Specific Radioactivity of Neutron Induced Radioisotopes: Assessment Methods and Application for Medically Useful 177Lu Production as a Case

The conventional reaction yield evaluation for radioisotope production is not sufficient to set up the optimal conditions for producing radionuclide products of the desired radiochemical quality. Alternatively, the specific radioactivity (SA) assessment, dealing with the relationship between the affecting factors and the inherent properties of the target and impurities, offers a way to optimally perform the irradiation for production of the best quality radioisotopes for various applications, especially for targeting radiopharmaceutical preparation. Neutron-capture characteristics, target impurity, side nuclear reactions, target burn-up and post-irradiation processing/cooling time are the main parameters affecting the SA of the radioisotope product. These parameters have been incorporated into the format of mathematical equations for the reaction yield and SA assessment. As a method demonstration, the SA assessment of 177Lu produced based on two different reactions, 176Lu (n,γ)177Lu and 176Yb (n,γ) 177Yb (β- decay) 177Lu, were performed. The irradiation time required for achieving a maximum yield and maximum SA value was evaluated for production based on the 176Lu (n,γ)177Lu reaction. The effect of several factors (such as elemental Lu and isotopic impurities) on the 177Lu SA degradation was evaluated for production based on the 176Yb (n,γ) 177Yb (β- decay) 177Lu reaction. The method of SA assessment of a mixture of several radioactive sources was developed for the radioisotope produced in a reactor from different targets.


Introduction
State-of-the-art radiopharmaceutical development requires radioisotopes of specific radioactivity (SA) as high as possible to overcome the limitation of in vivo uptake of the entity of living cells for the peptide and/or monoclonal antibody based radiopharmaceuticals which are currently used in the molecular PET/CT imaging and endo-radiotherapy. The medical radioisotopes of reasonable short half-life are usually preferred because they have, as a rule of thumb, higher SA. These radioisotopes can be produced from cyclotrons, radionuclide generators and nuclear reactors. The advantage of the last one lies in its large production capacity, comfortable targetry and robustness in operation. This ensures the sustainable supply and production of key, medically useful radioisotopes such as 99 Mo/ 99m Tc for diagnostic imaging and 131 I, 32 P, 192 Ir and 60 Co for radiotherapy. The high SA requirement for these radioisotopes was not critically considered with respect to their effective utilization in nuclear medicine, except for 99 Mo. The current wide expansion of targeting endoradiotherapy depends very much on the availability of high SA radionuclides which can be produced from nuclear research reactor such as 153 Sm, 188 W/ 188 Re, 90 Y and 177 Lu. As an example, as high as 20 Ci per mg SA 177 Lu is a prerequisite to formulate radiopharmaceuticals targeting tumors in different cancer treatments [1,2].
So far in radioisotope production, reaction yield has been the main parameter to be concerned with rather than SA assessment and unfortunately, the literature of detailed SA assessment is scarcely to be found [3,4]. The SA assessment of radioisotopes produced in a reactor neutron-activated target is a complex issue. This is due to the influence of the affecting factors such as target burn-up, reaction yield of expected radionuclide and unavoidable side-reactions. All these depend again on the available neutron fluxes and neutron spectrum, which are not always adequately recorded. Besides, the reactor power-on time and target self-shielding effect is usually poorly followed up. Certainly, the SA of target radionuclides has been a major concern for a long time, especially for the production of radioisotopes, such as 60 Co and 192 Ir, used in industry and radiotherapy. In spite of the target burn-up parameter present in the formula of reaction yield calculation to describe the impact of target depression, the SA assessment using the reaction yield was so significantly simplified that the target mass was assumed to be an invariable value during the reactor activation. Critically, this simplification was only favored by virtue of an inherent advantageous combination of the low neutron capture cross section (37 barns) of the target nuclide 59 Co and the long half-life of 60 Co (which keeps the amount of elemental Co unchanged during neutron bombardment) [4].
The targets used in the production of short-lived medical radioisotopes, however, have high neutron capture cross sections to obtain as high as possible SA values. This fact causes a high "real" burn-up of the target elemental content. Especially, the short half-life of the beta emitting radioisotope produced in the target hastens the chemical element transformation of the target nuclide and strongly affects the SA of the produced radioisotope. The triple factors influencing the production mentioned above (target, neutron flux and short half-life of produced radionuclide) are also critical with respect to the influence of the nuclear side-reactions and impurities present in the target. Moreover, the SA of a radionuclide produced in nuclear reactor varies with the irradiation and post-irradiation processing time as well. All these issues should be considered for a convincing SA assessment of the producible radioisotope for any state-of-the-art nuclear medicine application. As an example, a theoretical approach to the SA assessment reported together with an up-to-date application for 177 Lu radioisotope production is presented in this paper. This assessment can also play a complementary or even substantial role in the quality management regarding certifying the SA of the product, when it may be experimentally unfeasible due to radiation protection and instrumentation difficulties in the practical measurement of very low elemental content in a small volume solution of high radioactivity content.
High SA nuclides can be produced by (n, )  reaction using high cross section targets such as the 176 Lu (n, )  177 Lu reaction (б = 2,300 barns). 177 Lu is a radioisotope of choice for endo-radiotherapy because of its favorable decay characteristics, such as a low energy beta decay of 497 keV (78.6%) and half-life of 6.71 day. It also emits gamma rays of 113 keV (6.4%) and 208 keV (11%) which make it useful for imaging in-vivo localization with a gamma camera. 177 Lu can be produced by two different routes, a direct route with the 176 Lu (n, )  177 Lu reaction and an indirect route via the 176 Yb (n, )  177 Yb ( decay) 177 Lu nuclear reaction-transformation. The direct route could be successfully performed in high neutron flux nuclear reactors but these are available in only a handful of countries in the world. Additionally, large burn-up of the target nuclide during high neutron flux irradiation may cause a degradation of the SA value of the produced nuclide if the target contains isotopic impurities. No-carrier-added (n.c.a) radioisotopes of higher SA can be produced via an indirect route with a nuclear reaction-followed -by-radioactive transformation process, such as in the process of neutron capture-followed-by-decay , 176 Yb (n, )  177 Yb ( decay) 177 Lu. In this case, the same reduction in SA is also be experienced if the target contains isotopic and/or elemental Lu impurities. 177 Lu production has been reported in many publications [5][6][7][8][9], but until now the product quality, especially the evaluation of 177 Lu specific radioactivity in the product, has not been sufficiently analyzed. Based on the theoretical SA assessment results obtained in this report, the optimal conditions for the 177 Lu production were set up to produce 177 Lu product suitable for radiopharmaceutical preparations for targeting endo-radiotherapy.

