Response surface methodology for predicting optimal
conditions in very low-dose chest CT imaging

Eléonore Pouget, Véronique Dedieu, Marie Lemery Magnin, Marie Biard,
Guillaume Lienemann, Jean-Marc Garcier, Benoît Magnin

► To cite this version:

Eléonore Pouget, Véronique Dedieu, Marie Lemery Magnin, Marie Biard, Guillaume Lienemann, et al.. Response surface methodology for predicting optimal conditions in very low-dose chest CT imaging. Physica Medica European Journal of Medical Physics, 2025, 131, pp.104916. 10.1016/j.ejmp.2025.104916. hal-05110072

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Original paper

# Response surface methodology for predicting optimal conditions in very low-dose chest CT imaging

Eléonore Pouget<sup>a,b,*</sup> <sup>ID</sup>, Véronique Dedieu<sup>a,b</sup>, Marie Lemery Magnin<sup>c</sup>, Marie Biard<sup>c</sup>, Guillaume Lienemann<sup>c>ID</sup>, Jean-Marc Garcier<sup>c,d,e</sup>, Benoît Magnin<sup>c,d,e</sup>

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

### Keywords:

Response surface methodology

Model observers

Deep learning-based reconstructions

low-dose CT imaging

## ABSTRACT

**Objectives:** Dose reduction techniques, such as new reconstruction algorithms and automated exposure control systems vary with manufacturer and scanner models, complicating the optimization and standardization procedures. We investigated the feasibility of using the design of experiments in CT protocols optimization.

**Materials & Methods:** A Doehlert matrix was used to define the experiments to carry out. Measurements were conducted on a 128-slice CT scanner using an anthropomorphic chest phantom with a 5 mm diameter lesion that has a HU of –800. CT images were reconstructed using iterative (ASIR-V) and deep learning-based reconstruction techniques at low (DLIR-L) and high (DLIR-H) strengths. Lesion detectability was assessed using two self-supervised learning-based model observers and six human observers. Second-order polynomial functions have been established to model the combined effect of noise index (NI) and percentage of ASIR-V on dose and model observers’ performances. The analysis of agreement between model and human observers was performed using correlation coefficients and Bland-Altman test.

**Results:** The optimal conditions predicted by this method were NI = 64, % ASIR-V = 60 and DLIR-H reconstruction. They were found in good agreement with the experimental results obtained by the average human observer, as showed by the Bland-Altman plot with a mean absolute difference of –0.01 ± 3.16. Compared to 60 % ASIR-V, these results suggested an approximately 64 % dose reduction potential for DLIR-H without compromising lesion detection.

**Conclusion:** The proposed method can predict the optimal conditions that ensure diagnostic quality of low-dose chest CT examinations, while minimizing the number of experiments to carry out.

## 1. Introduction

The complexity of new x-ray computed tomography (CT) technologies has grown in the last decade. In particular, manufacturers have developed several techniques to maintain image quality at lower radiation dose to address the increasing focus on reducing patients’ radiation exposure. Iterative reconstruction (IR) algorithms and automatic exposure control (AEC) techniques are the most widely used dose management tools in clinical practice. The former aims to decrease image noise to facilitate use of decreased radiation dose levels [1], while the latter adapts the tube output according to the radiological properties

of the patient to achieve a predicted image quality [1]. The potential of these techniques for dose reduction has been largely reported in many studies, extending the use of low-dose CT as a modality for lung cancer screening [2–5]. The radiation dose optimization is becoming more challenging due to the increasing variability in reconstruction algorithms and AEC systems. An efficient method to replicate protocols across different scanners is thus needed in order to ensure the diagnostic quality of images required for the exam [6,7], yielding to consider mathematical models as a basis for CT protocols optimization.

