Generalized multi-energy model for computing radiation treatment parameters on image domain and automated model-selection framework

doi:10.4329/wjr.126287

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World Journal of Radiology

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1949-8470

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Baishideng Publishing Group Inc, 7041 Koll Center Parkway, Suite 160, Pleasanton, CA 94566, USA

- Abstract
- INTRODUCTION
- METHODOLOGY
- RESULTS
- DISCUSSION
- CONCLUSION
- Footnotes
- References

Minireviews Open Access

Copyright: ©Author(s) 2026. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial (CC BY-NC 4.0) license. No commercial re-use. See permissions. Published by Baishideng Publishing Group Inc.

World J Radiol. Oct 28, 2026; 18(10): 126287
Published online Oct 28, 2026. doi: 10.4329/wjr.126287

Generalized multi-energy model for computing radiation treatment parameters on image domain and automated model-selection framework

Charles E Chika

Charles E Chika, Department of Mathematics, University of Nigeria, Nsukka 410001, Nigeria

ORCID number: Charles E Chika (0000-0002-0428-224X).

Author contributions: Chika CE conceptualized and wrote the manuscript as well as completed all the research and manuscript processes.

AI contribution statement: No AI tool was used.

Conflict-of-interest statement: The author declares that there is no known conflict of interest.

Corresponding author: Charles E Chika, PhD, Department of Mathematics, University of Nigeria, Obukpa Road, Nsukka 410001, Nigeria. charles.chika@unn.edu.ng

Received: July 30, 2026
Revised: September 2, 2026
Accepted: October 8, 2026
Published online: October 28, 2026
Processing time: 90 Days and 9.8 Hours

Abstract

This study developed a general n-energy mathematical model that predicts relative electron density (ρ_e), effective atomic number (Z_ef), mean excitation energy (I) and stopping power ratio (SPR) using computed tomography image, and developed an algorithm for best model selection. Mathematical proof is provided. Some numerical examples were carried out and the results were compared. The proposed method includes most of the existing methods as a special case. All the examples of the proposed method considered have modeling error that’s less than or equal to 0.85% and 0.89% for SPR and ρ_e, respectively, with mean error close to 0.00% and low median absolute error. The method was able to achieve modeling root mean square error of 0.85% and 1.61% for I and Z_ef, respectively, with some flexibility for improvement. The mathematical model showed to be more general by including most of the other existing models and presents a direction to formulating new ones. It shows little or no bias on prediction since it has mean error that is close to zero in almost all predictions and low median absolute error. The algorithm gives the leverage to select the best model, improve the model and automation.

Key Words: Proton stopping-power ratio; Computed tomography image; Tumor treatment; Proton therapy; Dual-energy computed tomography; Multi-energy computed tomography; Mathematical model; Machine learning; Optimization

Core Tip: This study presents a mathematical model for improving the precision of relative electron density, mean excitation energy, effective atomic number and stopping power ratio using computed tomography image. It exposes the need for machine learning algorithm that will help with best model selection. Based on the observation that the performance of the models differ based on the type of linear attenuation used (i.e., spectrally averaged linear attenuation or linear attenuation at effective energy), this study presents machine learning algorithm for best model selection. It also discusses the mathematical proof of the presented generalized model.

Citation: Chika CE. Generalized multi-energy model for computing radiation treatment parameters on image domain and automated model-selection framework. World J Radiol 2026; 18(10): 126287
URL: https://www.wjgnet.com/1949-8470/full/v18/i10/126287.htm
DOI: https://dx.doi.org/10.4329/wjr.126287

INTRODUCTION

Treatment of tumors using ions requires planning and this planning includes dosimetry. Optimal dose should eliminate unhealthy cells as much as possible while sparing the healthy ones. In the case of proton therapy the need to take full advantage of the Bragg peak depth cannot be overemphasized and this includes minimizing the range uncertainty by accurate calculation of proton stopping power ratio (SPR).

The above reason has led to developing different methods for estimating SPR, which include projection and image domain methods, single and dual-energy methods as well as multi-energy methods. Many researches have been carried out on estimating different parameters like relative electron density (ρ_e), effective atomic number (Z_ef) and photon cross-sections using dual-energy[1-5]. This dual-energy idea has been adopted for SPR computation using different methods like deep learning[6], Monte Carlo simulation[7-9] and other methods that make use of segmentation, decomposition and parametrization[10-13] as well as some comparative studies[14]. The projection domain methods have been developed for more robustness to some factors like beam hardening and computed tomography (CT) number variation[14-17]. Uncertainties associated to these approaches have been studied[18-21] as well as impact of noise[22], and it was found that dual-energy CT approach gives a better result compared to single energy CT approach. This has led to different experimental studies confirming the improvement in the estimated results for SPR[23-27]. Other methods are still developing like the one presented by Chika and Hooshyar[28] which might help improve projection domain computation. For the case of energy, some studies have been carried out to find out if energy, kVp, affects the accuracy of SPR estimation, the best energy combination for dual-energy has been studied like the one presented by Alvarez et al[29], Lehmann et al[30], Yang et al[31] and Flohr et al[32]. There’s enough evidence that the number of energies used affects the accuracy of SPR and that accuracy can be improved with the right energy combination.

Image domain is simpler when it comes to computation and application but many times it’s prone to uncertainties like noise, composition, movement, etc. These demands approach that will minimize these uncertainties while keeping the simplicity. Based on this, machine learning algorithm will help choose the method that is least prone to uncertainties. The aim of this study is to present a general mathematical model for computing different parameters that are needed to compute treatment planning quantities using CT image, provide a mathematical proof and develop a machine learning algorithm for model selection.

This study is focused on developing and proving a generalized mathematical model for estimating the four parameters, establishing a machine learning approach for selecting the best performing model on any presented data. Most of the existing models belong to this common model class. The data considered in this study is attenuation coefficient at effective energy which showed a different performance on different models if compared with the performance when spectrally averaged attenuation is used as presented by previous studies. The best model is selected by comparing the modeling and testing errors as presented in numerical examples. The model that gives the lowest modeling and testing root mean square error (RMSE) is considered the best model; if two models have relatively close modeling error, the one with lowest testing error is considered to be the best model and vice versa.

More general mathematical model that uses n-energies CT images is presented with mathematical proof. Machine learning algorithm that helps in model/method selection which can lead to automation is presented. Numerical examples and comparisons are also shown using theoretically simulated data.

METHODOLOGY

Theoretical CT data acquisition

The images intensity HU_s of each pixel in CT images for any tissue are written as: $< \frac{μ}{μ_{w a t e r}} >_{s} = \frac{H U_{s}}{A_{s}} + A_{s}^{*}$ (1), where μ_water and μ are the linear attenuation coefficients of water and unknown tissue, respectively. Low and high energy spectra are represented by the subscript s = L,H. In quantitative CT, the parameters A_s and $A_{s}^{*}$ are estimated by maximizing the fit between measured HU_s values and spectrally averaged $< \frac{μ}{μ_{w a t e r}} >_{s}$ values for scanned samples of known composition and density.