Units of specific radioactivity, their conversion and SA of carrier-free radionuclide
The specific radioactivity is defined by different ways. In our present paper we apply the percentage of the hot atom numbers of a specified radioactive isotope to the total atom numbers of its chemical element present in the product as the specific radioactivity. This is denoted as atom %.
The following denotation will be used for further discussion. N Ri(A) is the hot atom numbers of radioisotope R i of the chemical element A and , its decay constant . N A is the atom numbers of the chemical element A and T 1/2 (sec) the half-life of radioisotope R i .
The SA unit of atom % is defined as follows: where M iA is the atomic weight of the target or radioactive material of given isotopic composition of the chemical element A. For a radioactive material containing n isotopes of the element A: Identifying eq.2 with eq.3 (individualizing M iA as the atomic weight of the concerned radioisotope), it is clear that the SA of a carrier-free radionuclide in unit atom % is 100%.

Theoretical Approach and Assessment Methods
Reactor-based radioisotope preparation usually involves two main nuclear reactions. The first one is the thermal neutron capture (n, )  reaction. This reaction doesn't lead to a radioisotope of another chemical element, but the following radioactive   decay of this isotope during target activation results in a decrease in both the reaction yield and atom numbers of the target chemical element. The second reaction is the thermal neutron capture followed by radioactive transformation S (n, ) This reaction leads to a carrier-free radioisotope of another chemical element than the target chemical element. The SA assessment in the radioisotope production using the first reaction (with a simple target system) is simple. Careful targetry could avoid the side reaction S (n, )  R x (   decay) R i which could result in the isotopic impurities for the radioisotope intended to be produced using the first reaction. In this case the SA assessment in (n, )  reaction based production process can be simplified by investigation of the SA degrading effect of target nuclide burn-up, chemical element depression due to radioactive decay and isotopic impurities present in the target. On the other hand the SA assessment in the radioisotope production using the second reaction (with complex target system) is more complicated. The complexity of the targetry used in S (n, )  R x ( decay) R i reaction based isotope production requires an analysis of the combined reaction system. This system is influenced by both (n, )  reaction and neutron-capture-followed-by-radioactive transformation S (n, )  R x ( decay) R i . So the effect of side nuclear reactions in this target system will be assessed in addition to the three above mentioned factors that are involved in the simple target system. In this case the SA assessment is best resolved by a method of SA calculation used for the mixture of several radioactive sources of variable SA, which is referred to as a radioisotope dilution process.
For the calculation of SA and reaction yield of the radioisotope R i in the two above mentioned reactions, the following reaction schemes are used for further discussion.
Reaction scheme 1: Reaction scheme 2: Reaction scheme 3: S 1,A is the target stable isotope of element A in the target; S g,A (with g ≥ 2) is the impure stable isotope of element A originally presented or produced in the target. S 1,B is the target stable isotope of element B in the target; S 2,B is the stable isotope of element B in the target. R i,A or R i is the wanted radioisotope of element A produced in the target from stable isotope S 1,A . R x and R y are the radioisotopes of element B produced in the target.

The (n, )
 reaction yield and the specific radioactivity calculated from it depends on the neutron flux and reaction cross-section which is variable with neutron energy ( n E ) or velocity ( n v ). In the thermal neutron region, the cross-section usually varies linearly as In practice the target is irradiated by reactor neutrons of in all the equations below has to be replaced by -named (n,γ)reaction. The detailed description of these eff  values can be found in the 'Notes on Formalism' at the end of this section.
