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This contribution deals with the Molecular Effect Model (MEM) Genetic Algorithms polynomial-dual optimization for High Temperature Superconductors (HTSCs) class of [ Sn-Sb-Te-Ba-Mn-Cu-O ] . Results comprise Tikhonov Regularization Functionals development and mathematical methods for this HTSCs group without using logarithmic changes. Findings for this MEM optimization, based on Genetic Algorithms polynomial-dual-method show acceptable theoretical Numerical and 2D/3D Graphical Optimization solutions and low residuals. Solutions comprise two parts, the modelling for TC Molecular Effect predictions equations, and 2D graphics series of results. Electronics Physics applications for Superconductors and High Temperature Superconductors are specified for Isotope Effect in BCS theory and for MEM and presented.

In subsequent study series for 3D High Temperature Superconductors (HTCSs), Inverse Least Squares (ILS) and 3D Numerical/Graphical Optimization is the objective for this Molecular Effect Model (MEM) contribution. HTSCs class selected is Thallium group of [ Tl-Sn-Pb-Ba-Si-Mn-Mg-Cu-O ], constrained/extended to [TC ? 0° , TC > 0° ]. Tetragonal Lattice and Amorphous compounds of this Type II Thallium HTSCs class are not included in this study. Results obtained for Matlab 3D Graphical Optimization techniques are proven/presented. Numerical calculations include 3D Tikhonov Regularization algorithms with mathematical developments for this Thallium class. Thus, results show two strands, 3D surfaces for Graphical Optimization MEM Thallium HTSCs modelling, and MEM analysis of numerical equations. 3D MEM and Numerical Optimization get low error and residuals. Electronics Physics applications emerge from allteh research.

Constrained algorithms for BED model (Biological Effective Dose) in Head and Neck tumors Hypofractionation TPO optimized with Pareto-Multiobjective (PMO) Genetic Algorithms (GA) software are obtained. The mathematical method for constrained GA is applied for a number of series of Pareto Functions. Results demonstrate PMO-AI imaging process sequences and extensive numerical values of PMO Head and Neck cancer parameters. Comparison and review with simple constrained GA Optimization is presented. Improved RT Head and Neckcancer TPO, and tumors in general for Fractional-dose photon dose delivery are explained in brief

Following High Temperature Superconductors (HTCSs) study series, Inverse Least Squares (ILS) 2D Numerical/Graphical Optimization is applied on Molecular Effect Model (MEM). Selected HTSCs class is Thallium group of [ Tl- Sn-Pb-Ba-Si-Mn-Mg-Cu-O ], constrained to [TC ? 0° , TC > 0° ]. Tetragonal Lattice and Amorphous compounds of this Type II Thallium HTSCs are not included at this research stage. Solutions with Matlab 2D Graphical Optimization techniques to validate primarily the MEM sinusoid-shape analytical geometry are presented. Programming-methods and imaging processing comparisons with GNU-Octave and Freemat software are also shown. Results comprise 2D Tikhonov Regularization algorithms with mathematical methods for this class. Findings obtain acceptable Numerical and 2D Graphical Optimization outcomes with low error-residuals. Results show two parts, the 2D Graphical Optimization MEM modelling, and MEM comparative imaging processing systems with examples and analysis of several numerical equations and predictions. Applications of MEM for the overview of the Multifunctional Transmission Line for SCs and HTSCs are included. Electronics Physics usages for Superconducting Multifunctional Transmission Lines, Superconductors in general, and HTSCs are presented.

