# Performance Analysis of Structure Factor Model-Based Scatterer Estimation Using the Cramér–Rao Bound

Lorena Leon, Adrian Basarab, Jonathan Mamou, Pauline Muleki-Seya

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Lorena Leon, Adrian Basarab, Jonathan Mamou, Pauline Muleki-Seya. Performance Analysis of Structure Factor Model-Based Scatterer Estimation Using the Cramér–Rao Bound. 2025 IEEE International Ultrasonics Symposium (IUS), Sep 2025, Utrecht, Netherlands. pp.1-4, 10.1109/IUS62464.2025.11201746. hal-05359469

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# Performance Analysis of Structure Factor
Model-Based Scatterer Estimation Using the
Cramér-Rao Bound

Lorena Leon¹, Adrian Basarab¹, Jonathan Mamou², Pauline Muleki-Seya¹,

¹ Université de Lyon, CREATIS, CNRS UMR 5220, Inserm U1044, INSA-Lyon, Université Lyon 1, Lyon, France

² Weill Cornell Medicine, Department of Radiology, New York, New York 10022, USA

lorena.leon@creatis.insa-lyon.fr, pauline.muleki-seya@creatis.insa-lyon.fr

***Abstract***—The ultrasound backscatter coefficient (BSC) provides insight into the properties of acoustic scatterers for the quantitative assessment of tissue microstructure. The structure factor model (SFM) is an advanced model, specifically adapted for concentrated media, that allows the extraction of key scatterer parameters from BSC measurements. These include the radius *a*, volume fraction *φ*, and relative impedance contrast $\gamma_Z$ which were found close to the values of cellular structures at high frequency (~40 MHz). In this study, we focused on deriving the Cramér-Rao bound and comparing it with the mean squared error of the BSC parameters estimated using the SFM to understand the fundamental performance limits of the estimation process. We evaluated the impact of different parameter values on estimation performance in the presence of a Gaussian noise. The results indicate that estimation becomes more challenging for low values of $\gamma_Z$, *a*, and $\phi$.

***Index Terms***—Quantitative Ultrasound, Cramér-Rao Bounds, Structure Factor Model, Scatterer Parameters.

## I. INTRODUCTION

Ultrasound backscatter coefficient (BSC) enables quantitative assessment of tissue microstructure by providing insight into scatterer properties. This information may be valuable in different applications when the tissue microstructure is affected, e.g., non-alcoholic fatty liver disease evaluation [1], cell-death quantification during anti-cancer therapy [2], comparison between cancerous and healthy tissues [3].

Using an inverse problem approach, it is possible to extract scatterer parameters from BSC as the scatterer's radius and the acoustic concentration, when the Gaussian model [4] is used for example. This model deals with identical randomly and independently distributed scatterers, making it valid only for diluted medium. It is generally difficult to correlate the scatterer parameters to tissue structures from histology [5]. The structure factor model (SFM) [6], [7], combining the contributions of the incoherent and coherent BSC, is adapted

for a concentrated medium. This model enables to extract the scatterer radius, volume fraction and relative impedance contrast. The use of SFM yielded scatterer parameters close to cellular structures: nuclei for canine liver (with a relative error on radius and volume fraction <7%) and whole cells for HT29 tumors (with a relative error on radius and volume fraction <15%) [8]. In another study [9], the mean relative errors on scatterer radius and volume fraction were <14% (17%, respectively) for LMTK (MAT, respectively) cell-pellet (CP) biophantoms and mean relative errors <5% on radius for 4T1, JC, LMTK and MAT ex vivo tumors.

However, for a reliable estimation of scatterer parameters using the SFM, it is crucial to understand the fundamental performance limits of the estimation process. The Cramér-Rao bound (CRB) [10]–[12] provides a theoretical lower bound on the variance of unbiased estimators under mild regularity conditions and it can be used to benchmark the performance of statistical estimators. In ultrasound imaging, CRB has been used to optimize the design of experiments or to evaluate the effect of measurement parameters on the results of ultrasound imaging using transverse oscillation [13], quantitative ultrasound tomography [14], quasi-static elastography [15], photoacoustics [16] and quantitative acoustic microscopy [17]. Chaturvedi et al. [18] investigated errors in estimating scatterer sizes using the Gaussian model. They aimed to understand the nature of errors in the measurement process, and noticed the importance of the analysis bandwidth size to obtain satisfactory results.

The aim of our study is to derive the CRB to establish, for the first time, a theoretical benchmark for the SFM-based scatterer parameter estimators and to provide a performance analysis with respect to noise and parameter values.

