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1 Q J NUCL MED MOL IMAGING 2;55:44-56 From imaging to dosimetry and biological effects The essential steps are explained in calculating a radiation dose for radionuclide therapy from imaging data. As the dose alone is a meaningless value, its consequences in tumour cell kill efficiency and normal tissue damage are explained. The influence of dose rate and inhomogeneous dose distributions are discussed. Key words: Radiation - Radiobiology - Radiopharmaceuticals. Radiation dosimetry and radiobiology are the sciences to estimate the amount of energy deposited in a given medium and to quantify biological effects caused by exposure to ionizing radiation. Radiation absorbed dose is the mean energy absorbed within the body or organ of interest. Dose is defined as the mean absorbed energy per unit of mass: D=E abs /m, expressed in the unit Gy, which corresponds to the SI units J/kg. A handy conversion factor is MeV/g =.622 ngy. The absorbed dose by a radiopharmaceutical is determined by measuring the radioactivity distribution pattern in the total body and organs with physiological uptake and calculating the absorbed energy in the total body and all major organs by simulating the radiation transport. The result of the radiation transport calculation is expressed as the absorbed energy E abs from a source organ or region r s to a target organ or region r t : E abs (r t r s ) per decay of Corresponding author: M. Konijnenberg, Klinisch Fysicus, Nucleaire Geneeskunde V-224, Erasmus MC, Postbus 24, 3CA, Rotterdam, the Netherlands. m.konijnenberg@erasmusmc.nl. M. KONIJNENBERG Department of Nuclear Medicine Erasmus MC, Rotterdam, The Netherlands the radionuclide and divided by the mass of the target m t the dose rate per unit of radioactivity or S-value is obtained: S(r t r s ) =.622 E abs(r t r s ), m t with the units for S in [mgy.mbq -.s - ], the conversion factor.622 in [ngy.mev -.g], E abs in [MeV. decay - ] and m t in [g]. Various papers and text books have been published describing radionuclide dosimetry in more and better detail., 2 Radiation transport calculations Radiation transport calculations are ideally performed for the actual geometry of the patient, both in anatomy as in the physiological uptake kinetics of the radiolabeled drug, to determine the absorbed energy in the volume region self and for any source to target combination. Four types of radiation transport calculation are possible:. use of a data-base with absorbed dose factors for pre-defined stylized models for source and target organs. This method relies on radiation transport calculations performed for mathematically defined phantoms, mimicking the geometry of the patients. For diagnostic radiopharmaceuti- 44 THE QUARTERLY JOURNAL OF NUCLEAR MEDICINE AND MOLECULAR IMAGING February 2

2 From imaging to dosimetry KONIJNENBERG cals a whole family of phantoms has been defined by the MIRD-committee, ranging from adult male and female to a newborn, based on standard man dimensions. 3 The most popular code Olinda/EXM works with data-bases of dose-factors, or S-values, specific for source-organ to target-organ combination within a specific phantom size for the radionuclide under investigation. 4 This method originally intended for use at low diagnostic doses is widely used, but has the disadvantage that the clinical reality may show large differences with the phantom simulation, either in the size and positions of the organs or in the source distribution by heterogeneity in uptake over the source organ. For dosimetry in radionuclide therapy this method is a good option, especially when the time-activity curve in the source organ has been based on planar conjugate view measurements. The lower accuracy by possible overlap from adjacent organs and activity inhomogeneity would not justify more sophisticated dosimetry. When the organ doses are corrected for their actual weight, based on volume estimation from CT-images, enables them to be used for prediction of toxicity, as shown with kidney dosimetry with 9Y-DOTA-octreotide. 5, 6 More realistic average man and woman and children type of phantoms are being developed, 7, 8 but up to now no radionuclide S-values have been published for these phantoms; 2. no radiation transport, by just assuming local absorption of the emitted radiation in the source. This has long been the practice and is acceptable for short-ranged (low energy) beta- and alpha-particle emitting radionuclides, but not for gamma-ray emitters. High energy beta-emitters like 9 Y with a maximum beta-energy of 2.29 MeV show ranges in soft tissue of. cm, so when the dose distribution is needed within the range of the beta-particles from the source one of the other methods should be applied. In therapy with 9 Y-ibritumomab it was shown that it was feasible to use the 9 Y bremsstrahlung SPECT images directly as a dose map in this way; 9 3. the Point-Kernel calculation, which calculates the transport of radiation through the geometry of interest by using a measured or calculated propagation function with distance from the point source in tissue. A volumetric source distribution is described by a combination of point sources, randomly chosen or regularly spaced over a grid within the source volume. This method is only applicable within one type of tissue without large density transitions, and would therefore fail in the thorax region with lung and normal tissue. A special point-kernel method has evolved: voxel S-values enabling calculation of the dose distribution for voxelized SPECT or PET images has been developed and published in MIRD pamphlet 7 for cuboid voxels and recently values have been published for any rectangular shaped voxel;, 2 4. the Monte Carlo method, which takes all relevant physics for attenuation, scatter and possible secondary nuclear reactions into account by randomly choosing the path of the radiation through the geometry and following the probabilities for any physical reaction defined and occurring according to the cross sections for the material it is passing through. This kind of calculations