# Breast Tumor Simulation and Parameters Estimation Using Evolutionary Algorithms

Authors: Manu Mital; Ramana M. Pidaparti
Journal: Modelling and Simulation in Engineering
 (2008)

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## Abstract

An estimation methodology is presented to determine the breast tumor parameters using the surface temperature profile that may be obtained by infrared thermography. The estimation methodology involves evolutionary algorithms using artificial neural network (ANN) and genetic algorithm (GA). The ANN is used to map the relationship of tumor parameters (depth, size, and heat generation) to the temperature profile over the idealized breast model. The relationship obtained from ANN is compared to that obtained by finite element software. Results from ANN training/testing were in good agreement with those obtained from finite element model. After ANN validation, GA is used to estimate tumor parameters by minimizing a fitness function involving comparing the temperature profiles from simulated or clinical data to those obtained by ANN. Results show that it is possible to determine the depth, diameter, and heat generation rate from the surface temperature data (with 5% random noise) with good accuracy for the 2D model. With 10% noise, the accuracy of estimation deteriorates for deep-seated tumors with low heat generation. In order to further develop this methodology for use in a clinical scenario, several aspects such as 3D breast geometry and the effects of nonuniform cooling should be considered in future investigations.


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## Body

## 1. Introduction

Breast cancer,  which is common among
women, is an international concern. There are about 200 000 cases a year, and
it is estimated that there are more than 1 million women with undetected breast
cancer.  Among various techniques to detect breast cancer, infrared (IR) imaging
has been widely used since the late 1950’s. In general, the temperature of the
human body on the surface of the skin depends on the metabolic activity, the
blood flow, and the temperature of the surroundings. Any abnormality in the
tissue, such as the presence of a tumor, alters the normal temperature on the
skin surface due to increased metabolic activity of the tumor. Therefore,
abnormal skin temperature profiles are an indication of diseases such as mastitis,
benign tumors, fibrocystic breast disease, and cancer. Lawson [1] was the first
to propose the use of thermographic detection of breast cancer, when he
observed that the local temperatures of the skin over a tumor were
significantly higher (about 2-3 degrees)
than normal skin temperatures. Lawson and Chughtai [2] established that the
regional temperature difference over an embedded tumor was due to convection
effects associated with increased blood perfusion, and increased metabolism around
the tumor.Thermography, also known as thermal or infrared imaging, is a procedure
to determine if an abnormality is present in the breast tissue surface
temperature distribution. This abnormality in temperature distribution might
indicate the presence of an embedded tumor. Amalu [3] and Keyserlingk et al.
[4] have reviewed the role of infrared thermography with respect to the role of
IR imaging in the detection of breast cancer and risk indicator. They [3, 4]
concluded that the IR imaging has a significant role to play in breast cancer
screening due to its ability to image metabolic aspects of the breast and along
with sensitivity, specificity, and prognostics. Further aspects of IR imaging techniques of breast cancer and detection
methods from infrared images are described in detail and compiled in a recent book
by Diakides and Bronzino [5].In order to complement the IR imaging techniques of breast tumor
detection, models have been proposed to estimate the temperature distributions
over breast with and without tumors. For
example, Sudarshan et al. [6] studied the temperature distributions  in breast
cancer by considering a two-dimensional model first proposed by Romrel and Bland [7].
In this model, Pennes bioheat equation was used to describe the flow of thermal
energy. The breast model consisted of a subcutaneous fat layer, followed by a
gland layer and a deep muscle layer adjacent to the thoracic wall. Using this
model, the effects of the tumor size, depth, and the blood perfusion rate on
the surface temperature magnitude and profile were determined. Evolutionary algorithms based on neural networks and genetic
algorithms have been proposed to detect abnormalities in several  applications (such
as ECG, tomography, ultrasound); see, for example, Diakides and Bronzino [5]. Recently,
Mital and Scott [8] developed a method to determine parameters of an embedded heat
source representing a tumor using infrared imaging. It appears that based on
literature  there are no methods available to estimate the tumor parameters
based on the temperature distribution obtained from IR imaging techniques.The objective of this study is the development of a methodology to
estimate the depth, size, and heat generation rate of an embedded tumor in the
breast, based on temperature profile on the surface that may be obtained by infrared
thermography. This study differs from other works related to breast cancer in
that inverse methods are used in the estimation methodology. The methodology is
demonstrated with a simple model in order to illustrate the details of the
procedures involved. Several cases of various embedded tumor conditions were
presented to illustrate the generality of the method.