Following High Temperature Superconductors (HTCSs) study series, Inverse Least Squares (ILS) 2D/3D Numerical/Graphical Optimization for Molecular Effect Model (MEM) in group of [ Tl- Sn-Pb-Ba-Si-Mn-Mg-Cu-O ], with [ TC > 0°], is developed. Additionally, a laboratory results-review of 2D/3D Optimization methods for this HTSCs class is included exclusively for [ TC > 0° ]. Solutions with 3D Graphical Optimization techniques to validate primarily the MEM are presented. Results comprise 2D/3D Tikhonov Regularization algorithms with mathematical methods for this class. Findings obtain acceptable Numerical and 2D/3D Graphical Optimization outcomes with low error-residuals. Results show two parts, the 3D Graphical Optimization MEM modelling and a review of 2D/3D Graphical Optimization findings. Electronics Physics applications for Superconductors and HTSCs are extrapolated.

In previous series of contributions, Inverse Least Squares (ILS) Numerical Optimization, 3D/4D Interior Optimization, 3D/4D Graphical Optimization software and algorithms-programming were presented. The BCS theory Isotope Effect applications on superconductors TC dual/multiobjective optimization was the main subject. This contribution deals with the Molecular Effect Model for High Temperature Superconductors (HTSCs), in primary hypothesis stages. Selected group of HTSCs is Sn-Sb-Te-Ba-Mn-Cu-O . Namely, classical Inverse Least Squares, 2D Interior Optimization, and 2D Graphical Optimization techniques are applied. Results comprise Tikhonov Regularization algorithms and mathematical methods for this HTSCs group. Mathematical-programming findings to optimize this Type II HTSCs are presented with ILS outcoming data in Matlab and GNU-Octave. Rulings present an hypothesis of new ‘Molecular Effect’ model/algorithm, intended be proven for this HTSCs group. In this Molecular Effect optimization, Inverse Least Squares and Inverse Least Squares polynomial methods are applied with acceptable Numerical and 2D Graphical Optimization solutions. Results show accuracy with low programming residuals that confirm these findings. Solutions comprise two strands, the modelling for Molecular Effect, and the Inverse Least Squares improved programming methods with confidence intervals and statistical images. Hypothesis in Electronics Physics applications for Superconductors and High Temperature Superconductors emerge from findings. Modelling-hypothesis applications in Superconductor Theory turn up from the numerical and imaging data obtained.

Ceramic THA constitutes an important group among the most frequent used implants in Biomedical Engineering and Medical Devices research field. A genetic algorithms computational nonlinear optimization is presented with two commonly ceramic materials for Ceramic-on-Ceramic (CoC) THA. This optimization is compared to a previously published Inverse Least_Squares one. Selected materials are Alumina (Al3O2), and Zirconium (ZrO2). Principal result is the numerical validation-verification of the K adimensional-constant parameter of the model with both methods. Results from previous Least-Squares algorithm and Genetic Algorithms show be closely with identical magnitude order. Numerical figures for both dual optimizations give acceptable model-parameter values with low residuals. These findings are demonstrated with series of 2D and 3D Graphical/Interior Optimization graphics also. 4D Interior Optimization method constitutes also the computational innovation of this study. The Genetic Algorithms dual-optimized ceramic-model parameters are mathematically proven/verified. Mathematical consequences are obtained for model improvements and in vitro simulation methodology. These confirmed wear parameters for in vitro determinations and efficacious Genetic Algorithms approach constitute the article novelty of both optimization methods. Results for in vitro tribotesting wear predictions with these parameters for laboratory experimental show be useful/effective. Applications for clinical Medical Physics and Bioengineering improvements in material/ceramic-THA and CAM constitute practical consequences.

BED model (Biological Effective Dose) for Hyperfractionation TPO was optimized with Pareto-Multiobjective Genetic Algorithms (GA) software. Secondly, the NEffective(Effective Tumor Population Clonogens Number) model optimization for breast cancer clonogens parameters determination in TPO (Treatment Planning Optimization) is carried out with 3D Graphical and Interior Optimization methods. BED model (Biological Effective Dose) for Hyperfractionation TPO was optimized with Pareto-Multiobjective GA software. Results comprise imaging process series and numerical values of NEffectivemodel for breast cancer parameters. Additional results demonstrate Pareto-Multiobjective GA BED model both with Pareto-Optimal Front graphics, charts and numerical dose fractionation datasets. For all these findings, supplementary new recent applications with 3D Isodoses TPO with AAA (Anisotropic Analytic Algorithm) model wedge filters dose delivery is shown. Modern RT treatment breast cancer, and tumors in general for Fractionation-dose protocols are explained.

