Experimental verification of ion stopping power prediction from dual energy CT data in tissue surrogates

Stopping power of swift ions is described by the Bethe equation. In Schneider et al (1996) the SPR_w of a medium relative to water was approximated by the following ratio without higher orders:

$$\begin{equation} {\rm SPR}_\mathrm{w} \approx \frac{\varrho _\mathrm{e}}{\varrho _\mathrm{e,w}}\cdot\frac{\mathrm{ln}\big(\frac{2m_{e}c^2\beta ^2}{I(1-\beta ^2)}\big)-\beta ^2}{\mathrm{ln}\big(\frac{2m_{e}c^2\beta ^2}{I_\mathrm{w}(1-\beta ^2)}\big)-\beta ^2}. \end{equation} \tag{ 1 }$$

The energy dependence of the SPR is considered very small (below 0.4%, Yang et al 2012) and is ignored by most analytical dose planning systems (e.g. there is only one Hounsfield look-up table which converts the entire CT image in WEPL). In this work the projectile energy was assumed to be 200 MeV/nucleon in accordance to experimental conditions. The I-value of water I_w was assumed to be 75 eV (ICRU 1993).

To predict the SPR_w of a medium for a specific ion energy, the ϱ_e/ϱ_{e, w} and the I-value of the medium has to be known (equation (1)). With DECT data the ϱ_e/ϱ_{e, w} of the medium can be derived directly. The I-value of the medium has to be parameterized through the Z_eff image in agreement with Yang et al (2010) by

$$\begin{equation} {\rm ln}(I)= a\cdot Z_\mathrm{eff} + b. \end{equation} \tag{ 2 }$$

The parameters a and b have to be defined in advance for an I-value prediction from measured Z_eff image data. Therefore, tabulated elemental compositions of 71 standard human tissues were taken from Woodard and White (1986) and White et al (1987) to calculate reference Z_effs and reference I-values. The same tissues were used in Schneider et al (2000). For Z_eff (equation (21)) the mass weight fractions were multiplied by the individual Z_i/A_i ratio of each element to obtain the electron density weight ($w_i\frac{Z_i}{A_i}$). I-values of the individual constituents were taken from ICRU (1993), table 2.11 and Bragg's additivity rule was applied for compounds to calculate the I-values. Thyroid was excluded because of its iodine amount which provokes a higher Z_eff compared to tissues in that range with 'usual' compositions.

The 71 calculated reference Z_eff and I-values were used for the linear parameterization and determination of a and b.

In summary the SPR_w could be subsequently predicted from the DECT ϱ_e/ϱ_{e, w} and Z_eff images by

$$\begin{equation} {\rm SPR}_\mathrm{w}= \frac{\varrho _\mathrm{e}}{\varrho _\mathrm{e,w}} \cdot \frac{12.77 - (a\cdot Z_\mathrm{eff} + b)}{8.45}. \end{equation} \tag{ 3 }$$

Through this equation it is clear that with DECT an almost direct imaging of SPR_w is possible. The ϱ_e/ϱ_{e, w} exhibits the main target contribution to the SPR_w for ions, the linearly parameterized I-value exhibits only a secondary order effect on predicted SPR_w due to the logarithmic term (equation (1)).

The x-ray attenuation spectrum for all body materials can be decomposed into a virtual pure photoelectric absorption material and a virtual pure Compton effect material:

$$\begin{equation} \mu (E)\approx a_1 \varrho _\mathrm{e} f(E)+a_2 \varrho _\mathrm{e} \frac{Z^n}{E^3} \end{equation} \tag{ 4 }$$

where n is approximately equal to 3, while f(E) is an almost flat function of the photon energy and a₁ and a₂ are proportionality factors that are approximately independent of the material and can be obtained from measured data or theoretical calculations. They characterize the relative strengths of the Compton effect and the photoelectric effect, respectively (Alvarez and Macovski 1976).