## 2. Estimation Methodology and Procedure

The estimation methodology involves evolutionary algorithms using
artificial neural network (ANN) and genetic algorithm (GA) as shown in Figure1.
The ANN is used to map the relationship of tumor depth, tumor size, and the
heat generation to the temperature profile over the idealized breast model. The
relationship obtained from ANN is compared and validated to that obtained by
standard numerical model-based finite element analysis software (ANSYS). GA is
used to estimate the tumor parameters (depth, size, and heat generation) by
minimizing a fitness function involving the temperature profiles obtained from
simulated or clinical data to those obtained by ANN. The fitness function is
given as a sum of the squares’ function that relates the data from the observed
thermographic temperature profile to the temperature profile for a given set of
estimated parameters. It is defined as
(1)S=[Y−T(β)]T[Y−T(β)], where Y and T are vectors containing the observed and estimated temperatures,
respectively, and superscript T denotes the transpose of the vector. The observed
temperature vector Y contains
surface temperatures that would be obtained experimentally from thermal
imaging. The estimated temperatures
contained in vector T are obtained
by the outputs of the trained neural network model using estimates for the
unknown parameters (from the population pool of the GA). The vector containing the estimates for unknown parameters is
denoted by β. In this study, the unknown parameter
vector β contains the depth, diameter, and the heat
generation rate of the source. Once the objective function was formulated, the
estimation scheme using GA was used to determine the unknown parameters.Figure 1
Overall estimation methodology
for prediction of breast tumor parameters.