## II. MATERIAL AND METHODS

### A. Structure Factor Model to Estimate Scatterer Parameters

SFM considers the scattering medium as an ensemble of identical fluid spheres. This model is adapted for concentrated media by taking into account the structure factor term

This work was supported by the ANR JCJC CARPUS project (ANR-21-CE19-0024) and the NIH grant R01GM143388. It was performed within the framework of the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program "Investissements d'Avenir" operated by the French National Research Agency (ANR).
<figure><img src="image_2.png" /><figcaption>Fig. 1. (A) Example of simulated BSC with and without noise ($a = 8.12$ µm, $\phi = 0.49$ and $\gamma_Z = 0.03$, $\sigma^2 = 0.22$). (B) SNR values obtained by varying $\gamma_Z$ and $\sigma^2$ and for $a = 8.12$ µm and $\phi = 0.49$. (C) Example of CRB and MSE of $\gamma_Z$ parameter for $a = 8.12$ µm, $\phi = 0.49$ and $\gamma_Z = 0.07$.</figcaption></figure>

$S(k, a, \phi)$ related to scatterer spatial positions. SFM is based on the assumption that, at high scatterer volume fractions, interference effects are mainly caused by correlations between the spatial positions of individual scatterers. By considering an ensemble of identical spheres of radius $a$, the theoretical BSC for the SFM formulation [6], [7] is given by:

$$BSC_{SFM}(k) = n \frac{k^4 V_s^2 \gamma_Z^2}{4\pi^2} \left[ \frac{3}{2ka} J_1(2ka) \right]^2 S(k, a, \phi), \quad (1)$$

where $k$ is the wavenumber, $V_s$ is the sphere volume and $n = \phi/V_s$ is the number density with $\phi$ the scatterer volume fraction, $\gamma_Z$ is the relative impedance contrast between the scatterer and the surrounding medium and $J_1$ is the spherical Bessel function of the first kind of order 1. $S(k, a, \phi)$ is the structure factor, which is analytically obtained as described in [7], [19]. The unknown parameters are the scatterer radius $a$, the volume fraction $\phi$, and the relative impedance contrast $\gamma_Z$.

In practice, we aim to estimate the parameter vector $\theta = (a, \phi, \gamma_Z) \in \mathbb{R}^3$ from a noisy observation $y \in \mathbb{R}^N$ of the theoretical BSC given by (1), at a set of discrete wavenumbers $k = k_1, \dots, k_N$. This is modeled as

$$y = g(\theta) + \epsilon, \quad (2)$$

where $g(\theta) = (BSC_{SFM}(k_1), \dots, BSC_{SFM}(k_N))$ is the predicted BSC vector for the given parameters, and $\epsilon$ denotes an additive noise.

## B. Cramér-Rao Bound

The Cramér-Rao bound (CRB) for $\theta$ in model (2) is given by the diagonal elements of the inverse of the Fisher Information Matrix (FIM) $F(\theta)$, i.e.,

$$CRB_{\theta_i} = [F^{-1}(\theta)]_{ii}, \quad i = 1, 2, 3.$$

Assuming the components of the error term $\epsilon$ in (2) are i.i.d. Gaussian with zero mean and variance $\sigma^2$, the FIM can be computed as

$$F(\theta) = \frac{1}{\sigma^2} \left( \frac{\partial g}{\partial \theta} \right)^T \left( \frac{\partial g}{\partial \theta} \right), \quad (3)$$

where $\frac{\partial g}{\partial \theta} \in \mathbb{R}^{N \times 3}$ denotes the Jacobian matrix of the model output $g(\theta)$ with respect to the parameter vector $\theta$.

However, due to the nonlinear and complex nature of the model in (1), computing $\frac{\partial g}{\partial \theta}$ and inverting the FIM analytically is extremely challenging. Therefore, the adopted strategy was to compute $\frac{\partial g}{\partial \theta}$ symbolically, evaluate it at given parameter values, and then compute the FIM and its inverse numerically. This approach is effective for overcoming the lack of a closed-form expression for the CRB, as it still allows us to study the dependence of the CRB on the parameter values by re-evaluating $F(\theta)$ as needed. The main drawback of this method is that the resulting FIM can be badly conditioned, making its inversion unstable and potentially leading to inaccurate CRB estimates. This issue is discussed further in the results section.

Finally, regardless of the estimation algorithm used, the mean squared error (MSE) of the estimator $\hat{\theta}_i$ satisfies the inequality $MSE_{\hat{\theta}_i} = E_{\theta}[(\hat{\theta}_i - \theta_i)^2] \ge CRB_{\theta_i}$, and the closer the MSE is to the CRB, the better the estimator performance.

