Comparing the biological washout of β⁺-activity induced in mice brain after ¹²C-ion and proton irradiation

The experiment was carried out at the Heidelberg Ion Beam Therapy Center (HIT). Ten 6–8 week-old female VM/DK mice (DKFZ animal core facility) were selected. Animal work was performed according to rules outlined by the local and governmental animal care committee instituted by the German government (Regierungspraesidium, Karlsruhe). For the first part of the experiment, the mice were anaesthetized, fixed in pairs and aligned at the isocenter of the irradiation site using a laser system. The irradiation target was the brain, where tumors (spontaneous murine astrocytoma (SMA)-560 cells) had developed in the right hemisphere. SMA-560 is characterized by the formation of tumor bulk with massive angiogenesis (perfusion), necrotic regions, invasive borders and a stroma-rich microenvironment. After irradiation, the mice were transported to a PET/CT scanner (Siemens Biograph mCT with UltraHD option) located approximately 100 m from the experimental treatment room. There, the induced activity as well as anatomic information about the mice was acquired via PET and CT measurements, respectively (figure 1). After a waiting time of at least two hours, so that residual activity could be considered negligible, the mice were sacrificed, irradiated and measured again under similar conditions. In total, the procedure was carried out for five pairs of mice.

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The beams were generated using the linac-synchrotron combination at the HIT facility. Comparable treatment plans were designed with protons and ¹²C-ion beams, in order to irradiate the brain with a homogeneous dose distribution from a spread-out Bragg peak (SOBP) target volume of 10 mm lateral and 5 mm longitudinal extension in water, delivering 30 Gy physical dose (figure 2). Using the active raster scan system, the two SOBPs were delivered at once to each mice pair, immobilized in a custom-designed plexiglas holder, with special care to adjust the lateral position of the treatment field to the middle of the mice skull and the beginning of the SOBP at the mice entrance skin surface using a 10 cm thick plexiglas degrader along the horizontal beam direction. Due to the synchrotron-based beam delivery, the irradiation consisted of a few particle spills of different energy. Specifically, four iso-energy slices (E = 241.8 − 246.6 MeV u⁻¹) with an average lateral beam size of 4.4 mm (FWHM) were delivered with carbon ions for each SOBP, whereas six energy slices (E = 125.7 − 128.7 MeV u⁻¹) with an average spot size of 12.8 mm (FWHM) were used for protons. Reproducibility of the lateral position of the irradiation field was in the order of ± 1 mm, which is somewhat worse than the achievable accuracy of ± 0.5 mm in the positioning of individual mice with the help of a holder and room laser. Relevant parameters for evaluation are total irradiation time (T_irr), elapsed time between the end of irradiation and beginning of PET-measurement (T_off) and the length of PET-measurement (T_PET) (table 1). The times between the end of irradiation and the beginning of PET-measurement were recorded using a stop watch and duration of PET-measurement was defined beforehand in the examination protocol, where intervals of 30 min were typically chosen, with 60 min and 90 min for some mice to test the influence of a longer reconstruction time.

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For diagnostic PET-investigations, the decay constant of the injected radionuclides is well-known. Hence, there is a radionuclide-specific correction implemented in the scanner. To prevent this correction for the unknown mixture of isotopes, ²²Na, with a half life of 2.6 y, was chosen as the applied isotope (Bauer et al 2013).

CT images (also used for attenuation and scatter correction of the PET data) were reconstructed with a pixel size of circa 0.605 mm and a slice thickness of 0.6 mm. The raw PET list mode data were reconstructed in two different ways: Static PET images were derived from averaging the activity values over the total measurement time. For dynamic PET images, the PET data sets were subdivided into shorter time frames over which the activity data were averaged. After testing a variety of configurations and algorithms, the time intervals (3 × 60 s, 4 × 180 s, 3 × 300 s) were chosen. All PET data were reconstructed using the filtered back projection (FBP) algorithm because of enhanced activity consistency between static and dynamic PET images (see section 2.3.1), with a pixel size of circa 1.06 mm and the same CT slice thickness of 0.6 mm. The resulting image data were saved in the digital imaging and communications in medicine (DICOM) format. Volumes of interest (VOI) containing the tumour and a large fraction of the brain were drawn by hand on the basis of the CT data using the scanner software syngo for quantitative evaluation. In this process, special care was taken to exclude regions close to the bone, resulting in VOI volumes of approximately 80 mm³ (figure 3).