In this study, the CT quantities are theoretically simulated and it’s assumed that phantom consisting of unknown tissues is scanned with the commercial CT scanner at low-energy (s = L) and high-energy (s = H) spectra, characterized by normalized X-ray energy fluence spectra Φ_s(E), where $\int_{E} ‍ Φ_{s} (E) d E = 1$ , $< \frac{μ}{μ_{w a t e r}} >_{s} = \frac{\int_{E} ‍ Φ_{s} (E) μ (x, E) d E}{\int_{E} ‍ Φ_{s} (E) μ_{w a t e r} (x, E) d E}$ (2).

Spectra of commercial CT scanner are generally not precisely known, and HU measurements can also be affected by some factors like beam hardening, scattering, noise and preprocessing corrections which leads to uncertainties in the measurement. Most of the times s(E) is approximated by a single effective energy, E_s, such that

\frac{μ_{k} (E_{s})}{μ_{w a t e r} (E_{s})} = < \frac{μ_{k} (E)}{μ_{w a t e r} (E)} >_{s} = μ_{e f, k} .

We got this μ_ef,k by maximizing the relationship between the theoretical CT number computed from Eq(2) and that of single energies. The effective energy is 43.5 keV for low energy and 89 keV for high energy, this is gotten by minimizing the difference between theoretical attenuation of the tissues, μ, and attenuation at different energy level i.e., $m i n | |μ - μ (E)| |$ where μ(E) is the attenuation of the tissues at different energy level. This is done using MATLAB and selecting the energy that gives the minimum value for $| |μ - μ (E)| |$ .

Linear attenuation coefficients are computed by applying the mixture rule to elemental mass attenuation coefficients gotten from the National Institute of Standards and Technology XCOM database[33-37] after which Eq(2) was used.

The spectra used in computing Eq(2) is illustrated in Figure 1 using its normalized form. 80 kVp is used as low energy while 150 kVp/Sn (with tin filter) is used as high energy. μ_L is attenuation at effective energy for low energy and μ_H is attenuation at effective energy for high energy[38,39]. These spectra were generated using SpekCalc and the software settings[38].

Open in New Tab Full Size Figure Download Figure

Figure 1 Normalized energy spectra used: The blue line is for 150 kVp/Sn and the red line is for 80 kVp.

Tissue classification

We used 33 ICRU human tissues as training data and 12 Gammex tissue inserts as our validation data[40-42]. We didn’t include cell nucleus in ICRU human tissues because we couldn’t access the composition as at the time of study. The 33 ICRU human tissues were classified into lung, soft and bone tissues while the Gammex inserts were grouped into soft and bone tissues. We also divided the soft tissues into two in the training data given us four (4) groups to study the effect of further classification.

All the classifications in this study were done with the transformation T_μL = μ_L which was presented by some studies[43,44]. The training data three (3) and four (4) group regions are illustrated in Figure 2. For the three (3) groups, tissues are classified as lung tissue if T_μL ≤ 0.3, as soft tissue if 0.3 < T_μL ≤ 1.4 and as bone tissue if T_μL > 1.4; the soft tissue is further divided into two to form four (4) groups with 0.3 < T_μL ≤ 1 classifying soft tissue 1 and 1 < T_μL ≤ 1.4 classifying soft group 2. There are one (1) lung tissue, twenty-one (21) soft tissues and eleven (11) bone tissues for the training data while seven (7) soft tissues and five (5) bone tissues were used for testing.

Open in New Tab Full Size Figure Download Figure

Figure 2 The training data three (3) and four (4) group regions. A: Tissues were classified into lung tissues, soft tissues and bone tissues contained in left, middle and right region, respectively; B: This is four (4) region classification that divided the middle region from the previous three (3) regions into two regions. This divides the soft tissues into two more classes. ρ_e: Relative electron density.

Proposed general method

Proposition: Given a radiological parameter, p, empirically related to the attenuation coefficient, μ, there exist transformations/maps T_p and T_μ such that $T_{p} = \sum_{i = - r}^{n} ‍ a_{i} T_{μ}^{(i)} + e r r$ (3), n ≥ 0, r ≥ 0, $p \approx T_{p}^{- 1} (T_{p})$ , $a_{i} \in ℝ$ , $T_{p} = f (p)$ , Tμ is some g(μ), where μ = ( μ₁, μ₂, μ₃, …, μ_n) for n-energies with at least one low energy and err is the estimation error.

Note: $T_{p} = \sum_{i = - r}^{n} ‍ a_{i} T_{μ}^{(i)} = a_{- r} T^{(- r)} + . . . + a_{- 1} T^{(- 1)} + a_{0} T^{(0)} + a_{1} T^{(1)} + . . . + a_{n} T^{(n)} .$ For example, a_-1T^(-1) does not mean $a_{- 1} T^{(- 1)} = \frac{a_{- 1}}{T}$ , for all T. Proof: If $\exists$ a relationship between $p$ and $μ$ , $p \approx T_{μ}$ where Tμ=g(μ). Take bijective map, T, of both sides i.e., $T (p) \approx T (T_{μ})$ , $T (p) = T (T_{μ}) + ϵ$ (4), where $ϵ$ is additive error.

If $T$ is analytic, we can expand T(Tμ) using Taylor or Laurent series. If not, it can be expanded about its singular points or within neighborhood that excludes the singular point (this might lead to piece-wise function). So, in general $T (T_{μ}) = \sum_{i = - r}^{n} ‍ a_{i} T_{μ}^{(i)} + ϵ^{*}$ . $T (p) = \sum_{i = - r}^{n} ‍ a_{i} T_{μ}^{(i)} + ϵ^{*} + ϵ$ . Therefore, $T (p) = \sum_{i = - r}^{n} ‍ a_{i} T_{μ}^{(i)} + e r r$ , where $e r r = ϵ^{*} + ϵ .$

For the case of multiplicative error; without loss of generality, let $p = ϵ^{†} T_{μ}$ , define $T_{μ} ∋ T_{μ} > 0$ (this is always possible since $μ \geq 0$ ), take $l o g$ of both sides $l o g (p) = l o g (ϵ^{†} T_{μ})$ , $l o g (p) = l o g (ϵ^{†}) + l o g (T_{μ}) .$

Since $l o g$ is a bijective function, this can be written as $T (p) = T (T_{μ}) + ϵ,$ following the same argument from equation (4) we have $T (p) = \sum_{i = - r}^{n} ‍ a_{i} T_{μ}^{(i)} + e r r$ .

We can keep splitting and adding terms that reduces the error. Denoting T(p) as T_p, we have $T_{p} = \sum_{i = - r}^{n} ‍ a_{i} T_{μ}^{(i)} + e r r,$ since T is bijective $p \approx T_{p}^{- 1} (T_{p})$ . QED.

The relationship is found using data observation approach, after which reasonable T_μs is constructed and a good method of estimating the constants is employed. The low energy requirement was empirically observed.

We considered one form of this class of model in equation (3) as our first study here using T_μ = g(μ) fixed, i.e., $T_{p} = \sum_{i = - r}^{n} ‍ a_{i} T_{μ}^{i} = a_{- r} T^{- r} + . . . + a_{- 1} T^{- 1} + a_{0} T^{0} + a_{1} T^{1} + . . . + a_{n} T^{n},$ where, $a_{- r} T^{- r} = \frac{a_{- r}}{T^{r}}$ .