## C. BSC simulations

Simulated BSC data were generated using (1). To reflect experimental BSCs from cell-pellet biophantoms or tumors (from study [9]), ground truth values ($a$ and $\phi$) were extracted from histology ($a = 8.12$ µm, $\phi = 0.49$ for cells). It corresponds to 4T1 cells from [9]. The parameter $\gamma_Z$ was manually adjusted to reflect experimental BSCs with $\gamma_Z = 0.03$ for these cell pellet biophantoms.

In order to study the effect of noise on scatterer parameter estimates, a zero-mean Gaussian noise with variance $\sigma^2$ was added for reproducing different signal-to-noise ratios (SNRs). The maximum noise variance was from 0.0025 and 1 (in 1/(sr.m)$^2$), corresponding approximately to 1 to 15% of the maximum of the BSC for cells. Examples of BSCs with and without adding noise ($\sigma^2 = 0.22$, SNR=17 dB) are shown in Fig. 1 (A).

For each input parameters $a$, $\phi$ and $\gamma_Z$, 200 random noise realizations were generated. From these simulated BSCs,
<figure><img src="image_3.png" /><figcaption>Fig. 2. Cramér–Rao bounds maps each parameters ($a$ on the left, $\phi$ at center and $\gamma_Z$ on the right) for $a = 8$ µm, $\phi = 0.49$, $\gamma_Z$ ranging from 0.001 to 0.1 and $\sigma^2$ from 0 to 1.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Difference between MSE maps and CRB for each parameters ($a$ on the left, $\phi$ at center and $\gamma_Z$ on the right). Top row: $a = 8.12$ µm, $\phi = 0.49$, $\gamma_Z$ ranging from 0.001 to 0.1 and $\sigma^2$ from 0 to 1. Middle row: $\phi = 0.49$, $\gamma_Z = 0.03$ and $a$ ranging from 3 to 12 µm and $\sigma^2$ from 0 to 1. Bottom row: $a = 8.12$ µm, $\gamma_Z = 0.03$ and $\phi$ ranging from 0.05 to 0.85 and $\sigma^2$ from 0 to 1. Dark blue on the right part of maps corresponds to inaccurate CRB estimates.</figcaption></figure>
scatterer parameters were estimated via a nonlinear least squares approach using (1). MSEs between the estimates and the ground truth were calculated and compared to the corresponding CRB. The closer the MSE to the CRB, the better. The impact of varying *a* ([3-12] µm), *φ* ([0.05-0.85]), γz ([0.001-0.1]), and noise level were assessed.

### III. RESULTS

Fig. 1 (B) shows the SNR values, ranging from -10 to 60 dB, obtained by varying γz and σ² for *a* and *φ* values from cells. An example of CRB and MSE for γz estimates is presented in Fig. 1 (C). As expected, MSE is close to and slightly higher than CRB. CRB maps for each estimate, cells parameters, and varying γz are presented in Fig. 2. As expected, CRB increases with increasing noise variance (decreasing SNR). Fig. 3 (top row) shows the difference between MSE and CRB as a function of γz. At low γz and poor SNR conditions, the MSE exceeds the CRB, reflecting the increased estimation difficulty and the potential for improvement of current estimators. This can be explained by the fact that low values of γz reduce the BSC amplitude and make estimation more sensitive to the presence of noise.

As mentioned earlier, the numerical computation of the FIM can result in badly conditioned matrices under certain parameter configurations. Specifically, for low values of *a* (i.e., < 7 µm) and high values of *φ* (i.e., > 0.65), the reciprocal condition number of *F*(*θ*) in the 1-norm approaches zero (we observed values smaller than 10<sup>-14</sup>). This leads to unstable matrix inversion and, consequently, inaccurate CRB estimates. In the following figures, whenever the CRB estimation is unreliable due to such instability, the result is plotted as missing value. The difference maps between MSE and CRB for each parameter estimate, across varying values of *a* and *φ*, are shown in Fig. 3 middle and bottom row. When both *a* and *φ* are small, estimation performance deteriorates even under moderate SNR conditions. This is expected because smaller values of *a* make estimation more difficult due to the absence of a distinct peak in the BSC curve. Additionally, when *φ* is low, the structure factor tends toward 1, which reduces sensitivity to both *φ* and γz. As a result, it becomes challenging to estimate these parameters simultaneously from the BSC.

### IV. CONCLUSION

In this study, we focused on estimating the CRB and comparing the MSEs of the different BSC parameters estimated by the structure factor model. The effect of the different parameters on these estimates in the presence of Gaussian noise was evaluated. We found that parameter estimates were more challenging for low values of γz, *a* and *φ*. In future work, the frequency range of the BSC will be studied. We will also investigate CRB and MSE performance under frequency-dependent noise to better reflect experimental conditions. Finally, we will explore the impact of noise inherent in scatterer positions. Specifically, we aim to compute the structure

factor directly from scatterer positions rather than relying on statistical approximations.

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