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To handle the image files, networks created in the software environment MeVisLab (MeVis Medical Solutions AG) were used, whose basic scaffold was developed at HIT (Unholtz et al 2011). The networks provided the possibility of overlapping and evaluating PET and CT images as well as the VOIs (figure 3). Using this software, the activity values of each voxel within the VOI were extracted from the PET images, summed and averaged. Hence, every time frame was assigned one value of activity, which was not dependent on location within the VOI, but only on the average activity of the VOI within the selected time frame. All evaluations of numerical data as described in section 2.3 were implemented using Matlab software (The MathWorks, Inc.). Additionally, the elemental composition of the tissue within the VOI was quantified by converting the Hounsfield units (HUs) of the DICOM CTs using a stoichiometric calibration table (Schneider et al 2000).

According to the method proposed by Mizuno and co-workers (Mizuno et al 2003), the experiment was evaluated by first estimating the activity induced purely by irradiation from the data of the dead mice (further referenced as physical activity) and subsequently evaluating the biological washout from the data on the living mice. In both cases, the desired quantities were obtained by fitting the averaged activities in the VOI of the dynamic PET as a function of time using a weighted least squares fit with the assumption of an activity signal governed by Poisson statistics.

Different fit scenarios were evaluated, which are discussed in the following.

2.3.1. Physical activity in the dead mice.

The physical activity A_phys of radionuclides as a function of time is described by the formula

$$\begin{eqnarray}&&{{A}_{\text{phys}}}\left(x\right)=\underset{i}{\sum} {{A}_{i}}\left({{t}^{*}}\right)\cdot \text{exp} \left(-{{\lambda}_{i}}x\right),\end{eqnarray} \tag{ 1 }$$

where λ_i is the half life of the isotope i, x = t − t^*, t is the time, t^* is the time at the beginning of data acquisition (with respect to time 0 at the end of irradiation) and A_i is the fitting parameter, estimating the activity of the isotope i. Using the data of the dead mice, A_i was determined. A_i(t^*) can be expressed in normalized form as

$$\begin{eqnarray}&&{{P}_{i}}\left({{t}^{*}}\right)=\frac{{{A}_{i}}\left({{t}^{*}}\right)}{\underset{i}{\sum} {{A}_{i}}\left({{t}^{*}}\right)}.\end{eqnarray} \tag{ 2 }$$

To obtain a valid estimation for the physical activity, the dominant isotopes i as well as appropriate time intervals for the dynamic PET reconstruction had to be chosen correctly. The isotopes chosen for A_phys were ¹¹C and ¹⁵O in all cases, as additional isotopes could not be detected properly by the fit, most likely due to their negligible contribution. In fact, implementations attempting to account for additional isotopes such as ¹³N or ³⁸K gave physical inconsistencies as, for example, negative A_i values. Hence, referring to formula (2), P(t^*) denotes the relative share of ¹¹C isotopes at the start of the scan while the share of ¹⁵O is deduced as 1 − P(t^*).

For the choice of time intervals, a compromise had to be found between increasing the information density for the time evaluation of the data and decreasing the error due to low counting statistics in short time intervals. For verification of the chosen settings, the mean activity < A > was calculated as

$$\begin{eqnarray}&&< A>=\underset{i}{\sum} \frac{{{A}_{i}}\left({{t}^{*}}\right)}{{{\lambda}_{i}}{{T}_{\text{PET}}}}\cdot \left(1- \text{exp} \left(-{{\lambda}_{i}}{{T}_{\text{PET}}}\right)\right),\end{eqnarray} \tag{ 3 }$$

with T_PET being the whole interval of measurement and A_i(t^*) the results from the dynamic data analysis. The result for < A > was then compared with the value obtained from the static PET.