A series of improved imaging-computational and algorithmic methods for new/different lumbar cadaveric specimens was obtained. These are based on previous publications [3,3.1], with an improved-imaging research line. Results show a systematic study of each lumbar cadaveric specimen. Enhanced imaging findings and resolution for vertebral facets/positioning, contrast, anatomical parts separation and visualization of lumbar spines are demonstrated. Medical Physics and clinical bioengineering advances related to previous contributions are proven with imaging processing, programming codes/patterns, and computer vision tools. Findings constitute computational imaging methods which are appropriate for sharp and detailed anatomical-clinical analysis and comparisons among cadaveric specimens. These processing solutions are useful for lumbar spine computational study and anatomical dissection. Applications on Medical Physics, Biomedical Engineering, and Computational-Forensic Diagnosis are obtained from this cadaveric imaging systematic comparison and software methods.

Th e research objective was to obtain a fast/simple analytic solution for previously published Radiotherapy Tumor Control Probability (TCP) Integral Model by other authors . Finding, firstly, is a new/simpler Probability Function than Poisson distribution of survival clonogens which is developed from classical biological model algebraic/numerically modified equation. Solution obtained is t his Probability Function alternative based on a Binomial distribution equation approximation from literature. Result, t hus, is its use of for developing a fast/analytic solution of the 2D [ α and β biological modelling parameters ] Integral Model Cumulative TCP. The Gaussian Normal distribution convolution is implemented with th is P robability F unction into the integral model to sort longer/programming numerical methods and determine an analyt ical solution expression. As a complete result, a formulation for TCP based on Erf functions is obtained. Applications for radiotherapy treatment planning optimization improvements with biological models are explained.

In a previous contribution a total hip arthroplasty optimization for metal materials was presented [5]. THA constitutes an important group among the most frequent used implants in Biomedical Engineering and Medical Devices research field. In this further multiobjective study, modelling and nonlinear optimization is performed with four commonly used ceramic materials for CoC hip arthroplasty—among the most surgically utilized currently. These are ZTA Biolox, ZTA Biolox-Delta, Alumina (Al3O2), and Zirconium (ZrO2). Numerical results for dual optimization show acceptable figures with low residuals. Results with algorithm of 2D Graphical Optimization and 3D Interior Multiobjective Optimization are proven, explained, shown acceptable. According to these optimal findings and calculations, the model parameters are mathematically demonstrated, and verified. Results of 2D graphics and 3D Interior Multiobjective Optimization to obtain the local minima are sharp. Optimization rising numbers match the model design because the hardness and experimentally-published erosion intervals of Alumina, Zirconia, ZTA Biolox and ZTA Biolox-Delta intervals overlap one another approximately. Medical-Mathematical Physics consequences emerge for new nonlinear multiobjective optimization algorithms. Based on these numerical multifunctional data results, applications in Biomedical Engineering devices and future Bioengineering/Biomaterials designs are guessed.

Previous contributions dealt with demonstration of software-computational methods (CAD) to digitalize cloud data, and filter numerically them. Subsequently, simulate and fit mathematically the anterior and lateral facet vertebral body morphometry of cadaveric lumbar spines. This advanced study presents three computational dissection methods for new/different lumbar cadaveric specimens. Namely, Region of Interest (ROI) software-method, Programming-dissection selection technique, and Plane-Segmentation dissection. Results demonstrate computational images of several ROIs from L2 to L5 anterior facets, programming-dissection of complete specimen. Third findings group prove planar dissections for a long cadaveric specimen. Additional results/outcomes emerge from anatomical analysis of the cadaveric specimens. Applications to orthopedics tools and prostheses, surgical robotics, spinal stabilization instrumentation, spine-rehabilitation apparatus and Forensic Robotics [Casesnoves, 2020] comprise additional findings. Results in pre-hypotheses theory for spine-biomechanical evolution are indicated.