For the isotope production based on (n,γ) reactions the neutron bombardment is normally carried out in a well-moderated nuclear reactor where the thermal and epithermal neutrons are dominant. The fast neutron flux is insignificant compared to thermal and epithermal flux (e.g. <10 7 n.cm −2 .s −1 fast neutron flux compared to >10 14 n.cm −2 .s −1 thermal one in the Rigs LE7-01 and HF-01 of OPAL reactor-Australia). Besides, the milli-barn cross-section of (n,γ) ,(n,p) and (n,α) reactions induced by fast neutrons is negligible compared to that of (n,γ) reaction with thermal neutron [11]. So the reaction rate of the fast neutron reactions is negligible. Nevertheless, for the generalization purposes the contribution of the fast neutron reaction is also included in the calculation methods below described. It can be ignored in the practical application of SA assessment without significant error.

The specific radioactivity of radionuclide R i in the simple target system for the (n, )
 reaction based radioisotope production 2.1.1. Main characteristics of the simple target system The simple target system contains several isotopes of the same chemical element. Among them only one radioisotope R i is intended to be produced from stable isotope S 1,A via a (n, )  reaction i = 1 as described above in reaction scheme 1. Other stable S g,A isotopes ( with g ≥ 2) of the target are considered as impure isotopes.

The target burn-up for each isotope in simple target system
The burn-up of the isotope S 1,A is the sum of the burn-up caused by different (n,γ) and (n, particle) reactions from reaction i = 1 to i = k, the cross sections of which are different б 1,i values. This total burn up rate could be formulated as follows: ratios into eq.4, the following is deduced. and: the above differential equation is simplified as follows: The un-burned atom numbers of the isotope S 1,A at any t irr values ( ) is achieved by the integration of eq.7 with the condition of From this equation, the burned-up atom numbers of the isotope S 1,A ( The same calculation process is performed for any isotope S g,A .
Half-burn-up time of the target nuclide. At half-burn-up time T 1/2-B a half of the original atom numbers of the isotope S 1,A are burned.
into eq. 8, the T 1/2-B value is achieved as follows: 2.1.1.2. Reaction yield of radioisotope R i in the simple target system By taking into consideration the un-burned atom numbers of the isotope S 1,A (eq. 8) , the reaction rate of any isotope in reaction scheme 1 will be evaluated as follows. In this reaction process the depression of the atom numbers of radioisotope R i is caused by beta radioactive decays and (n, γ)/(n, particle) reaction-related destruction. The depression factor i R  of the radioisotope R i in reaction scheme 1 is formulated as follows: Taking into account eq.5, R i radioisotope formation rate is the following: , this equation is converted into the following form: ) (  By integrating this equation and assuming N Ri = 0 at t irr = 0, the yield of radioisotope R i at the irradiation time t irr is the following: The R i atom numbers (N Ri ): The R i isotope radioactivity (A Ri ): These equations can be deduced from the well known Bateman equation [3,12]. The R i atom numbers and radioactivity at the post-irradiation time t c ( N Ri,tc and A Ri,tc , respectively) are calculated by multiplying eqs.13 and 14 with the factor ). By differentiating eq.14 and making it equal to zero: the t irr-max is deduced as follows: Equation (15) is useful for irradiation optimization to produce R i radioisotope of highest yield. By introducing the value t irr-max into eqs.13 and 14, we achieve the maximum yield of radioisotope R i ( N Rimax and A Ri-max ) as follows: The maximum atom numbers N Ri-max is: The maximum radioactivity A Ri-max is: As shown the maximum yield of radioisotope R i is a function of the variable D.