2.3.2. Mathematical representation of the biological washout.

The model used by Mizuno and co-workers estimates the washout in terms of so called biological half times. The model to describe the effective decay for the living mice A_eff can be written as

$$\begin{eqnarray}&&{{A}_{\text{eff}}}\left(t\right)={{A}_{\text{phys}}}\left(t\right) \cdot {{C}_{\text{bio}}}\left(t\right),\end{eqnarray} \tag{ 4 }$$

with

$$\begin{eqnarray}&&{{C}_{\text{bio}}}\left(t\right)={{M}_{f}} \cdot \text{exp} \left(-{{\lambda}_{f}}t\right)+{{M}_{m}} \cdot \text{exp} \left(-{{\lambda}_{m}}t\right)+{{M}_{s}} \cdot \text{exp} \left(-{{\lambda}_{s}}t\right),\end{eqnarray} \tag{ 5 }$$

where M_f, M_m and M_s are the fractions of the respective fast, medium and slow components and correspond to fast $\left({{\lambda}_{f}}=\frac{\ln 2}{{{T}_{f}}}\right)$, medium $\left({{\lambda}_{m}}=\frac{\ln 2}{{{T}_{m}}}\right)$ and slow $\left({{\lambda}_{s}}=\frac{\ln 2}{{{T}_{s}}}\right)$ decay constants, which are speculated to be due to fast blood flow, microcirculation and trapping of the isotope by the stable molecules of the organs, respectively (Mizuno et al 2003). For a functioning model, the fractions must satisfy the constraint

$$\begin{eqnarray}&&{{M}_{f}}+{{M}_{m}}+{{M}_{s}}=1.\end{eqnarray} \tag{ 6 }$$

Thus, if there is no biological decay, all biological decay constants are zero (i.e. the corresponding half times are infinite) and A_eff does not differ from A_phys. As the fast component was estimated to be a few seconds by Mizuno et al, while T_off varied between 2.5 min and 4.5 min in the HIT experiments (table 1), the expression M_f·exp(−λ_ft) was set to zero and hence neglected in equation (5), giving a two component function.

2.3.3. Physical activity in the living mice.

As can be seen in table 1, there are fluctuations in the irradiation times; in particular, the transfer times T_off differ for dead and live mice. The effect on the live mice is due to a more time-consuming procedure required when handling live animals. As it was desired to transfer the result for A_phys from the dead to the living mice, effects due to the different time courses of irradiation and imaging must be compensated. This can be achieved by using the concept of a production rate R_i of isotopes i during irradiation. Unfortunately, it was not possible to recalculate exactly the real applied field, due to missing logged information and varying beam energy over the few delivered spills. Hence, approximations were used. To estimate the uncertainties associated with these approximations, the production rate was calculated twice. The first calculation used the approximation of continuous irradiation with constant beam parameters (Parodi et al 2002)

$$\begin{eqnarray}&&{{R}_{i,\text{cont}}}=\frac{{{A}_{i}}\left({{T}_{\text{irr}}}\right)}{1- \text{exp} \left(-{{\lambda}_{i}}{{T}_{\text{irr}}}\right)},\end{eqnarray} \tag{ 7 }$$

with T_irr being the time at the end of irradiation. A more refined approximation took into account the pulsed structure of the synchrotron irradiation by estimating a repetition of an equal number of spills and pauses with fixed t_spill and t_pause:

$$\begin{eqnarray}&&{R_{i,{\rm{spill}}}} = \frac{{{A_i}\left( {{T_{{\rm{irr}}}}} \right)}}{{\left( {1 - {\rm{exp}}\left( { - {\lambda _i}{t_{{\rm{spill}}}}} \right)} \right)\cdot\sum\nolimits_{k = 0}^{N - 1} {{\rm{exp}}\left( { - {\lambda _i}k\left( {{t_{{\rm{spill}}}} + {t_{{\rm{pause}}}}} \right)} \right),} }}\end{eqnarray} \tag{ 8 }$$