This contribution deals with demonstration of software-computational methods (CAD) to digitalize, simulate, and fit mathematically the anterior vertebral body morphometry of cadaveric lumbar spines. From previous research publications, computational techniques are developed and explained. Based on anatomical-cadaveric spinal specimens, experimental data was implemented to obtain practical surgical applications. With these anatomical samples, large sets of surface digital points were generated. Complete anterior vertebral body morphologies were visualized and analyzed with designed software-engineering. Selection method for extraction of Regions of Interest (ROIs) is presented. Applications of Lumbar anterior, vertebras, and intervertebral disks surfaces, are explained. Further biomedical anatomical analysis applications are included. Results comprise realistic anatomical representations of lumbar spine. Important applications in computational design, surgical robotics, and optimization of surgery tools and techniques are presented.

GNU-Octave System software was designed to obtain 3D/2DGraphical Optimization for Molecular EffectModel (MEM)in [ Tl-Sn-Pb-Ba-Si-Mn-Mg-Cu-O ]High Temperature Superconductors (HTSCs) class [exclusively forTC > 0° ]. The study results show two parts, namely, GNU-Octave 3D Graphical Optimization and 2D/3D MEM optimization review. Series of 2D MEM GNU-Octave graphs with increasing ILS polynomial degree are developed to demonstrate the equaling of the MEM for this HTSCs class.Comparisons to Matlab solutionsfromprevious contributions were also presented. Results comprise Inverse Tikhonov Regularization algorithms and mathematical methods for this HTSCs class. Solutions prove Numerical/Graphical coincidence between MEM and experimental data is proven. ILS norm-residuals for MEM show be acceptable and low. Electronics Physics applications for Superconductors and HTSCs and optimal TCselection/constraints are explained.Invention of Superconducting Multifunctional Transmission Line is proposed [Casesnoves, 2021]

Interior Optimization (IO) software and algorithms-programming were primarily presented in previous articles [3,4]. The mathematical framework of this new method, [Casesnoves, 2018-2020], was also proven [3,4]. The links among Interior Optimization, Graphical Optimization [Casesnoves, 2016-7], and classical methods in Nonlinear Equations Systems were developed. This paper is focused on software engineering with mathematical methods implementation in Multiobjective Interior Optimization programming as a primary subject. Second subject is Electronics applications of software in the field of BCS Superconductivity. These applications not only constitute a proof of the method, but also an useful BCS electronics numerical framework. They comprise a series of new BCS Equation optimization for multiple Type I superconductors, based on previous research for other different Type I ones previously published [3,4]. A Dual Optimization for two superconductors is also simulated. Several deductions/findings in computational technique for Multiobjective IO are guessed. Results are acceptable with low errors and 3D imaging demonstrations of the Interior Optimization utility.

The nonlinear multiobjective optimization problem constitutes a difficult task to find/determine a global minimum, approximated-global minimum, or a convenient local minimum whith/without constraints. Nonlinear systems of equations principles set the base for further development of Interior Optimization and Interior-Graphical Optimization [Casesnoves, 2016-7]. Mathematical proofs with mathematical definitions, algorithms, and nonlinear optimization equations are presented. From Graphical Optimization 3D optimization stages, it is proven that solution of nonlinear systems of equations is not unique in general. Software-engineering and computational simulations are shown with electronics superconductors and electronics physics applications. Extensions to similar applications for materials-tribology models and Biomedical Tribology are explained.

