TEMPERATURE MODELS OF A HOMOGENEOUS MEDIUM UNDER THERAPEUTIC ULTRASOUND

C. A. Teixeira*, G. Cortela**, H. Gomez**, M. G. Ruano*, A. E. Ruano*, C. Negreira** e W. C. A. Pereira***

* Universidade do Algarve/Centro de Sistemas Inteligentes, Faro, Portugal
** Universidad de la República /Lab. de Acústica Ultrasonora, Montevideo, Uruguay
*** Universidade Federal do Rio de Janeiro/COPPE, Rio de Janeiro, Brasil
cateixeira@ualg.pt

Abstract: Temperature modelling of human tissue subjected to ultrasound for therapeutic use is essential for an accurate instrumental assessment and calibration. Prior studies developed on a homogeneous medium are hereby reported. Non-linear punctual temperature modelling is proposed by means of Radial Basis Functions Neural Network (RBFNN) structures. The best-performed structures are obtained using a Multiobjective Genetic Algorithm (MOGA). The best performed neural structure presents a Root Mean Square Error (RMSE) of one order magnitude less than the one presented by the best behaved linear model - the AutoRegressive with eXogenous inputs (ARX); The maximum absolute error achieved with the neural model was 0.2 ºC.
Keywords: Temperature modelling, neural networks, multi-objective genetic algorithms, ultrasound.

Introduction

Accurate determination of human tissue temperature is a fundamental aspect concerning the secure application of therapeutic ultrasound instrumentation in cancer treatment (Hyperthermia). The main constrain for the generalised clinical use of hyperthermia is the lack of accurate knowledge of localised temperature patterns in time and space, which would enable a lesion-less treatment.

Having in mind non-invasive temperature estimation in time and space, previous work in the field relates the changes in sound velocity and medium expansion as linearly dependent of changes in temperature. The temperature range considered was between 20 and 24 ºC, using a therapeutic transducer to induce heat, and a diagnostic transducer to extract characteristics (sound velocity and medium expansion) from the medium [1].

The main objective of the work hereby presented is to compare the performance of neural networks versus linear ARX models, in punctual temperature estimation in a homogeneous medium radiated by therapeutic ultrasound. The temperature variation indicators considered were the amplitude of the fundamental component of the intensity spectrum, and the measured past temperature values. The work reported on this paper represents a first step towards the use of a neural model as an estimator and controller of temperature variations in biological tissues under therapeutic ultrasound.

Materials and Methods

Data acquisition: The real data used in this work are temperature and acoustic intensity signals, collected in a point 48 mm distant (axial distance) from an ultrasonic therapeutic transducer, in a glycerine (homogeneous medium) tank. Data was acquired during approximately 110 min. At each 10 seconds a temperature value was recorded, as well as a window of 5µs of the acoustic intensity signal, corresponding to 2000 points of the intensity waveform.

Figure 1: Experimental setup

Ultrasound in continuous operating mode was used, at 3 MHz. Three sets of signals were acquired according to the signal intensities applied: 1 Watt/cm², 1.5 Watt/cm², and 2 Watt/cm². The temperature ranges recorded are presented in Table 1. Glycerine was insonated only in the first 60 min, by the therapeutic device (Ibramed Sonopulse, Sãão Paulo). During the remaining 50 minutes the acoustical energy was maintained at a zero level, while temperature variations were observed and recorded.

Data processing: The construction of neural and ARX models requires the computation of features from the intensity signals in order to estimate temperature. In this paper only the fundamental component of the intensity spectrum, located at 3 MHz was computed.

Table 1: Temperature ranges

Accurate development of neural and ARX models requires pre-processing of data. Usually this phase encompasses a filtering operation followed by a normalisation operation. The filtering operation reduces the high frequency noise introduced by instrumentation. The normalisation is required to enable correct training of the models. That is, the attainment of well conditioned models with a high capacity of generalisation (good performance in different situations). In this work the measured temperature and the fundamental component of the intensity spectrum present smooth waveforms, making unnecessary the filtering process. The temperature and the fundamental component of the intensity waveforms were normalised in amplitude between 0 and 1.

In general, the construction of a neural model encompasses a training phase and a test phase. During the training phase the neural network parameters are computed, while during the test phase the generalisation capacities of the neural model obtained are accessed, using a different data set. In this paper the data collected at 1 Watt/cm² was used for training, while the data collected at 1.5 Watt/cm² was used for test.