The design of experiments (DOE) method can be used to analyse multi-parametric problems, such as CT protocols optimization. DOE

<footnote><sup>a</sup> Department of Medical Physics, Jean Perrin Comprehensive Cancer Center F-63000 Clermont-Ferrand, France</footnote>

<footnote><sup>b</sup> Clermont-Ferrand University, UMR 1240 INSERM IMoST, 58 rue Montalembert F-63000 Clermont-Ferrand, France</footnote>

<footnote><sup>c</sup> CHU Estaing, Service de radiologie F-63000 Clermont-Ferrand, France</footnote>

<footnote><sup>d</sup> Institut Pascal, UMR 6602 CNRS, Université Clermont Auvergne, Clermont-Ferrand, France</footnote>

<footnote><sup>e</sup> DI2AM, DRCI, Clermont University Hospital, Clermont-Ferrand, France</footnote>

<footnote>* Corresponding author at: Jean Perrin Comprehensive Cancer Center, 58 Rue Montalembert, 63011 Clermont-Ferrand Cedex 1, France.</footnote>

<footnote>E-mail address: eleonore.pouget@clermont.unicancer.fr (E. Pouget).</footnote>
were successfully applied in industry and medicine [8–11], and consist in a set of algebraic and statistical methods aiming to establish the relationship between the studied quantities (responses) and their supposed sources of variations (factors). This analysis can be qualitative or quantitative and is referred to as screening study or response surface methodology (RSM) respectively. The former aims to determine the more influential factors while the latter allows for searching optimal levels of factors to achieve the desired level of response. The optimization process can be formulated as the search for a minimum dose under the constraint of diagnostic image quality [8]. This can be mathematically translated into a response to be minimized, reflecting the patients' exposure level and a response to be maximized, reflecting the image quality [8]. Since traditional physical metrics may not be adequate to evaluate IR and other nonlinear noise-reduction algorithms [7,12], the literature favours a task-based approach for assessing image quality in CT [1,13]. In this context, there has been a growing interest in developing and utilizing model observers for imaging systems optimization and comparison [12,14].

Among the models described in the literature, the Bayesian ideal observer (IO) is optimal by definition for signal detection tasks [15]. However, the IO is often intractable and the common alternative is to consider the Hotelling observer (HO) whose computation is also not without challenges [15–17]. Recent works have demonstrated the potential of supervised learning-based methods to provide accurate estimates of HO performances, that is however dependent upon the availability of sufficient amount of labelled data [17–20]. Previous works have reduced the reliance on labelled data by using pre-trained autoencoders (AEs) as feature extractor in classification problems. This self-supervised learning (SSL) model achieved competitive performances with the state-of-the-art supervised approaches [19,21–23].

In this work, we were interested in developing a predictive mathematical model able to identify the optimal conditions in chest CT imaging. For this purpose, the RSM was applied for modelling the combined effect of IR algorithm and AEC technique on dose level and on SSL-based model observers' performance in a binary signal-detection task.

## 2. Materials and methods

### 2.1. Task-based image quality assessment

#### 2.1.1. Model observers

Unsupervised feature learning algorithms were becoming widespread to address the lack of large-scale annotated data [19,21]. In particular, convolutional auto-encoders (CAEs) have performed well in medical image classification tasks [19,22]. The CAEs' architecture is similar to classic AEs, except that it uses 2D convolution operations [19,24]. Considering a 2D input image g, a layer with k convolutional filters maps g into k hidden features maps, h, which can be represented as follows [25]:

$$h_j = \sigma(g^*W_j + b_j), j = 1, 2, \dots, k, \quad (1)$$

where * denotes a 2D convolution, $W_j$ and $b_j$ are the weights and bias of the jth kernel filter respectively, $\sigma$ is a nonlinear activation function. The parameters (i.e. W and b) are learned by minimizing the cross-entropy loss function. In this work, a denoising CAE (CDAE) was used to generate more robust features than classic CAEs by introducing noise into original data [26,27]. The CDAE was then trained to reconstruct the corrupted input data by minimizing the loss function between reconstructed and original data.