The SA assessment of radionuclide R i in the simple target for (n, )
 reaction based radioisotope production 2.1.2.1. General formula of SA calculation for the simple multi-isotope target The simplification in the calculation is based on the fact that the target isotope S i,A captures neutrons to form the wanted radioisotope R i and the isotopic impurities in the target don't get involved in any nuclear reactions whatsoever. The isotopic impurities may participate in some nuclear reactions to generate either stable isotopes of the target element or an insignificant amount of the isotopes of other chemical element than the target one. This simplified calculation process is supported by a careful targetry study regarding minimizing the radioactive isotopic impurities in the radioisotope product. The following is the SA of radioisotope R i formed in a target composed of different stable isotopes: is the sum of the remaining (unburned) atom numbers of g different stable isotopes of the same chemical element in the target. By placing the values If the target contains impure isotopes of another chemical element, more stable isotopes of chemical element A generated via reaction scheme 3 above could be present in the denominator of this formula. This amount may cause additional depression of irr i t R SA , . This small impurity will, however, bring about an insignificant amount of stable isotope S g,A and its depression effect will be ignored. The eq.19 is set up with an ignorance of insignificant amount of not-really-burned impure stable isotope which captures neutron, but not yet transformed into the isotope of other chemical element via a radioactive decay).
If the impure isotope S g,A doesn't participate in any nuclear reaction or its neutron capture generates a stable isotope of the target element, then zero value will be given to the parameter From the practical point of view, the target composed of two stable isotopes is among the widely used ones for radioisotope production. For this case the SA calculation is performed as follows: Maximum SA of radioisotope R i in the simple two-isotope target. Rendering the differential of eq. 21 equal to zero offers the way to calculate the irradiation time at which the SA of nuclide R i reaches maximum value ( The irradiation time where the SA reaches maximum is denoted as value, which is derived from the above differential equation, is the following: These parameters are identical to that of the eq. (17) and (21).
SA of radioisotope R i in the target which is considered as a simple two-isotope target. It is also a matter of fact that another very commonly used target system contains more than two stable isotopes (simple multi-isotope target system, g ≥ 2). Except S 1,A as shown in reaction scheme 1, all the impure isotopes of the same chemical element in the target don't get involved in any nuclear reactions or they may participate in with very low rate giving insignificant burn-up ( This system is considered as a special two-isotope target system for which the non-depression of impure isotopes 2.1.2.3. SA of radioisotope R i in the simple one-isotope target system By introducing P 2 = 0 into eq.(21), the SA of radioisotope R i in the simple one-isotope target is the following: It is shown that the achieved differential eq. (27) has no solution with the variable t irr and gives a correct solution when t irr value approaches to infinity .When then a = 1 (as shown in eq.25), hence the differential value is not defined, so the specific radioactivity has no maximum value at any time. This means that the SA of nuclide R i in the stable isotope target of 100% isotopic purity never reaches maximum at any irradiation time.
It is also worth mentioning that when the value of  is very large, eq. (21) is converted to eq. (26). It means that the high burn-up of impure stable isotope S g,A makes a multi-isotope target system change to a one-isotope target one. So, no maximum SA will be expected with this type of multi-isotope target system too.
As shown in eq. (26) the SA of these target systems increases with t irr . This fact teaches us that a compromise between maximum yield achievable at t irr,max and favorable higher SA at the time t irr> t irr,max is subject to the priority of the producer.

