A hybrid time-frequency parametric modelling of medical ultrasound signal transmission

Chiara Razzetta; Valentina Candiani; Marco Crocco; Federico Benvenuto

doi:10.3934/acse.2023011

Article Contents

2023, Volume 1, Issue 3: 249-270. Doi: 10.3934/acse.2023011

This issue Previous Article Next Article Simulating two-dimensional optimal control problem of fractional partial differential equations

A hybrid time-frequency parametric modelling of medical ultrasound signal transmission

1.
Department of Mathematics, University of Genova, via Dodecaneso 35, IT-16146 Genova, Italy
2.
Esaote S.p.A., Via E. Melen 77, IT-16152 Genova, Italy

^*Corresponding author: Chiara Razzetta
^*Corresponding author: Chiara Razzetta

Received: May 2023

Revised: July 2023

Early access: August 2023

Published: September 2023

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Abstract

Medical ultrasound imaging is the most widespread real-time non-invasive imaging system and its formulation comprises signal transmission, signal reception, and image formation. Ultrasound signal transmission modelling has been formalized over the years through different approaches by exploiting the physics of the associated wave problem. This work proposes a novel computational framework for modelling the ultrasound signal transmission step in the time-frequency domain for a linear-array probe. More specifically, from the impulse response theory defined in the time domain, we derived a parametric model in the corresponding frequency domain, with appropriate approximations for the narrowband case. To validate the model, we implemented a numerical simulator and tested it with synthetic data. Numerical experiments demonstrate that the proposed model is computationally feasible, efficient, and compatible with realistic measurements and existing state-of-the-art simulators. The formulated model can be employed for analyzing how the involved parameters affect the generated beam pattern, and ultimately for optimizing measurement settings in an automatic and systematic way.

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Mathematics Subject Classification: Primary: 92C55, 94A08; Secondary: 68U10, 94A12.

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Figure 1. Depiction of a beam pattern resulting from a linear-array probe with $ 2N $ elements of area $ A $, of which only $ M $ are activated, symmetrically with respect to the probe central axis

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Figure 2. Behaviour of the sinc function with increasing number of cycles $ C $

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Figure 3. Depiction of the geometry adopted for the calculation of a spatial impulse response for a fixed field point $ \vec{r} $ and a probe element. The element discretization grid is represented by the red dots, each of which is associated with the distance $ d_j $ and the angle $ \phi_j $

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Figure 4. Average BP error $ D(BP, BP^F) $ on the $ 12 $ wideband and $ 12 $ narrowband cases, as a function of the number of points along depth. The error is computed for $ 5 $ different resolution sizes along depth, with a fixed $ dx = \frac{pitch}{4} $

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Figure 5. Comparison of generated wideband BPs in different settings. On each column from the left: BPs generated with our simulator parUST, BPs from FIELD II, their absolute difference, and the distribution of pixel value differences. The unity of measure is dB

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Figure 6. Comparison of generated narrowband BPs in different settings. On each column from the left: BPs generated with our simulator parUST, BPs from FIELD II, their absolute difference, and the distribution of pixel value differences. The unity of measure is dB

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Figure 7. Comparison of generated narrowband BPs using a $ 9 MHz $ pulse and different apertures. On each column from the left: BPs generated with our simulator parUST, BPs from FIELD II, their absolute difference, and the distribution of pixel value differences. The unity of measure is dB

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Figure 8. Examples of the parameters' domain for a linear-array probe. From the top, each row corresponds to a different fixed frequency $ f_0 = 4, \, 4.5, \, 5, \, 5.5 \, MHz $, where $ M = 8 $ ($ \bar{M} = 4 $) elements are active, with the two central ones not delayed. On each row the axes $ x, y, z $ represent the first three components of the PCA analysis, respectively. In all plots, each point corresponds to a BP represented through the three components instead of each pixel of which it is composed, while the color codifies a delay, one for each column

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Table 1. Quantitative measure $ D $ for comparing $ 12 $ experiments in the wideband case. Each row contains the values for a different choice frequency, number of active elements M, and depth of focus F

	Frequency 3.0 MHz		Frequency 4.5 MHz
F (mm)	$ M = 20 $	$ M = 50 $	$ M = 20 $	$ M = 50 $
10	4.91e-4	9.11e-5	2.17e-4	6.43e-5
25	1.05e-3	9.20e-5	3.57e-4	4.25e-4
35	1.21e-3	2.83e-3	4.51e-4	1.58e-3

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Table 2. Quantitative measure $ D $ for comparing $ 12 $ experiments in the narrowband case. Each row contains the values for a different choice of frequency, number of active elements M, and depth of focus F

	Frequency 3e6 Hz		Frequency 4.5e6 Hz
F (mm)	$ M = 20 $	$ M = 50 $	$ M = 20 $	$ M = 50 $
10	5.83e-4	1.10e-4	9.03e-6	1.13e-5
25	1.37e-5	3.05e-5	1.04e-5	1.10e-5
35	1.27e-3	1.15e-4	1.75e-5	7.28e-5

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Table 3. Quantitative measure $ D $ for comparing $ 9 $ experiments for the evaluation of the grating lobes. Each row contains the values for a different choice of number of active elements M and depth of focus F, for a fixed frequency $ 9 \, MHz $

	Frequency 9e6 Hz
F (mm)	$ M = 20 $	$ M = 24 $	$ M = 28 $
15	6.12e-4	5.14e-4	4.35e-4
25	1.1e-3	1.2e-3	1.1e-3
35	6.4e-4	6.8e-4	7.7e-4

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Table 4. Comparison of average computational times needed for the computation of maps and BPs on a laptop and on a workstation (WS), with both our simulator (first $ 4 $ columns) and FIELD II (last $ 2 $ columns). Once the maps are computed and stored, the running time for BP generation is significantly advantageous with respect to FIELD II

	Maps Computing		BPs Computing		FIELD II
	Laptop	WS	Laptop	WS	Laptop	WS
Wide	720 s	50 s	8 s	5 s	80 s	76 s
Narrow	246 s	16 s	0.007 s	0.004 s	115 s	105 s

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