In order to relate this equation to the CT-value, it is useful to define the weighted average 〈μ(E)〉, which depends on the detected x-ray energy spectrum S(E) as

$$\begin{equation} \langle \mu (E)\rangle = \frac{\int {\mu (E) S(E) \,{\rm d}E}}{\int {S(E)\,{\rm d}E}}. \end{equation} \tag{ 5 }$$

For a thin absorber inside a water phantom of arbitrary size it is known that the commonly used CT-value x is related to this average attenuation coefficient of the absorber 〈μ(E)〉 and the average attenuation coefficient of water 〈μ_w(E)〉 by (Brooks 1977)

$$\begin{equation} x= \left( \frac{ \langle \mu (E)\rangle }{\langle \mu _\mathrm{w}(E)\rangle }-1\right) \times 1000\,{\rm HU} \end{equation} \tag{ 6 }$$

and this can be solved for 〈μ(E)〉:

$$\begin{equation} \langle \mu (E)\rangle =\langle \mu _w(E)\rangle \times \left(\frac{x}{1000\,{\rm HU}} +1\right) . \end{equation} \tag{ 7 }$$

In order to extract the electron density, two spectra are used as indicated by the indices 1 and 2:

$$\begin{eqnarray} &&\langle \mu (E)\rangle _1\approx a_1 \varrho _\mathrm{e} \langle f(E)\rangle _1+a_2 \varrho _\mathrm{e} Z^n \left\langle \frac{1}{E^3}\right\rangle _1\nonumber\\ &&\langle \mu (E)\rangle _2\approx a_1 \varrho _\mathrm{e} \langle f(E)\rangle _2+a_2 \varrho _\mathrm{e}Z^n \left\langle \frac{1}{E^3}\right\rangle _2. \end{eqnarray} \tag{ 8 }$$

The first term depends on electron density, while the second term depends on electron density times some power of the atomic number. In order to get the electron density, we solve for the first term:

$$\begin{equation} a_1 \varrho _\mathrm{e} = \frac{ \langle \mu (E)\rangle _1 \times \big\langle \frac{1}{E^3}\big\rangle _2 - \langle \mu (E)\rangle _2 \times \big\langle \frac{1}{E^3}\big\rangle _1}{\langle f(E)\rangle _1 \times \big\langle \frac{1}{E^3}\big\rangle _2 - \langle f(E)\rangle _2 \times \big\langle \frac{1}{E^3}\big\rangle _1}. \end{equation} \tag{ 9 }$$

To simplify this, two constants b₁ and b₂ can be introduced, which only depend on the spectrum, but not on the studied material:

$$\begin{equation} a_1 \varrho _\mathrm{e} = b_1\langle \mu (E)\rangle _1 + b_2 \langle \mu (E)\rangle _2. \end{equation} \tag{ 10 }$$

And after using the definition of the CT-value

$$\begin{equation} a_1 \varrho _\mathrm{e} = b_1\langle \mu _w(E)\rangle _1 \times \left(\frac{x_1}{1000}+1\right) + b_2 \langle \mu _w(E)\rangle _2 \times \left(\frac{x_2}{1000}+1\right). \end{equation} \tag{ 11 }$$

Dividing by a₁ϱ_{e, w} leads to

$$\begin{equation} \frac{\varrho _\mathrm{e}}{\varrho _\mathrm{e,w}} = \frac{b_1\langle \mu _w(E)\rangle _1}{a_1 \varrho _\mathrm{e,w}} \times \left(\frac{x_1}{1000}+1\right) + \frac{b_2 \langle \mu _w(E)\rangle _2}{{a_1 \varrho _\mathrm{e,w}}} \times \left(\frac{x_2}{1000}+1\right). \end{equation} \tag{ 12 }$$

Now a new constant c_e can be introduced:

$$\begin{equation} c_e = \frac{b_1\langle \mu _w(E)\rangle _1}{a_1 \varrho _\mathrm{e,w}}. \end{equation} \tag{ 13 }$$