is possible to calculate along media transitions, as in the thorax or from bone to soft tissue. Various Monte Carlo software codes are available: MCNP5/MCNPX, EGSnrc, Penelope, Geant/Gate, Fluka. Most models are well validated in nuclear physics and the choice of the code is directed by its user interface friendliness as well as calculation time. Dose calculation The absorbed dose to a target organ can be calculated by integration over time of the product of the activity distribution function in the source region A s and the S-value: D(r t ) = A(r s,t) S(r t r s,t)dt, usually the geometry does not alter during the irradiation therefore the S-value factor is time-independent and taken out of the integration. The dose is then defined as the product of the S-value and the cumulated activity Ã(r s ) = A(r s,t)dt in the source organ. The endpoint of integration is infinity ( ) for convenience in integrating exponential functions, but 99% of the dose is actually delivered within seven half-lives. For multiple source regions in the body the dose to the target region is calculated by summing over all source regions: D(r t ) = Σ S(r t r s,t) Ã(r s ). s The source activity distribution function is determined by quantitative imaging of the radioactivity in the source region over a series of time-points. This source region can be a voxel or a combination of voxels (Volume of Interest) from PET or SPECT im- Vol No. THE QUARTERLY JOURNAL OF NUCLEAR MEDICINE AND MOLECULAR IMAGING 45

3 KONIJNENBERG From imaging to dosimetry In the clinical reality it is not always possible to get enough data at well chosen time-points to enable fitting of multiexponential time-activity curves. The trapezoid integration method is in that case the most straightforward method to calculate Ã, it simply calculates the area under the curve drawn in straight lines connecting the data in order of time: m A Ã(r s ) = i+ Σ i- + A i (ti t i- ). A problem with this method 2 arises when the measured activity at the last time-point m is well above ; how does the activity disappear beyond this point, just with physical decay of the radionuclide or with a combination of physical decay and continued biological clearance. In the case of exponential curve fitting, the biological clearing is assumed to proceed beyond the last time-point. Compartmental modelling forms a powerful tool for determining the pharmacokinetics of the radiolabelled compound, by using physiological interactions between organs, blood and urinary (and possibly) fecal excretion pathways. The pharmacokinetics is described as a set of coupled differential equations, which can be solved either analytically for a limited set or numerically for larger sets. The result of the compartmental analysis can usually be expressed as a sum of exponentials, which enables easy integration. Several software codes exist for numerically solving the differential equations describing the pharmacokinetics. A much used code is SAAMII (Saam Institute Inc, Washington, DC, USA). The choice of fitting methods is best illustrated with an example. For the dosimetry in therapy with 9Y-DOTA-octreotide two diagnostic options were evaluated in patients: quantitative conjugate view planar imaging with In-DTPA-octreotide (Octreoscan) or quantitative PET imaging with 86 Y-DOTAoctreotide. 5 Radioactivity uptake in the kidneys (organ at risk) was measured at 24 and 48 h with In and 4, 24 and 48 h with 86 Y. The extra time-point at 4 h for 86 Y was chosen because of its much shorter half life in comparison to 9 Y ( 86 Y: 4.7 h, In: 67.3 h and 9 Y: 64. h). Additionally blood samples were taken at regular intervals up to 24 h and urine was collected the first 48 h. The radioactivity amounts of these blood samples and samples of the urine were measured. The decay corrected kidney uptake data are shown in figure. With 86 Y later time-points than 48 h were clinically not feasible, due to its short halfages or a region of interest (ROI) drawn in conjugate view planar images. The timing of the images has an high impact on the quality of the derived cumulated activity. When the first time-point is taken too late a high initial uptake may be missed, or when the last time-point is taken too early, a residual uptake in the target region may be overseen and could easily lead to errors of 3% or more in the cumulated activity. Pharmacokinetic compartment analysis may form a good aid in choosing the right time-points and give indication for the distribution of the radiopharmaceutical beyond the last time-point, recommendations can be found in MIRD pamphlet 6 3 like most of the items discussed here and in the following paragraph. Time activity curve fitting Making up the time-activity curve A(t) through the activity measurement data is needed to calculate the time-integrated activity à in the source volume. The time-integrated activity à is the sum of the total number of nuclear decays, which is the integral over time of the time-activity curve, or more simply the area under the time-activity curve. Sometimes the residence time, which has now been renamed to the time-integrated activity coefficient ã, 4 is used to indicate the ratio of the time-integrated activity and the injected activity A : ã =Ã/A, hence the integration over time of the fractional activity uptake curve results into the activity coefficient ã. Multiplying this coefficient ã with the S-values yields the absorbed dose per injected activity: d=d/a =ã S. Ideally the time-activity curve is obtained by fitting an exponential curve to the data points by the least squares method. Depending on the number of data A(t) can then be written as a sum of exponentials (three time points per exponential minimally needed): A(t,r s ) = Σ A i (,r s )exp( (λ i + λ)t), with λ the physical decay constant of the radionuclide with half life T /2 i (λ =ln(2)/t /2 ) and λ i the biological clearing or uptake (when A i < ) constant for the compartment with biological half life T i. Integration over time of A(t) from t= to t=t D yields: Ã(r s,t D ) = T A DΣA i (,r s )exp( (λ i + λ)t)dt = i Σ i i λ + λ i ( exp( (λ i + λ)t D ), when the dose-integration period T D is very large or goes to infinity: Ã(r s,t D ) = Σ i A i. λ + λ i 46 THE QUARTERLY JOURNAL OF NUCLEAR MEDICINE AND MOLECULAR IMAGING February 2

