution–both theoretical advancement in geometric function theory and practical application to medical imaging–demonstrates the versatility of the introduced subclass. Future work may explore further applications in other imaging modalities and extend the framework to higher-dimensional settings.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

S.Manoj - Conceptualization, Investigation, Methodology, Visualization, Writing - original draft, Writing, review editing. B.Sruthakeerthi- Supervision, Validation.

Funding

Open access funding provided by Vellore Institute of Technology.

Data availability

The following links are provided for the MRI images and datasets used in this study. Source file of TCGA-LUAD MRI dataset: The Cancer Imaging Archive (TCIA), TCGA-LUAD collection TCGA-LUAD data set https://www.cancerimagingarchive.net/collection/tcga-luad/. The underlying code for this study is available at 10.17632/wwhbrwz78y.1.

Declarations

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-39523-2.