ARITHMETIC METHODS IN MATHEMATICAL PHYSICS AND BIOLOGY

A Quantitative Characterization of Blood Vessels by Complexity Measures in Diabetic Retinopathy and Cystopathy

M. Karaszewski¹, A. Andrzejak¹, P. Waliszewski^2,3

Background:
Diabetic retinopathy is a highly prevalent vascular complication of diabetes mellitus present in 1/3 out of 360 millions of patients worldwide. It is the most frequent cause of blindness among adults. The condition is caused by microvascular retinal changes, hyperglycemia-induced intramural pericyte death and thickening of the basement membrane. This leads to incompetence of the blood-retina barrier, the increased permeability of blood vessels, and retinal ischemia. Disease progresses from the nonproliferative to proliferative retinopathy due to growth of pathological vascular network and fibroblasts.

Similarly, diabetic cystopathy is a condition caused by a damage of nerves to the bladder. Disease is marked by insidious onset and mild symptoms resulting from decreased detrusor contractility, such as impairment of bladder sensation, diminished urinary flow, or increased post-void residual volume. Detrusor areflexia leads to problems with bladder emptying. There are no specific morphological changes in urinary bladder mucosa that would define the beginning of the bladder decompensation. In addition, recording of the signals from the urinary bladder nerves is technically difficult and invasive. The characterization of the vascular network complexity in urinary bladder mucosa and its comparison with the retinal network known for distinct morphological changes in the course of the disease is particularly important from the practical point of view.

The objectives of this pilot study were to characterize grey images of both normal and diabetic retina or normal and diabetic mucosa of urinary bladder by complexity measures, i.e. the capacity dimension D0, the information dimension D1, the capacity dimension D2, and entropy S. In addition, we investigated changes in multifractal structure of the images in the course of well-controlled or advanced diabetes mellitus.

Material:
The color images of retina or urinary bladder mucosa representing all stages of diabetes mellitus. The images were obtained during the fundoscopy or cystoscopy using lens 70o, were converted, and analyzed by the open free software Image J. All patients with diabetic retinopathy had severe vision impairment. All patients with diabetic cystopathy had small prostate glands < 30 ml, and no other pathological conditions in the low urinary tract. They were examined by urologists for the first time owing to acute urine retention.

Results:
Multifractal structure was identified in all cases. The mean value of the capacity fractal dimension D₀ for retina was 1.8793, and for urinary bladder mucosa 1.8593. The values of that dimension remained constant for all the diabetic cases. The statistically significant differences were observed in the mean values of the information fractal dimension D₁ as well as the capacity fractal dimension D₂ characterizing complexity of the cases with diabetic retinopathy or diabetic cystopathy vs. well-controlled diabetic or normal counterparts. The values for the information fractal dimension D₁ in retina were 1.8714 vs 1.8541 vs 1.8532, and for the capacity fractal dimension D₂ were 1.8514 vs. 1.8231 vs. 1.8109, respectively. The values for the information fractal dimension D₁ in urinary bladder mucosa were 1.8523 vs 1.8154 vs 1.7603, and for the capacity fractal dimension D₂ were 1.8371 vs. 1.7692 vs. 1.6196, respectively. In the cases of diabetic retinopathy as well as diabetic cystopathy, entropy was decreased. Results of the ROC analysis defined the cut-off values of both fractal dimensions D₁ and D₂. Those values allow a separation of the advanced cases of diabetic retinopathy or cystopathy from the early alterations.

Conclusions:
Multifractal structure is seen in normal or diabetic tissues of both organs. The changes of that structure in eye retina share similarities with the changes in urinary bladder as demonstrated by the evolution of the mean values of the complexity measures. Those values define a frame of reference for more extensive study.

Fractal Geometry Enables Differentiation between Patients with Pulmonary Hypertension and Healthy Controls

Martin Obert^1,3, Jan-Christoph Asbach^1,2, Hannah Schröer¹, Klara Franzki¹, Janine Wolf², Hossein Arde-schir Ghofrani³, Werner Seeger³, Gabriele A. Krombach^2,3, Henning Tiede³

Objective:
We try to find numerical tools that enable a differentiation between the lungs of healthy subjects and the lung diseases pulmonary hypertension, emphysema, fibrosis, and different tumor variants. The hy-pothesis of this study was that the variances of Gaussian fit parameters, applied to the Hounsfield Unit (HU) frequency distribution of a segmented Computed Tomography (CT) lung image, varies larger in a disease affected image slice compared to a healthy image slice of a subject. This hypothesis is based on the assumption that a disease does not necessarily affect each image slice similarly within a 3D image stack. Upon success, such numerical tools could support the radiologist to find the adequate diagnosis. Furthermore, such methods would be cost saving, strongly reproducible, easy to perform, and user inde-pendent.