where t_spill is the average length of the spill and t_pause is the average time between. Although this mathematical formalism neglects in both cases that the spills may have different energies, the model can be regarded as a reasonable approximation since the number of spills is small (five to 15) and the extension in depth of the treatment field is very limited. A_i(T_irr) can be calculated easily from the fit of the dead mice data using the decay law, thus yielding the production rate R_i for the dead mice data. This allows estimation of the physical activity in the living mice by rearranging equations (7) or (8), respectively, to solve for A_i, plugging in T_irr of the living mice (table 1) and the isotope production rates R_i deduced from the dead mice data. As the same approximation was performed in two steps, calculating backwards from the production rate (for the dead mice) and then projecting this forwards to the live mice, the effect of the different energy spills is additionally reduced. Hereby, the unknown biological decay during the relatively short irradiation is neglected. For verification of this approximation, the values for biological decay, once determined with the proposed analysis, can be inserted in the following formula to evaluate their impact with respect to the calculation that neglects the biological decay during the short irradiation time:

$$\begin{eqnarray}&&{{A}_{\text{phys},\text{corrected}}}\left({{T}_{\text{irr}}}\right)=\underset{i}{\sum} \underset{b}{\sum} \frac{{{\lambda}_{i}}{{R}_{i}}{{M}_{b}}}{\left({{\lambda}_{i}}+{{\lambda}_{b}}\right)}\left(1- \text{exp} \left(-\left({{\lambda}_{i}}+{{\lambda}_{b}}\right){{T}_{\text{irr}}}\right)\right),\end{eqnarray} \tag{ 9 }$$

with λ_b being the biological washout coefficient determined from the fit and and M_b the corresponding fraction for the accessible medium and slow components. This is a reasonable approximation, as a fast washout component of only a few seconds as reported by Mizuno et al would give an already negligible contribution for the shortest irradiation time of circa 30 s (see table 1). The values for A_phys · (M_m + M_s) and A_{phys,corrected}, derived from equations (7)–(9) were finally compared at the end of irradiation T_irr.

2.3.4. Fit approaches.

Several types of function showed to be relevant for evaluation. Most important, a two component fit function A_fit(x) of the form

$$\begin{eqnarray}&&{{A}_{\text{fit}}}\left(x\right)={{\bar{A}}_{\text{phys}}}\left(x\right)\cdot {{A}^{*}} \cdot \left[{{a}_{1}} \cdot \text{exp} \left(-\frac{ \text{ln} 2\cdot \left(x+{{t}^{*}}\right)}{{{b}_{1}}}\right)+{{a}_{2}} \cdot \text{exp} \left(-\frac{ \text{ln} 2\cdot \left(x+{{t}^{*}}\right)}{{{b}_{2}}}\right)\right],\end{eqnarray} \tag{ 10 }$$

where x = t − t^* and the fitting parameters a₁ = M_s, a₂ = M_m, b₁ = T_s and b₂ = T_m with the biological half times T_s and T_m. ${{\bar{A}}_{\text{phys}}}\left(x\right)$ is the normalized (see equation (2)) representation of A_phys (equation (1)),

$$\begin{eqnarray}&&{\bar A_{{\rm{phys}}}}\left( x \right) = P\left( {{t^*}} \right)\cdot{\rm{exp( - x}}\cdot{\lambda _{{\rm{1}}{{\rm{1}}_{\rm{C}}}}}{\rm{) + }}\left( {{\rm{1 - P}}\left( {{{\rm{t}}^{\rm{*}}}} \right)} \right)\cdot{\rm{exp}}\left( {{\rm{ - x}}\cdot{\lambda _{{\rm{1}}{{\rm{5}}_{\rm{O}}}}}} \right).\end{eqnarray} \tag{ 11 }$$

The total physical activity (i.e. the activity that would exist without any biological decay) at the beginning of measurement (A_phys(t = t^*)) is referred to as A^*, which was deduced by adjusting the data of the dead mice to the living mice as described in section 2.3.3.