The specific radioactivity of radionuclide R i in a complex target system for the S(n, )
 R x ( decay) R i reaction based radioisotope production 2.2.1. Main characteristics of the complex target system The complex target system contains several isotopes of different chemical elements. Among them only one radioisotope R i is intended to be produced from stable isotope S 1,B of chemical element B via a S 1,B (n, )  R x ( decay) R i reaction i = 1 as described above in reaction scheme 2. Other stable S g,B isotopes ( with g ≥ 2) of the element B are considered as impure isotopes and they could be transformed into other isotopes (except R i ) of the chemical element A as described above in reaction scheme 3. Besides, the target could contain different isotopes of the element A as impure isotopes which could be involved in different nuclear reactions during target irradiation.

The yield of S 1B (n, )
 R x ( decay) R i reaction This reaction generates a carrier-free radioisotope R i . The SA Ri value is 100 atom %. As shown in reaction scheme 2, the atom numbers (N Ri ) and the radioactivity (A Ri ) of R i radioisotope of chemical element A are calculated based on the general Bateman equation [3,12]. This is detailed in the following equation: The R i atom numbers  R y ( decay) S g , A reaction Referred to reaction scheme 3 involving the impure stable isotope S 2,B in the S 1,B target , reaction S 2,B (n, )  R y ( decay) S g,A generates an amount of stable isotope S g,A of the same chemical element to the wanted radionuclide R i,A . This fact makes the SA of radionuclide R i,A produced from stable isotope S 1,B lower, so the atom numbers of the stable isotope S g,A should be evaluated for the purpose of SA assessment. The atom numbers of S g,A is determined based on the activity of radioisotope R y . Identifying eqs. (13) and (14) described for reaction scheme 1 with the process of reaction scheme 3, we get the following equations.
The atom numbers (   (14) ) formed during neutron activation of the impure stable isotope S 2,B is calculated by integrating R y nuclide radioactivity for the neutron irradiation time t irr as below.
) ( , then . Putting C value into the above equation we get: of nuclide R y at the end-of-neutron-bombardment (E.O.B) time t irr is independent on the variable t c , we get the following integral:  ) formed in the target from the S 2,B impure stable isotope is composed of a partial amount formed during neutron activation ( , As a result of the analysis of the above equations, minimizing post-irradiation cooling/processing time is recommended to reduce the SA-degradation effect of the impure isotopes.

SA-degradation effect of impure isotopes of the chemical element A
The assessment of SA in system containing these impure isotopes can be found in the Section 2.1 for the simple target system.

The SA assessment of radionuclide R i in a complex target system
The radioisotope dilution is involved in SA depression in a complex target system in which both the wanted radioisotope R i and its unfavorable stable isotope are generated from different nuclear reactions of both the target isotope and impurities. The complex target system is considered as a mixture of several radioactive sources of variable SA. The method of SA assessment for this mixture is formulated as below.
SA j,Ri is the SA of R i in the radioactive source S j the R i radioactivity of which is A j,Ri .The radioactive source S j is produced in the target from a given nuclear reaction such as S (n, )  R x ( decay) R i reaction (reaction scheme 2) or (n, )  reaction (reaction scheme 1). There are n different radioactive sources S j (j=1…n) in the target. So the target is a mixture of radioactive sources. The SA of this radioactive source mixture (SA Mix,Ri ) is calculated as follows:   reaction which generates a stable brother isotope S g,A of radioisotope R i in the target system ( Reaction scheme 3). To solve this problem we have to combine the atom numbers of this stable brother isotope with the atom numbers of one specified radioactive source of the mixture to generate a new radioactive source of SA ≠ 0, e.g. the combination of radioactive sources produced from the reactions in the scheme 2 and 3. This treatment will be detailed in a practical application for the 176 Yb target system in the following section.