This paper presents an extension both in software optimization with simulations and detailed mathematical theory of Numerical Reuleaux Method based on previous publications. In the literature, there are a number of papers in Numerical Reuleaux Method and its applications (Aerospace, Helicopter Dynamics in Turbulence Conditions, Biomedical Engineering, Biomechanics, etc) since 2007. This contribution is a detailed presentation of the mathematical framework that constituted the basis for those articles along 2007-2020. The Classical Reuleaux Method (CRM) is frequently used in Physical Dynamics, Engineering Mechanics and Bioengineering to determine the Instantaneous Rotation Center (IRC) of a rigid body in arbitrary movement. The generic mathematical CRM only can be applied on rigid bodies, whose shape remains constant during the movement. If the solid in movement is a Pseudo-Rigid Body (PRB), the CRM has to be modified numerically to conform the shape changes and adapt on the density distribution variations of the PRB (we denominate it, in this case1, The Numerical Reuleaux Method, NRM). This Geometrical-Numerical Approximation Method is based on the division of the Pseudo-rigid body into small volume parts called voxels (roughly speaking parallelepipedic), namely, voxelization of the body subject to dynamics. The theoretical basis of the method is strictly shown in complementary details, with the necessary Theorems and Propositions of the model. Nonlinear Optimization Techniques that support the initial theory have been developed, and the Error boundaries with Error reduction techniques are determined. Computational Simulations have been carried out to prove the NRM Theoretical Model feasibility and numerical veracity of the Propositions, Theorems, and Error Boundaries. Appropriate software was made to carry out these simulations conveniently. The initial results agree to the theoretical calculations, and the IRC computation for 2 voxels shows to be simple and easy. Some initial guidelines for a theoretical development of this algorithm applied on large pseudo-rigid bodies, by using Monte-Carlo techniques, are sketched. Recent applications, Aerospace and Biomechanics, are also shown.

Total hip metal arthroplasty (THA) constitutes an important proportion of the standard clinical hip implant usage in Medical Physics and Biomedical Engineering. A computational nonlinear optimization is performed with two commonly metal materials in Metal-on-Metal (MoM) THA. Namely, Cast Co-Cr Alloy and Titanium. The principal result is the numerical determination of the K adimensional-constant parameter of the model. Results from a new more powerful algorithm than previous contributions, show significant improvements. Numerical standard figures for dual optimization give acceptable model-parameter values with low residuals. These results are demonstrated with 2D and 3D Graphical/Interior Optimization also. According to the findings/calculations, the standard optimized metal-model parameters are mathematically proven and verified. Mathematical consequences are obtained for model improvements and in vitro simulation methodology. The wear magnitude for in vitro determinations with these model parameter data constitute the innovation of the method. In consequence, the erosion prediction for laboratory experimental testing in THA adds valuable information to the literature. Applications lead to medical physics improvements for material/metal-THA designs.

Analytical-convoluted and numerical Gaussian models have been used in recent decades for radiotherapy treatment planning software/calculations, to perform accurately radiation dose delivery –numerical, analytical, or numerical-analytical. The objective of this evoluted-contribution was to obtain an exact dose delivery, 3D analytical-integral-equation solution, for the triple Gaussian model of wedge filters, since previous/initial 2D approximations of other authors, although correct, were not completely exact. Additionally, to set conceptual and mathematical-geometrical differences between the beam modification created by Multi-Leaf Collimator and Wedge Filters, either standard or Conformal. Ever the precision, from mathematical theory algorithms to real laboratory measurements, a series of simulations are presented. The generic triple Gaussian model of Ulmer and Harder sets an Attenuation Exponential Factor, AEF, well approximated in 2 variables, namely, u and z. This evoluted contribution of the research contribution was specially focused on numerical methods and approximation analysis of the integral equation resolution –with extent details about numerical data, Appendix 3. In this paper we set a detailed spatial-spherical geometry discussion/proof towards the determination of a 3D integral form of the delivery dose in water. In other words, with an AEF for magnitude-values of variables u,v, and z. Simulations, based on these new determinations were shown with sharp presentation of the numerical-computational software and functional programming series development. Computing encode techniques are explained with some practical examples for numerical radiotherapy calculus.

