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# The formalism of fractal aggregation phenomena of colloidal drug delivery systems

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## Abstract

Classical Newtonian Physics and Euclidean Geometry are currently used to describe biological phenomena and the processes of drug formulation, which are characterized by homogeneity and linearity. On the other hand, at the mesoscopic level, the principles and the laws of physics are quite different from the Classical Newtonian Physics and Euclidean approach especially at nanoscale dimension. The investigation of the aggregation process of liposomes is of paramount importance due to their applications in pharmaceutical nanotechnology as drug delivery systems and as membrane models, in biosciences. The physical stability and the aggregation process of colloidal systems as well as the surface phenomena are described using the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. The elucidation of the dimensionality of liposome aggregates obeys the fractal approach because the aggregation phenomena are irreversible. This approach can be correlated with the extended DLVO theory, which includes the hydration energy, too.

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... Advanced Drug Delivery Systems (aDDSs) have presented fractal dimensions [6][7][8][9][10][11][12]. Liposomes, polymeric nanoparticles and micelles are in the focus of the nanotechnology field for treatment and diagnostics of chronic and incurable diseases following Mandelbrot geometry principles. ...
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Unlabelled: Colloidal nanocarriers, in their various forms, have the possibility of providing endless opportunities in the area of drug delivery. The current communication embodies an in-depth discussion of colloidal nanocarriers with respect to formulation aspects, types, and site-specific drug targeting using various forms of colloidal nanocarriers with special insights to the field of oncology. Specialized nanotechnological approaches like quantum dots, dendrimers, integrins, monoclonal antibodies, and so forth, which have been extensively researched for targeted delivery of therapeutic and diagnostic agents, are also discussed. Nanotechnological patents, issued by the U.S. Patent and Trademark Office in the area of drug delivery, are also included in this review to emphasize the importance of nanotechnology in the current research scenario. From the clinical editor: Colloidal nanocarriers provide almost endless opportunities in the area of drug delivery. While the review mainly addresses potential oncological applications, similar approaches may be applicable in other conditions with a requirement for targeted drug delivery. Technologies including quantum dots, dendrimers, integrins, monoclonal antibodies are discussed, along with US-based patents related to these methods.
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The letter discusses the indispensable importance of Nanotechnology for the scientific and economical revival of the developing world. Similar to the nuclear age, and maybe far more so, the nanoage will be something of a Hemingway line of demarcation between the have and the have nots.
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"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature
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Drug delivery has metamorphosed from the concept of a pill to molecular medicine in the past 100 years. Better appreciation and integration of pharmacokinetic and pharmacodynamic principles in design of drug delivery systems has led to improved therapeutic efficacy. A greater understanding of the molecular transport in relation to physico-chemical properties has led to the evolution of a biopharmaceutics classification system, which should be a future road map, governing drug design, development and delivery. While drugs belonging to class I and II will be delivered by established platform technologies, novel delivery strategies will evolve and mature to realize the potential of 'new generation' biotech and non biotech drugs belonging to class III and IV, respectively.
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In the present study, the influence of vesicle size on the penetration of two fluorescently labeled substances into the human skin was investigated. For the measurements either a hydrophilic fluorescent compound [carboxyfluorescein (CF)] or a lipophilic one [1,1'-dioctadecyl-3,3,3',3'-tertramethylindocarbo-cyanine perchlorate (DiI)] were encapsulated into vesicles. Liposomal formulations were prepared by extruding the vesicles through polycarbonate membrane filters with pores of different sizes. In vitro penetration studies into human abdominal skin were performed by using the Franz diffusion cell and a standardized skin stripping technique in attempt to find an optimum size for topical drug delivery by liposomes. Confocal laser scanning microscopy (CLSM) was used to visualize the effect of penetration ability of liposomal DiI. The maximum DiI fluorescence in the skin was observed with smaller liposomes of 71 nm diameter. The liposomes with a size of 120 nm diameter showed statistically enhanced penetration of CF into the skin as compared to larger ones. The results indicated that the CF penetration was inversely related to the size of the liposomes, which was confirmed by the data of the confocal laser scanning microscopy studies.