As principal component analysis (PCA) can be thought as a simplified form of AEs, PCA was computed here for comparison. PCA can be mathematically defined as an orthogonal linear transformation that projects the original data into another space composed of the principal components (PC) that successively maximize variance. Finding such PC

reduces to solving an eigenvalue/eigenvector problem [28]. The computational complexity of PCA is only based on data matrix decomposition, potentially leading to greater numerical stability of the solution than that of AEs on limited-size datasets. Both CDAE and PCA were trained on a training dataset constructed from numerical simulations [29]. This dataset comprised 20,000 images of the Lungman chest phantom [30]. Implementation details of unsupervised feature learning algorithms can be found in a preliminary work [31].

The relationship between learned features and image classification was established with a support vector machine (SVM), who used a radial basis function as the kernel function. The signal-to-noise ratio (SNR) was computed for measuring observer performance. This figure of merit (FOM) was preferred to the common area under an receiver operating characteristic curve (AUC) because the latter can become saturated [32] (i.e. converged to 1) limiting the detection of some differences occurring between tested conditions. As the SNR requires Gaussian response of the observer in order to be used as FOM [33,34], it is important to check that the decision variable is approximately Gaussian before computing the SNR for all the experiments in this study.

#### 2.1.2. Human observer experiments

A proper calibration of the model observer performances with actual human observer performances is necessary to validate the model as a meaningful metric for identifying optimal imaging conditions and establish to what extent dose can be reduced without compromising diagnostic information. To do so, a two-alternative forced choice (2-AFC) study was constructed. An application written in Python was developed in house to facilitate the human observer study [12]. The samples containing the signal were generated by extracting regions of interest around the lesion (signal) implanted into the phantom. The background samples were extracted from the signal-absent images at the same locations as signal samples.

Within each 2-AFC study, the images with signal were displayed randomly at left or right [7] of the software interface and the human observer had to identify the image containing the signal. The average human observer performance was defined as the mean of the proportion of correct (PC) responses obtained by the readers recruited to perform the detection task (i.e. three radiologists and three medical physicists) [7]. The reason of using medical physicists alongside radiologists as observers is that, for purposes of simple phantom experiments, such as those involved in this study, physicists can reasonably be considered as expert observers [7,35,36]. The PC was then converted into the detectability $d'$ applying the equation [7,34]:

$$d = \sqrt{2\Phi^{-1}(PC)}, \quad (2)$$

where $\Phi^{-1}(PC)$ is the inverse of the standard cumulative normal distribution function [7,34]. To relate the average human observer performance to the model observer [7], an efficiency factor ($\eta$) was determined from the slope of the linear fit obtained for human observer performances as a function of model observer performances. The agreement between these observers was assessed using the Bland-Altman test [7,34]. Additionally, quantitative analysis of the correlation between model and human observers was performed with both Spearman's rank and Pearson's product-moment correlation coefficients, which assess a monotonic and a linear relationship respectively [7,12].

### 2.2. Response surface methodology

A key point of the RSM is the choice of experimental design which allows minimizing the number of experiments by excluding non-relevant ones [8]. A good selection of the experiments reduces the variance of the coefficients in the mathematical model, making the response surface more reliable. A two-factor Doehlert matrix was used here to model the combined effect of IR algorithm and AEC technique,
managed by the percentage of ASIR-V ($x_2$) and by the noise index (NI, $x_1$) respectively, on dose and image quality. The experimental domain was explored through seven uniformly distributed points, as summarized in the Table 1.

This design allows the exploration of quadratic response surfaces described by a second-order polynomial function [37]:

$$y = a_0 + a_1x_1 + a_2x_2 + a_{11}x_1^2 + a_{22}x_2^2 + a_{12}x_1x_2, \quad (3)$$

where y is the predicted response, $a_i$ are the linear coefficients and $x_i$ are the values of the two factors, expressed as coded variables. The response to maximize, $y_1$, was defined as the model observers' performance, while the response to minimize, $y_2$, was defined as the volumetric computed tomography dose index (CTDI\_vol). An analysis of variance (ANOVA) was then conducted to test the models adequacy [8].