RBFNN construction: A RBFNN is a three fully connected layers neural network. The first layer is a set of inputs, the second (hidden) layer is formed by a set of processing elements, called neurons. The outputs of the hidden layer are linearly combined to produce the last layer, where the overall network output is computed. The input/output relation for a RBF is given by:

where n is the number of neurons in the hidden layer, b is the bias term, ||.|| is a norm (an Euclidean norm was employed), and φ (|| x_j - c_i || ) is a set of non-linear radial basis functions weighted by {a_i}ⁿi=1. The basis functions are centred at {c_i}ⁿi=1 (centres) and are evaluated at points x_j. Usually the basis functions are Gaussian [2]:

In the construction of a RBFNN several questions arise [3]: What is the appropriate number of neurons in the hidden layer? Which are the important input variables to achieve a good model? What is the range of lag variability to consider such that the truly important lags where considered for each determinant feature? Which is the maximum lag to be considered?

The answers to these questions depend on the problem under study, being neither unique nor easy to determine. The number of possible structures can be enormous, disabling an exhaustive search due to the computational time involved on its implementation. To solve this problem a multi-objective genetic algorithm (MOGA) [4] [5] was applied to select the best-fitted RBFNN structures.

The input variables considered for the RBFNN were the fundamental component of the intensity spectrum (Im), and the measured temperature (T). The number of model inputs was restricted to the interval [2, 30], while the possible number of neurons is an integer in the interval [2, 15].

In this work the MOGA was defined to have 100 generations, of 100 RBFNN (individuals) each. The objectives to minimise were: training RMSE, test RMSE, maximum of the error auto-correlation (R_ee), and the maximum of the cross-correlation between the inputs and the error (R_ue). Having in mind the attainment of models with a high generalization capacity, the test RMSE was defined as a performance metric, where an optimum value 0.003 was seek. The maximum of the correlation tests (R_ee e R_ue) was also considered as a performance goal, with value CI = 1.96 / √N where N is the number of points in the training set. If R_ee and R_ue have values less than CI then the models are considered adequate with 95% of confidence [3]. In this work N = 377 and CI is approximately 0.1.

When applying the MOGA, the Levenberg-Marquardt (LM) algorithm was used for the training of each network (individual), using the Early-Stopping termination criteria [3].

The MOGA parameters were: 10% of random immigrants (RBFNN), selection pressure (probable number of copies of the best individual) of 2, crossover rate of 0.7, and mutation rate of 0.5.

After the MOGA selection, the validation of the best individuals was performed with a third data set, called validation data. These data was the one collected at 2 Watt/cm² of intensity. Such as in the training and test data, normalization between 0 and 1 was performed.

ARX construction: The most used ARX model is defined by the following difference equation:

This model relates the actual output y[t] with a finite number of past values of the output y[t-k], and of the input u[t-k]. The structure of the model is completely defined by three parameters: number of poles (na), number of zeros (nb-1), and time delay of the system (nk).

In this work and coefficients were determined using the least squares strategy.

Results

This section presents the best models obtained with the MOGA, called preferable models [3], as well as the best-performed ARX model (linear model). Both meth- ods were subject to the same data sets.

The preferable set is composed of 11 models. The inputs and other characteristics of these models are presented in Tables 2 and 3, respectively.

Table 2: Inputs of the preferable models

The numbers in Table 2 are represents I_m and T lags. For example model number 10 has as inputs: I_m(k-19), I_m(k-45), T(k-1), T(k-8), T(k-13), and T(k-22).

The columns R_ee, R_ue, ||W||, #Neu, and Val. RMSE in Table 3 present the maximum of the error autocorrelation, the maximum of the cross-correlation between the inputs and the error, the linear weights
() norm, the number of neurons, and the RMSE in the validation set, respectively. The values typed in bold indicate that the goal associated with the characteristic was fulfilled.

Table 3: Other preferable model characteristics, and associated goals defined in the MOGA.
(NA=Not Attributed).

The best ARX structure was computed considering a scanning of 48 lags (na=1,...,48, nb=1,...,48) for each variable, and a null delay for the in puts (nk=0). This model presents a RMSE in the validation set of 0.0253, and a maximum absolute error of 2.1 ºC.