## 3. Artificial Neural Network Model to Map Breast Temperature Profiles

A multilayer feed-forward ANN with backpropagation learning algorithm is
developed to represent the surface temperature profile of the breast as a
function of depth, diameter, and heat generation rate of the tumor. The network
is developed with the MATLAB Neural Network Toolbox using the “newff” function.
The learning rate was set reasonably low at 0.05 to ensure convergence of the
algorithm. There are many transfer functions available in MATLAB software.
After some experimentation, the “logsig” transfer function was chosen for the
hidden layers and the output neurons, due to nature of the desired
outputs. A schematic of the network is shown
in Figure2. The network has 3 input nodes for the three parameters, and 31
output nodes that define one half of the symmetrical temperature profile on the
skin surface. There are 2 hidden layers in the network, the first with 21
nodes, and the second with 19 nodes. This configuration was reached after a few
iterations of a single hidden layer network proved to give less than adequate
results. Two layer deep-nested
loops were used in an exhaustive search that varied the number of neurons in
the first and second hidden layers,
and found the appropriate combination (21-19) that best trained the network.
Therefore, a 3-21-19-31 neural network architecture was developed and trained
and tested to validate the model. More details about the neural networks and
their concepts can be found in Russell and Norvig [9].Figure 2
Schematic of the neural network.In
this study, a simplified 2D model of the cross-section of the breast similar to
that used by Sudershan et al. [6] was considered, as shown in Figure 3. The
Pennes bioheat equation has to be solved to estimate the temperature
distribution over the breast model. Due to the complexity of the breast
geometry, there are no known analytical solutions to the bioheat equation.
Therefore, solutions are usually obtained using computational techniques such
as the finite element method. A
finite element model created using a commercially available finite element
package, ANSYS, solves the Pennes bioheat equation to find the surface
temperatures, and this information is used in training/validating ANN model. In
this study, the breast is assumed to be of 0.072 m radius hemispherical shape,
with a subcutaneous fat layer of 0.005 m uniform thickness, followed by gland,
muscle, and the thoracic wall. The tumor
is situated symmetrically in the gland layer and assumed to be circular in
shape for simplicity. Note that the entire breast is modeled even though the 2D
geometry is symmetrical about the Y axis. This is done for future studies where
an offcenter tumor location may be considered. The heat transfer in each layer
is governed by (1). The values of thermal conductivity, metabolic heat
generation rate, and the perfusion source term for each layer, adopted from
Werner and Buse [10], are presented in Table 1. The thoracic wall is assumed to
be maintained at a constant core body temperature of 37°C. Based on estimate by
Osman and Afify [11], the surface of the breast is subjected to a combined heat
transfer coefficient (convection, radiation, and evaporation) of 13.5 W/m2-K,
a reasonable value in the absence of forced convection. The clinical
environment under which the thermographic measurements are made is assumed to
be at a constant temperature of 21°C.Table 1
Values of thermal conductivity, metabolic heat generation, and perfusion term.
LayerThermal
conductivity
(W/m-K)Metabolic
heat generation
(W/m3)Perfusion
(W/
m3-C)Subcutaneous
fat0.21400800Gland0.487002400Muscle0.487002400Tumor0.4814000–7000048000Figure 3
A 2D model of the breast based on
Sudershan et al. [6].Using the finite element analysis of the breast model, a total of 17
different cases were analyzed and used for training the ANN. The inputs
parameters are scaled between 0 and 1 using upper and lower bounds for each
parameter. The diameter of the tumor is allowed to vary between 0.005 and 0.015 m so that it is in mammary gland region. The bounds for depth are 0.014 and
0.041 m (based on the assumption that the tumor will most likely be present in
the gland). The lower and upper bounds on the heat generation rate of the tumor
are 20 times (14,000 W/m3) and 100 times (70,000 W/m3)
that of normal tissue. The output surface temperatures of the network are
scaled values between 0 and 1, using a lower limit of 26°C and upper limit of 33°C.
The network was trained for 400 epochs or training cycles using the data
generated from finite element analysis results. Three different cases were used
for validating the trained network.


## 4. Genetic Algorithms in Breast Tumor Parameters Estimation

In
the implementation of the GA,  the potential solution (chromosome) contains the depth, diameter, and the tumor heat
generation rate as the 3 genes and 10 bits are used to represent the value
of each gene in binary format.
Therefore, each chromosome has 30
bits. The population size has been
chosen to be 20. The initial population pool was generated using a
random number generator.The
parameters contained in eachchromosome are changed from their binary representation to floating point representation
using (for the ith parameter)
(2)Pi=a+b−a2nb−1(m), where Pi is the decimal value of ith parameter, a and b are the lower and
upper limits of the search interval, respectively, nb is the number of bits used to represent the parameter (e.g., 10
in this case), and m is the decimal value of the parameter in binary form. The
lower limit a and upper limit b for each of the three parameters were
mentioned  earlier.Thefitness of each chromosome is evaluated using
the sum of the squares objective function given by (2). The fitness f of a chromosome is related to the sum of the squares function using
(3)f=SMax−S(β), where SMAX was arbitrarily chosen
to be 10 so it always stays greater than
the maximum value of sum of squares possible, and S is the sum of the squares generated using a parameter set (chromosome) β.Thefitness is determined for each parameter set (chromosome)
and then ranked. Chromosomes of
higher fitness are more likely to be
selected to reproduce and contribute their genetic material to the next generation. A probability of mutation (random changing of a chromosome) of 0.10 and a probability of crossover (formation of children) equal to 0.90 are used to
determine how members of the population will reproduce to bring forth the next generation. A flowchart of the solution method is
presented in Figure 4. More details about GA procedures and applications can be
found in Goldberg [12] and Davis [13].Flowchart of the GA estimation
procedure.
(a)(b)Several cases were simulated in order to validate the ability of the
estimation procedure to accurately predict the depth, size, and heat generation
rate of the embedded tumor. For each parameter set, two runs of the GA were
conducted by introducing 5% and 10% noises in the simulation data to see the
sensitivity of the prediction of breast tumor parameters.