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Topical administration of cosmetics and pharmaceuticals involves a variety of different formulations of which colloidal drug carrier systems are currently of particular interest. After a short introduction of reverse micellar solutions, liquid crystals, vesicles and nanoparticles, appropriate methods of physicochemical characterization are introduced including X-ray diffraction, laser light scattering, electron microscopy, and differential scanning calorimetry. Emphasis is laid on topical applications of the colloidal drug delivery systems (DDS) covered, with the main objective of both sustained drug release and improved stability of DDS.
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An "extended DLVO" approach that includes Lifshitz-van der Waals, Lewis acid-base, and electrostatic double layer interactions is used to describe interaction energies between spherical particles and rough surfaces. Favorable, unfavorable, and intermediate deposition conditions are simulated using surface properties representing common aquatic colloids and polymeric membranes. The surface element integration (SEI) technique and Derjaguin's integration method are employed to calculate interaction energy. Numerical simulations using SEI demonstrate that nanometer scale surface roughness features can produce a distribution of interaction energy profiles. Local interaction energies are statistically analyzed to define representative interaction energy profiles-minimum, average, and maximum-for various combinations of simulated particles and surfaces. In all cases, the magnitude of the average interaction energy profile is reduced, but the reduction of energy depends on particle size, asperity size, and density of asperities. In some cases, a surface that is on average unfavorable for deposition (repulsive) may possess locally favorable (attractive) sites solely due to nanoscale surface roughness. A weighted average of the analytical sphere-sphere and sphere-plate expressions of Derjaguin reasonably approximates the average interaction energy profiles predicted by the SEI model, where the weighting factor is based on the fraction of interactions involving asperities.
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The fractal dimension (FD) was used to reveal brain structure irregularities in patients with schizophrenia. FD provides a unique way of quantifying the shape complexity of cortical folding of the human brain. MR images were obtained from seven patients with schizophrenia that were compared with six healthy control subjects. The MR images were first segmented, and the FD was calculated for the grey/white matter boundary for the whole brain and the hemispheres separately, using the box-counting and Minkowski-Bouligand methods. The results showed that the patients had larger FD values than the controls, for the whole brain volume and right hemisphere.
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Living cells can be viewed as complex adaptive systems that exhibit non-linear dynamics and fractal features. We investigated the fractal qualities of normal and malignant hematological cells and their potential as a tool for characterizing cell phenotype and clinical behavior. A mathematical algorithm and an optic tool for fractal analysis of nuclei were developed. A total of 4,713 lymphoid cells derived from 66 patients of five distinct diagnostic groups (normal and reactive lymphocytes, low-grade lymphomas and an aggressive lymphoma) were assessed for their fractal dimension. In addition, in 19 patients fractal analysis of leukemia cells was compared to clinical endpoints. After validating our method, hematological cells possessed fractal dimensions corresponding to their clinical entity. There was a highly significant overall difference in fractal dimensions between various types of hematological malignancies. A preliminary correlation was found between the fractal dimension and the clinical outcome of leukemia patients. Hematological cells possess fractal dimensions that correlate with their biological properties. Measurement of fractal dimension seems to be a sensitive method to assess the hematological cell phenotype and to define a clinical group. This tool may be potentially useful for the evaluation of clinical behavior of hematological diseases.
Article
Growing interest is being dedicated to soft matter because of its potential in delivering any type of drugs. Since hydrophilic, lipophilic, small and big molecules can be loaded into these colloidal systems and administered through the parenteral or nonparenteral route, soft matter systems have been used to solve many biomedical and pharmaceutical problems. In fact, they make possible to overcome difficulties in the formulation and delivery of poorly water-soluble drug molecules, settle some stability issues typical of biological drug molecules, design parenteral sustained release forms and provide functionalized soft particles that are very effective in drug targeting. This review deals with the important role that colloids play in the drug delivery and targeting, with particular attention to the more currently used systems such as microemulsions, organogels, liposomes, micelles, and dendrimers. Though significant progress has been made in drug targeting, some challenges still remain. Further efforts will be required to better understand the characteristics of targets and to discover new ones. In-depth knowledge of the physico-chemical structure and properties of the systems used for targeting is fundamental for understanding the mechanism of interaction with the biological substrate and the consequent drug release.
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Fractal dimension analysis of MR images reveals grey matter structure irregularities in schizophrenia
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