Radial Basis Function Artificial Neural Network Research Articles

Ponceau 4R elimination from fruit juice: An integrated optimization strategy utilizing artificial neural networks, least squares, and chitosan-nickel ferrite Nano Sorbent

The goal of present work is to examine the efficiency of aminated-chitosan/NiFe2O4 nanoparticles (AmCs/NiFe2O4 NPs) produced for removing Ponceau 4R (P4R) from fruit juice through an adsorption process. The resulting nanoparticles were characterized using various techniques. The modeling of results was done using least squares (LS) and Radial basis function-artificial neural network (RBF-ANN). The optimum removal of P4R (91.43 %) was obtained at the following optimum conditions: pH 4.47, adsorbent dosage 0.047 g/L, contact time approximately 57.78 min, and initial concentration P4R 26.89 mg/L. The highest adsorption capacity (qm) was found to be 208.33 mg g−1. The P4R adsorption mostly followed the Freundlich and pseudo-second-order isotherm kinetic models. Both LS-based models and RBF-ANN provided good predictions for independent variables. The dye elimination efficacy for juice samples were approximately 90.34 %. Therefore, based on the obtained results, it can be claimed that the prepared AmCs/NiFe2O4 NPs can be used to remove P4R.

Primary investigation of an innovative method for milk authenticity using two handheld spectrometers and chemometrics: Comparison with portable ATR-FTIR

This study advocates the use of handheld spectrometers, spanning the visible-near-infrared (Vis-NIR, 400–1000 nm) and near-infrared (NIR, 900–1700 nm) ranges, alongside a portable attenuated total reflection-Fourier transform infrared (ATR-FTIR) instrument, for discerning water adulteration in raw bovine milk. Samples with water adulteration levels ranging from 0 % to 30 % underwent direct analysis without additional preparation. Principal component analysis was utilized for differentiation, while quantification was achieved using partial least squares regression (PLSR) and radial basis function-artificial neural network (ANN) models. The ATR-FTIR instrument displayed remarkable performance, achieving regression coefficients of prediction (R2p) of 0.99 and 0.99, with root mean square error of prediction (RMSEP) values of 0.20 % and 0.53 % in PLS and ANN regression models, respectively. Similarly, handheld NIR spectrometer yielded robust results, with R2p values of 0.99 and 0.99 and RMSEP values of 1.28 % and 1.12 % in PLS and ANN models, respectively. Vis-NIR spectrometer also demonstrated commendable performance, with R2p values of 0.92 and 0.94 and RMSEP values of 4.53 % and 4.83 % in PLS and ANN models, respectively. ATR-FTIR showed the best performance followed by handheld NIR and Vis-NIR spectrometers. All instruments can be used potentially for screening and/or quality control in the dairy industry.

Using radial basis artificial neural networks to predict radiation hazard indices in geological materials.

The estimation of exposures to humans from the various sources of radiation is important. Radiation hazard indices are computed using procedures described in the literature for evaluating the combined effects of the activity concentrations of primordial radionuclides, namely, 238U, 232Th, and 40K. The computed indices are then compared to the allowed limits defined by International Radiation Protection Organizations to determine any radiation hazard associated with the geological materials. In this paper, four distinct radial basis function artificial neural network (RBF-ANN) models were developed to predict radiation hazard indices, namely, external gamma dose rates, annual effective dose, radium equivalent activity, and external hazard index. To make RBF-ANN models, 348 different geological materials' gamma spectrometry data were acquired from the literature. Radiation hazards indices predicted from each RBF-ANN model were compared to the radiation hazards calculated using gamma spectrum analysis. The predicted hazard indices values of each RBF-ANN model were found to precisely align with the calculated values. To validate the accuracy and the adaptability of each RBF-ANN model, statistical tests (determination coefficient (R2), relative absolute error (RAE), root mean square error (RMSE), Nash-Sutcliffe Efficiency (NSE)), and significance tests (F-test and Student's t-test) were performed to analyze the relationship between calculated and predicted hazard indices. Low RAE and RMSE values as well as high R2, NSE, and p-values greater than 0.95, 0.71, and 0.05, respectively, were found for RBF-ANN models. The statistical tests' results show that all RBF-ANN models created exhibit precise performance, indicating their applicability and efficiency in forecasting the radiation hazard indices of geological materials. All the RBF-ANN models can be used to predict radiation hazard indices of geological materials quite efficiently, according to the performance level attained.