For reasons of fit stability, in some fits a two component function was used, where the slow decay constant was approximated to be zero (i.e. half life time going to infinity), giving

$$\begin{eqnarray}&&{{A}_{\text{fit}}}\left(x\right)={{\bar{A}}_{\text{phys}}}\left(x\right) \cdot {{A}^{*}} \cdot \left[{{a}_{1}}+{{a}_{2}} \cdot \text{exp} \left(-\frac{ \text{ln} 2\cdot \left(x+{{t}^{*}}\right)}{{{b}_{1}}}\right)\right],\end{eqnarray} \tag{ 12 }$$

with a₁ = M_s, a₂ = M_m, and b₁ = T_m.

A one component function of the form

$$\begin{eqnarray}&&{{A}_{\text{fit}}}\left(x\right)={{\bar{A}}_{\text{phys}}}\left(x\right) \cdot {{A}^{*}} \cdot \left[{{a}_{1}}\cdot \text{exp} \left(-\frac{ \text{ln} 2\cdot \left(x+{{t}^{*}}\right)}{{{b}_{1}}}\right)\right],\end{eqnarray} \tag{ 13 }$$

with a₁ = M_s and b₁ = T_s, was also implemented for cases where no medium component was evident. When necessary, constraints were set in the form of upper and lower boundaries for the fitting parameters to guarantee physically plausible results.

A third approach, overcoming the uncertainties of the ${{\bar{A}}_{\text{phys}}}$ evaluation at the expense of decreasing fit stability, is to approximate the function A_eff with an additional free fit parameter under the assumption of the two dominant isotopes ¹¹C and ¹⁵O

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{A}_{\text{eff}}}\left(t\right)={{M}_{s}}{{A}_{{{0}^{11}}\text{C}}} \cdot \left[\text{exp}\left(-{{\lambda}_{{{11}_{\text{C}}}}}\left(t-{{t}^{*}}\right)\right)+\frac{{{A}_{{{0}^{15}}\text{O}}}}{{{A}_{{{0}^{11}}\text{C}}}}\text{exp}\left(-{{\lambda}_{{{15}_{\text{O}}}}}\left(t-{{t}^{*}}\right)\right)\right] \\ \cdot \left[\text{exp}\left(-{{\lambda}_{s}}t\right)+\frac{{{M}_{m}}}{{{M}_{s}}} \cdot \text{exp}\left(-{{\lambda}_{m}}t\right)\right]. \\ \end{array}\end{eqnarray} \tag{ 14 }$$

The fit-function can be written in the form

$$\begin{eqnarray}&&\begin{array}{*{20}{l}} {{A_{{\rm{fit}}}}(x) = {a_1}\,\cdot\left[ {{\rm{exp(}} - {\lambda _{{{11}_{\rm{C}}}}}x{\rm{)}} + {a_2}\,\cdot\,{\rm{exp(}} - {\lambda _{{{15}_{\rm{O}}}}}x{\rm{)}}} \right]}\\ {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\cdot\left[ {{\rm{exp}}\,[ - \frac{{\,{\rm{ln}}\,2}}{{{a_3}}}(x + {t^*})] + {a_4}\,\cdot\,\,{\rm{exp}}\,( - \frac{{\,{\rm{ln}}\,2}}{{{a_5}}}(x + {t^*}))} \right],} \end{array}\end{eqnarray} \tag{ 15 }$$

resulting in five fit parameters a₁ = M_sA_0¹¹C, ${{a}_{2}}=\frac{{{A}_{{{0}^{15}}\text{O}}}}{{{A}_{{{0}^{11}}\text{C}}}}$, a₃ = T_s, ${{a}_{4}}=\frac{{{M}_{m}}}{{{M}_{s}}}$, a₅ = T_m, with A₀ corresponding to the activity at the beginning of the PET measurement t^*. To enhance the probability of getting physically plausible results, upper and lower boundaries for the parameters were set. As the additional free parameter is leading to a higher fit instability, this fit was only used for verification.

Additionally, the approach of fitting the slow component first and afterwards the medium component, as implemented by Mizuno and colleagues, was tested. For this successive approach the data was first fitted at late times, to estimate solely the slow decay component with a stable one component fit. In a second step, the slow decay constant was fixed to the estimated value and the whole data set was fitted again using a one component fit.