4 From imaging to dosimetry KONIJNENBERG Radioactivity in kidneys (%IA) Time (h) Time (h) 75 Figure. Radioactivity uptake in the kidneys of a patient, injected with In-DTPA-octreotide (left graph) and with 86 Y-DOTA-octreotide (right graph). Three curve fitting methods were used for establishing the time-activity curve: the trapezoid method, a single exponential and by compartmental modelling. Other Input Blood In-DTPA-octreotide k in k off Trapezoid Exponential Compartmental Urine k k 2 Kidneys Figure 2. Pharmacokinetic model for renal clearance of radiolabelled octreotides linking data from blood and urine sample measurements to the data for radioactivity uptake in the kidneys. life, an additional time-point with In would have been beneficial for the dosimetry (Figures, 2). In this example both the trapezoid method as the mono-exponential fitting leave no or just one degree of freedom for the fit, whereas with the compartmental it is possible to make a fit with 7 parameters for a total of 4 ( In) to 8 ( 86 Y) data points, leaving 7 to degrees of freedom. In the trapezoid method it is assumed that from t= to the first time-point t=4 or 24 the renal uptake is equal to the first uptake value. Hence the activity shows a linear relation connecting the next data points and after 48 hours the renal activity clears by physical decay only. As the measurements were performed with different radionuclides the time-integrated activity (and its time-integrated activity coefficient ã = Ã/A ) for 9 Y is then calculated by: k cl Radioactivity in kidneys (%IA) 3 2 Ã(r s ) = (A + A exp( λt )) t + A exp( λt ) + 2 A 2 exp( λt 2 )) t 2 t A m exp( λtm ), with m=2 2 λ for the In-DTPA-octreotide data, m=3 for the 86Y-DOTA-octreotide data and the decay constant λ is for 9 Y (λ=.693/t /2 ( 9 Y) =.8 h - ). The trapezoid model produces the highest values for the time-integrated activity coefficient, due to the most probably underestimation of the renal clearance after 48 h and overestimation of the uptake at t= (Table I). With just two data points the mono-exponential curve is just the connecting curve without any need for least-squares approximate fitting: A(r s,t)=a(r s,) exp( λ b t), with the initial activity ( A(r s,) = A A ) t A Y-DOTA-octreotide Trapezoid Exponential Compartmental t t and the biological clearance rate λ b = In(A ) In(A 2 ). With two or more data points a t 2 t squares approximation fit can be performed, using any curve fitting software, like Prism, or CurveExpert, but also within the Olinda/EXM code. The single-exponential fit assumes instantaneous uptake in the kidneys, but does provide an indication for the clearance after the last data point. The time-integrated activity is easily calculated: Ã(r s ) = ( s,)) A(r Up to 48 λ b +λ Vol No. THE QUARTERLY JOURNAL OF NUCLEAR MEDICINE AND MOLECULAR IMAGING 47

5 KONIJNENBERG From imaging to dosimetry Table I. Kidneys time-integrated activity coefficients ã (in h) for 9Y-DOTA-octreotide therapy, based on In-DTPA-octreotide or 86Y-DOTA-octreotide uptake data in one patient, using three models for the time-activity curve: ) trapezoid model; 2) exponential curve fit and 3) compartmental model. Imaging analogue h, 74% to 6% of the dose is built up in this example and when after 48 h just physical decay is assumed the residence time raises with 4% and %, respectively. Using the compartment model not only enables to get an estimate of the clearance but also of the uptake kinetics. The compartment model shown in Figure 2 is just a first order model with two compartments for the kidneys. The model is more complete if implemented with compartments for liver and spleen, first used for the dosimetry of octreoscan. 6 In a pharmacokinetic model for use in therapy a tumor distribution compartment should be added, as the tumor distribution will influence the overall kinetics. The result from the compartmental model can be integrated within the SAAMII software, by adding a physical decay rate with λ from each compartment. Also it is possible to fit a sum of exponentials to the result, in this example the uptake is bi-exponential and the clearance single-exponential. The cumulated activity and residence times is then calculated as in the exponential fits. Based on classic statistics reasoning the F-test favors the compartmental model and the models without degrees of freedom are considered non-statistical. A more elegant and valid method based on the Aikake Information Criterion has been proposed to choose between different models for the timeactivity curve by Glatting et al. 7 For small ratios of data size N and model parameters K a corrected AIC (AICc) is defined by: AICc = N ln ( SS ) Time-integrated activity coefficient for 9Y-DOTA-octreotide (h) Trapezoid Exponential Compartmental In-DTPA-octreotide Y-DOTA-octreotide (K + )(K + 2) + 2(K + ) + N N K 2 with SS the weighted sum of squares in the distances between the data and the model outcome. The AICc is only relevant for evaluations using the same number of data and therefore it is of little value in the used example, as the compartment model used much more data than the other fit models. The AICc was used to determine in a complex pharmacokinetic model for the biodistribution of a 9 Y labelled anti- CD66 monoclonal antibody measured with a In labelled analog what parameters in the model can be best be set as global (mean) values and what parameters need to be taken patient-specifically by using probability functions based on the AICc. 8 In the example of the dosimetry for 9 Y-DOTA-octreotide the AICc is the lowest for the trapezoid fitting for both the In as the 86 Y based cases. As the associated systematic uncertainty with this method is large, more data-points are needed. The dosimetry for the actual therapy, however, was based on trapezoid integration of the 86 Y data together with an exponential extrapolation beyond the last time-point of 48 h. 9 From dose to effect The absorbed dose is just a measure of the energy absorbed per unit mass in some organ that is exposed to ionizing radiation. Does it also yield a measure of the amount of damage this radiation exposure will cause? The answer to this question lies in the cell survival curves after radiation. The number of cells lost is proportional to the dose and the initial number of cells before radiation exposure. One Gy of X-ray exposure produces approximately 25 DNA double-strand breaks (DSBs) and about 9 single-strand breaks (SSBs). 