We implemented one of the simplest examples of the model of this form which was presented by Chika[43]. Two energy spectra were used for this study commonly known as dual-energy CT. Two transformations T_μL = μ_L and T_μm = μ_L*μ_H were used as presented in his paper and a new transformation:

$T_{μ m r} = \frac{μ_{L} μ_{H}}{μ_{L} + μ_{H}}$ was introduced to improve results based on the data we used here.

The transformations/maps used are: T_p = p for ρ_e and SPR, $T_{p} = l n (p)$ for mean excitation energy (I) and Z_ef. We implemented the simple approximation:

T_{p} = \sum_{i = - 2}^{2} ‍ a_{i} T_{μ}^{i} .

Symbol definition: T_p,r,n here means T for parameter p, low index r and top index n example:

T_{p, 1,2} = \sum_{i = - 1}^{2} ‍ a_{i} T_{μ}^{i} .

In the following subsections we presented four major fundamental quantities used to compute other quantities needed for radiation treatment with examples of the models that are in the class of the proposed general form.

SPR

Bethe equation is used to approximate the stopping power at a given energy E_k typically used in proton therapy. The equation is given as:

(5),

S (E_{k}) = - \frac{d E_{k}}{d x} = ρ_{e} \frac{τ_{1}}{β^{2}} l n [\frac{1}{2} l n \frac{τ_{2} β^{2} T_{m a x}}{I^{2} (1 - β^{2})} - β^{2}]

where $τ_{1}$ and $τ_{2}$ are products of physical constants; $β$ = $\frac{v_{p}}{c}$ is the proton speed relative to that of light; T_max is the maximum energy transferred from the proton to a single electron; ρ_e and I are the electron density and mean excitation energy of the medium. To get SPRof the tissue, the tissue SP is divided with that of water.

This can be written in a simpler approximated form known as Bethe-Bloch equation:

S P R = ρ_{e} \frac{l n \frac{2 m_{e} c^{2} β^{2}}{1 - β^{2}} - β^{2} - l n I}{l n \frac{2 m_{e} c^{2} β^{2}}{1 - β^{2}} - β^{2} - l n I_{w}},

where m_e is the rest mas of an electron, c is the speed of light, β is the velocity of the proton in vacuum relative to the speed of light, and I_w is the I of water.

We present some examples of the generalized method: Method presented by Chika[43]: We considered the following T_SPR,r,n for comparison and discussion of the methods.

T_{S P R, 1,2} = \{\begin{array}{l} 0.2579, & l u n g \\ 1.3419 - \frac{0.3119}{T_{μ_{L}}}, & s o f t \\ - 0.006 T_{μ_{L}}^{2} + 0.24 T_{μ_{L}} + 0.731 + \frac{0.0478}{T_{μ_{L}}}, & b o n e \end{array},

T_{S P R, 1,2} = \{\begin{array}{l} 0.2582, & l u n g \\ 1.2482 - \frac{0.2223}{T_{μ_{m}}}, & s o f t \\ - 0.0023 T_{μ_{m}}^{2} + 0.0919 T_{μ_{m}} + 1.0536 - \frac{0.1615}{T_{μ_{m}}}, & b o n e \end{array},

T_{S P R, 2,1} = \{\begin{array}{l} 0.2579, & l u n g \\ \frac{1.2509}{T_{μ_{m r}}} - \frac{0.3679}{T_{μ_{m r}}^{2}}, & s o f t \\ - 0.6783 T_{μ_{m r}} + 0.8117 - \frac{0.1894}{T_{μ_{m r}}} + \frac{0.1894}{T_{μ_{m r}}^{2}}, & b o n e \end{array},

T_{S P R_{s o f t}, 2,1} = \{\begin{array}{l} 0.2579, & l u n g \\ 16.8345 T_{μ_{m r}} - 23.9081 + \frac{12.4929}{T_{μ_{m r}}} - \frac{2.117}{T_{μ_{m r}}^{2}}, & s o f t \\ - 0.6783 T_{μ_{m r}} + 0.8117 - \frac{0.1894}{T_{μ_{m r}}} + \frac{0.1894}{T_{μ_{m r}}^{2}}, & b o n e \end{array},

T_{S P R_{4 g}, 2,1} = \{\begin{array}{l} 0.2579, & l u n g \\ \frac{1.2054}{T_{μ_{m r}}} - \frac{0.3450}{T_{μ_{m r}}^{2}}, & s o f t 1 \\ \frac{1.4047}{T_{μ_{m r}}} - \frac{0.4493}{T_{μ_{m r}}^{2}}, & s o f t 2 \\ - 0.6783 T_{μ_{m r}} + 0.8117 - \frac{0.1894}{T_{μ_{m r}}} + \frac{0.1894}{T_{μ_{m r}}^{2}}, & b o n e \end{array} .

Stoichiometric method, T_SPR,0,1: This method is widely used in clinical implementation. It’s also referred to as single energy CT method. Single energy calibration method estimates SPR using its linear piece-wise empirical relationship with CT numbers[45]. i.e.,

S P R = \{\begin{array}{l} a_{0, l}, & l u n g \\ a_{1, s} μ + a_{0, s}, & s o f t \\ a_{1, b} μ + a_{0, b}, & b o n e \end{array},

Where a_0,l = 0.2579, a_1,s = 0.319, a_0,s = 0.7091, a_1,b = 0.2071, and a_0,b = 0.8158, in this study. This can be represented in the form of the generalized model as follows: $T_{S P R} = a_{1} T_{μ} + a_{0}$ , i.e., $T_{S P R} = \sum_{i = 0}^{1} ‍ a_{i} T_{μ}^{i}$ , where T_SPR = SPR and T_μ = μ.

Taasti method: Empirical parametrization model was presented by Taasti et al[11] using dual-energy CT as stated below:

(6),

S P R_{s o f t}^{e s t} = [(1 + a_{1}) μ_{H} - a_{1} μ_{L}] + a_{2} μ_{L}^{2} + a_{3} μ_{H}^{2} + a_{4} (μ_{L}^{3} + μ_{H}^{3})

(7).

S P R_{b o n e}^{e s t} = [(1 + a_{1}) μ_{H} - a_{1} μ_{L}] + a_{2} \frac{μ_{L}}{μ_{H}} + a_{3} (μ_{L}^{2} - μ_{H}^{2}) + a_{4} (μ_{L}^{3} + μ_{H}^{3})

For soft tissues, a_l = 1.1859, a₂ = 0.4637, a₃ = -0.4153, a₄ = -0.0223 and for bone a_l = 0.2866, a₂ = 0.0167, a₃ = 0.0167, a₄ = -0.0009 for the data used in this study. If we use the following representation $T_{μ}^{1} = μ_{L}$ , $T_{μ}^{2} = μ_{H}$ , $T_{μ}^{3} = μ_{L}^{2}$ , $T_{μ}^{4} = μ_{H}^{2}$ , $T_{μ}^{5} = μ_{L}^{2} - μ_{H}^{2}$ , $T_{μ}^{6} = μ_{L}^{3} + μ_{H}^{3}$ , $T_{μ}^{7} = \frac{μ_{L}}{μ_{H}}$ . One way of writing Taasti method in the form of Eq(3) is $T_{S P R} = \sum_{i = 0}^{7} ‍ a_{i} T_{μ}^{i}$ , here, r = 0, n = 7, a₂ = a₁+1, a₀ = 0 or $T_{μ}^{0} = 0$ . a₅ = a₇ = 0 for soft tissues and a₃ = a₄ = 0 for bone tissues. Note: This representation is not unique.

