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The frequency of the cancer cases is continuously increasing. The golden standard for the malignant tumor diagnosis is the biopsy, but this is often a dangerous method. We aim to develop computerized, non-invasive techniques for the automatic diagnosis of the abdominal malignant tumors, based on ultrasound images. We take into consideration the hepatocellular carcinoma (HCC) and the colo-rectal tumors for this purpose. The texture is an important property of the internal organ tissues, providing subtle information about the pathology. We previously defined the textural model of HCC, consisting in the exhaustive set of the relevant textural features, appropriate for HCC characterization and in their specific values. In this work, we analyze the role that the third order Gray Level Cooccurrence Matrix (GLCM) has on the characterization and automatic diagnosis of the abdominal malignant tumors. We also determine the best spatial relation between the pixels that leads to the highest performances.