Sub and ultraharmonic (SUH) ultrasound contrast imaging is an alternative modality to the second harmonic imaging, since, in specific conditions it could produce high quality echographic images. This modality enables the contrast enhancement of echographic images by using SUH present in the contrast agent response but absent from the nonperfused tissue. For a better access to the components generated by the ultrasound contrast agents, nonlinear techniques based on Hammerstein model are preferred. As the major limitation of Hammerstein model is its capacity of modeling harmonic components only, in this work we propose two methods allowing to model SUH. These new methods use several Hammerstein models to identify contrast agent signals having SUH components and to separate these components from harmonic components. The application of the proposed methods for modeling simulated contrast agent signals shows their efficiency in modeling these signals and in separating SUH components. The achieved gain with respect to the standard Hammerstein model was 26.8 dB and 22.8 dB for the two proposed methods, respectively.
Introduction of contrast agents in ultrasound medical imaging has strongly improved the image contrast leading to a better medical diagnosis [
Block diagram of second harmonic imaging.
Although the second harmonic imaging possesses undoubted advantages compared to standard echographic imaging, contrast harmonic imaging, however, has image contrast limitations related to the presence of harmonic components of nonlinear nonperfused tissues [
Block diagram of subultraharmonic imaging.
To extract such SUH components from the whole spectrum, a certain number of approaches called “black box methods” has been proposed such as those based on the multiple input and single output (MISO) Volterra filtering [
In order to reduce the complexity of such methods and to extract SUH components from all spectral components, we propose two new original approaches neither based on the Volterra filtering but based on the Hammerstein filtering.
This paper is organized as follows: after recalling standard Hammerstein model, the new methods are presented. To validate our methods, we propose realistic simulations of contrast agents signals. Then a quantitative and qualitative comparison is made between the two proposed methods with respect to standard Hammerstein model. Finally a discussion completed by a conclusion closed the paper.
Polynomial Hammerstein model is a special type of nonlinear filters in which a static nonlinear system is followed by a dynamic linear system [
Block diagram of Hammerstein model.
As for the Volterra decomposition, the Hammerstein decomposition is able to model harmonic components, but it was unable to model sub and ultraharmonic (SUH) components. Before explaining how it was possible to model SUH components, we recall the Hammerstein decomposition.
The Hammerstein modeled signal
In each branch, the signal
with
Consequently, identifying a nonlinear system of input
As previously mentioned, Hammerstein model is able to model harmonic components only. This is justified by the steady state theorem reported in [
Block diagram of method 1, modeling by input frequency downshifting.
Thus, modeling SUH components of frequency
Each of the two proposed methods consists of two steps: one step for harmonic modeling and another step for SUH modeling. The first method is based on the modification of the input frequency, while the second one is based on the modification of the output frequency.
As previously mentioned, this method consists of two steps; each step uses one Hammerstein model as presented in Figure
(1) Harmonic modeling: harmonic modeling is done by identifying the system of input
(2) sub and ultraharmonic modeling: the SUH information is found in the difference signal
The spectral content of
The modified input signal
with
Then, the SUH signal
Finally, the total modeled signal
For this method, note that the maximal frequency that could be modeled is limited by the order
This method also consists of two steps, on step dedicated for harmonic modeling and another step dedicated for SUH modeling. Each step uses one Hammerstein model as presented in Figure
Block diagram of method 2: modeling by output frequency shifting.
(1) Harmonic modeling: this step is the same as the first step of the method 1. The signal
(2) Sub and Ultraharmonic modeling: based on the same idea as reported in method 1, SUH components could be modeled when they are considered as integer multiples of the input frequency. In this method, we propose to keep the input signal
In vector form,
with
Then the signal
has SUH components upshifted by
The final signal
To validate the two proposed methods and to quantify their performance in ultrasound medical imaging, realistic simulations are proposed. To achieve these simulations, the free simulation program Bubblesim developed by Hoff [
The incident burst is a sinusoidal wave composed of
The parameters of contrast agent.
Resting radius 

Shell thickness 

Shear modulus 

Shear viscosity 

In this research work, the performances of the different methods are evaluated qualitatively and quantitatively.
Figure
(a) Comparison between the signal backscattered by the contrast agent
Figure
Figure
(a) Comparison between the backscattered difference signal between the backscattered signal by the contrast agent (black) and the SUH signal (green) modeled with (top) method 1 and (bottom) method 2. (b) Spectra of different signals presented in (a). Here
To quantify the performance of each method and to know which method provides the best performance, a quantitative study is necessary. The relative mean square error EQMR defined as
is evaluated for different noise levels at the system output. The noise level adjusted as the function of SNR (signal to noise ratio) is Gaussian and white. Ten realizations are made to evaluate the fluctuations of RMSE. RMSE for
Variation of RMSE in dB between the backscattered signal by the contrast agent and that modeled with (blue) the standard Hammerstein model, (black) method 1, and (green) method 2 as a function of the memory of Hammerstein model in presence of output noise:
These simulations show that the RMSE achieved with the two proposed methods 1 and 2 is always less than the RMSE achieved with the standard Hammerstein model for the different SNR values.
The gap between the standard model and the two methods 1 and 2 decreases when the SNR value increases. A gap ranging from
Table
RMSE between the signal backscattered by the contrast agent and that modeled with the Hammerstein model, method 1, and method 2.
Standard Hammerstein  Method 1  Method 2  

RMSE (dB)  −8.3  −30.5  −31.1 
In this research work the problem of modeling sub and ultraharmonics with Hammerstein model is presented. Usually, the standard Hammerstein model is able to model harmonics only, which are integer multiples of the input frequency. Sub and ultraharmonics could not be modeled.
In this work, we propose for the first time two new methods that use Hammerstein filters that model sub and ultraharmonics. The two methods are based on the same idea stipulating that modeling SUH with Hammerstein model is possible if the input signal or the output signal is modified In such a way that the SUH components become in the position of integer multiples of the input frequency.
Each method uses two Hammerstein models successively. The first one is dedicated to model harmonic components and the second one to model the SUH components.
The first method (method 1) applies a spectral downshifting of
The two proposed methods are characterized by its simplicity. The originality of these methods is that they allow for the first time both the modeling of contrast agents signals and the separation of SUH components of the contrast agent response.
However, the two methods do not present the same advantages and disadvantages.
Method 1, which is based on the modification of the input signal, is less sensitive to the noise compared to method 2, which is based on the modification of the output signal. This is due to the fact that all the noise generated in the different parts of the non linear procedure are added to the output.
On the other hand, although method 1 has a more simple structure, it is slower than method 2. This is due to the fact that the second step of method 1 requires an order higher than method 2, the order of the first step being fixed. And as the computation time is related to the order of the model, higher the order, the slower the method.
The application of the proposed methods for modeling the contrast agents response shows their efficiency in modeling and in separating SUH components from other harmonic components. Gains of
The two proposed methods find theirs applications in the field of sub and ultraharmonic contrast imaging in order to produce high contrast images. This work opens a new research axis for new modeling techniques of SUH using Hammerstein model or any other non linear models.
The authors would to thank the Lebanese Council of Scientific Research (CNRSL) for financing this work.