Notes on formalism
During neutron bombardment of the target in the nuclear reactor of   1 / 1 n E epithermal neutron spectrum, the rate of (n,γ) reactions is calculated based on either Westcott or Hogdahl formalism [13] depending on the excitation function of the target nuclide ( the dependence of the reaction cross section on the neutron energy). For the " / 1 " n v non  -named (n,γ) reactions the modified Westcott formalism can be used to improve the accuracy of reaction yield calculation and the reaction rate in both thermal and epithermal neutron region for a diluted sample( both the thermal and epithermal neutron self-shielding factors are set equal to 1 or very close to unity) is: .Under this condition, the reaction rate is: f as mentioned above, the reaction rate for the " / 1 " n v non  -named (n,γ) reactions can be calculated as: For the " / 1 " n v -named (n,γ) reactions, the reaction rate in both thermal and epithermal neutron region calculated based on the Hogdahl convention ion is: it is written as: For the above equations, is for the cadmium cut-off energy eV E Cd 55 . 0  correction,  is epithermal flux distribution parameter, its extreme values −0.15 <  < +0.3, is Hogdahl convention effective cross-section. n n is total neutron density.
n v is neutron velocity and 0 v is the most probable neutron velocity at 20 °C (2200 m/s). ) ( n T g is Westcott's g-factor for neutron temperature n T , r is a measure for the epithermal to total neutron density ratio in the Westcott formalism, tail subtracted) to the thermal cross-section 0  ,

Reagents and materials
The isotopically enriched 176 Yb 2 O 3 and 176 Lu 2 O 3 targets for neutron activation were purchased from Trace-Sciences International Inc.USA [10].

Targets, reactor irradiations, chemical separation, elemental analysis and radioactivity calibration
The radioactive 177 Lu + 175 Yb solutions were obtained by the reactor thermal neutron irradiation of 176 Yb 2 O 3 and/or 176 Lu 2 O 3 targets. A quartz ampoule containing an adequate amount of 176 Yb 2 O 3 or 176 Lu 2 O 3 target was irradiated with a thermal neutron flux in HIFAR reactor (Australia). A 24-hour cooling period was needed to let all 177 Yb (T 1/2 = 1.911 hours) radionuclides (which formed via 176 Yb (n, γ) 177 Yb) to be transformed to 177 Lu via beta particle decay. The irradiated target was then dissolved in HCl solution and the radiochemical separation of 177 Lu from the target solution was performed as reported in our previous papers [8,9]. The radioactivity of the different radioisotopes was calibrated using a CAPINTEC Dose calibrator and gamma-ray spectrometer coupled with ORTEC HP Ge detector. The gamma ray energy and counting efficiency calibration of this analyzer system were performed using a radioactive standard source of 152 Eu solution. Lutetium element and other metal content in the completely decayed 177 Lu solutions (at least > 10 half-lives) was analyzed using ICP-MS instrument.