Modeling of the biosorption of Fe(III) ions by olive‐stone activated carbon

This study deals with the optimization of Fe(III) ion removal using activated carbon from olive stone waste using advanced machine learning models. The main objective is to evaluate and compare the performance of machine learning models, specifically multilayer perceptron artificial neural network (MLP‐ANN), general regression artificial neural network (GR‐ANN), radial basis function artificial neural network (RBF‐ANN), and particle swarm optimization artificial neural network (PSO‐ANN) in predicting Fe(III) removal efficiency. Experimental data on adsorption parameters were used to train and test the models. Techniques such as tuning hidden layer neurons, optimizing propagation values, and using a Taguchi approach PSO algorithm were applied to improve the models. For the MLP‐ANN model, the optimal configuration contains 13 neurons in the hidden layer. Concerning the parameters involved in the PSO‐ANN model, the coefficient C2 and the particle have the main effect on the reduction of the error. Their contributions are respectively 49% and 19%. The PSO‐ANN model showed superior performance with the highest regression coefficient (0.9997) and remarkable prediction accuracy, surpassing other models such as MLP‐ANN and GR‐ANN. This research suggests that innovative optimization techniques, particularly using PSO algorithms, significantly enhance the predictive capabilities of machine learning models in complex adsorption processes, contributing to more accurate Fe(III) removal models.

A novel and intelligent chemometric-electrochemical-enzymatic biosensing procedure and mimicking a clinical condition environment to trick the red blood cells for counting them under physiological conditions: A new connection among chemometry, electrochemistry and hematology

Here, a novel electrochemical biosensing procedure has been developed for determination of the number of red blood cells (RBCs) under physiochemical conditions based on chemometric modeling of hydrodynamic differential pulse voltammetric (HDPV), and amperometric data as responses of a modified edge plane pyrolytic graphite electrode (EPPGE). In order to obtain a good sensitivity from the EPPGE, its surface was modified with a thin layer of multiwalled carbon nanotubes-ionic liquid (MWCNTs-IL). Catalase (CAT) was immobilized onto the surface of MWCNTs-IL/EPPGE with help of nafion. The response of the biosensor was based on electrochemical reduction of oxygen of the blood samples which was enhanced by a trick based on addition of hydrogen peroxide (H2O2) to blood samples which can be reduced by the CAT to produce extra oxygen. Prior to experiments, the solution in electrochemical cell was bubbled with pure N2 to purge the oxygen in the solution, but in order to increase the selectivity of the biosensor towards detection of the oxygen obtained from the red blood cells, voltammetric responses of the biosensor were modeled by multivariate chemometric calibration methods with the help of radial basis function-partial least squares (RBF-PLS), least squares-support vector machines (LS-SVM), recursive weighted partial least squares (rPLS), ant colony optimization-mathematical pre-processing selection by genetic algorithm-sample selection through a distance-based procedure-partial least squares-1 (ACO-GA-SS-PLS1), and radial basis function-artificial neural networks (RBF-ANN) to select the best method for determination of the number of the RBCs. The results confirmed the amperometric methods modeled by RBF-ANN showed the best performance for supporting the biosensor in determination of the number of the RBCs with a performance which had an excellent compatibility with the results of a hemocytometer. The results of this study as the newest application of chemometric-electrochemical methods can make a strong connection among electrochemists, chemometricians and hematologists to expand their collaborations on determination of blood factors.

Unraveling hot deformation behavior and microstructure evolution, flow stress prediction of powder metallurgy BCC/B2 Al1.8CrCuFeNi2 HEA

In this work, hot compress tests were conducted at the strain rate range of 1–0.001 s−1 and the temperature range of 900–1050 °C to unravel the hot deformation behavior of a powder metallurgy (PM) Al1.8CrCuFeNi2 high entropy alloy with BCC/B2 structure. Arrhenius model and radial basis function artificial neural network (RBF-ANN) machine learning (ML) method were used to model flow stress. RBF-ANN ML method possesses superior predictability. Elongated grains almost change to equiaxed grains under the low strain rate of 0.001 s−1 due to the activity of continuous dynamic recrystallization (CDRX). There are no significant changes in the distribution, size and content of fine B2 and BCC eutectic phases during deformation. Stress index ranges form 2.2–2.6 at various strains. Dislocation viscous gliding and grain-boundary sliding (GBS) are predominant deformation modes. The contribution of GBS to deformation is more significant at low strain rate. For dislocation slip, multiple slip systems are activated,{123̅}<111> slip system has higher Schmid factor. Hot processing ability is improved at high temperature and low strain rate. Absorbing of dislocation at grain boundaries and the dominance of GBS deformation mode under high temperature and low strain rate condition can avoid instability.