By definition the CT-value of water is zero, which means that c_e is the only unknown parameter and the electron density relative to water can be written as

$$\begin{equation} \frac{\varrho _\mathrm{e}}{\varrho _\mathrm{e,w}} = c_e \times \left(\frac{x_1}{1000}+1\right) + (1-c_e)\times \left(\frac{x_2}{1000}+1\right), \end{equation} \tag{ 14 }$$

where c_e depends on the two employed effective spectra, while x₁ and x₂ denote the two CT-values in Hounsfield units, respectively. A single calibration material, which is scanned in a single Dual energy scan, is therefore sufficient to determine the parameter c_e. This is more practical than the fundamental approach to start from measured x-ray spectra. In the same way the effective atomic number can be determined by eliminating the first term in equation (4):

$$\begin{equation} a_2 \varrho _\mathrm{e} Z^n = \frac{\langle f(E)\rangle _1 \times \langle \mu (E)\rangle _2 - \langle f(E)\rangle _2 \times \langle \mu (E)\rangle _1}{\langle f(E)\rangle _1 \times \big\langle \frac{1}{E^3}\big\rangle _2 - \langle f(E)\rangle _2 \times \big\langle \frac{1}{E^3}\big\rangle _1}, \end{equation} \tag{ 15 }$$

so that in a more compact notation using the abbreviations c₁ and c₂ we obtain:

$$\begin{equation} a_2 \varrho _\mathrm{e} Z^n = c_1\langle \mu (E)\rangle _1 + c_2 \langle \mu (E)\rangle _2. \end{equation} \tag{ 16 }$$

Dividing by a₂ϱ_e and inserting the definition of the Hounsfield unit results in

$$\begin{eqnarray} &&\fl Z^n = \left(\frac{\varrho _\mathrm{e}}{\varrho _\mathrm{e,w}}\right)^{-1}\bigg( \frac{c_1}{a_2\varrho _\mathrm{e,w}}\langle \mu _w(E)\rangle _1 \times \left(\frac{x_1}{1000\,{\rm HU}}+1\right)\nonumber\\ &&+ \frac{c_2}{a_2\varrho _\mathrm{e,w}}\langle \mu _w(E)\rangle _2 \times \left(\frac{x_2}{1000\,{\rm HU}}+1\right) \bigg). \end{eqnarray} \tag{ 17 }$$

Now a new variable d_e can be introduced as:

$$\begin{equation} d_e = \frac{c_1 \langle \mu _w(E)\rangle _1}{a_2 \varrho _\mathrm{e,w}}. \end{equation} \tag{ 18 }$$

Applying the equation to water shows that d_e is the only unknown parameter and hence the effective atomic number can be obtained from:

$$\begin{equation} \fl Z_\mathrm{eff}= \left( \left(\frac{\varrho _\mathrm{e}}{\varrho _{e,w}}\right)^{-1}\left( d_e \left( \frac{x_1}{1000\,{\rm HU}}+1 \right) + (Z_{{\rm eff},w}^n-d_e)\left( \frac{x_2}{1000\,{\rm HU}}+1 \right) \right) \right)^{1/n}. \end{equation} \tag{ 19 }$$

This procedure automatically fixes the equation to calculate the effective atomic number. In order to have the same photoelectric absorption for a mixture of atoms as for a hypothetical material with the same electron density and a certain effective atomic number, the constrain is

$$\begin{equation} \varrho _\mathrm{e} Z_\mathrm{eff}^n = \sum _i{\varrho _\mathrm{e,i} Z_i^n}, \end{equation} \tag{ 20 }$$

where ϱ_{e, i} denotes the contribution of atom type i to the total electron density. This can be rewritten in terms of the number density n_i of atoms of type i, i.e. the number of atoms per unit volume of type i, where ϱ_{e, i} = n_i × Z_i:

$$\begin{equation} Z_\mathrm{eff}=\left(\frac{\sum _i{n_i Z_i^{n+1}}}{\sum _i{n_i Z_i}}\right)^{1/n} \end{equation} \tag{ 21 }$$

and this definition is also used to calculate the effective atomic number in the equation above.