Today, artificial implants (AI) industry depends strongly on tribological constitution of the material (s) of the implant. Erosion, corrosion, tribocorrosion and biocorrosion are essential factors to determine both functionality and lifetime of the AIs. Histo-Biocompatibility is also an additional constraint, indispensable for implant manufacturing process. The prediction of durability, based on the computational and experimental study of constituents of AI material (s) are key factors to obtain objective data of any AI characteristics. This contribution deals with a computational comparative analysis of materials for hip implants using Archard’s model mainly. Selected hip implant material hardness are Co-Cr alloy and Titanium types. Method is carried out with specific material data, e.g., hardness or wear constants, nonlinear optimization and graphical subroutines. Results presented are both numerical and graphical. Particular interest is focused on application of the 3D Graphical Optimization method.

Stepping forward from a previous conference contribution, the article focuses on extension of inverse problem algorithms to integral-differential modelling and formal/strict demonstration of graphical-optimization method. It shows evident-mathematical and 3D-imaging proofs of the graphical optimization method with L1 Norm simulations and algorithms. At present, Linear/Nonlinear Optimization mathematical methods constitute the choice of preference in getting improvements for erosion and corrosion simulations- determinations in general tribology, biotribology and tribocorrosion. The method(s) developed are classical numerical optimization settings for objective functions, programming optimization and simulations, and special software for imaging in 3D. Results are diverse and the range of their applications is wide. First, the article provides a definite formal demonstration of the nonlinear graphical optimization both in numerical results and in imaging. Then, the authors propose the development of programming optimization and mathematical proofs-algorithms of the integral-differential model for various models. Subsequently, an overview of stochastic erosion methods based on Markov Chain is presented in the article. Finally, the second generation of tribology models is defined and conceptually explained. To summarise, the article comprises new findings towards modernization of tribology, biotribology and tribocorrosion models, gathering innovative research branches for future extension of the mathematical modelling progress. The results can be applied to both general techniques and mechanical engineering. The analytical and numerical demonstration of the integral-differential model constitutes a key point and essential result of the research. Extension to electromagnetic and electronic models of these methods is also considered feasible and practical

This contribution follows previous publications in Integral-Differential Model for mechanical systems of power plants. The objective of research was to obtain practical mathematical algorithms to optimize the probability of cracks/breaks of the mechanical system by impact erosion wear. Probability of failure concept and its mathematical formulation are developed to set algorithms for lifetime determination of mechanical systems .In other words, to prevent unexpected mechanical failures by severe damage of mechanical parts that could lead to power plant stops for repair. Firstable, the article presents the fundamental formulation/mathematical-definitions for lifetime determination and probability of failure of a mechanical system—with focus on further applications for Integral Differential model. Subsequently, a series of modeling optimization and simulations are shown with Gaussian models and stochastic ones. Methods applied in programming are outlined in brief and software development is explained. Formal results are a group of Bayesian formulas, computational graphs and numerical database charts to support the initial hypotheses and mathematical framework. Practical engineering-tribological consequences for optimal operation of power plants derive from the basics of the study.

Surface damage in power plants causes economic loss and waste of operational time in modifications or repair. The wear is caused mainly by solid particles and water droplets. Erosion and corrosion of protective coatings constitute a number of significant engineering difficulties. “Trial and error” type methodology to improve the materials design is rather expensive, inaccurate and time consuming. Mathematical modelling through optimization methods overcome these engineering complications/difficulties, and reduce the experimental/tribotesting period. In this paper we provide a brief review of the current classified erosion and/or corrosion models and, additionally, detailed modern optimization methods for precise modelling of given applications. A concise computational optimization example for an erosion model is presented with brief software details/approximations.