The minimum $y_1$ response value was defined as the performance obtained under local reference conditions (i.e. NI = 28 and % ASIR-V = 60). Thus, the optimization process rejects factor levels leading to a degradation of signal detection. Similarly, set the maximum $y_2$ response value at the exposure level associated with local reference conditions warrants a radiation dose reduction. The maximum $y_1$ response value was chosen as the SNR value upon which the classification is assumed to be perfect (AUC convergence towards 1). Thus, a higher SNR value can be interpreted as an improvement in image quality that does not contribute to an increased classification performance. The minimum $y_2$ response value was set close to zero. The simultaneous optimization of both responses was achieved through the computation of the Derringer's desirability function (D) [8,37,38]. The optimal conditions were then determined by the levels of $x_1$ and $x_2$ leading to the maximal value of D.

## 2.3. Experimental design

The multipurpose chest phantom Lungman® (NI “LUNGMAN”, Kyoto Kagaku [30]) was scanned on a 128-slice CT scanner (Revolution CT, GE Healthcare). In order to simulate a signal-known-exactly and background-known-exactly (SKE/BKE) signal detection task, a lesion of 5 mm diameter and -800 HU, mimicking the attenuation of low attenuation ground-glass opacities [3], was implanted. A routine clinical adult chest protocol was used with the acquisition and reconstruction settings listed in Table 2. As concerns the experimental space investigated, the range of factors levels was [28, 72] and [20, 100] for the NI and the percentage of ASIR-V respectively. The domain of each variable was defined according to the reference-dose and the very low-dose chest CT described in the literature [2-4]. The relationship between coded values, given by the Doehlert matrix (Table 1), and real values results in the experimental conditions summarized in Table 2.

For this study, each scan was repeated 20 times and 30 times to generate signal-present and signal-absent image category respectively. Images were also reconstructed with a deep learning-based reconstruction algorithm, True Fidelity, at low (DLIR-L) and high (DLIR-H) strengths yielding to 21 experiments (seven acquisitions × three reconstructions). The strength levels manage the amount of noise reduction, without impacting reconstruction speed [39]. All CT images were used to create the database of scanned images.

<table><caption>**Table 1**

Doehlert matrix for two factors.</caption><thead><tr><th rowspan="2">Run</th><th colspan="2">NI (x<sub>1</sub>)</th><th colspan="2">% ASIR-V (x<sub>2</sub>)</th></tr><tr><th>Coded values</th><th>Real values</th><th>Coded values</th><th>Real values</th></tr></thead><tbody><tr><td>1</td><td>0</td><td>50</td><td>0</td><td>60</td></tr><tr><td>2</td><td>1</td><td>72</td><td>0</td><td>60</td></tr><tr><td>3</td><td>0.5</td><td>61</td><td>0.866</td><td>100</td></tr><tr><td>4</td><td>-0.5</td><td>39</td><td>0.866</td><td>100</td></tr><tr><td>5</td><td>-1</td><td>28</td><td>0</td><td>60</td></tr><tr><td>6</td><td>-0.5</td><td>39</td><td>-0.866</td><td>30</td></tr><tr><td>7</td><td>0.5</td><td>61</td><td>-0.866</td><td>30</td></tr></tbody></table>

<table><caption>**Table 2**

Scan settings used to image the Lungman phantom.</caption><thead><tr><th>Settings</th><th>Revolution CT</th></tr></thead><tbody><tr><td>kVp</td><td>100</td></tr><tr><td>Automatic tube current modulation algorithm</td><td>Smart mA</td></tr><tr><td>mA range (min/max)</td><td>10/400</td></tr><tr><td>Noise Index</td><td>28, 39, 50, 61, 72</td></tr><tr><td>Scan mode</td><td>Helical (pitch = 1.531)</td></tr><tr><td>Reconstruction algorithms (blending level)</td><td>ASIR-V (30 %, 60 %, 100 %)</td></tr><tr><td>Convolution kernel</td><td>STD</td></tr><tr><td>Slice thickness (mm)</td><td>1.25</td></tr><tr><td>Reconstruction intervals (mm)</td><td>1.25</td></tr><tr><td>Reconstructed FOV (mm)</td><td>360</td></tr><tr><td>Matrix size (pixels)</td><td>512 × 512</td></tr></tbody></table>

Both model and human observers performed the SKE/BKE task after extracting 128 × 128 pixels regions of interest (Fig. 1). The test dataset was successively composed of the images related to each of the 21 experiments performed, while the training dataset comprised 200 images randomly selected from the remainder of the whole dataset. The performance obtained for each setup served as the response to maximize (i. e. $y_1$).