2 The fractional cell survival shows a logarithmic dose response in a linear and quadratic relation with absorbed dose D: SF = exp(-ad - bd 2 ). This Linear-Quadratic model equation describes two different ways leading to cell kill by radiation. 2, 22 In the first way the cell is lethally damaged, e.g., by complex DSBs or two chromosomes, caused by a single radiation track. This process depends linearly on the dose: with the coefficient a, in [Gy - ] units. The second way of cell kill is described by combinations of sub-lethal damages, e.g., SSB or simple DSB, caused by two or more radiation tracks within one cell. The single damaged lesion as such is repairable and this process starts rapidly, but when a second damage occurs within some base-pairs of the first lesion, which has not been 48 THE QUARTERLY JOURNAL OF NUCLEAR MEDICINE AND MOLECULAR IMAGING February 2

6 From imaging to dosimetry KONIJNENBERG Tumour tissue (α/β= Gy) Tumour tissue (α/β= Gy) Surviving fraction of cells Surviving fraction of cells... x Gy 2 x 5 Gy 5 x 2 Gy EBRT dose (Gy) Normal tissue (α/β=.5 Gy) x Gy 2 x 5 Gy 5 x 2 Gy The radiation effect E can be expressed by the logarithm of the cell survival SF, when we consider the problem of radiation damage to be a matter of damage to cells, by which they become unable to replicate and Poisson statistics describes the probability of this process. Traditionally and most probrepaired yet, this may become unrepairable and the cell is doomed to die. This process depends on the probability of two independent events, i.e., quadraticly on the dose by the term b, in [Gy -2 ] units. An important parameter is the dose at which the linear term is equal to the quadratic term; this occurs at a dose of a/b (in [Gy]). This a/b is a measure of the intrinsic radiation-responsiveness of the tissue. In general, early-responding tissues like tumors, skin, testes hardly show a shoulder in the dose-effect curve for cell kill, hence the quadratic term (b) is low and inversely a/b is high 5~25 Gy. Late-responding tissues,however, like lung, liver, kidneys, show a clear shoulder in their dose effect curves, indicating a higher contribution from the quadratic Surviving fraction of cells.. h 6 h 6 h RNT dose (Gy) EBRT dose (Gy) RNT dose (Gy) Figure 3. Theoretical (LQ model) surviving fractions of cells after a dose up to Gy, given by external beam radiotherapy (EBRT: left graphs) or by radionuclide therapy (TRT: right graphs). The EBRT dose was either delivered in one dose, by 2 or by 5 fractions and the TRT dose was delivered by a radionuclide with either, 6 or 6 hours half life. The repair half life of sublethal damage in tumor was h and in normal tissue 2 h. 25 Surviving fraction of cells. h 6 h 6 h Normal tissue (α/β=.5 Gy) term and inversely a lower a/b than early-responding tissue (range -5 Gy). Tissues having a low a/b will show a higher sparing capacity by using lower doses per fraction or lower dose rates than tissues with high a/b. Biologically equivalent dose (BED) Vol No. THE QUARTERLY JOURNAL OF NUCLEAR MEDICINE AND MOLECULAR IMAGING 49

7 KONIJNENBERG From imaging to dosimetry ably also intuitively the effect is always considered to show a linear (logarithmic) relation with dose, so therefore the effect is divided by the linear term a. This defines the biologically equivalent dose (BED), as the dose times the relative effectiveness factor RE specific for the type of exposure: E = log(sf) = ad + bd 2, BED = E = D( + b D) = D RE a a Fractionation of the dose delivery with external beam radiotherapy is usually performed by giving daily doses of 2 Gy to the tumor/target area, instead of delivering the prescribed dose all at once. In between the fractions there is enough time for repairing all remaining sublethal lesions, which are more predominant in normal tissue (with low a/b) than in tumor (with high a/b). The higher sparing effect of fractionation in normal tissue in comparison to the tumor response is shown in Figure The curvature of the single dose exposure, which is much larger in normal tissue than in tumor, is straightened out by delivering the dose in fractions of 2 Gy. In this example for Gy in normal tissue the fractionation reduces the damage by a factor 6, while the tumor cell killing is reduced by only a factor 7. When the interval between the fraction is large enough to allow full repair: RE = + b/a d, with the dose per fraction d = D/N with N the total number of fractions. Dose rate effect In radionuclide therapy sparing of normal tissue can be achieved by using radionuclides with longer half lives, as the repair of sub-lethal damage sets in while the dose is still being delivered. In contrast to the situation in external beam radiotherapy, the repair half life is of great importance in radionuclide therapy. The probability that a radiation track produces a lesion that makes a lethal combination with a lesion from another track, while this is still being repaired, is given by an additional time factor G in the linear-quadratic model equation: 2 S = exp( ad GbD 2 ), with G = 2 R (t)dt exp( m(t t ))R (t )dt D 2 where m is the repair constant R (t), is the dose rate as a function of time. The median repair half life of several tumor cell lines has been determined to be about h, as the accuracy of the results of these cell experiments are obscured by effects from proliferation and cell-cycle re-distribution. The repair mechanism in normal tissue show half-lives longer than h and late effects quite often show bi-exponential repair kinetics. 