The engineering design of surgical instrumentation to apply mechanical forces and linear moments on the human bones during the operations constitutes a rather difficult task. This is due both to the natural and pathological irregularities of the human bone morphology and surfaces and also to the individual variations from one patient to another. Usually, the forces are applied by the surgeon only on a determined part of the bone surfaces. This paper describes an innovative computational design method to digitalize, simulate, and fit mathematically the anterior vertebral body facet. We used real experimental data from 17 human cadaveric specimens to get and store a large amount of numerical surface digital values. The complete anterior vertebral body side was visualized and analyzed with grid data Subroutine, which was also used first to select the so-called natural regions of interest (ROIs). These ROIs correspond to those parts of the surface in contact with the surgical instrumentation, where the mechanical forces are applied. Subsequently, a numerical mathematical fitting-model was implemented for these ROIs. This was carried out with the development of a 3D geometrical least-squares optimization algorithm and appropriate software designed according to the proper numerical method selected. In doing so, the 3D superficies equations of the anterior vertebral body (L3, L4, L5, and S1) were determined after these fittings were mathematically checked as appropriate. Statistical parameters and determination coefficients that define the error boundaries and the goodness of this optimal fitting-model were calculated and NURBS error data in similar studies were commented. It was proven that the principal source of error was the micro- and macro-irregularities of human bone facets. The final surface equations, and their geodesics, were used to obtain accurate data for the spinal surgery instrumentation manufacturing. The industrial bioengineering result was the application of these equations for the design of a new spinal vertebral surgical distractor. This innovative distractor separates two adjacent vertebrae while keeping them parallel. That is, at their natural inclination, avoiding hammering the vertebrae to make the intervertebral space wider. The device mechanics also minimizes the necessary force to be carried out by the surgeon during the operation.

Purpose: Static wedge filters (WF) are commonly used in radiation therapy, forward and/or inverse planning. We calculated the exact 2D/3D geometrical pathway of the photon-beam through the usual alloy WF, in order to get a better dose related to the beam intensity attenuation factor(s), after the beam has passed through the WF. The objective was to provide general formulation into the Anisotropic Analytical Algorithm (AAA) model coordinates system (depending on collimator/wedge angles) that also can be applied to other models. Additionally, second purpose of this study was to develop integral formulation for 3D wedge exponential factor with statistical approximations, with introduction for the limit angle/conformal wedge.Methods: The radiotherapy model used to develop this mathematical task is the classical superposition-convolution algorithm, AAA (developed by Ulmer and Harder). We worked with optimal geometrical approximations to make the computational IMRT calculations quicker/reduce the planning-system time. Analytic geometry/computational-techniques to carry out simulations (for standard wedges) are detailed/developed sharply. Integral developments/integral-statistical approximations are explained. Beam-divergence limit Angle for optimal wedge filtration formulas is calculated/sketched, with geometrical approximations. Fundamental trigonometry is used for this purpose.Results: Extent simulation tables for WF of 15º, 30º, 45º, and 60º are shown with errors. As a result, it is possible to determine the best individual treatment dose distribution for each patient. We presented these basic simulations/numerical examples for standard manufacturing WF of straight sloping surface, to check the accuracy/errors of the calculations. Simulations results give low RMS/Relative Error values (formulated) for WF of 15º, 30º, 45º, and 60º.Conclusion: We obtained a series of formulas of analytic geometry for WF that can be applied for any particular dose delivery model. Simulations results gave acceptable trigonometrical approximations/data that can be used for LINAC applications/planning-system software. The integral formulas presented are practical for dose delivery calculations/3D-approximations when using WF/other similar types of beam modification devices. Limit angle formulation and conformal wedge concept was also presented...................................................Cite this article as: Casesnoves F. Geometrical determinations of IMRT photon pencil-beam path in radiotherapy wedges and limit divergence angle with the Anisotropic Analytic Algorithm (AAA). Int J Cancer Ther Oncol 2014; 2(3):02031. DOI:10.14319/ijcto.0203.1