# 3. Results

## 3.1. Response surface methodology

A model can be considered adequate within 95 % confidence level if the calculated *F-statistic* of the regression model is larger than the specified value of 230.2, in our case [8]. In Table 3, the results of ANOVA suggest that the quadratic models differ from the observed models, except for the CDAE/SVM observer response in DLIR-L-reconstructed images. This lack of fit suggests the need of introducing an error term into Eq. (3) that translates the difference between postulated and observed models. It can be noted that the $y_1$ response was first normalized by the one obtained under local reference conditions in order to express both observer performances on a common scale.

Analysis of the mathematical models obtained does, however, make it possible to predict how changes in the combination of NI and % ASIR-V will affect observers performance and exposure level.

### 3.1.1. $y_1$ response modeling

Consideration of the planes such as % ASIR-V = 60 (i.e. $x_2 = 0$) and NI = 50 (i.e. $x_1 = 0$) allows us to plot the evolution of observers performance as a function of NI and percentage of ASIR-V respectively, as depicted in Fig. 2.

As expected for scanners based on a constant noise paradigm [40], the detectability values increased with decreasing the NI. This trend was approximately linear for the PCA/SVM observer, whereas the CDAE/SVM observer seemed less sensible to low noise levels. For both observers, the detection performance was higher in DLIR-H- than in ASIR-V-reconstructed images. Similarly, Fig. 2 shows a significant improvement in signal detection as the percentage of ASIR-V in the main reconstruction decreased. This trend was more pronounced for the DLIR-H algorithm regardless of the model observer.

### 3.1.2. $y_2$ response modeling

As showed in Fig. 3, increasing the percentage of ASIR-V resulted in a reduced radiation dose required to achieve the prescribed NI. This is due to the AEC scheme owned to Revolution CT systems, for which the percentage of ASIR-V in the main reconstruction acts on the tube current required to meet the NI objective. As a consequence, the percentage of ASIR-V is able to manage the dose level for this CT system.

It can be noted that only the main reconstruction (i.e. ASIR-V) has an impact on the patient's exposure level, therefore only one model has been established for the $y_2$ response.
<figure><img src="image_3.png" /><figcaption>Fig. 1. CT images of the anthropomorphic Lungman phantom acquired at 0.15 mGy and reconstructed with (a) DLIR-L, (b) DLIR-H and (c) 100 % ASIR-V.</figcaption></figure>

<table><caption>**Table 3**
ANOVA of the regression models.</caption><thead><tr><th rowspan="2"></th><th colspan="3">y<sub>1</sub> (CDAE/SVM)</th><th colspan="3">y<sub>1</sub> (PCA/SVM)</th><th rowspan="2">y<sub>2</sub></th></tr><tr><th>ASIR-V</th><th>DLIR-L</th><th>DLIR-H</th><th>ASIR-V</th><th>DLIR-L</th><th>DLIR-H</th></tr></thead><tbody><tr><td>R-squared</td><td>0.993</td><td>0.999</td><td>0.995</td><td>0.994</td><td>0.989</td><td>0.995</td><td>0.984</td></tr><tr><td>Adjusted R-squared</td><td>0.985</td><td>0.996</td><td>0.988</td><td>0.966</td><td>0.937</td><td>0.968</td><td>0.783</td></tr><tr><td>F-statistic</td><td>28.63</td><td>271.9</td><td>39.22</td><td>34.66</td><td>18.72</td><td>36.25</td><td>5.332</td></tr><tr><td>Prob(F-statistic)</td><td>0.141</td><td>0.046</td><td>0.121</td><td>0.128</td><td>0.174</td><td>0.125</td><td>0.317</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>Fig. 2. Effect of (a) NI and (b) percentage of ASIR-V on model observers' performance.</figcaption></figure>

### 3.1.3. Desirability function

Table 4 summarizes the maximum *D* values (*D*<sub>max</sub>) obtained for each reconstruction algorithm, together with the associated factor levels.