Finally, a collective fit was used in order to attempt determining global coefficients that would give a reasonable fit for multiple mice, instead of optimizing data for individuals. For this purpose, the two component fit (equation (10)) was used. A collection of N mice-datasets was chosen (see section 3.2). For each step of the fit-iteration, a curve with identical parameters a₁, a₂, b₁, b₂ was plotted for every dataset and only A_phys was adapted for each curve according to the individual mice measurements. The weighted squared distances of the N curves to the data points of the N datasets were summed up and this collective sum was then minimized by the weighted least squares iteration. To ensure equal influence of every dataset, all individual squared distances were divided by their minimum squared distance from the individual fits, before being summed up.

2.3.5. Comparison to static PET for alive mice.

For verification, the average activity deduced from the fit results of the dynamic acquisitions were compared to the static PET of the living mice, using the formula

$$\begin{eqnarray}&&< A>=\underset{p}{\sum} \underset{b}{\sum} \frac{{{A}_{\text{irr},p}}{{M}_{b}}}{{{T}_{\text{PET}}}\left({{\lambda}_{p}}+{{\lambda}_{b}}\right)} \text{exp} \left(-\left({{\lambda}_{p}}+{{\lambda}_{b}}\right){{T}_{\text{off}}}\right)\cdot \left(1- \text{exp} \left(-\left({{\lambda}_{p}}+{{\lambda}_{b}}\right){{T}_{\text{PET}}}\right)\right),\end{eqnarray} \tag{ 16 }$$

where the index p denotes the physical and b the biological parameters, while A_irr,p refers to the estimated activity at the end of irradiation.

2.3.6. Sensitivity study on the P-values.

2.3.7. Estimation of standard errors of the fit parameters.

The error of each fit parameter σ_p was estimated by the diagonal elements of the covariance matrix C_pp, weighted with the reduced χ² of the fit, assuming Poisson distribution of the measured data. This gave the expression

$$\begin{eqnarray}&&{{\sigma}_{p}}=\sqrt{{{\mathbf{C}}_{pp}}}\sqrt{{{\chi}^{2}}/\text{dof}},\end{eqnarray} \tag{ 17 }$$

where

$$\begin{eqnarray}&&{{\chi}^{2}}=\underset{i}{\sum} {{\left(\frac{y_{i}^{\text{data}}-y_{i}^{\text{fit}}}{\sqrt{y_{i}^{\text{data}}}}\right)}^{2}}.\end{eqnarray} \tag{ 18 }$$

The degrees of freedom (dof) were obtained by subtracting the number of fit parameters from the number of data points.

The dynamic PET reconstruction gave a good consistency with the static PET activity values for the dead mice (table 2). Only the first 30 min of PET data were used for all mice, because no improvement could be seen in the results when taking longer decay intervals.

Table 2. Mean activity determined from the static PET and from the dynamic PET via equation (3) for the dead mice, labelled additionally according to position in the CT (l: left; r: right). The relative deviation ΔS (in %) is given in the last row.

Mouse (12C)	1de,l	1de,r	2de,l	2de,r	3de,l	3de,r
$< A {{>}_{\text{stat}}}\left[\frac{Bq}{ml}\right]$	703.5	654.6	618.1	672.2	418.0	344.9
$< A {{>}_{\text{dyn}}}\left[\frac{Bq}{ml}\right]$	699.7	650.0	619.7	675.1	416.4	339.4
ΔS	0.5%	0.7%	0.3%	0.4%	0.4%	1.6%
Mouse (protons)	4de,l	4de,r	5de,l	5de,r

$< A {{>}_{\text{stat}}}\left[\frac{Bq}{ml}\right]$	153.2	148.0	223.2	218.3
$< A {{>}_{\text{dyn}}}\left[\frac{Bq}{ml}\right]$	152.0	146.6	216.5	215.5
ΔS	0.7%	0.9%	3.0%	1.3%