It is especially important that the same coefficient n shows up in equations (19) and (21) and that n_i is the number density and not the mass density of the atom i. This difference matters for all hydrogen rich materials. The coefficient n has been determined by x-ray absorption simulations which indicate that for the range Z = 1, ..., 20 and the used CT spectra the optimum coefficient is n = 3.1. There are alternative definitions of the effective atomic number, which vary depending on the considered attenuation processes as well as on the studied materials. It should be noted that this method for calculating the electron density and effective atomic number from clinical CT-images is not applicable for high Z materials above iodine as the K-edge of these material will move into the low energy spectrum and thus invalidate equation (4). For our study we have chosen a single reference material for all measurements and we have taken the effect of beam-hardening into account, by correcting for the dependence of the measured CT-values of the calibration sample on the patient diameter. In addition a low pass filter (2D boxcar) was applied on both CT HU images to suppress noise.

For this study different materials were measured in a DECT scanner to

• assess the imaged based calculation of ϱ_e/ϱ_{e, w} (previous section) and Z_eff and compare them to reference data
• predict SPR_w from ϱ_e/ϱ_{e, w} and Z_eff images (2.1) and compare them to measured WEPL.

2.3.1. Measurements of various materials in a dual energy CT scanner

2.3.2. Accuracy of derived electron density and effective atomic number

2.3.3. Accuracy of stopping-power-ratio prediction from DECT images

Figure 1 shows the I-value parameterization through the Z_eff from the 71 tabulated tissue compositions. Two distinct regions were observed (soft and bony tissue) and the SPR_w was predicted from the DECT ϱ_e/ϱ_{e, w} and Z_eff with equation (3) as follows:

$$\begin{eqnarray} &&a= 0.125, b= 3.378\,\,\, \mbox{for tissues with\,\,} Z_\mathrm{eff} \le 8.5 \nonumber\\ &&a=0.098, b= 3.376\,\,\, \mbox{for tissues with\,\,} Z_\mathrm{eff} > 8.5. \end{eqnarray} \tag{ 22 }$$

The reason for the division in two distinct regions is the difference of the effective to the mean atomic number (〈Z〉 = ∑_iw_iZ_i). Z_eff represents a photon interaction quantity and is—due to the elevated photo electric interaction with high Z materials—weighted exponentially with n = 3.1 (equation (21), $Z_\mathrm{eff}=(\sum _i \frac{n_i Z_i^{n+1}}{n_i Z_i})^{1/n}$). Bony tissues containing calcium and phosphorus have a significant higher Z_eff than 〈Z〉 since the high atomic Z constituents are weighted higher. Hence, the slope of lnI to Z_eff is smaller for bony than for soft tissue due to the rapid increase of Z_eff compared to 〈Z〉 with the I-value.

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Parameters a and b derived from the linear fitting with tabulated tissue compositions were then applied to the measured Z_eff images for the I-value prediction.

Here, I-value residuals for the investigated measured tissue surrogates (table 3) emerge from the I-value parameterization through the Z_eff which was conducted with 71 tabulated tissues compositions (figure 1). As mentioned in Hünemohr et al (2013) tissue surrogates have a different composition compared to real tissue, especially the carbon and oxygen are interchanged in the epoxy bases surrogates.

Measured mean values of ϱ_e/ϱ_{e, w} and Z_eff derived from both DECT images are given in table 2 along with mean CT numbers for both image acquisitions (80/140Sn kVp and 100/140Sn kVp). For the measured tissue surrogates the ϱ_e/ϱ_{e, w} was determined with mean accuracy of 0.4% from the DECT images of both voltage pairs. An overall mean accuracy of 1.7% (1.9%) for the Z_eff was achieved with the 80/140Sn kVp (100/140Sn kVp) DECT images of the tissue surrogate (excluding the lung surrogate). Here an absolute mean accuracy of 0.14 (0.15) is obtained for the Z_eff at 80/140Sn kVp (100/140Sn kVp). A Z_eff calculation for the lung material was not possible due to the instability caused by the division by the low ϱ_e/ϱ_{e, w} (equation (19)). We notice a higher difference (outside the 2-σ range) between the measured and the reference Z_eff of the brain surrogate (6%) which has been observed also by Goodsitt et al (2011). The tissue surrogates used in this article are part of the Gammex electron density CT phantom providing high precision in the reference ϱ_e/ϱ_{e, w} of the materials (batch specific), whereas less control of the material composition (and hence Z_eff) in the production process might be a possible explanation for the higher Z_eff residuals.