Research on artificial neural networks to accurately predict element concentrations in nutrient solutions

Calcium, potassium, nitrogen, magnesium, and phosphorus, the main elements of the nutrient solution, are absorbed by plants and play an important role in plants. By measuring Ca2+, K+, Mg2+, NH4 +, NO3 −, HPO4 2−, the artificial neural networks (ANNs) were used in this study to accurately calculate the concentrations of these elements. Firstly, the error sources of the calculating element concentration were analyzed based on the data of six-ion measurement experiments. Subsequently, various optimization algorithms were compared to optimize back propagation and radial basis function ANNs. Finally, the results of mean relative errors (MREs) and recovery values show that ANNs can effectively reduce the measurement error of ion sensors. From the perspective of recovery values, the prediction error of all elements can be controlled within 15%. From the perspective of MRE, except for magnesium and phosphorus elements, the improved model prediction errors of other elements were also less than 10%.

The combination of multiple linear regression and adaptive neuro-fuzzy inference system can accurately predict trihalomethane levels in tap water with fewer water quality parameters

Artificial Neural Network (ANN) models are accurate in predicting the levels of disinfection by-products (DBPs) in drinking water. However, these models are not yet practical due to the large number of parameters involved, which should take a significant amount of time and cost to detect. Developing accurate and reliable prediction models of DBPs with fewest parameters is essential in the management of drinking water safety. This study used the adaptive neuro-fuzzy inference system (ANFIS) and radial basis function artificial neural network (RBF-ANN) to predict the levels of trihalomethanes (THMs), the most abundant DBPs in drinking water. Two water quality parameters identified by multiple linear regression (MLR) models were used as model inputs, and the quality of the models was assessed based on criteria such as correlation coefficient (r), mean absolute relative error (MARE), and the percentage of predictions with absolute relative error less than 25% (NE<25%) and over than 40% (NE>40%), etc. The results showed that the ANFIS models had higher correlation coefficients (r = 0.853–0.898) and prediction accuracy (NE<25% = 91%–94%) compared to RBF-ANN models (r = 0.553–0.819; NE<25% = 77%–86%) and traditional MLR models (r = 0.389–0.619; NE<25% = 67%–77%). Conversely, the prediction error, as indicated by MARE and NE>40%, showed the opposite trend: ANFIS models (MARE = 8%–11%; NE>40% = 0–5%) < RBF-ANN models (MARE = 15%–18%; NE>40% = 5%–11%) < MLR models (MARE = 19%–21%; NE>40% = 11%–17%). The present study provided a novel approach for constructing high-quality prediction models of THMs in water supply systems using only two parameters. This method holds promise as a viable alternative for monitoring THMs concentrations in tap water, thereby contributing to the improvement of water quality management strategies.

Comparison and error evaluation of Arrhenius model and typical machine learning algorithms for high-temperature flow stress prediction of GH3536 superalloy

In this work, the Arrhenius model and four typical machine learning (ML) algorithms including random forest (RF), support vector machine (SVM), back propagation artificial neural network (BP-ANN) and radial basis function artificial neural network (RBF-ANN) were used to forecast the high-temperature flow stress of GH3536 superalloy. The prediction accuracy order is RBF＞ BP＞SVM＞Arrhenius model＞RF. The accuracy of SVM, BP and RBF algorithms is significantly higher than that of Arrhenius model, and the error distribution range is much smaller than that of Arrhenius model and RF. For all ML algorithms, the error distributions of test and training set are basically similar. The RBF-ANN model presents extremely excellent prediction performance, the correlation coefficient R2/ root mean square error (RMSE)/ average absolute relative error (AARE) of reaches 0.9998/0.63 MPa/0.3%. Except RF, the prediction performance of test set is basically equivalent to that of training set.

Klasifikasi Jenis Tanaman Fast Growing Species Menggunakan Algoritma Radial Basis Function Berdasarkan Citra Daun

Indonesia has vast forests, even ranked as the third largest forest in the world. However, currently many forest areas have been deforested or the phenomenon of losing tree cover and forest areas. Forest rehabilitation programs develop by prioritizing plant or tree species that have fast growth or are called fast growing species. However, many people do not know about these fast growing species. Even though knowledge about the types of fast growing plant species is very important for the community to have so that the community can find out which plants can accelerate forest rehabilitation. Fast growing species of plants can actually be identified from the shape of the leaves. This study aims to build a classification model for fast growing species plant images based on leaf images by applying the Radial Basis Function (RBF) artificial neural network algorithm with morphological feature extraction. Morphological feature extraction is used to identify the shape of an object in order to obtain feature values based on predetermined parameters. These features then become input for the RBF artificial neural network to obtain learning patterns. The RBF network has three layers that are feedforward so that it can support solving classification or pattern recognition problems. Based on the results of accuracy testing, an accuracy value of 87.50% was obtained. This means that the Radial Basis Function (RBF) neural network is able to classify fast growing plant species based on leaf images.