Hunemohr-saito method: The model below was developed by Hunemohr et al[46], and Saito[47] developed a similar formula for ρ_e. This is referred to as Hunemohr-Saito (H-S) method in this work.

(8).

\frac{ρ_{e}}{ρ_{e, w}} = c_{H} μ_{L} + (1 - c_{H}) μ_{H}

(9).

Z_{e f} = ((\frac{ρ_{e}}{ρ_{e, w}})^{- 1} (d_{H} μ_{L} + (Z_{e, w}^{n} - d_{H}) μ_{H}))^{\frac{1}{n}}

The two constants c_H and d_H depend on specific dual-energy CT scanning protocol. The relationship found by some studies[48,49] was used to calculate I-value from Z_ef after which Bethe-Bloch equation was used to calculate SPR. The empirical relationship is as stated below.

l n (I) = \{\begin{array}{l} a_{0, l}, & l u n g \\ a_{1, s} Z_{e f} + a_{0, s}, & s o f t \\ a_{1, b} Z_{e f} + a_{0, b}, & b o n e \end{array} .

Based on this study, c_H = -0.2087, d_H = 15 and n = 1.0715 for soft tissues while c_H = -0.2, d_H = 15 and n = 1.3411 for bone tissues. On estimation of $I$ , a_0,l = 4.3197, a_1,s = 0.0998, a_0,s = 3.5524, a_1,b = 0.1105, and a_0,b = 3.327.

From our notation in Taasti method, we have:

T_{ρ_{e}} = \frac{ρ_{e}}{ρ_{e, w}} = a_{1} T_{1} + a_{2} T_{2},

where a₂ = 1-a₁.

ρ_e

The true ρ_e can be computed with the following formula:

(10),

ρ_{e} = \frac{ρ_{e, x}}{ρ_{e, w}} = \frac{ρ_{x} \sum_{j = 1}^{J} ‍ \frac{ω_{j} Z_{j}}{A_{j}}}{ρ_{e, w}}

where ρ_x denotes the mass density and ρ_e,w is the electron density of water. ω_j, Z_j, and A_j are the mass fraction, atomic number and atomic mass of the i-th element in the tissue, respectively.

Method presented by Chika[43]: The following function are considered from the method proposed by Chika[43].

T_{ρ_{e}, 1,2} = \{\begin{array}{l} 0.2579, & l u n g \\ 1.4170 - \frac{0.3927}{T_{μ_{L}}}, & s o f t \\ - 0.0063 T_{μ_{L}}^{2} + 0.2805 T_{μ_{L}} + 0.6859 + \frac{0.0497}{T_{μ_{L}}}, & b o n e \end{array},

T_{ρ_{e}, 1,2} = \{\begin{array}{l} 0.2582, & l u n g \\ 1.2955 - \frac{0.2761}{T_{μ_{m}}}, & s o f t \\ - 0.0026 T_{μ_{m}}^{2} + 0.1050 T_{μ_{m}} + 1.0449 - \frac{0.1811}{T_{μ_{m}}}, & b o n e \end{array},

T_{ρ_{e}, 2,1} = \{\begin{array}{l} 0.2553, & l u n g \\ \frac{1.2984}{T_{μ_{m r}}} - \frac{0.3932}{T_{μ_{m r}}^{2}}, & s o f t \\ 0.7872 T_{μ_{m r}} + 0.7368 - \frac{0.1877}{T_{μ_{m r}}} + \frac{0.0660}{T_{μ_{m r}}^{2}}, & b o n e \end{array},

T_{ρ_{e_{4 g}}, 2,1} = \{\begin{array}{l} 0.2553, & l u n g \\ \frac{1.2355}{T_{μ_{m r}}} - \frac{0.3624}{T_{μ_{m r}}^{2}}, & s o f t 1 \\ \frac{1.4323}{T_{μ_{m r}}} - \frac{0.4640}{T_{μ_{m r}}^{2}}, & s o f t 2 \\ 0.7872 T_{μ_{m r}} + 0.7368 - \frac{0.1877}{T_{μ_{m r}}} + \frac{0.0660}{T_{μ_{m r}}^{2}}, & b o n e \end{array} .

Z_ef

The true Z_ef is computed as:

(11),

Z_{e f}^{n} = \frac{\sum_{i} ‍ \frac{ω_{i} Z_{i}}{A_{i}} Z_{i}^{n}}{\sum_{i} ‍ \frac{ω_{i} Z_{i}}{A_{i}} Z_{i}}

where ω_i, Z_i, and A_i are the mass fraction, atomic mass and the atomic number of the $i$ -th element in the tissue, respectively and n = 3.3. The equation above is called the Mayneord’s equation.

Method presented by Chika[43]: The following $T_{Z, r, n} = l n (Z_{e f})$ are presented:

T_{Z, 1,1} = \{\begin{array}{l} 2.0272, & l u n g \\ 0.1987 T_{μ_{m}} + 2.1654 - \frac{0.3968}{T_{μ_{m}}}, & s o f t \\ 0.0075 T_{μ_{m}} + 2.6059 - \frac{0.5963}{T_{μ_{m}}}, & b o n e \end{array},

T_{Z, 2,0} = \{\begin{array}{l} 2.0302, & l u n g \\ 3.4915 - \frac{0.9277}{T_{μ_{m r}}} + \frac{0.0827}{T_{μ_{m r}}^{2}}, & s o f t \\ 2.8458 - \frac{0.3175}{T_{μ_{m r}}} - \frac{0.0314}{T_{μ_{m r}}^{2}}, & b o n e \end{array},

T_{Z_{4 g}, 2,0} = \{\begin{array}{l} 2.0302, & l u n g \\ 18.6634 - {\frac{15.4103}{T}}_{μ_{m r}} + \frac{3.5308}{T_{μ_{m r}}^{2}}, & s o f t 1 \\ 7.1234 - {\frac{4.8661}{T}}_{μ_{m r}} + \frac{1.1493}{T_{μ_{m r}}^{2}}, & s o f t 2 \\ 2.8458 - \frac{0.3175}{T_{μ_{m r}}} - \frac{0.0314}{T_{μ_{m r}}^{2}}, & b o n e \end{array} .

Bourque’s model: Spectrally averaged elemental electronic cross sections are fit to polynomial function,

< σ_{e} > (Z) = \sum_{k}^{K} ‍ a_{k} Z^{k - 1},

where Z is their atomic number.

Then, the spectrally-dependent Z_ef for an arbitrary mixture is given as $Z_{e f} = < σ_{e} >^{- 1} (σ_{e, m i x})$ [50].

The model is given by

Z_{e f} = \sum_{k = 1}^{K} ‍ a_{k} (\frac{μ_{L} - μ_{H}}{μ_{L} + μ_{H}})^{(k - 1)},

\frac{ρ_{e}}{ρ_{e, w}} = \frac{μ_{L / H}}{\sum_{m = 1}^{M} ‍ a_{m, L / H}^{*} Z_{e f}^{m - 1}},

where $a_{k}$ and $a_{m, L / H}^{*}$ are scanner-specific model parameters. The original study used K = M = 6 which was used here.