According to the maximal value of *D*<sub>max</sub> related to both model observers (in bold face in Table 4), DLIR-H algorithm enabled improved image quality and, as a consequence, dose reduction. The optimal
<figure><img src="image_5.png" /><figcaption>Fig. 3. Effect of (a) NI and (b) percentage of ASIR-V on the CTDI<sub>vol</sub>.</figcaption></figure>

**Table 4**

Values of NI and percentage of ASIR-V associated with the maximum desirability, $D_{max}$.

<table><thead><tr><th rowspan="2"></th><th colspan="3">CDAE/SVM observer</th><th colspan="3">PCA/SVM observer</th></tr><tr><th>ASIR-V</th><th>DLIR-L</th><th>DLIR-H</th><th>ASIR-V</th><th>DLIR-L</th><th>DLIR-H</th></tr></thead><tbody><tr><th>$D_{max}$</th><td>0.42</td><td>0.36</td><td><strong>0.91</strong></td><td>0.28</td><td>0.43</td><td><strong>0.50</strong></td></tr><tr><th>NI ($x_1$)</th><td>46</td><td>28</td><td><strong>64</strong></td><td>39</td><td>35</td><td><strong>32</strong></td></tr><tr><th>% ASIR-V ($x_2$)</th><td>50</td><td>60</td><td><strong>60</strong></td><td>50</td><td>70</td><td><strong>80</strong></td></tr></tbody></table>

conditions identified by the CDAE/SVM model yielded to an exposure level of 0.2 mGy and an image quality equivalent to that obtained for a dose of 0.55 mGy with 60 % ASIR-V. Similarly, the optimal conditions identified by the PCA/SVM model conducted to an exposure level of 0.32 mGy and an image quality equivalent to that obtained for a dose of 0.48 mGy with 60 % ASIR-V. Therefore, compared to 60 % ASIR-V, the CDAE/SVM and PCA/SVM model estimated an approximately 64 % and 33 % dose reduction potential for DLIR-H respectively.

## 3.2. Analysis of agreement between model and human observers

In a 2-AFC experiment, the curve representing $d'$ as a function of PC saturates above three (PC ≈ 0.98) [7,34]. Therefore, only the values below this threshold were used to investigate the correlation between human and model observer performance. The Pearson and Spearman's correlation coefficients between human and CDAE/SVM performances were 0.82 and 0.95 across the most challenging conditions that verified PC < 0.98. These values reflected a monotonic, non-perfectly linear relationship between these two observers, as depicted in Fig. 4.

Similarly, the correlation coefficients calculated between human and PCA/SVM performances were 0.52 and 0.59, indicating a non-significant correlation. This discrepancy suggests that the human observer's decision-making strategy deviates from that adopted by the PCA/SVM model.

In Fig. 5, the Bland-Altman plot showed an excellent agreement between human and CDAE/SVM (after the $\eta$ correction) observers, with a mean absolute difference of -0.01 and a range of differences about ± 3.16. Due to the lack of linear relationship between human and PCA/SVM observer, the Bland-Altman test was not performed here.

Although not presented in this work, a Bland-Altman test was

<figure><img src="image_6.png" /><figcaption>Fig. 4. Squared detectability index (d'2) of the average human observer as a function of the CDAE/SVM model observer for images related to the most challenging conditions. The efficiency tallies the slope of the linear fit [34].</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 5. Bland-Altman plot of $d'^2$ difference between human and CDAE/SVM (after correction by $\eta$) observers for images related to the most challenging conditions. The straight line represents the average absolute difference ($\Delta$) and the dash lines represent $\Delta \pm 2\sigma$, where $\sigma$ is the standard deviation of the differences [7,34].</figcaption></figure>

performed for the detection performance obtained by both model observers. The results showed an overall good agreement, with a mean absolute difference of 0.21, 1.05 and -1.63 for ASIR-V-, DLIR-L- and DLIR-H-reconstructed images respectively. These trends were consistent with the results extracted from Table 4.