23 All values used for the LQ-model parameters are based on external beam exposure. With the development of patient-conformal therapies, like image guided radiotherapy with external beams, there is a development of giving higher dose fractions, in a hypo- or oligo-fractionated treatment, as this reduces the normal tissue dose, but it inevitably cause doses with steep gradients to normal tissue adjacent to the target volume (Figure 3). 24, 25 It was suggested that at doses per fraction of -2 Gy the LQ model describing tumor cell kill should be straightened out above the transition dose D T of 5 Gy, using the LQ(L) model. 26 This can be interpreted as that above D T the dose-effect curve becomes linear, whether the repair mechanism is turned off or saturated is unclear in this model. Higher values for a/b than the conventional Gy, however, can also explain the observed loglinear dose relationship. 27 The doses obtained with peptide receptor radionuclide therapy also follow an oligo-fractionation scheme by giving the total therapy in three to four fractions, 28, 29 though at much lower dose rates and consequently longer irradiation times. The flattening of the dose-effect curve during longer irradiation times is caused by the increased possibility for sub-lethal damage repair, thus reducing the quadratic term in the LQ-model equation. For mono-exponential dose delivery by radionuclides with effective decay constant l the time factor G as a function of irradiation time T becomes: G(T D ) = 2 TD R exp( λt)dt t exp( m(t t ))R exp( λt )dt D 2 { 2λ exp( (λ + m)td + λ + m exp( 2λT D } ) λ λ m λ m = λ + m ( exp( λt D ))2 lim G(T D ) = λ T D λ + m 5 THE QUARTERLY JOURNAL OF NUCLEAR MEDICINE AND MOLECULAR IMAGING February 2

8 From imaging to dosimetry KONIJNENBERG More complex functions of time and position for G have been derived by Baechler et al. 3 For instance, a bi-exponential repair process with is described by: G,2 (T D )= mλ { λ m 2λ exp( λ +m )T D ) (λ+ m ) exp( 2λT D )} + λ+m (λ m )( wxp( λ T D )) 2 ( m)λ λ+m 2 (λ m 2 )( wxp( λ T D )) 2 lim G,2 (T D ) = T D { λ m 2 2λ exp( λ +m 2 )T D ) (λ+ m 2 ) exp( 2λT D )} mλ (l m)λ + λ + m λ + m 2 Calculation of the G-factor for any dose rate function, including trapezoidal fit functions, have been derived by Hobbs et al., 3 but only show huge variation in the initial high dose rate exposure part. In therapeutic exposures with radionuclides there is especially a clear need for accurate repair kinetics models in contrast to external beam radiotherapy, where the repair mechanism is only of importance with too short interfraction intervals and possibly only to low dose per fraction when the LQ(L) model holds. The influence of repair is most prominent in normal tissue with low a/b and usually with longer repair half lives than tumor tissue, as depicted in Figure 4. In tumor with a high a/b value of Gy variation of the repair half life between minute and 2 hours hardly changes the cell survival at a dose of Gy. At a 5 Gy dose, however, given in one fraction, a factor less cell kill is expected at T rep of minute in comparison to the median T rep of hour. The bi-exponential repair kinetics in this example shows a comparable dose response curve to that with a single repair half life of h (F-test: 98.5%). Repair half lives for tumors can be derived from in-vitro experiments with irradiation of cell cultures. The curvature (or a/b) of the dose effect curve found in vitro is mostly equal to the in-vivo results, 32, 33 and this also seem to hold for the repair half lives (Figure 4). 25 containing initially a total of N * clonogenic cells TCP is: TCP = exp ( N* ST) = exp( N* exp(gt) exp( ad GbD 2 ) ), where g is the tumor repopulation rate (g=ln(2)/ T D ; with T D the tumor doubling time) and T is the effective treatment time, during which the cell kill by radiation outweighs the growth by repopulation. For a mono-exponential build-up of the dose D(t) = R λ (l exp( λt)) and with T rep much lower than the effective half life T eff the effective treatment time T can be solved from E = : t t = T T = + ln( a a2 + 8bg ) m λ λ 4bR m λ or when the quadratic term is neglected: T l ln ( g ). λ ar Curability of tumors by beta-ray emitting radionuclides depends strongly on the energy of the b-rays and their range in relation to the tumor size. Each radionuclide has therefore an optimal tumor size for curability as theoretically derived by O Donoghue. 34 Experiments with radiolabelled peptides in rats are in accordance with this postulate; it was shown that 9Y with high energy beta s is capable of curing larger size tumors than the low-energy b-emitter 77 Lu. 35, 36 Also the decay half life of the radionuclide used is of importance. It was deduced for brachytherapy that shorter half lives (-5 days) are optimal for fast repopulating tumors ( g a ~ Gy/d) and that longer half lives (5- days) are optimal for slowly growing tumors ( g a ~.5-. Gy/d).37 A comparable theoretical evaluation was performed for radiolabeled antibodies by Howell et al.; 38 they reported more advantage for the longer half-life 32 P in comparison with 9 Y in fast growing tumors ( g a =.83 Gy/d). The reason for this contradictory result is that the gradual uptake of the antibodies in the tumors together with the immediate exposure of the bone marrow by radioactivity circulating in the blood yield a better tumor -to-marrow dose ratio for longer half-life radionuclides. Tumor control probability (TCP) The efficacy of radiation therapy is expressed by the tumor control probability (TCP), a mechanistic model based on Poisson statistics for surviving tumor cells capable of tumor regeneration. In a tumor Vol No. THE QUARTERLY JOURNAL OF NUCLEAR MEDICINE AND MOLECULAR IMAGING 5

