Estimation of tumor parameters using neural networks for inverse bioheat problem

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Highlights

  • These simulations resulted in a temperature distribution profile that could be used to locate and determine the parameters of the cancerous tumor within the breast.

  • A mathematical neural network model was used to estimate the required parameters by computing the data obtained from the bioheat transfer simulations.

  • The architecture of the NN Model in use was optimized by using a specific number of neurons in the hidden layer that would result in the lowest RMSE values.

  • The prediction accuracy showed the capacity of the trained Feed Forward Neural Network to estimate the unknown parameters within an acceptable range of error.

  • A non-intrusive method for the diagnosis of breast cancer was modelled, which yields conclusive results for the estimation of the tumor parameters.

Abstract

Background and objective

Some types of cancer cause rapid cell growth, while others cause cells to grow and divide at a slower rate. Certain forms of cancer result in visible growths called tumors. This work proposes an inverse estimation of the size and location of the tumor using a feedforward Neural Network (FFNN) model.

Methods

The forward model is a 3D model of the breast induced with a tumor of various sizes at different locations within the breast, and it is solved using the Pennes equation. The data obtained from the simulation of the bioheat transfer is used for training the neural network. In order to optimize the neural network architecture, the work proposes varying the number of neurons in the hidden layer and thus finding the best fit to create a relationship between the temperature profile and tumor parameters which can be used to estimate the tumor parameters given the temperature profile.

Results

These simulations resulted in a temperature distribution profile that could thus be used to locate and determine the parameters of the cancerous tumor within the breast. The prediction accuracy showed the capacity of the trained Feed Forward Neural Network to estimate the unknown parameters within an acceptable range of error. The model validations use the Root Mean Square Error method to quantify and minimize the prediction error.

Conclusions

In this work, a non-intrusive method for the diagnosis of breast cancer was modelled, which yields conclusive results for the estimation of the tumor parameters.

Introduction

Cancer is a broad term describing the disease that is a consequence of cellular changes causing uncontrolled growth and division of cells. Certain types of cancer cause rapid cell growth, while others cause cells to grow and divide at a slower rate. Certain forms of cancer result in visible growths called tumors, while others, such as leukemia [1]. Breast cancer starts in the breast, usually in the inner lining of the milk ducts or lobules of the breast. There are various types of breast cancer, having different stages, aggressiveness, and genetic makeup. Treatment includes surgery, radiation, and drugs (hormone therapy and chemotherapy) [2].

Cancer diagnosis, at its earliest stages, often provides better chances for a cure. During a physical exam for diagnosis, the doctor may look for abnormalities like changes in the skin colour or enlargement of an organ that may perhaps indicate the presence of cancer. Laboratory tests, such as urine and blood tests, may also help doctors identify abnormalities that can be caused by cancer. Imaging tests are a non-invasive way to diagnose cancer and include bone scintigraphy, computerized tomography (CT) scan, positron emission tomography (PET) scan, magnetic resonance imaging (MRI), ultrasound, and X-ray, among others. During a biopsy, the doctor collects a sample of cells for testing the presence of cancerous cells in the laboratory [3].

A significant limitation of the current approaches is that the subjective morphological examination of cytological specimens alone is highly prone to error. It becomes notoriously difficult to differentiate between normal and dysplastic cells at an early stage of malignant transformation. Also, the use of intrusive methods to estimate the size and position of the tumor is a relatively cumbersome and annoying process. Non-intrusive methods have an advantage in this aspect [4].

A neural network is a set of algorithms that seeks to understand underlying relationships in a dataset through methods that closely resemble how the human brain works. Neural networks model itself based on the various intricacies present in the dataset so the results that the network generates do not have to change according to the changes in data rather it takes into account the changes and tries to create the best possible output based on the parameters that are used while designing the network. A “neuron” in a neural network is a mathematical function that processes and tries to find correlations between the various aspects of the data that enters it. The Neural network finds a correlation between the input and output signals. Hidden layers refine and tweak the input weights until the neural network's margin of error reaches close to acceptable levels throughout training the model. It is hypothesized that hidden layers give more importance to certain features in the input data that have a greater say on the output that is predicted [5]. For the experiment that is undertaken in this paper, there is a need to find patterns from the thermograph readings of the tumor. Since it is a task that involves the usage of vast amounts of data and finding patterns, applying neural networks would be the best option to get the desired results.