We implemented the Z_ef as presented by Chika[43], it is represented as T_B,0,1. $T_{B, 0,1} = \sum_{k = 1}^{K} ‍ a_{k} T_{μ}^{(k - 1)},$ where

T_{μ} = \frac{μ_{1} - μ_{2}}{μ_{1} + μ_{2}} .

For soft tissue, a_l = 7.4765, a₂ = 26.4841, a₃ = -4.0747, a₄ = 101.6414, a₅ = 18.2317, a₆ = 99.97; for bone tissues, a_l = 7.4368, a₂ = 22.7045, a₃ = -11.3634, a₄ = -89.412, a₅ = 453.7994, a₆ = -522.3422 while a_l = 7.5975 and a₂ = a₃ = a₄ = a₅ = a₆ = 0 for lung in this study.

True I of the tissues was computed from elemental I-values using the Bragg additivity rule:

l n (I) = \frac{\sum_{i} ‍ ω_{i} (\frac{Z}{A})_{i} l n (I_{i})}{\sum_{i} ‍ ω_{i} (\frac{Z}{A})_{i}},

where ω_i, Z_i, A_i and I_i are the mass fraction, atomic number, atomic mass and I of the i-th element in the tissue, respectively.

Method proposed by Chika[43]: The following simple form of the proposed model are presented below:

T_{I, r, n} = l n (I),

T_{I, 2,0} = \{\begin{array}{l} 4.3216, & l u n g \\ 3.4088 + \frac{1.3189}{T_{μ_{m r}}} - \frac{0.4438}{T_{μ_{m r}}^{2}}, & s o f t \\ 5.2708 - \frac{0.9619}{T_{μ_{m r}}} + \frac{0.2323}{T_{μ_{m r}}^{2}}, & b o n e \end{array},

T_{I_{4 g}, 2,0} = \{\begin{array}{l} 4.3216, & l u n g \\ 11.3950 - \frac{6.4434}{T_{μ_{m r}}} + \frac{1.4350}{T_{μ_{m r}}^{2}}, & s o f t 1 \\ 13.0218 - \frac{9.0908}{T_{μ_{m r}}} + \frac{2.3717}{T_{μ_{m r}}^{2}}, & s o f t 2 \\ 5.2708 - \frac{0.9619}{T_{μ_{m r}}} + \frac{0.2323}{T_{μ_{m r}}^{2}}, & b o n e \end{array} .

Accuracy analysis

M E = \frac{1}{N} \sum_{i = 1}^{N} ‍ e_{i}, R M S E = \sqrt{\frac{\sum_{i = 1}^{N} ‍ e_{i}^{2}}{N}}, e = \frac{p_{t r u e} - p_{p r e d}}{p_{t r u e}},

where $p_{t r u e}$ is the true value of the parameter and $p_{p r e d}$ is the estimated value.

The mean error measures the overall bias of the values estimates while RMSE measures the systematic estimation error for different tissues. Also, maximum absolute error and median absolute error are included to provide more insight on worst estimation and biasedness respectively.

Computation and software

Model parameters (constants) estimations were done using MATLAB while estimation of SPR, ρ_e, Z_ef and I were computed using Excel sheet. All the errors were computed using Excel sheet while the figures were plotted using MATLAB plot function and scatter diagram. Linear model parameters were estimated using fitlm in MATLAB which creates linear regression model by fitting to data while nonlinear model parameters were estimated using lsqnonlin in MATLAB which solves nonlinear least squares problems. The effective energy is derived using MATLAB by selecting the energy that gives the minimum value for norm(μ-μ_k(E)), where the norm is 2-norm. The spectrum was generated using SpekCal, this can be seen in reference[38] and mass attenuation was simulated using XCOM, this can be seen in reference number[33]. Interpolation was used to match the generated spectrum energy levels with simulated attenuation energy levels using MATLAB function.

RESULTS

SPR

Table 1 shows that Taasti method has the best prediction performance based on all the methods considered with training RMSE 0.15% and testing RMSE 0.67%.Tables 2 and 3 indicate that all methods have relatively low bias on training data since their mean errors are less than 0.04% except H-S method and median absolute error less than or equal 0.37%, though these are relatively higher on testing data. Table 4 shows that T_SPR,1,2; T_μL has the highest maximum absolute error of 22.35% which occurred at cortical bone. This might be improved by further partitioning the bone tissues since it shows an outlier behavior. Figure 3 showed that the methods have close performance on bone tissues with relative errors of less than 1% and T_SPR,1,2 performs slightly better than stoichiometric method in terms of single energy.

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Figure 3 Showed that the methods have close performance on bone tissues with relative errors of less than 1% and T_SPR,1,2 performs slightly better than stoichiometric method in terms of single energy. A: Stopping power ratio modeling relative error for each human tissue using single energy computed tomography (CT) methods and one dual-energy CT method that performs well. It errors are as presented in the graph for individual tissues; B: Stopping power ratio modeling relative error for each human tissue using dual-energy CT methods. SPR: Stopping power ratio; H-S: Hunemohr-Saito.

Table 1 Modeling and testing root mean square error (%) for stopping power ratio.

SPR modeling RMSE	Total	Lung	Soft	Bone	Testing RMSE	Total	Soft	Bone
T_SPR,2,1; T_μmr	0.85	0.00	1.03	0.40		2.69	2.72	2.65
T_SPRsoft,2,1 bone; T_μmr	0.73	0.00	0.87	0.40		2.63	2.62	2.65
T_SPR,1,2; T_μm	0.76	0.13	0.90	0.42		2.66	2.66	2.65
T_SPR,1,2; T_μL	0.83	0.02	0.96	0.53		7.26	3.42	10.49
Stoichiometric	0.93	-0.01	1.10	0.56		3.40	3.38	3.42
Taasti	0.15	0.01	0.15	0.15		0.67	0.76	0.60
H-S	0.39	0.28	0.41	0.35		0.85	0.91	0.77
T_SPR,2,1; T_μmr 4 g	0.69	0.00	0.81	0.40		2.37	2.14	2.65

RMSE: Root mean square error; SPR: Stopping power ratio; H-S: Hunemohr-Saito.

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Table 2 Modeling and testing mean error (%) for stopping power ratio.

SPR modeling ME	Total	Lung	Soft	Bone	Testing ME	Total	Soft	Bone
T_SPR,2,1; T_μmr	-0.01	0.00	-0.01	0.00		-0.59	-0.32	-0.98
T_SPRsoft,2,1 bone; T_μmr	0.00	0.00	0.00	0.00		-0.70	-0.50	-0.98
T_SPR,1,2; T_μm	-0.01	-0.13	-0.01	-0.01		-0.74	-0.56	-1.00
T_SPR,1,2; T_μL	0.04	-0.02	0.07	-0.02		1.02	-0.73	3.46
Stoichiometric	-0.01	-0.01	-0.01	0.00		-1.19	-1.00	-1.45
Taasti	0.02	-0.01	0.00	0.08		0.34	0.28	0.43
H-S	0.29	0.28	0.30	0.29		0.74	0.76	0.71
T_SPR,2,1; T_μmr 4 g	0.01	0.00	0.02	0.00		-0.51	-0.18	-0.98

ME: Mean error; ρ_e: Relative electron density; SPR: Stopping power ratio; H-S: Hunemohr-Saito.