Materials and Methods:
High-resolution lung data sets of 286 patients (24 healthy, 50 emphysema, 78 fibrosis, 77 pulmonary hypertension, 57 tumor suspects) were retrospectively investigated. The study was supported by the University’s ethics committee. Lungs were segmented according to standard methods. Histograms of the frequencies of the HU distributions, which are the grey-level values that correlate linearly with the tissue density in a CT image, were fitted using a four parameter Gaussian-algorithm. Parameter A₀ is the height, A₁ the center, A₂ the width (standard deviation), and A₃ is the constant term of the Gaussian. Hence, A₁ and A₂ may especially contain disease related information. The four fit parameters were estimated sepa-rately on each sectional image slice of a 3D thorax scan of a patient. Finally, the variances of each of the four parameters of a patient’s data set were estimated. Variance analysis and post hoc tests were per-formed to estimate significances of differences of mean values of the different patient groups being healthy or sick.

Results:
Levene homogeneity test of variances indicates different variances for parameters A₀, A₁, and A₂. There-fore, Tamhane post hoc tests were applied for patient group comparisons (conf. lev. = 0.05%). A₁ enables a differentiation between: healthy and emphysema; healthy and fibrosis; healthy and suspected tumor; emphysema and fibrosis. A₂ enables a differentiation between: healthy and fibrosis; healthy and sus-pected tumor; emphysema and fibrosis; fibrosis and suspected tumor. Other group combinations cannot be differentiated based on A₁ or A₂.

Conclusions:
The application of the Gaussian fit approach to the description of the HU-frequency distribution of dif-ferent lung diseases enables very helpful group differentiations. This is an important step towards an automatic classification scheme between different lung diseases. In further investigations we shall try to revise the fit equation so that the mathematical model matches better to the HU-frequency distributions in segmented CT lung images.

Preliminary Statistical Analysis of Hounsfield Unit Distributions of Segmented CT Images in Order to Differentiate Between Healthy Human Lungs and Various Diseases Thereof

Martin Obert¹, Sina Bergmann², Jan-Christoph Asbach², Hannah Schröer², Klara Franzki², Theresa Ohlwärther², Pia Crtalic², Regina Moritz², Gabriele A. Krombach²

Objective:
We try to find numerical tools that enable a differentiation between the lungs of healthy subjects and the lung diseases pulmonary hypertension, emphysema, fibrosis, and different tumor variants. The hy-pothesis of this study was that the variances of Gaussian fit parameters, applied to the Hounsfield Unit (HU) frequency distribution of a segmented Computed Tomography (CT) lung image, varies larger in a disease affected image slice compared to a healthy image slice of a subject. This hypothesis is based on the assumption that a disease does not necessarily affect each image slice similarly within a 3D image stack. Upon success, such numerical tools could support the radiologist to find the adequate diagnosis. Furthermore, such methods would be cost saving, strongly reproducible, easy to perform, and user inde-pendent.

Materials and Methods:
High-resolution lung data sets of 286 patients (24 healthy, 50 emphysema, 78 fibrosis, 77 pulmonary hypertension, 57 tumor suspects) were retrospectively investigated. The study was supported by the University’s ethics committee. Lungs were segmented according to standard methods. Histograms of the frequencies of the HU distributions, which are the grey-level values that correlate linearly with the tissue density in a CT image, were fitted using a four parameter Gaussian-algorithm. Parameter A₀ is the height, A₁ the center, A₂ the width (standard deviation), and A₃ is the constant term of the Gaussian. Hence, A₁ and A₂ may especially contain disease related information. The four fit parameters were estimated sepa-rately on each sectional image slice of a 3D thorax scan of a patient. Finally, the variances of each of the four parameters of a patient’s data set were estimated. Variance analysis and post hoc tests were per-formed to estimate significances of differences of mean values of the different patient groups being healthy or sick.

Results:
Levene homogeneity test of variances indicates different variances for parameters A₀, A₁, and A₂. There-fore, Tamhane post hoc tests were applied for patient group comparisons (conf. lev. = 0.05%). A₁ enables a differentiation between: healthy and emphysema; healthy and fibrosis; healthy and suspected tumor; emphysema and fibrosis. A₂ enables a differentiation between: healthy and fibrosis; healthy and sus-pected tumor; emphysema and fibrosis; fibrosis and suspected tumor. Other group combinations cannot be differentiated based on A₁ or A₂.

Conclusions:
The application of the Gaussian fit approach to the description of the HU-frequency distribution of dif-ferent lung diseases enables very helpful group differentiations. This is an important step towards an automatic classification scheme between different lung diseases. In further investigations we shall try to revise the fit equation so that the mathematical model matches better to the HU-frequency distributions in segmented CT lung images.

Detection of cell nuclei on prostate tissue samples

Rainer Schoof

To this moment, the diagnosis of prostate cancer is mainly based upon a method called gleason grading. While the method relies on the visual comparison of stained tissue samples with defined patterns by pathologists, the quality of the diagnosis depends on their perception and experience, both being subjective human traits.

We present an algorithm, developed to support this currently used method by the computation of an objective measurement of stained tissue samples. The algorithm uses image processing parameters to identify the central points of cell nuclei, especially aiming at clustered cells. Therefore, special points on the boundary of such clusters are identified to be used as anchor points for the segmentation of the clusters.

A configurable Java program was written, implementing the algorithm. Furthermore, a testing environment was established, serving the purpose of producing qualified configuration parameters for a defined set of images.

With the evaluated configuration parameters, the software is capable of segmenting a good part of representative test files. Advancements of the developed software and further systematic parameter evaluations are needed to increase the quality of the segmentation.