4] have shown that a means of solving problems concerning the existence and uniqueness of eigenvectors of positive operators is given by introducing a suitable metric on a subset of the cone with respect to which the operators are contractions. More specifically, they have proved the Perron theorem for matrices with positive elements by intersecting the positive quadrant with a hyperplane and by using the Hubert metric (see, for example, [3]) on this section. Birkhoff was able to extend this method (using the same metric) to certain positive linear operators in a more general setting. Since the contraction mapping principle is essentially nonlinear it seemed likely that this method could be used for a class of nonlinear operators. In this paper we use a slightly different distance function the domain of definition of which is not restricted to such a section of the cone and we obtain a theorem for a class of nonlinear mappings which contract this metric.After giving necessary preliminaries the metric is defined in §2 and the completeness of certain subsets is proved. This is followed by a theorem on nonlinear operators and two examples.
A few studies have indicated that repeated dosing of acidic fibroblast growth factor (FGF-1) is essential to be effective in modulating the wound-healing response. However, little investigation has been done to determine the effective dosing regimen of FGF-1 or the appropriate carrier vehicle for this growth factor. The main objective of this study was to determine the effective angiogenic stimulatatory dose of FGF-1 delivered through a modified fibrin matrix, using a rabbit ear ulcer model. Specifically, the aim was to test the effects of FGF-1 on the angiogenic, fibroblastic, and epithelial responses in a wound model. Five 6-mm diameter ulcers to the depth of bare cartilage were created on each rabbit ear. Four different combinations (0.8, 8, 80, and 800 micrograms/ml) of the growth factor were examined across two periods of study. Pooled modified fibrin was used to deliver the growth factor. Histomorphometrical quantification was conducted after routine histological processing of the ulcers sites. Data analysis indicated a strong correlation between concentration and the histomorphometric response. In general, the growth factor treatments affected the healing response and exhibited a dose-dependent behavior. The addition of FGF-1 led to an increase in the angiogenic and fibroblastic responses, as well as an increase in the epithelialization rate. The preferred dose of 8 micrograms initiated a high epithelialization rate, fibroblastic, and angiogenic responses, and was the lowest dose required to initiate these responses.
In this study the wound healing response of full-thickness skin defects was enhanced through the use of acidic fibroblast growth factor using a collagen scaffold. The objectives of this study were (1) to compare the facilitatory effects of acidic fibroblast growth factor on the cellular response (fibroblast, neutrophil, macrophage, epithelial), tissue response (angiogenesis, collagen) and mechanical properties in a healing wound; and (2) to determine the effect of a collagen scaffold on wound healing when compared with the control. Three 3 x 3 cm full-thickness defects were created on the dorsi of 15 New Zealand White rabbits. Each rabbit had a control (no treatment), collagen scaffold, and collagen scaffold with FGF-1 (100 microg/cm2). All the wounds were covered with a transparent polyurethane dressing. There were three periods of study (1, 2, and 3 weeks) with five rabbits in each period. The volume fraction of the tissue and cells was histomorphometrically determined for each wound. The acidic fibroblast growth factor/collagen system showed promise in enhancing the healing process. Acidic fibroblast growth factor/collagen treatment increased angiogenesis, enhanced epithelialization, and reduced contraction rate over the control. A higher inflammatory response was indicated in the collagen scaffold treated group. Hence acidic fibroblast growth factor delivered through a collagen scaffold shows promise in future clinical applications.
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