A novel eco‐friendly methods for simultaneous determination of aspirin, clopidogrel, and atorvastatin or rosuvastatin in their fixed‐dose combination using chemometric techniques and artificial neural networks

AbstractIn this study, the simultaneous determination of aspirin, clopidogrel, and either atorvastatin or rosuvastatin in their fixed‐dose combination (FDC) formulations has been reported. As a straightforward substitute for employing distinct models for each component, UV spectrophotometry was applied with chemometric approaches and artificial neural networks to achieve this. Three chemometric techniques, including principal component regression (PCR), partial least‐squares (PLS), and classical least‐squares (CLS), were applied in addition to the radial basis function‐artificial neural network (RBF‐ANN). The validation of a set of laboratory‐prepared combinations of aspirin, clopidogrel, and atorvastatin in one ternary mixture and aspirin, clopidogrel, and rosuvastatin in a second ternary mixture was assessed, and the results from the use of these approaches were recorded and compared. The absorbance data matrix matching the concentration data matrix in CLS, PCR, and PLS was created using measurements of absorbances in the range of 250–280 nm at intervals of 0.2 nm in their zero‐order spectra. Then, in order to forecast the unknown concentrations, calibration or regression was created utilizing the concentration and absorbance data matrices. Using RBF‐ANN for the simultaneous determination of aspirin, clopidogrel, and atorvastatin or rosuvastatin in their formulations was achieved by providing the input layer with 151 neurons; there are 2 hidden layers and 3 output neurons were obtained. The green profile of the developed methods has been assessed and compared with previously reported spectrophotometric methods. The suggested techniques were effectively applied to FDC dosage forms that contained the cited medications.

Mechanical Dispatch Reliability Prediction for Civil Aircraft Considering Operational Parameters

To effectively predict the mechanical dispatch reliability (MDR), the artificial neural networks method combined with aircraft operation health status parameters is proposed, which introduces the real civil aircraft operation data for verification, to improve the modeling precision and computing efficiency. Grey relational analysis can identify the degree of correlation between aircraft system health status (such as the unscheduled maintenance event, unit report event, and services number) and dispatch release and screen out the most closely related systems to determine the set of input parameters required for the prediction model. The artificial neural network using radial basis function (RBF) as a kernel function, has the best applicability in the prediction of multidimensional, small sample problems. Health status parameters of related systems are used as the input to predict the changing trend of MDR, under the artificial neural network modeling framework. The case study collects real operation data for a certain civil aircraft over the past five years to validate the performance of the model which meets the requirements of the application. The results show that the prediction quadratic error Ep of the model reaches 6.9 × 10−8. That is to say, in the existing operating environment, the prediction of the number of delay ＆ cancel events per month can be less than once. The accuracy of RBF ANN, BP ANN and GA-BP ANN are compared further, and the results show that RBF ANN has better adaptability to such multidimensional small sample problems. The efforts of this paper provide a highly efficient method for the MDR prediction through aircraft system health state parameters, which is a promising model to enhance the prediction and controllability of the dispatch release, providing support for the construction of the civil aircraft operation system.

Klasifikasi Citra Tanaman Perdu Liar Berkhasiat Obat Menggunakan Jaringan Syaraf Tiruan Radial Basis Function

Wild plants or what are usually called weeds are plants that are considered harmful because they grow in unwanted places. But it turns out that some wild plants have many benefits for the health of the human body. Wild plants have many forms of vegetation, one of which is often encountered is shrubs. There are many wild herbaceous plants that are efficacious as medicine. However, most of the people who do not have knowledge about the types of wild shrubs that have medicinal properties. This study aims to implement the Radial Basis Function (RBF) algorithm for the classification of wild herbaceous plant species with medicinal properties by extracting color and texture features. The color feature extraction is based on the average RGB value, while the texture feature extraction uses a Gabor filter with the mean, entropy, and variance parameters of the magnitude image. The result of feature extraction becomes input data which will be managed by the RBF artificial neural network. RBF is a neural network that has three layers that have feedforward properties that can assist in solving classification or pattern recognition problems. Based on the test results, the precision value is 91%, recall is 89% and accuracy is 90%. These results show that the Radial Basis Function (RBF) algorithm with color and texture feature extraction can classify wild shrubs with medicinal properties well.