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Table 3 Modeling and testing median absolute error (%) for stopping power ratio.

SPR modeling MedAE	Total	Lung	Soft	Bone	Testing MedAE	Total	Soft	Bone
T_SPR,2,1; T_μmr	0.37	0.00	0.58	0.20		2.25	1.63	2.87
T_SPRsoft,2,1 bone; T_μmr	0.30	0.00	0.31	0.20		2.48	2.26	2.87
T_SPR,1,2; T_μm	0.29	0.13	0.29	0.18		2.20	1.81	2.91
T_SPR,1,2; T_μL	0.30	0.02	0.43	0.27		3.22	2.93	4.22
Stoichiometric	0.37	0.01	0.37	0.35		3.08	2.81	3.35
Taasti	0.08	0.01	0.08	0.08		0.67	0.79	0.54
H-S	0.22	0.28	0.22	0.25		0.62	0.64	0.53
T_SPR,2,1; T_μmr 4 g	0.27	0.00	0.29	0.20		1.47	1.22	2.87

MedAE: Median absolute error; SPR: Stopping power ratio; H-S: Hunemohr-Saito.

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Table 4 Modeling and testing maximum absolute error (%) for stopping power ratio.

SPR modeling MaxAE	Total	Lung	Soft	Bone	Testing MaxAE	Total	Soft	Bone
T_SPR,2,1; T_μmr	2.57	0.00	2.57	1.04		4.76	4.76	3.74
T_SPRsoft,2,1 bone; T_μmr	2.51	0.00	2.51	1.04		4.83	4.83	3.74
T_SPR,1,2; T_μm	2.44	0.13	2.44	1.16		5.05	5.05	3.67
T_SPR,1,2; T_μL	2.91	0.02	2.91	1.39		22.35	5.87	22.35
Stoichiometric	2.93	0.01	2.91	1.38		5.94	5.94	4.88
Taasti	0.40	0.01	0.40	0.28		1.04	1.04	1.02
H-S	1.21	0.28	1.21	0.67		1.31	1.31	1.13
T_SPR,2,1; T_μmr 4 g	2.26	0.00	2.26	1.04		4.49	4.49	3.74

MaxAE: Maximum absolute error; SPR: Stopping power ratio; H-S: Hunemohr-Saito.

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ρ_e

All the methods presented here achieved training RMSE of less than 0.90% for prediction of ρ_e. H-S method achieved the least training and testing error which are 0.04% and 0.11% respectively, this can be seen in Table 5, and Table 6 shows low training bias since the absolute mean error is less than 0.03% and that of testing data is not that high since it’s less than $0.90 %$ except for single energy which is 1.29%. This is further justified by the modeling median absolute error which is less than or equal to 0.37%, though relatively higher in testing data as can be seen in Table 7. Table 8 shows that H-S has the least maximum absolute error while the single energy model has the highest modeling maximum absolute error of 2.72% which occurred at cortical bone. Just as in SPR, Figure 4 shows that the models fit better to bone tissues compared to soft tissues and the training absolute relative error of most bone tissues are less than 1%.

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Figure 4 Predicts relative electron density modeling relative error for each human tissue using dual-energy computed tomography methods. The errors for each method for individual tissues are as presented on the graph. ρ_e: Relative electron density; H-S: Hunemohr-Saito.

Table 5 Modeling and testing root mean square error (%) for relative electron density.

ρ_e modeling RMSE	Total	Lung	Soft	Bone	Testing RMSE	Total	Soft	Bone
T_ρe,2,1; T_μmr	0.82	0.99	0.96	0.44		2.48	2.55	2.37
T_ρe,1,2; T_μm	0.70	0.16	0.81	0.47		2.32	2.25	2.41
T_ρe,1,2; T_μL	0.89	0.04	1.02	0.59		3.03	3.00	3.08
H-S	0.04	0.01	0.04	0.04		0.11	0.13	0.08
T_ρe,2,1; T_μmr 4 g	0.69	0.99	0.77	0.44		2.31	2.26	2.37

RMSE: Root mean square error; ρ_e: Relative electron density; H-S: Hunemohr-Saito.

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Table 6 Modeling and testing mean error (%) for relative electron density.

ρ_e modeling ME	Total	Lung	Soft	Bone	Testing ME	Total	Soft	Bone
T_ρe,2,1; T_μmr	-0.02	-0.99	0.01	0.00		0.73	0.41	1.18
T_ρe,1,2; T_μm	0.01	0.16	0.00	0.02		0.89	0.65	1.22
T_ρe,1,2; T_μL	0.01	0.04	0.01	0.00		1.29	1.09	1.56
H-S	0.02	0.01	0.02	0.02		0.10	0.11	0.07
T_ρe,2,1; T_μmr 4 g	-0.03	-0.99	0.00	0.00		0.89	0.69	1.18

ME: Mean error; ρ_e: Relative electron density; H-S: Hunemohr-Saito.

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Table 7 Modeling and testing median absolute error (%) for relative electron density.

ρ_e modeling MedAE	Total	Lung	Soft	Bone	Testing MedAE	Total	Soft	Bone
T_ρe,2,1; T_μmr	0.37	0.99	0.46	0.27		2.33	2.40	2.25
T_ρe,1,2; T_μm	0.26	0.16	0.29	0.23		1.90	1.71	2.31
T_ρe,1,2; T_μL	0.35	0.04	0.37	0.35		2.41	2.40	2.92
H-S	0.02	0.01	0.02	0.03		0.11	0.12	0.05
T_ρe,2,1; T_μmr 4 g	0.24	0.99	0.20	0.27		2.09	2.04	2.25

MedAE: Median absolute error; ρ_e: Relative electron density; H-S: Hunemohr-Saito.

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Table 8 Modeling and testing maximum absolute error (%) for relative electron density.

ρ_e modeling MaxAE	Total	Lung	Soft	Bone	Testing MaxAE	Total	Soft	Bone
T_ρe,2,1; T_μmr	2.27	0.99	2.27	1.11		3.69	3.69	3.35
T_ρe,1,2; T_μm	2.15	0.16	2.15	1.24		3.97	3.97	3.29
T_ρe,1,2; T_μL	2.72	0.04	2.72	1.49		5.07	5.07	4.28
H-S	0.10	0.01	0.10	0.06		0.16	0.16	0.11
T_ρe,2,1; T_μmr 4 g	2.27	0.99	2.27	1.11		3.62	3.62	3.35

MaxAE: Maximum absolute error; ρ_e: Relative electron density; H-S: Hunemohr-Saito.

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Z_ef

T_B,0,1 gives the best performance here with training and testing RMSE of 0.32% and 0.39%. Others have reasonable errors though relatively high on soft tissues and low in bone tissues except H-S method that behaves in the opposite direction as can be seen in Table 9. Similarly, T_B,0,1 shows least bias with training mean error of 0.00% as in Table 10 and has absolute median training and testing errors of 0.10% and 0.27%, respectively as can be seen Table 11. T_Z,2,0; T_μmr 4 g shows good performance. Table 12 shows that T_Z,2,0; T_μmr has highest maximum absolute error which occurred at brain tissue. Though, the course of this is yet to be determined but further classification reduced the error. Figure 5 shows the individual tissue performance which indicates that other methods perform better than H-S method on bone tissues.