Table 2. Measured mean values of CT numbers, ϱ_e and Z_eff dependent on two different dual energy tube voltage pairs.

Voltage pair	Material	Class	Pixels	HUlow kVp	HUhigh kVp	$\varrho _\mathrm{e,w}^\mathrm{meas}$	$Z_\mathrm{eff}^\mathrm{meas}$
80 140	Lung	Gammex	31 331	−556 ± 27	−559 ± 28	0.442 ± 0.027	NA
100 140	Lung	Gammex	31 331	−559 ± 27	−560 ± 27	0.441 ± 0.027	NA
80 140	Adipose	Gammex	39 576	−124 ± 8	−85 ± 8	0.932 ± 0.005	6.30 ± 0.20
100 140	Adipose	Gammex	39 576	−109 ± 5	−85 ± 6	0.931 ± 0.004	6.33 ± 0.18
80 140	Breast	Gammex	37 927	−59 ± 8	−39 ± 9	0.970 ± 0.005	6.91 ± 0.16
100 140	Breast	Gammex	37 927	−51 ± 5	−39 ± 6	0.970 ± 0.004	6.93 ± 0.14
80 140	True water	Gammex	18 139	−2 ± 8	−1 ± 9	1.000 ± 0.005	7.40 ± 0.14
100 140	True water	Gammex	18 139	0 ± 5	0 ± 6	1.000 ± 0.004	7.44 ± 0.12
80 140	Solid water	Gammex	28 033	8 ± 8	−1 ± 9	0.994 ± 0.006	7.67 ± 0.13
100 140	Solid water	Gammex	28 033	5 ± 6	−1 ± 6	0.994 ± 0.004	7.69 ± 0.11
80 140	Muscle	Gammex	39 576	39 ± 9	29 ± 10	1.025 ± 0.006	7.67 ± 0.13
100 140	Muscle	Gammex	39 576	36 ± 7	29 ± 7	1.024 ± 0.005	7.69 ± 0.12
80 140	Brain	Gammex	39 576	−8 ± 8	32 ± 9	1.049 ± 0.005	6.41 ± 0.17
100 140	Brain	Gammex	39 576	7 ± 5	32 ± 6	1.048 ± 0.004	6.44 ± 0.16
80 140	Liver	Gammex	39 576	78 ± 8	69 ± 9	1.065 ± 0.006	7.65 ± 0.12
100 140	Liver	Gammex	39 576	76 ± 6	68 ± 7	1.063 ± 0.004	7.69 ± 0.11
80 140	Inner bone	Gammex	39 576	351 ± 10	176 ± 9	1.098 ± 0.006	10.11 ± 0.09
100 140	Inner bone	Gammex	39 576	286 ± 7	176 ± 7	1.097 ± 0.004	10.13 ± 0.08
80 140	B200	Gammex	37 927	360 ± 10	185 ± 10	1.107 ± 0.006	10.08 ± 0.09
100 140	B200	Gammex	37 927	295 ± 7	185 ± 7	1.106 ± 0.005	10.10 ± 0.08
80 140	CB30	Gammex	39 576	657 ± 11	389 ± 10	1.269 ± 0.007	10.73 ± 0.08
100 140	CB30	Gammex	39 576	557 ± 8	389 ± 7	1.269 ± 0.005	10.74 ± 0.07
80 140	CB50	Gammex	37 927	1212 ± 14	695 ± 11	1.463 ± 0.008	12.16 ± 0.08
100 140	CB50	Gammex	39 576	1014 ± 9	694 ± 8	1.463 ± 0.006	12.14 ± 0.07
80 140	Cortical bone	Gammex	39 576	1823 ± 21	1045 ± 12	1.696 ± 0.009	13.05 ± 0.08
100 140	Cortical bone	Gammex	41 225	1515 ± 13	1044 ± 9	1.700 ± 0.008	12.97 ± 0.07
80 140	Tecapeek	Polymer	28 033	160 ± 10	210 ± 12	1.232 ± 0.008	6.32 ± 0.21
100 140	Tecapeek	Polymer	29 682	180 ± 7	209 ± 9	1.230 ± 0.006	6.41 ± 0.18
80 140	Tecaform	Polymer	28 033	324 ± 12	350 ± 13	1.362 ± 0.008	6.94 ± 0.19
100 140	Tecaform	Polymer	29 682	336 ± 8	350 ± 9	1.361 ± 0.007	6.99 ± 0.21
80 140	Tecadur	Polymer	29 682	595 ± 20	432 ± 17	1.359 ± 0.012	9.59 ± 0.11
100 140	Tecadur	Polymer	28 033	536 ± 16	432 ± 15	1.358 ± 0.012	9.63 ± 0.09
80 140	PMMA	Polymer	28 033	100 ± 11	139 ± 12	1.157 ± 0.007	6.52 ± 0.21
100 140	PMMA	Polymer	28 033	116 ± 7	140 ± 8	1.157 ± 0.005	6.56 ± 0.18
80 140	Teflon	Polymer	29 682	972 ± 15	899 ± 15	1.865 ± 0.011	8.29 ± 0.13
100 140	Teflon	Polymer	26 384	949 ± 10	900 ± 10	1.865 ± 0.008	8.35 ± 0.13
80 140	Aluminum	Metal	193	2751 ± 20	1761 ± 13	2.280 ± 0.016	13.55 ± 0.05
100 140	Aluminum	Metal	193	2365 ± 15	1764 ± 11	2.285 ± 0.013	13.62 ± 0.04
80 140	Titanium	Metal	193	12338 ± 166	6011 ± 20	3.438 ± 0.069	22.99 ± 0.33
100 140	Titanium	Metal	193	9530 ± 100	6029 ± 22	3.496 ± 0.017	23.11 ± 0.16