9 KONIJNENBERG From imaging to dosimetry Surviving fraction of cells.. min 3 min 6 min 2 min Tumour tissue dose (Gy) Normal tissue dose (Gy) Figure 4. Surviving fraction of cells after a dose up to Gy by radionuclide therapy at an effective decay half life of 2 hours, in tumor (left graph) and normal tissue (right graph) for various sublethal damage repair half lives. A bi-exponential repair mechanism according to 23 was assumed in one of the normal tissue response curves. Normal tissue complication probability (NTCP) The philosophy in planning treatment with radionuclide therapy is usually based on minimizing radiation toxicity in normal organs. Many models have been developed to describe late normal toxicity mathematically. The Lyman-Kutcher-Burman (LKB) model is most commonly used; it describes the normal tissue complication probability (NTCP) as a sigmoid shaped curve: 39 NTCP LKB = l exp( c2 )dx, with t = D TD5 (ν), 2p -c 2 m TD 5 and TD 5 (ν) = TD 5(). ν n α/β= Gy T½=2 h This model uses a normal distribution of complications as function of dose around the mean value TD 5 with standard deviation m TD 5 where radiation toxicity occurs with 5% probability at this dose. Inhomogeneous dose distribution in the organs is incorporated in the model by correction of the uniform whole organ irradiation TD 5 () with the fractional organ volume ν that receives the dose (the complement volume -ν gets no dose). The conversion factor n for compensating inhomogenous dose distributions gives an indication of the response type of the organ. The values for TD 5, m and n have to be determined from clinical data. A classical paper summarizing these data for various organs obtained for external beam exposures was written by Emami et al. 4 Almost 2 years later the dose limits have been Surviving fraction of cells.. 9 min 3 min 6 min 66% 9 min + 34% 5 h 2 min 8 min updated by the Quantec group, 4 mostly concerning improvements in dosimetry calculations and some additional clinical data, for kidneys, 42 liver, 43 lungs 44, 45 and the data is shown in Table II. The dose limit of 8 Gy for the kidneys is lower than the 23Gy from the Emami paper and the published constraints on the dose-volume limits do not follow the LKB model e.g. the total kidney volume with a dose above 2 Gy V 2 < 55% with GFR as end-point for late toxicity. Unfortunately no additional data has been gathered on bone marrow toxicity after radiation. The threshold doses for external beam exposures are based on the normal fractionation scheme of 2 Gy per fraction. In order to use these parameters in radionuclide therapy applications the threshold doses should be transformed to BEDs. For renal radiation toxicity after [ 9 Y]DOTA-octreotide therapy this has been performed and it was possible to explain the observed 7 Gy shift in threshold compared to external beam radiotherapy with the LQ model parameters with a/b = 2.5 Gy and T rep = 2.8 h. 46 This was actually quite surprising when the uncertainties in all dose and radiobiology parameters are considered. Equivalent uniform dose (EUD) α/β= Gy T½=2 h A problem with both the external beam and the radionuclide exposure is how to compensate for inhomogeneous dose distributions. Three-dimensional radiation dose distributions can be visualized by 52 THE QUARTERLY JOURNAL OF NUCLEAR MEDICINE AND MOLECULAR IMAGING February 2

10 From imaging to dosimetry KONIJNENBERG Table II. Threshold doses for late toxicity effects in normal organs TD 5 (5% probability) and TD 5 (5% probability), the slope parameters m (with dose) and n (with irradiated fractional volume) and the LQ parameters α/β and sub-lethal damage repair half life T rep. isodose graphs along surfaces through the central axes of the organ. Downgrading to two dimensions is obtained by calculating the Dose Volume Histogram; a histogram is binned by summing the fractional volumes of an organ or tumor within each dose bin. 47 Usually a cumulative DVH is derived by summing all fractional volumes with doses greater than or equal to the dose of the bin. This approach assumes that the organ consists of a large number of identical functional sub-units (FSU). In tumors the FSU is the tumor cell, and in kidneys the nephron. 42 The arrangement of the FSUs within the organ tissue is of importance to determine its response to dose. 4 Parallel arrangements of the FSUs within an organ indicates that subvolumes function independently and consequently it is possible to cause damage in a substantial part of this organ without clinical signs of toxicity. Only when the damage exceeds a critical volume complications arise and the TD 5 for partial irradiation is inversely proportional with the fractional volume (n ). Examples of organs with parallel architecture are liver, lungs and renal cortex. Organs with a serial structure of their FSUs show complications already after a small part has been damaged and TD 5 for partial irradiation is almost equal to the TD 5 for homogeneous irradiation (n<.). Examples for serial architecture organs are optic nerve, spinal chord and stomach. Further reduction of the DVH data to one single value for parallel arranged organs can be obtained by calculating the equivalent uniform dose (EUD). The EUD is defined as the dose when given uniform over the whole organ or tumor would produce the same effect E (NTCP or TCP): 48, Σ i E(D i ) V i E(EUD) EUD = E - { V tot EUD LKB = ( Σ i D i n - νi)n Organ TD 5 (Gy) TD 5 (Gy) m n α/β (Gy) Trep (h) Kidneys Liver (normal) Liver (diseased) Lungs Σ i E(D i ) V i V to } when n= EUD is simply the mean organ dose. Without the quadratic term in the LQ model EUD can be written out attractively by: EUD = ln(e). 49 For tua mors, a is usually known 32, 33 but for normal tissue it will be problematic to obtain reliable values, as the dose-effects in normal tissues are not visible until the deterministic threshold for damage has been exceeded. Three-dimensional dosimetry was used to analyze tumor shrinkage after therapy with the 3 I labelled antibody tositumomab. 5 In this study with B-cell lymphoma patients (N.=9) EUD showed a slightly better correlation with tumor shrinkage than the mean dose to the tumor did. In normal tissue examples for the use of EUD within nuclear medicine are limited to radiation dosimetry in the largest organ: the liver, as resolution of SPECT is not high enough to distinguish large inhomogeneities. 