In the data analysis mouse 1l and mouse 3r were excluded due to missing stability of the fit results under change of P(t^*) (section 2.3.6). Mouse 4r was excluded due to bad consistency with the static PET for the living mice. For mouse 2l, the most consistent fit was the multiplication of A_phys with a constant. Inspection of the datapoints did not indicate a clear activity decay and inspection of the overlapped PET, VOI and CT showed that irradiation had taken place mostly outside the VOI. Hence, mouse 2l was excluded due to lack of information correlated with the irradiation-induced activity. Therefore, a total of six mice were considered in the final evaluation. The collective fitting was done separately for the ¹²C and proton irradiated mice data, with 3 datasets for each.

The results for the washout parameters are reported in table 3. The model of continuous irradiation is used for the reported results. All of the selected ¹²C mice data sets were best fitted by an unconstrained two component fit. Fits that converged to very high T_s are marked by >100 000 and >18 000 in one case. The data sets of mice irradiated with protons were best fitted by an unconstrained one component fit. They gave physically inconsistent values when fitted with an unconstrained two component fit (negative M_s/M_m), while the constrained two component fit converged to the listed one component fit. The successive fit proved to be inconclusive due to the decrease in statistics when starting the fit at least 500 s after irradiation, as required to suppress the medium component. The result of the collective fit gave instead satisfactory results (figure 4) and its comparison to the average and one standard deviation from the separate fits of the individual mice data sets can be seen in table 4.

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Using the results of the elemental analysis of the CT images (see tables A1 and A2 in appendix A), we tried to determine a correlation between different fit results and differences in the elemental composition, but none could be found. Hence it is unlikely that the observed differences between proton and ¹²C-ion irradiated mice are due to a different mixture of various β⁺-emitting target fragments in the considered VOIs.

The comparison of the dynamic with the static PET gave very good consistency for almost all living mice, except mouse 4l,a. Here, the deviation was higher with 6.4% (table 3), likely due to low counting statistics (see figure 4d).

The effects resulting from neglecting the biological decay during irradiation, estimated via equation (9), mostly affected the data sets of mouse 1r and mouse 2r by a raise of M_m + M_s of 3.4% and 2.4%, respectively. The other mice were affected by changes smaller than 1.5% . In all cases, the estimation of P(t^*) did not change significantly due to biological washout during irradiation (<0.1%). Comparing the examined approximations for irradiation, the model of continuous irradiation (equation (7)) and the model of pulsed delivery of equal spills and pauses (equation (8)) gave nearly identical results under the condition that the sequence of t_s + t_p matched the irradiation time.

The change of the M_s + M_m fraction under manipulation of P(t^*), according to section 2.3.6, can be seen in table 5. For reasons of stability, the carbon mice data sets were manipulated with a two component fit with one constant (which gave the same M_s + M_m values). As can be seen, only mouse 3l changes overproportionally under lowering of the P(t^*) value, which corresponds to the raising of the ¹⁵O fraction. As T_m for mouse 3l is relatively large, it can be seen as unlikely that P(t^*) is overestimated (i.e. the ¹⁵O fraction is underestimated), as the faster half time of ¹⁵O would show up in the result. In addition, inspection of the washout components corresponding to the lowered P(t^*) value showed a drastic increase of T_m, supporting the thesis that the lowering of P(t^*) for mouse 3 left corresponds to an unrealistic overestimation of the ¹⁵O fraction. The M_s + M_m values for the ¹²C irradiated mice are additionally verified by comparison to the result of the fit without A_phys (equation (15)), yielding similar values differing in the range of ± 0.04.

The influence of the explicit form of the VOI on the main result of this study was tested by manually cutting the peripheral part of the VOI for one carbon ion mouse (3l) and one proton mouse (4l), hereby drastically reducing the examined volume and increasing the distance to the skull bone (to exclude partial volume effects). Under these manipulations, the individual parameters changed, but the M_s + M_m value remained the same within a range of ± 0.03. Hence, a clear distinction between ¹²C-ion and proton irradiation could still be made, regardless of the considered VOI.