Prediction Performance Comparison of Risk Management and Control Mode in Regional Sites Based on Decision Tree and Neural Network.

The traditional risk management and control mode (RMCM) in regional sites has the defects of low efficiency, high cost, and lack of systematism. Trying to resolve these defects and explore the application possibility of machine learning, a characteristic dataset for RMCM in regional sites was established. Three decision tree (DT) algorithms (CHAID, EXHAUSTIVE CHAID, and CART) and two artificial neural network (ANN) algorithms [back propagation (BP) and radial basis function (RBF)] were implemented to predict RMCM in regional sites. The results showed that in the aspects of accuracy (ACC), precision (PRE), recall ratio (REC), and F1 value, CART–DT was superior to CHAID–DT and EXHAUSTIVE CHAID–DT (E-CHAID–DT); and BP–ANN was superior to RBF–ANN. However, CART–DT was inferior to BP–ANN in ACC, PRE, REC, and F1 value. BP–ANN model is good at non-linear mapping, and it has a flexible network structure and a low risk of over-fitting. The case study of a typical county demonstration area confirmed the extensibility of the method, and the method has great potential in RMCM prediction in regional sites in the future.

Machine Learning Approaches to Estimate Flow Rate of Drip Irrigation Emitter Based on Structure Parameters and Pressure

The agricultural water-saving irrigation systems need to be adapted to local conditions, and irrigation emitters with different flow rates need to be designed to meet actual needs. In this study, machine learning approaches were used to predict the flow rate of the labyrinth emitters. The structural parameters of the emitter [number of channel units (N), channel unit length (L), channel width (W), tooth height (H) and channel depth (D)] and working pressure (P) were considered as independent variables. The applied machine learning models included k-nearest neighbor (KNN), multi-layer perceptron (MLP), support vector machine (SVM) and radial basis function artificial neural network (RBF-ANN). The accuracy of the machine learning model was evaluated by the mean square error (MSE) and coefficient of determination (R 2). The results show that the MSE and R2 of the RBF-ANN model were higher than the corresponding parameters of the other three models, indicating the RBF-ANN model can predict the flow rate of the labyrinth emitter more accurately. The research provides a reference for the rapid design of emitters with different flow rate. It is helpful to realize the automation of agricultural irrigation, and then realize the construction of a smart agricultural irrigation system.

The importance of permafrost in the steady and fast increase in net primary production of the grassland on the Qinghai–Tibet Plateau

Permafrost affects soil water and soil temperature regimes; however, its effects on net primary production (NPP) remain unknown. Here, we examined temporal-spatial changes in grassland NPP during 2000–2018 in permafrost and permafrost-free areas on the Qinghai–Tibetan Plateau using the random forest (RF) and radial basis function artificial neural network (RBF-ANN). Our results indicated that the areas that showed increasing, decreasing, and non-significant trends for NPP accounted for 13.88%, 1.90%, and 84.22% of the permafrost area, respectively. For the permafrost-free areas, these NPP trends accounted for 22.25%, 2.68%, and 75.07% of the permafrost-free area, respectively. The mean NPP in the permafrost regions showed a faster and steadier (1.520 g C/m2/yr, p < 0.05) increase than in non-permafrost regions (1.224 g C/m2/yr, p < 0.05). The Biome-BGC model confirmed that these spatial NPP patterns could be attributed to differences in soil water and soil temperature between permafrost and permafrost-free areas. Both the soil temperature and soil water content in permafrost sites exhibited relatively lower variance than in permafrost-free sites. Although many factors may be attributed to these patterns, our results suggest that there is a possibility that the relatively stable change in permafrost NPP can be explained by the fact that permafrost can regulate soil water and temperature regimes. Therefore, climate warming can increase NPP in cold regions, and permafrost degradation may destabilize the grassland ecosystem, which may cause NPP values to exhibit greater interannual changes in the future.