2, 5 Even then, the voxel size used in voxelised point kernel method is too large to calculate the full dose inhomogeneity by beta-emitters like 9 Y. Input of the structures in kidney radioactivity uptake found in autoradiographs after injection with In-DTPA-octreotide into a Monte Carlo code did give a good indication of the inhomogeneities and led to the conclusion that short-ranged betaemitters as 77 Lu will induce less damage. 52 For a mean dose to the cortex of 23 Gy, V 2 for 77 Lu is 37%, well below the 55% limit 42 which should additionally be corrected on a BED basis. It is, however, arguable whether to use all irradiated volume with radionuclide exposure indiscriminately, as the uptake of the drug in the organ is usually of physiological nature and inevitably involving the functional structure of this organ. In external beam radiotherapy where heterogeneities in the dose distribution to normal tissue are of geometric origin by entrance fields or regions adjacent Vol No. THE QUARTERLY JOURNAL OF NUCLEAR MEDICINE AND MOLECULAR IMAGING 53

11 KONIJNENBERG From imaging to dosimetry to the target area, it is realized that the functional status of the irradiated normal tissue should be considered. Functional dose volume histograms (DFH) were proposed for lung irradiations taking lung perfusion SPECT images into account. 53 The treatment plan based on the DFH was able to spare the wellfunctioning lung regions. Recommendations for treatment planning of targeted radionuclide therapy Steps for dosimetry-based safe and efficacious treatment planning of radionuclide therapy:. read MIRD pamphlet 6 3 and EANM guidelines for good dosimetry reporting 54 carefully and design the dosimetry assessment protocol accordingly; 2. measure the radioactivity at different timepoints depending on (pre-)clinical experience, use appropriate attenuation, scatter and background correction; 3. for agents with potential radiation-induced bone-marrow toxicity determine radioactivity in blood samples and urine sample at regular timepoints, or use an external probe to measure total body accumulation and clearance; 4. perform least square fit of (multi-)exponential curve through data, with three data points per exponential (any reliable fitting software). Select the best fit model by using e.g., the Aikake information criterion; 5. integrate time-activity curve either analytically or by Simpson s trapezoid integration method. Consider pharmacokinetic modelling for distribution after last time-point (SAAMII software or any mathematical code for solving differential equations); 6. determine what dose model is needed and calculate S-value according to this model: for planar conjugate view quantification Olinda/EXM, or equivalent S values source (RADAR website) could be used with correction for the actual volume of the critical organ, derived from CT, ultrasound or MRI; for quantitative SPECT/CT quantification, or combination of SPECT at initial time point with planar at later time points, use a 3D dosimetry code like 3D/ID, Geant, MCNPX. Commercially available codes have been developed by several companies (Philips, GE, Hermes, ). Only doses inside the tho- rax region or in or near bone structures need the complexity of Monte Carlo calculations, else Point Kernel based methods fulfil, or even assuming local energy absorption within the source voxel. When the voxel geometry is only used for better delineation of the organ the same dosimetry model as for planar imaging could be used; 7. determine the mean organ dose by multiplication of time-integrated activity and S-value preferably within a validated dosimetry code. When 3 dimensional activity distribution is available calculate the 3-D dose distribution within tumors and doselimiting organs; 8. determine maximum tolerable dose to the critical organ at risk, taking dose rate and fractionation in account, possibly adjust the EBRT-derived dose limits to TRT-equivalent limits via BED when a/b and T rep are known; 9. determine maximum tolerated activity from tolerable dose. For 3-D dose distributions derive the dose-volume histogram in organs at risk and large tumors at this activity;. when the dose distribution in the critical organ shows a large level of heterogeneity correct for it using the appropriate model, depending on the serial or parallel structure of the functional subunits within this organ;. calculate the radiation dose to the tumor and metastases at maximum tolerable activity and consider the tumor cure probability and ways to optimise the therapy. For highly heterogenic dose distributions inside the tumor reduction of the DVH to an EUD can give a better prediction of therapy efficacy than mean dose. References. Stabin MG. Fundamentals of nuclear medicine dosimetry. New York, NY: Springer; Sgouros G. Dosimetry of internal emitters. J Nucl Med 25;46:8S-27S. 3. Cristy M, Eckerman KF. Specific absorbed fractions of energy at various ages for internal photon sources. ORNL/TM-838. Oak Ridge, TN: Oak Ridge National Lab; Stabin MG, Siegel JA. Physical models and dose factors for use in internal dose assessment. Health Physics 23;85: Barone R, Borson-Chazot F, Valkema R, Walrand S, Chauvin F, Gogou L et al. Patient-specific dosimetry in predicting renal toxicity with (9)Y-DOTATOC: relevance of kidney volume and dose rate in finding a dose-effect relationship. J Nucl Med 25;46(Suppl ):99S-6S. 6. Bodei L, Cremonesi M, Ferrari M, Pacifici M, Grana CM, Bartolomei M et al. Long-term evaluation of renal toxicity after 54 THE QUARTERLY JOURNAL OF NUCLEAR MEDICINE AND MOLECULAR IMAGING February 2