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Figure 5 Z modeling relative error for each human tissue using dual-energy computed tomography methods. The errors for each method for individual tissues are as presented on the graph; H-S: Hunemohr-Saito.

Table 9 Modeling and testing root mean square error (%) for effective atomic number.

Z_ef modeling RMSE	Total	Lung	Soft	Bone	Testing RMSE	Total	Soft	Bone
T_Z,2,0; T_μmr	9.34	0.24	11.71	0.45		10.63	13.77	2.45
T_Z,1,1; T_μm	4.33	0.06	5.42	0.45		6.28	7.88	2.79
T_B,0,1	0.32	0.00	0.40	0.01		0.39	0.49	0.08
H-S	5.27	0.00	0.53	9.11		5.63	1.00	8.64
T_Z,2,0; T_μr 4 g	1.61	0.24	1.99	0.45		5.55	6.97	2.45

RMSE: Root mean square error, Z_ef: Effective atomic number; H-S: Hunemohr-Saito.

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Table 10 Modeling and testing mean error (%) for effective atomic number.

Z_ef modeling ME	Total	Lung	Soft	Bone	Testing ME	Total	Soft	Bone
T_Z,2,0; T_μmr	7.23	0.24	11.34	0.01		6.15	11.42	-1.23
T_Z,1,1; T_μm	3.12	-0.06	4.91	0.00		1.25	3.20	-1.49
T_B,0,1	0.00	0.00	0.00	0.00		-0.17	-0.29	0.04
H-S	3.10	0.00	0.10	9.10		3.25	-0.55	8.57
T_Z,2,0; T_μr 4 g	0.01	0.24	0.01	0.01		-0.79	-0.48	-1.23

ME: Mean error; Z_ef: Effective atomic number; H-S: Hunemohr-Saito.

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Table 11 Modeling and testing median absolute error (%) for effective atomic number.

Z_ef modeling MedAE	Total	Lung	Soft	Bone	Testing MedAE	Total	Soft	Bone
T_Z,2,0; T_μmr	10.19	0.24	10.74	0.19		5.60	9.04	2.24
T_Z,1,1; T_μm	3.93	0.06	4.95	0.33		1.78	1.00	2.22
T_B,0,1	0.10	0.00	0.15	0.00		0.27	0.27	0.05
H-S	0.65	0.00	0.25	8.98		1.83	0.20	7.85
T_Z,2,0; T_μr 4 g	0.47	0.24	0.53	0.19		3.50	7.20	2.24

MedAE: Median absolute error; Z_ef: Effective atomic number; H-S: Hunemohr-Saito.

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Table 12 Modeling and testing maximum absolute error (%) for effective atomic number.

Z_ef modeling MaxAE	Total	Lung	Soft	Bone	Testing MaxAE	Total	Soft	Bone
T_Z,2,0; T_μmr	18.39	0.24	18.39	1.19		29.28	29.28	3.53
T_Z,1,1; T_μm	10.29	0.06	10.29	0.97		19.97	19.97	4.17
T_B,0,1	1.17	0.00	1.17	0.01		0.97	0.97	0.12
H-S	9.64	0.00	1.10	9.64		10.54	1.99	10.54
T_Z,2,0; T_μmr 4 g	6.32	0.24	6.32	1.19		11.04	11.04	3.53

MaxAE: Maximum absolute error; Z_ef: Effective atomic number; H-S: Hunemohr-Saito.

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Table 13 shows that T_I,2,0; T_μmr 4 g has the best training performance with RMSE of 0.85% and close testing performance with H-S method. They have relatively low bias on training data with an increased bias on testing data as presented in Tables 14 and 15. T_I,2,0; T_μmr 4 g also has the least maximum absolute error in training and testing which are 2.01% and 7.09%, respectively, as can be seen in Table 16. Figure 6 shows that the model fit better on bone tissues compare to soft tissues.

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Figure 6 Mean excitation energy modeling relative error for each human tissue using dual-energy computed tomography methods. The errors for each method for individual tissues are as presented on the graph. I: Mean excitation energy; H-S: Hunemohr-Saito.

Table 13 Modeling and testing root mean square error (%) for mean excitation energy.

I modeling RMSE	Total	Lung	Soft	Bone	Testing RMSE	Total	Soft	Bone
T_I,2,0; T_μmr	3.94	0.19	4.93	0.49		5.93	6.93	4.14
H-S	1.95	0.00	2.23	1.41		4.71	5.13	4.05
T_I,2,0; T_μmr 4 g	0.85	0.19	1.00	0.49		4.92	5.40	4.14

RMSE: Root mean square error; I: Mean excitation energy; H-S: Hunemohr-Saito.

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Table 14 Modeling and testing mean error (%) for mean excitation energy.

I modeling ME	Total	Lung	Soft	Bone	Testing ME	Total	Soft	Bone
T_I,2,0; T_μmr	-0.71	0.19	-1.13	0.01		3.46	4.56	1.91
H-S	-0.04	0.00	-0.06	0.00		-3.14	-3.26	-2.97
T_I,2,0; T_μmr 4 g	0.02	0.19	0.02	0.01		1.19	0.68	1.91

ME: Mean error; I: Mean excitation energy; H-S: Hunemohr-Saito.

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Table 15 Modeling and testing median absolute error (%) for mean excitation energy.

I modeling MedAE	Total	Lung	Soft	Bone	Testing MedAE	Total	Soft	Bone
T_I,2,0; T_μmr	2.54	0.19	3.22	0.39		4.59	4.45	4.73
H-S	0.86	0.00	1.01	0.61		2.18	2.29	1.31
T_I,2,0; T_μmr 4 g	0.57	0.19	0.73	0.39		4.75	6.13	4.73

MedAE: Median absolute error; I: Mean excitation energy; H-S: Hunemohr-Saito.

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Table 16 Modeling and testing maximum absolute error (%) for mean excitation energy.

I modeling MaxAE	Total	Lung	Soft	Bone	Testing MaxAE	Total	Soft	Bone
T_I,2,0; T_μmr	12.73	0.19	12.73	0.97		13.15	13.15	5.22
H-S	7.52	0.00	7.52	3.09		7.67	7.67	7.03
T_I,2,0; T_μmr 4 g	2.01	0.19	2.01	0.97		7.09	7.09	5.22

MaxAE: Maximum absolute error; I: Mean excitation energy; H-S: Hunemohr-Saito.

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DISCUSSION

We are able to present a generalized model with proof. The model provides a sense of direction for construction of functions that will improve the prediction of quantities of interest in radiation treatment. Most of the existing empirical and none empirical parametric methods for image domain approach can be represented in the form of the presented model. To include some other models such as estimating ρ_ewith Bourque’s model, we can state the general form as: $T_{p} = \sum_{i = - r}^{n} ‍ a_{i} T^{(i)} (μ, p^{*})$ (12), $n \geq 0$ , $r \geq 0$ , $p = T_{p}^{- 1} (T_{p})$ , $a_{i} \in ℝ$ , $T_{p} = f (p)$ , where, $μ = (μ_{1}, μ_{2}, μ_{3}, . . ., μ_{n})$ for n-energies with at least one low energy and $p^{*} = (p_{1}^{*}, p_{2}^{*}, p_{3}^{*}, . . ., p_{m}^{*})$ for m-parameters.