Results and Discussion
The methods developed in the above sections were evaluated and used for the assessment of SA values of two typical isotope target systems, enriched 176 Lu and 176 Yb targets. 177 Lu produced from these targets is a representative for the state-of-the-art radioisotopes of high specific radioactivity used in targeted endo-radiotherapy.  Table 1. The production of 177 Lu radioisotope is based on the reaction Lu-1. As shown in this data list, the target composes of two stable isotopes, 176 Lu and 175 Lu. In the reactions Lu-6 and Lu-3 the neutron captures yield the isotopes of another chemical element, so these reactions may cause a depression in elemental Lu atom numbers of the target during neutron bombardment. Lu-6 (y = 3) * 177 Lu depression caused by possible (n, γ) and/or (n, p) reactions is ignored compared to radioactive decay rate of 177 Lu isotope However, the effect of these reactions is ignored due to their low cross sections. The reactions Lu-2, Lu-4 and Lu-5 yield long -lived and stable isotopes of the Lutetium element, so the elemental Lu atom numbers of these isotopes are likely to be unchanged during neutron irradiation. This condition shows a similarity between the 176 Lu enriched target and the multi-isotope target system of depression factor 0 , 2   A S as described in Section 2.1.2.2 (Third bullet). So, eqs. 20-25 are adopted for the 177 Lu specific radioactivity assessment of the Lu target. For this SA assessment process the relevant parameters should be identified to individualize the selected equations for the above mentioned 177 Lu production reaction. These parameters are the following: S 1,A is 176 Lu and re-denoted as S 1,Lu . S 2,A is 175 Lu and re-denoted as S 2,Lu . R i is 177 Lu.  Table   1), which is tabulated for neutron of E= 0.0253 eV and υ n = 2,200 m/s, should be multiplied with a so called k-factor which is based on the Westcott convention equation as discussed in Section 2.3 [5,6].
The k values ranging from 1.67 at 10 °C to 1.9 at 40 °C were calculated for the Munich reactor [5]. As generally accepted, we use the value k = 1.74 tabulated in reference [11] for our further calculation.   177 Lu specific radioactivities as a function of the target isotopic composition, neutron flux and irradiation time were formulated and calculated (as shown in Figures 1 and 2). The maximum values of 177 Lu specific radioactivity and irradiation time were evaluated. These were used as optimal conditions for carrier-containing 177 Lu production.
As shown in Figure 1a, the irradiation time for maximum yield (t irr,Yield-max ) and that for maximum specific radioactivity (t irr,SA-max ) are different. The results presented in Figure 3a state that the higher the 176 Lu enrichment of the target, the bigger the difference between values t irr,Yield-max and t irr,SA-max. . The ratio of these times varies with thermal neutron flux applied and reaches a maximum value at the flux value of around plays an important role in the creation of this maximum value of t irr,Yield-max and t irr,SA-max / t irr,Yield-max ratio at a specified neutron flux value specific for a given target system.. Although the higher neutron flux irradiation gives the higher SA as shown in Figure 3b, the bigger difference between values t irr,Yield-max and t irr,SA-max. makes the outcomes of maximum yield and maximum SA incompatible.   could be awkward. Hence the production yield and desired SA should be compromised to achieve a cost effective production of clinically useful 177 Lu product. The t irr,SA-max values increase with the 176 Lu enrichment on the target ( Figure 2) and the 100% purity 176 Lu target showed no-maximum SA value during neutron activation as seen in Figure 1b. This is confirmed by the analysis of differential equation 27, which was described in Section 2.
The relevant parameters should be identified to individualize this equation for the reaction Yb-1 mentioned in Table 2 ). The maximum SA is predicted based on the opposite tendency of SA variation of elementally pure and Lu-contaminated Yb targets. This argument is supported by the results obtained below.
The SA of 177 Lu radioisotope produced by the 176 Yb (n, )  177 Yb ( decay) 177 Lu process as a function of the elemental and isotopic impurities of the 176 Yb enriched target and the times t irr and t c is shown in Figures 4 and 5. The experimental results reported in our previous publications agree well with the theoretical calculation results shown in Figure 5. The maximum SA value present on the curve D of Figure 4 shows a combined effect of 174 Yb-and elemental Lu-impurities on the SA degradation. This tells us that the irradiation time should be optimized to obtain the highest SA for 177 Lu produced via 176 Yb (n, )  177 Yb ( decay) 177 Lu reaction. While being the best theoretical way to produce carrier-free 177 Lu, with this reaction we always obtain a 177 Lu product of much lower SA due to the use of an isotopically/elementally impure target. The maximum SA value mentioned above is characterized for a specific target composition and its neutron irradiation conditions, so the theoretical assessment of SA developed in this paper is important before starting the neutron activation process. This avoids over-bombardment destroying the SA of 177 Lu product and wasting expensive reactor operation time. Moreover, the post-irradiation processing time should be minimized to keep the SA as high as possible, the effect of which is shown in Figure 5.

Conclusions
Several factors affect SA of the radionuclide product which can be produced either by neutron capture reaction or by neutron-capture-followed-by -radioactive transformation processes. Among them the target composition (elemental and isotopic impurities), target nuclide and produced radioisotope depression causes (including target nuclide burn-up, reaction rate of target nuclide and the decay property of produced radionuclide) and the activation or post-irradiation time are most accounted for. With the method of SA assessment of multi radioactive source system the SA of a radioisotope produced in a reactor from different targets can be evaluated. The theoretical SA assessment of a radioactive nuclide has definitely given us a firm basis to set up an optimized process for the production of clinically useful radioisotopes and to evaluate the quality of the radionuclide product. A useful computer code based on the above developed SA assessment methods can be set up for a convenient daily use in the reactor-based production of medical radioisotopes such as 177 Lu, 153 Sm, 169 Yb, 165 Dy, 153 Gd... This evaluation plays a complementary or even substantial role in the quality management system regarding certifying the SA of the product which may be experimentally inaccessible due to radiation protection and instrumentation difficulties in the practical measurement of very low elemental content (<0.01 μg/mCiμL) of a radioactive solution of very high specific volume and specific radioactivity.