Improving Plot-Level Model of Forest Biomass: A Combined Approach Using Machine Learning with Spatial Statistics

Estimating the aboveground biomass (AGB) at the plot level plays a major role in connecting accurate single-tree AGB measurements to relatively difficult regional AGB estimates. However, AGB estimates at the plot level suffer from many uncertainties. The goal of this study is to determine whether combining machine learning with spatial statistics reduces the uncertainty of plot-level AGB estimates. To illustrate this issue, this study evaluates and compares the performance of different models for estimating plot-level forest AGB. These models include three different machine learning models [support vector machine (SVM), random forest (RF), and a radial basis function artificial neural network (RBF-ANN)], one spatial statistic model (P-BSHADE), and three combinations thereof (SVM &amp; P-BSHADE, RF &amp; P-BSHADE, and RBF-ANN &amp; P-BSHADE). The results show that the root mean square error, mean absolute error, and mean relative error of all combined models are substantially smaller than those of any individual model, with the RF &amp; P-BSHADE combined method generating the smallest values. These results indicate that a combined approach using machine learning with spatial statistics, especially the RF &amp; P-BSHADE model, improves the accuracy of plot-level AGB models. These research results contribute to the development of accurate large-forested-landscape AGB maps.

Physics-Informed Neural Network for High Frequency Noise Performance in Quasi-Ballistic MOSFETs

A physics-informed neural network (PINN) model is presented to predict the nonlinear characteristics of high frequency (HF) noise performance in quasi-ballistic MOSFETs. The PINN model is formulated by combining the radial basis function-artificial neural networks (RBF-ANNs) with an improved noise equivalent circuit model, including all the noise sources. The RBF-ANNs are utilized to model the thermal channel noise, induced gate noise, correlation noise, as well as the shot noise, due to the gate and source-drain tunneling current through the potential barriers. By training a spatial distribution of the thermal channel noise and a Fano factor of the shot noise, underlying physical theories are naturally embedded into the PINN model as prior information. The PINN model shows good capability of predicting the noise performance at high frequencies.

Recognition of Human Emotion using Radial Basis Function Neural Networks with Inverse Fisher Transformed Physiological Signals

Emotion is a complex state of human mind influenced by body physiological changes and interdependent external events thus making an automatic recognition of emotional state a challenging task. A number of recognition methods have been applied in recent years to recognize human emotion. The motivation for this study is therefore to discover a combination of emotion features and recognition method that will produce the best result in building an efficient emotion recognizer in an affective system. We introduced a shifted tanh normalization scheme to realize the inverse Fisher transformation applied to the DEAP physiological dataset and consequently performed series of experiments using the Radial Basis Function Artificial Neural Networks (RBFANN). In our experiments, we have compared the performances of digital image based feature extraction techniques such as the Histogram of Oriented Gradient (HOG), Local Binary Pattern (LBP) and the Histogram of Images (HIM). These feature extraction techniques were utilized to extract discriminatory features from the multimodal DEAP dataset of physiological signals. Experimental results obtained indicate that the best recognition accuracy was achieved with the EEG modality data using the HIM features extraction technique and classification done along the dominance emotion dimension. The result is very remarkable when compared with existing results in the literature including deep learning studies that have utilized the DEAP corpus and also applicable to diverse fields of engineering studies.

Mathematical simulation and prediction of tumor volume using RBF artificial neural network at different circumstances in the tumor microenvironment.

Uncontrolled proliferation of cells in a tissue caused by genetic mutations inside a cell is referred to as a tumor. A tumor which grows rapidly encounters a barrier when it grows to a certain size in presence of preexisting vasculature. This is the time when it has to find a way to go on the growth. The tumor starts to secrete tumor angiogenic factors (TAFs) and stimulate preexisting vessels to grow new sprouts. These new sprouts will find their way to the tumor in the extracellular matrix (ECM) by the gradient of TAF. As these new capillaries anastomose and reach tumor, fresh oxygen is available for the tumor and it will reinitiate the growth. Number of initial sprouts, distance of initial tumor cells from the vessel(s) and initial density of the tumor at the time of sprout formation are questions which are to be investigated. In the present study, the aim is to find the response of tumor cells and vessels to the reciprocal effects of each other in different circumstances in the tissue. Together with a mathematical formulation, a radial basis function (RBF) neural network is established to predict the number of tumor cells at different circumstances including size and distance of initial tumors from the parent vessel. A final formulation is given for the final number of tumor cells as a function of initial tumor size and distance between a parent vessel and a tumor. Results of this simulation demonstrate that, increasing the distance between a tumor and a parent vessel decreases the number of final tumor cells. Specially, this decrement becomes faster beyond a certain distance. Moreover, initial tumors in bigger domains must become much bigger before inducing angiogenesis which makes it harder for them to survive.