Calculations of the aluminum and titanium material suffer from scattering provoked by the elevated photon attenuation of the high Z materials. Figure 2 shows the individual ϱ_e/ϱ_{e, w} and Z_eff residuals for each tissue surrogate compared to reference values (table 1). Table 3 lists all results in absolute and percentage residuals to reference values.

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Especially the ϱ_e/ϱ_{e, w} which plays an essential role in radiotherapy can be calculated with an excellent accuracy from DECT images. We do not see a dependency of the ϱ_e/ϱ_{e, w} residuals on the mean atomic number or density as well as we do not see a statistical significant difference between both voltage pairs. Tube currents for both voltage pairs were not similar and therefore it is impossible to evaluate the noise level. It is not the scope of this paper to evaluate noise, filter and voltage pairs and tube currents and we would like to refer the reader to the publications of Saito (2011, 2012). For dose issues from dual source CT compared to single energy CT the reader might be referred to Henzler et al (2012).

The SPR_w (table 1) could be predicted (equation (3)) for tissue surrogates within a mean accuracy of 0.6% (0.7%) from the measured 80/140Sn kVp (100/140Sn kVp) DECT images (table 3). To predict the SPR_w of the lung surrogate only the electron density image was used. This is plausible since the I-value of the lung surrogate is similar to water. For polymers the SPR_w was predicted within 2.0% (1.9%) of the 80/140Sn kVp (100/140Sn kVp) images. The SPR_w for aluminum was predicted within 1.7% (1.8%), for titanium within 10.8% (9.4%) from the 16-bit reconstructed 80/140Sn kVp (100/140Sn kVp) images. I-values were estimated for the tissue surrogates within a mean accuracy of 3 eV. This might translate into a range uncertainty in the order of ∼0.3% (Andreo 2009, Paganetti 2012).

Figure 2 shows no clear dependence of the SPR_w residual to the ϱ_e/ϱ_{e, w} residual. This might be because the residuals of the ϱ_e/ϱ_{e, w} and I-value can compensate each other or add up. Also as mentioned before, elemental compositions are not specifically given for our measured tissue surrogates and there might be an uncertainty in the production process and a subsequent high uncertainty in the reference I-value.