Pennes [6] proposed a simplified bioheat model describing the effects of metabolic heat generation and blood perfusion on heat transfer within living tissues. Kumar et al. [7] use the Pennes equation to govern bioheat transfer within the mathematical model of a breast tissue embedded with a tumor. Ströher et al. [8] advocate the use of heat as a powerful tool in diagnosing cancers, with an added advantage of being a non-intrusive or non-invasive method of examination. N. M. Sudharsan et al. [9] found that the surface temperature distribution of the breast provides information on the presence of a tumor. This distribution has a relation to the size and location of the tumor and can be seen using thermography, where the infrared radiation emitted from the surface of the breast is recorded and a thermal pattern obtained. Subhadeep et al. [10] proposed the estimation of the position and size of a spherical tumor in a human breast using temperatures obtained on the surface of the breast through a breast thermogram in conjunction with artificial neural networks. Agnelli et al. [11] have presented an estimation methodology to determine the unknown thermophysical or geometrical parameters of the tumor region using the temperature profile on the skin surface obtained by infrared thermography. The bioheat Pennes equation is solved for and the parameters are determined from the temperature profiles obtained from simulated data. Umadevi et al. [12] proposed a framework for estimation of tumor size using clever algorithms and the radiative heat transfer model. They incorporated the more realistic Pennes bio-heat transfer model and Markov Chain Monte Carlo (MCMC) method and established the capacity of this method to give quick, dependable results against erroneous data in tune with the accuracy standards today. Mital et al. [13] presented a framework to determine the breast tumor parameters using the profile of surface temperature obtained by infrared thermography. The estimation methodology involves evolutionary algorithms with an artificial neural network (ANN) and genetic algorithm (GA) model. Convergence plots of the mean squared error of the neural network output during training were obtained. Kumar et al. [7] estimated the unknown heat flux at the surface of a solid body. Hafid et al. [14] aimed to develop an inverse heat transfer approach to predict the time-varying freezing front and temperature distribution of tumors during cryosurgery. The direct model rests on a one-dimensional Pennes bioheat equation. Recently some interesting works were published in field of application of Neural Networks which can be interesting for future works [15], [16], [17], [18].

This work proposes to solve the problem of intrusive canker tumor detection by using a method that can decide the cancer tumor parameters from the temperature profile of the cancerous tissue. The process involves the generation of cancer tumor models that are based on the penne's bioheat equation and using this data to train a neural network model that can detect the cancer tumor parameters from the temperature profile distribution from the surface of the body.

Section snippets

Problem statement

A 3D model of the human breast embedded with a spherical tumor is simulated according to the Pennes bioheat transfer equation. The temperature distribution of the breast tissue from the simulation generated serves as the basis of data for training the ANN model. Multiple training cases are simulated by varying the size and position of the tumor inside the breast to generate sufficient data for training the neural network created in the present work. The optimum neuron count in the hidden layer

Modelling of breast tissue and material selection

The software chosen for modeling and simulation purposes is COMSOL Multiphysics® and the parameters such as shape, size, and material of the base model are decided based on the following reasons. Anderson et al. [19] have reported that the average woman in almost every country, except in the US, has pear-shaped (semi-ellipsoid) breasts rather than the conventional round (hemispherical) shape. Contijoch et al. [20] also advocate ellipsoid fitting in the shape analysis of the anatomy of the

RMSE values and the selection of hidden layer neuron

The simulations using the neural networks had found out that the Root Mean Square Error is very high when the neurons are less and it decreases with increasing the number of neurons in the hidden layer. There is a general trend of decreasing Root Square Mean Error with the increase in the number of neurons until the optimum number of neurons with the least amount of error is reached, except for the odd outlier in the hidden layer. In the simulation, the findings show that the least error is

Conclusion

The trained model was tested on 7 entirely different models and simulations. The results have been tabulated in Table 2 and the following observations are made. For models embedded with tumors whose size lies in categories T1a, the prediction of tumor parameters is noticeably different from the actual parameters. This is especially the case when such tumors are located in breasts of comparatively larger sizes. However, for tumors larger than 0.5cm (size category > T1a), the model predicts the

Declaration of Competing Interest

There is not any conflict of interest for all authors of this manuscript.

Acknowledgments

The authors would like to thank Deanship of Scientific Research at Majmaah University for supporting this work under Project Number No. R- 2021-79.

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