Discussion

From the analysis of Table 2 it is possible to realise that the MOGA selects few lags of I_m (intensity) and many lags of T (temperature). The input T(k-1) appears in all preferable RBFNN, demonstrating the high dependency of the actual temperature on the temperature of the past 10 seconds. This fact indicates that the MOGA converges to a set of models with a physical meaning. The temperature 6 samples in the past (T(k-6)), that is 1 minute in the past appears with an absolute frequency of 9, disclosing its importance. The medium lags (between 20 and 40) appear with some frequency, in special the lags 20 and 35 with an absolute frequency of 6 and 5, respectively. The dependence of T(k) on the higher lags of the temperature is also visible. This affirmation is based on the existence of lags with values between 39 and 47. The MOGA selection of medium and higher lags is probably due to the thermal capacity of the glycerine tank. However, MOGA indicates that T(k) is dependent on the actual intensity value (Im(k)), and in the intensity 32 samples in the past (I_m(k-32)). The presence of few lags of Im, and the presence of many lags of T can be related with the application of a constant intensity in the 60 minutes of heating, and the reduction of the intensity to zero between 60 and ap- proximately 110 minutes. In other words, the dynamics of the system is completely dependent on the past values of the temperature.

From the analysis of Table 3 one may conclude that model 11 fulfils 2 out of 3 goals (test RMSE less than 0.003 and R_ue less than CI=0.1) defined in MOGA. This model has also a reduced number of neurons and presents the smallest RMSE in the validation set, being as so considered the best-performed model. The maximum absolute error for this model is 0.2 °C. The temperature estimated by this model and estimated by the best ARX model are presented in Figure 2, as well as the measured temperature at 2 Watt/cm2.

In the preferable set there are also models (1, 2, 4, 6, 7, and 9) that fulfil one or none of the goals defined, but those that are not ful- filled are close to the desired values, and these models are also considered good models. A common characteristic of these models is the smallest number of neurons, showing that the estimation of T(k) in a glycerine medium is well performed with a low order model (reduced complexity). The remaining models (3, 5, 8, and 10) present higher values for the RMSE in the validation set, showing a bad generalization capacity. All of these models present a high number of neurons, which can be explained by the MOGA attempt to fulfil the goals defined for R_ee and R_ue. Models with a high number of neurons tend to model the noise of the inputs, decreasing R_ee and R_ue values. For example the model number 3 has 15 neurons (maximum value defined), fulfilling the goals defined for R_ee and R_ue, but presents a high RMSE in the test and validation sets.

Figure 2: Estimated output by the model 11, estimated output by the best ARX, and the measured output (validation data).

Comparing the RMSE in the validation set obtained with both strategies and observing Figure 2, it can be stated that the modelling of the temperature dynamics can only be attained with success using non-linear procedures, such as the RBFNN. The increase in performance obtained, justifies the computational time spend in the selection of the best RBFNN structures.

Conclusion

This paper presents a preliminary study of the neural networks applicability in temperature estimation in a homogeneous medium, having in mind the determination of safe conditions for the application of ultrasound in therapy, such as, in hyperthermia. The results reveal that this kind of modelling can be accomplished with success (maximum absolute error less than 0.2 °C) using RBFNN models (non-linear modelling). The best RBFNN attains an increase in performance of one order of magnitude when in comparison with the best ARX model tested. Despite the real data used in this work was collected in a in-vitro environment, the results obtained point out that the RBFNN might bring great improvements on temperature estimation of biological tissues and eventually in-vivo, in real time.

Acknowledgements

The authors gratefully acknowledge the financial support of: Fundação para a Ciência e a Tecnologia (scholarship SFRH / BD / 14061 / 2003), and Conselho Nacional de Desenvolvimento Cientifico e Tecnológico (CNPq/CYTED 490.013/03-1), Brazil.

References

1. SIMON, S., VANBAREN, P. and EBBINI E. S., "Two-dimensional temperature estimation using diagnostic ultrasound", IEEE Trans. on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 45, pp. 1088-1099, 1998.
2. TEIXEIRA, C. A., RUANO, M. G. and RUANO, A. E. "Emboli classification using RBF neural networks", Proc. Sixth Portuguese Conference on Automatic Control, vol. 2, pp. 631-635, Faro, Portugal, 2004.
3. FERREIRA, P. M., RUANO A. E. and FONSECA, C. M., "Genetic assisted selection of RBF model structures for greenhouse inside air temperature prediction", Proc. IEEE Conference on Control Applications, Vols 1 and 2, pp. 576-581, Istanbul, Turkey, 2003.
4. FONSECA, C. M. and FLEMING, P. J., .Non-linear system identification with multiobjective genetic algorithms., Proc. 13th IFAC World Congress, vol C, San Francisco, California, E. U. A., pp. 187-192, 1996.
5. FONSECA, C. M. and FLEMING, P. J., .Multiobjective optimization and constraint handling with evolutionary algorithms I: A unified formulation., IEEE Trans. on Systems, Man and Cybernetics-Part A: Systems and Humans, vol. 28, no. 1, pp. 26-37, 1998.