Classification, in terms of grouping the tissues, plays a big role in improving the accuracy of the models. Though, this idea has been proposed by different authors which rallies around empirical relationship between the I and Z_ef considering their chlorine composition[48], we used the CT image data for our classification. Similar grouping has been done using ρ_e[51] and was compared with results in[52]. The importance of this is observed in one of the T_μ we tried that’s not presented here which gives reasonable RMSE but gives a high error for spongiosa and sacrum with more than 6% and 3% respectively; when these two tissues were classified as soft tissues instead of bone tissues, their error reduced drastically with that of spongiosa reducing to -0.04%.

Table 9 shows that dividing the soft tissues into two gives better result as it reduces the overall training RMSE of T_Z,2,0 from 9.34% to 1.61% and the mean error from 7.23% to 0.01% while reducing the soft tissue RMSE from 11.71% to 1.99% and mean error from 11.34% to 0.01%. It also exhibits similar effect on testing error. So, we found here that careful classification and grouping of tissues can improve both training and testing accuracy. The classification can be done with any preferred T_μ.

Based on the data we used here which is theoretical CT image at effective energy, we obtained slightly different result from what was presented by Chika[43] using spectrally averaged theoretical CT image attenuation computed using the energy spectrum. The method presented by Chika[43] works better using the spectrally averaged theoretical attenuation when compared to that of effective energy while that of Taasti and H-S works better with effective energy attenuation when estimating SPR and ρ_e, respectively.

The I and Z_ef seems to be more technical to predict but has better result presented here and hope for improvement when compared to other results. There is no known popular way of estimating I from CT image attenuation, which makes it difficult to present comparison on that. What we used for comparison is the popular Z-I empirical relationship first presented by Yang et al[48]. This is kind of crude comparison since we are doing less work on the method proposed by Chika[43] because we are predicting I using the data that’s given to us directly.

Among all the methods presented, the one presented by Chika[43] has more opportunity for improvement; apart from improvement that comes with more grouping, the degree and function of the model can be adjusted based on the available data which directly or indirectly includes the spectrum information as well as other phantom and machine information which is the reason it works better with the poly-energy attenuation compared to mono-energy attenuation. The method is also for n-energies just as the one proposed here unlike others that are for single or dual energy.

The limitation of this study is limited data as machine learning approaches need more data for robustness. We used 33 of the 34 ICRU 44 human tissues as our training data, this is because we couldn’t find the cell nucleus composition as at the time of this study. This study can be considered ambitious since we used Gammex tissues that are not that closely related to the one we used for training. The common idea that is normally used in other studies is to change composition of few tissues by small percentage. So, the testing error difference we are getting here is expected.

The general machine learning algorithm for model selection is presented below: Decide the parameter p = SPR, ρ_e, Z, I …you need: (1) Given CT image data for different energies; (2) Compute μ; (3) Construct different T_μ; (4) Formulate different T_p; (5) Classify the tissues into n-groups; (6) Compute the training and testing error; (7) Check other uncertainties; and (8) Choose the method/model with the optimal training and testing errors given priority to testing error.

Algorithm 4.1

Considering N models and a threshold c > 0; For i = 1:N construct T_μ(i); formulate T_p(i); compute training and testing RMSE for different tissue groups on T_p(i); end, compare the modeling and testing RMSE of T_p(i); For i = 1:N, if (modeling RMSE of T_p(i) < modeling RMSE T_p(i+1) & testing RMSE of T_p(i) < testing RMSE of T_p(i+1)); T_p = T_p(i); Else If (modeling RMSE of T_p(i+1) < modeling RMSE T_p(i) & testing RMSE of T_p(i+1) < testing RMSE of T_p(i))); T_p = T_p(i+1); Else If ((|modeling RMSE of T_p(i) - modeling RMSE T_p(i+1)| < c & testing RMSE of T_p(i) < testing RMSE of T_p(i+1) or (modeling RMSE of T_p(i)< modeling RMSE T_p(i+1) &|testing RMSE of T_p(i)-testing RMSE of T_p(i+1)|< c)); T_p = T_p(i); Else if ((|modeling RMSE of T_p(i) - modeling RMSE T_p(i+1)|< c & testing RMSE of T_p(i+1) < testing RMSE of T_p(i) or (modeling RMSE of T_p(i+1) < modeling RMSE of T_p(i))&|testing RMSE of T_p(i) - testing RMSE of T_p(i+1) < c)); T_p = T_p(i+1); else, reformulate T_p(i) or T_p(i+1) whichever is with bigger modeling RMSE. End, best model =T_p.

Implementing the algorithm in the above example using c = 0.5% gives the best model for SPR as Taasti model, for ρ_e as H-S model, for Z_ef as T_B,0,1, and for I as T_I,2,0; T_μmr 4 g. In practical sense when this is applied in clinic, the clinic will be able to adopt the best model for estimating each of these parameters using calibration or patients’ data.

This study will also improve calculation of SPR using Bethe-Bloch equation since the best estimation approaches for calculating ρ_e and I can be adopted without restriction to one specific method of estimating both quantities. Improved precision of these parameters will improve the accuracy of Bethe-Bloch equation estimation.

Uncertainty measurements can also be used as a benchmark for chosen best model in this algorithm but this study is not focused on detail uncertainty measurements; this will be a point of focus on future studies. Lung tissue was not considered in testing because it was modelled as a fixed point i.e., a fixed value; future studies will consider ways of addressing this such as simulating more lung tissues artificially, adjusting compositions or searching for more data on lung tissue.

The presented model selection algorithm will also help to prevent overfitting as training and testing errors are considered before selecting a model. This will help to decide the best degree for the model presented by Chika[43].

CONCLUSION

The generalized mathematical model estimates ρ_e, Z_ef, I and proton SPR with good precision especially for ρ_e and SPR where all presented examples of the model have total modeling RMSE less than 0.9%; though, comparison shows that the performance is data dependent as well as parameter dependent. Numerical examples showed that the model is effective, easy to implement and have little or no bias since the mean error for all the quantities are near zero and absolute median errors are very low. This generalized model includes most of the existing model and therefore provides insight on developing new simpler models that will give improved performance. The mathematical proof for the proposed model gives more details on the technicalities of the model. The algorithm provided helps to choose the best model for a given data and problem, thereby making the process flexible and effective, and can as well lead to automation.

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Footnotes

Peer review: Externally peer reviewed.

Peer-review model: Single blind

Specialty type: Radiology, nuclear medicine and medical imaging

Country of origin: Nigeria

Peer-review report’s classification

Scientific quality: Grade B, Grade C

Novelty: Grade B, Grade C

Creativity or innovation: Grade B, Grade C

Scientific significance: Grade B, Grade B

P-Reviewer: Yan YQ, Assistant Professor, China; Zhang ZW, MD, PhD, Postdoc, China S-Editor: Liu JH L-Editor: A P-Editor: Zhao YQ