12 From imaging to dosimetry KONIJNENBERG peptide receptor radionuclide therapy with 9 Y-DOTATOC and 77Lu-DOTATATE: the role of associated risk factors. Eur J Nucl Med Mol Imaging 28;35: Menzel HG. Adult reference computational phantoms, ICRP Publication. Ann ICRP 29;39: Lee C, Lodwick D, Hurtado J, Pafundi D, Williams JL, Bolch WE. The UF family of reference hybrid phantoms for computational radiation dosimetry. Phys Med Biol 2;55: Minarik D, Sjögreen-Gleisner K, Linden O, Wingårdh K, Tennvall J, Strand SE et al. 9 Y Bremsstrahlung imaging for absorbed-dose assessment in high-dose radioimmunotherapy. J Nucl Med 2;5: Bolch WE, Bouchet LG, Robertson JS, Wessels BW, Siegel JA, Howell RW et al. MIRD pamphlet No. 7: the dosimetry of nonuniform activity distributions radionuclide S values at the voxel level. J Nucl Med 999;4:S-36S.. Fabbri C, Sarti G, Cremonesi M, Ferrari M, Di Dia A, Agostini M et al. Quantitative analysis of 9Y Bremsstrahlung SPECT-CT images for application to 3D patient-specific dosimetry. Cancer Biother Radiopharm 29;24: Dieudonné A, Hobbs RF, Bolch WE, Sgouros G, Gardin I. Fineresolution voxel S values for constructing absorbed dose distributions at variable voxel size. J Nucl Med 2;5: Siegel JA, Thomas SR, Stubbs JB, Stabin MG, Hays MT, Koral KF et al. MIRD pamphlet no. 6: Techniques for quantitative radiopharmaceutical biodistribution data acquisition and analysis for use in human radiation dose estimates. J Nucl Med 999;4:37S-6S. 4. Bolch WE, Eckerman KF, Sgouros G, Thomas SR. MIRD pamphlet No. 2: a generalized schema for radiopharmaceutical dosimetry -- standardization of nomenclature. J Nucl Med 29;5: Pauwels S, Barone R, Walrand S, Borson-Chazot F, Valkema R, Kvols LK et al. Practical dosimetry of peptide receptor radionuclide therapy with (9)Y-labeled somatostatin analogs. J Nucl Med 25;46:92S-8S. 6. Stabin MG, Kooij PP, Bakker WH, Inoue T, Endo K, Coveney J et al. Radiation dosimetry for indium--pentetreotide. J Nucl Med 997;38: Kletting P, Kull T, Reske SN, Glatting G. Comparing time activity curves using the Akaike information criterion. Phys Med Biol 29;54:N Kletting P, Kull T, Bunjes D, Mahren B, Luster M, Reske SN et al. Radioimmunotherapy with anti-cd66 antibody: improving the biodistribution using a physiologically based pharmacokinetic model. J Nucl Med 2;5: Barone R, Borson-Chazot F, Valkema R, Walrand S, Chauvin F, Gogou L et al. Patient-specific dosimetry in predicting renal toxicity with (9)Y-DOTATOC: relevance of kidney volume and dose rate in finding a dose-effect relationship. J Nucl Med 25;46:99S-6S. 2. Hall EJ. Radiobiology for the radiobiologist. 5 th ed. Philadelphia, PA: JB Lippincott Co.; Dale RG. The application of the linear-quadratic dose-effect equation to fractionated and protracted radiotherapy. Br J Radiol 985;58: Brenner DJ, Hlatky LR, Hahnfeldt PJ, Huang Y, Sachs RK. The linear-quadratic model and most other common radiobiological models result in similar predictions of time-dose relationships. Radiat Res 998;5: Millar WT, Jen YM, Hendry JH, Canney PA. Two components of repair in irradiated kidney colony forming cells. Int J Radiat Biol 994;66: Tepper JE. Hypofractionation. Sem Radiat Oncol 28;8(issue 4: October). 25. Ling CC, Gerweck LE, Zaider M, Yorke E. Dose-rate effects in external beam radiotherapy redux. Radioth Oncol 2;95: Borst GR, Ishikawa M, Nijkamp J, Hauptmann M, Shirato H, Bengua G et al. Radiation pneumonitis after hypofractionated radiotherapy: evaluation of the LQ(L) model and different dose parameters. Int J Radiat Oncol Biol Phys 2;77: Fowler J. 2 Years of biologivcally effective dose. Br J Radiol 2;83: Bushnell DL Jr, O Dorisio TM, O Dorisio MS, Menda Y, Hicks RJ, Van Cutsem E et al. 9 Y-edotreotide for metastatic carcinoid refractory to octreotide. J Clin Oncol 2;28: Kwekkeboom DJ, de Herder WW, Kam BL, van Eijck CH, van Essen M, Kooij PP et al. Treatment with the radiolabeled somatostatin analog [77 Lu-DOTA,Tyr3]octreotate: toxicity, efficacy, and survival. J Clin Oncol 28;26: Baechler S, Hobbs RF, Prideaux AR, Wahl RL, Sgouros G. Extension of the biological effective dose to the MIRD schema and possible implications in radionuclide therapy dosimetry. Med Phys 28;35: Hobbs RF, Sgouros G. Calculation of the biological effective dose for piecewise defined dose-rate fits. Med Phys 29;36: Malaise EP, Fertil B, Chavaudra N, Guichard M. Distribution of radiation sensitivities for human tumor cells of specific histological types: comparison of in vitro to in vivo data. Int J Radiat Oncol Biol Phys 986;2: Björk-Eriksson T, West C, Karlsson E, Mercke C. Discrimination of human tumor radioresponsiveness using low-dose rate irradiation. Int J Radiat Oncol Biol Phys 998;42: O Donoghue JA Bardiès M, Wheldon TE. 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13 KONIJNENBERG From imaging to dosimetry combined analysis of published clinical data. Phys Med Biol 28;53: Wessels BW, Konijnenberg MW, Dale RG, Breitz HB, Cremonesi M, Meredith RF et al. MIRD pamphlet No. 2: the effect of model assumptions on kidney dosimetry and response implications for radionuclide therapy. J Nucl Med 28;49: Kutcher GJ, Burman C, Brewster L, Goitein M, Mohan R. Histogram reduction method for calculating complication probabilities for three-dimensional treatment planning evaluations. Int J Radiat Oncol Biol Phys 99;2: Seppenwoolde Y, Lebesque JV, de Jaeger K, Belderbos JS, Boersma LJ, Schilstra C et al. Comparing different NTCP models that predict the incidence of radiation pneumonitis. Normal tissue complication probability. Int J Radiat Oncol Biol Phys 23;55: O Donoghue JA. Implications of nonuniform tumor doses for radioimmunotherapy. J Nucl Med 999;4: Dewaraja YK, Schipper MJ, Roberson PL, Wilderman SJ, Amro H, Regan DD et al. 3 I-tositumomab radioimmunotherapy: initial tumor dose-response results using 3-dimensional dosimetry including radiobiologic modeling. J Nucl Med 2;5: Strigari L, Sciuto R, Rea S, Carpanese L, Pizzi G, Soriani A et al. Efficacy and toxicity related to treatment of hepatocellular carcinoma with 9Y-SIR spheres: radiobiologic considerations. J Nucl Med 2;5: Konijnenberg MW, Bijster M, Krenning EP, De Jong M. A stylized computational model of the rat for organ dosimetry in support of preclinical evaluations of peptide receptor radionuclide therapy with (9)Y, ()In, or (77)Lu. J Nucl Med 24;45:26-9. Erratum in: J Nucl Med 29;5: Marks LB, Sherouse GW, Munley MT, Bentel GC, Spencer DP. Incorporation of functional status into dose-volume analysis. Med Phys 999;26: Lassmann M, Chiesa C, Flux G, Bardiès M; EANM Dosimetry Committee guidance document: good practice of clinical dosimetry reporting. Eur J Nucl Med Mol Imaging 2;38: THE QUARTERLY JOURNAL OF NUCLEAR MEDICINE AND MOLECULAR IMAGING February 2