## 5. Results and Discussion

A convergence plot of the mean squared error of the neural network output
during training is shown in Figure5. In general, the results obtained from ANN
training were in good agreement with those obtained from finite element
analysis model. After the ANN is
trained, the network is validated for three different cases as shown in Figure 6.
It can be seen from Figure 6 that, for each validation case, the output of the
neural network model shows good agreement with the simulation results with R2 values of 0.936, 0.982, and 0.988, respectively. To further validate the ANN
results, the temperature profiles
obtained for a specific case 3 (depth of 0.029 m, diameter of 0.008 m, and heat
generation of 30000 W/m3) are compared with the finite element and ANN models as shown in
Figure 7. Once again a good comparison is seen.Figure 5
Convergence plot with mean
squared error of the network output during training.Figure 6
Comparison of output of trained
ANN with finite element simulation.Comparison of temperature profile
of trained ANN with direct numerical simulation.
(a)(b)The
parameter estimates for the depth, diameter, and heat generation rate of the
tumor for the four cases, corresponding to 5% and 10% random noise in the
temperature rise on the skin surface, are presented in Tables2 and 3,
respectively. MATLAB built in function “rand”
was used for generating random noise in the temperature rise on the skin
surface. For each case, the GA was run twice with different random initial
populations yielding similar results. With 5% random noise, the results show
good agreement between actual and predicted parameters. In this case, the absolute
errors in depth and size were within 0.001 m, and absolute errors in heat
generation rate were within 3000 W/m3. With 10% noise, the accuracy
of the estimated parameters deteriorates for deep seated tumors with low heat
generation rate. In this case, the absolute errors in depth were within 0.005 m,
error in size within 0.002 m, and in heat generation rate within 4870 W/m3.Table 2
Results from parameter estimation
procedure to determine embedded tumor parameters using temperature data with 5% noise.
CaseRunActualEstimatedDepth (m)Diameter (m)Heat
generation
(W/m3)Depth
(m)Diameter (m)Heat
generation
(W/m3)110.0150.008250000.0150.0072544020.0150.00727302210.0150.012400000.0140.0113912620.0150.01239619310.0350.008250000.0350.0082702820.0340.00728396410.0350.012400000.0360.0114033020.0360.01242082Table 3
Results from parameter estimation
procedure to determine embedded tumor parameters using temperature data with 10% noise.
CaseRunActualEstimatedDepth (m)Diameter (m)Heat
generation
(W/m3)Depth
(m)Diameter (m)Heat
generation
(W/m3)110.0150.008250000.0160.0072368920.0140.01021000210.0150.012400000.0150.0123900020.0140.01140987310.0350.008250000.0400.0102013020.0390.01020980410.0350.012400000.0370.0123562220.0380.01436428


## 6. Conclusions

An estimation methodology-based
evolutionary algorithm
using neural networks and genetic algorithms was developed to estimate the
breast tumor parameters based on surface temperature profile that may be
obtained by infrared thermography. The methodology is demonstrated with a
simple model in order to illustrate the details of the procedures involved. For
the cases studied, results from simulations show that it is possible to
determine the depth, diameter, and heat generation rate from the surface temperature
data (with 5% random noise) with good accuracy for the simplified 2D model.
With 10% noise, the accuracy of the estimated parameters deteriorates for deep-seated tumors with
low heat generation rate. With the recent advances in the accuracy of IR images
and noise reduction techniques, it may be possible to reduce noise to within
acceptable levels (5%) for the methodology to be effective. In general, the
breast shape is complex, and this should be considered for realistic studies.
However, the estimation methodology is general and can be applied even to realistic
breast geometry. This may increase the
computation time but the methodology can be easily extended. In order to further
develop the proposed methodology so that it can be used in a clinical scenario,
several aspects such as complexities of 3D breast geometry and the effects of
nonuniform cooling should be considered in future investigations.



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*Source: 756436-2008-05-25.xml*