## 4. Discussion

In this work, we proposed a mathematical method, based on the RSM, to optimize and standardize image quality among CT protocols. The performance of SSL-based model observers was chosen as the response to maximize in order to make the diagnostic information reliably rendered on reconstructed images [40].

The results obtained suggested that, compared to current IR algorithm, DLIR-H allows an additional dose reduction for low-contrast diagnostic tasks in chest CT. These results are consistent with other phantom-based studies using different anatomical localizations [41,42]. Despite an overall good agreement between CDAE/SVM and PCA/SVM models, a discrepancy was observed on DLIR-H sets. This is most likely due to the fact that the amount of noise reduction allowed by advanced nonlinear reconstruction techniques is not consistent across the entire image, when containing structures such as pulmonary vessels [43,44]. This results in higher noise near high-contrast features and lower noise in uniform regions, as stated in previous published works [43,44]. Because of the way they operate, the CDAE-based model extracted the high-contrast edges of structures and was therefore strongly affected by the high-noise level, while the linear PCA tended to extract the overall noise level. This property was found more pronounced in DLIR-H-reconstructed images over the very low-dose range (i.e., associated with 100% ASIR-V in the main reconstruction). As a consequence, the CDAE/SVM model produced lower decision variable than the PCA/SVM model in this case. This result points out the limits of the DLIR dose reduction potential in images with complex background features [43].

As expected, both SSL-based models outperformed the average human observer. Nevertheless, the ability of the CDAE/SVM model to reproduce the evolution of the average human performance supports the hypothesis of using this observer as the response to maximize in the optimization process. This point is of interest because it suggests that the features learned by the CDAE are closer to the information used by

human observers to make a detection decision than those learned by the PCA. In this study, the minimum performance to be achieved was set arbitrarily as that obtained under local reference conditions. However, a visibility threshold could be defined, set as the PC value of the average human observer upon which, the lesion would be considered visible [7]. The minimum value of the response to maximize would be accordingly set as the corresponding CDAE/SVM model value in order to establish to what extent dose can be reduced without compromising diagnostic information.

Some limitations need to be mentioned. First, as the curve of $d'$ as a function of PC saturates above three [7,34], only the $d'$ values below this threshold were used to relate the average human observer to the model observer performance [7]. As a result, the data related to the seven most challenging conditions were taken into account in the plot of Fig. 4. The related $R^2$ value of the linear fit obtained from these experiments, which was 0.52, requires careful interpretation of the derived $\eta$ value ($R^2 = 1$ for a perfect fit). Second, the analysis of agreement between model and human observers being based on these seven experiments, the generalizability of the results obtained was limited. Third, each human observer scored once the pairs of images presented in 2-AFC tests, meaning that the intra-observer variability was not analysed here. Therefore, the resulting estimates of differences between human and model could be optimistically biased with respect to the true differences on both average and range [7]. Finally, we intend to extend this study to other anatomical localizations, phantom sizes and diagnostic tasks to generalize the conclusions drawn from this work.

## 5. Conclusion

The optimal conditions of low-dose CT protocols have been identified as the conditions leading to the maximal value of the desirability function, which translated the aim of delivering diagnostic quality of images at the lowest dose achievable. The excellent agreement between the human and CDAE/SVM observer demonstrated the potential for using this model to provide insights into reconstruction techniques performance in chest CT examinations. Particularly, it highlighted a higher dose reduction potential for DLIR algorithm on one hand, and the limits of this potential in complex textured backgrounds on the other hand. These findings can be explained by the non-stationarity property
of noise, which is focused in a near future work. We anticipate that the proposed mathematical method can represent a usefulness tool for optimization purposes with the increasing variety and complexity of CT protocols.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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