Dynamic estimation of a biological effect of a variable composition of non-photon radiation

Abstract

A method for dynamically estimating a biological effect of a variable combination of non-photon radiation in accordance with a relative biological effectiveness, RBE, model including at least one biological effect multiplier δ(T, E) which depends on particle type T and/or particle energy E, the method comprising: obtaining one or more non-photon radiation contributions D.sup.(i)(T, E), 1 ≤ i ≤ N, at least one of said contributions including multiple particle types and/or multiple particle energies; storing per-contribution dose-weighted averages δ̅.sup.(i), 1 ≤ i ≤ N, of said at least one biological effect multiplier with respect to each of the one or more contributions; and in response to obtaining an assignment Π of the combination, the assignment being in terms of non-negative coefficients k.sub.1, k.sub.2, ..., k.sub.N ≥ 0 to be applied to the one or more contributions, determining a biological effect of the combination, including computing a combined dose-weighted average δ̅.sup.Π of said at least one biological effect multiplier on the basis of the stored per-contribution dose-weighted averages.

Claims

1. A method for dynamically estimating a biological effect of a variable combination of non-photon radiation in accordance with a relative biological effectiveness, RBE, model including at least one biological effect multiplier δ(T, E) which depends on particle type T and/or particle energy E, such that its contribution to the biological effect is δ(T,E)D(T,E).sup.p, where p > 0 is a characteristic power of the biological effect multiplier, the method comprising: obtaining one or more non-photon radiation contributions D.sup.(i)(T,E), 1 ≤ i ≤ N, at least one of said contributions including multiple particle types and/or multiple particle energies; storing per-contribution dose-weighted averages δ.sup.(i), 1 ≤ i ≤ N, of said at least one biological effect multiplier with respect to each of the one or more contributions; and in response to obtaining an assignment Π of the combination, the assignment being in terms of non-negative coefficients k.sub.1, k.sub.2, ..., k.sub.N ≥ 0 to be applied to the one or more contributions, determining a biological effect of the combination, including computing a combined dose-weighted average δ.sup.Π of said at least one biological effect multiplier on the basis of the stored per-contribution dose-weighted averages.

2. The method of claim 1, wherein the combined dose-weighted average δ.sup.Π of said at least one biological effect multiplier is computed as a power mean with exponent 1/p of each stored per-contribution dose-weighted average δ.sup.(i) weighted by the respective coefficient k.sub.i and a total dose $D_{tot}^{\left(i\right)}={\displaystyle {.Math.}_{T,E}{D}^{\left(i\right)}\left(T,E\right)}$ of that contribution: ${\left({\overline{\delta }}^{\text{Π}}\right)}^{1/p}=\frac{{\displaystyle {.Math.}_i{\left({\overline{\delta }}^{\left(i\right)}\right)}^{1/p}{k}_i{D}_{tot}^{\left(i\right)}}}{{\displaystyle {.Math.}_i{k}_i{D}_{tot}^{\left(i\right)}}},$ where p > 0 is the characteristic power of the biological effect multiplier.

3. The method of claim 1, wherein determining the biological effect of the combination includes multiplying a p.sup.th power of the total dose $D_{tot}^{.Math.}$ of the combination Π with the combined dose-weighted average δ.sup.Π of said at least one biological effect multiplier.

4. The method of claim 1, further comprising storing a total dose $D_{tot}^{\left(i\right)}={\displaystyle {.Math.}_{T,E}{D}^{\left(i\right)}\left(T,E\right)}$ of the contribution for use in the determination of the biological effect of the combination Π.

5. The method of claim 1, further comprising computing the per-contribution dose-weighted average of the at least one biological effect multiplier using the equation: ${\left({\overline{\delta }}^{\left(i\right)}\right)}^{1/p}=\frac{{\displaystyle {.Math.}_{T,E}{\left(\delta \left(T,E\right)\right)}^{1/p}{D}^{\left(i\right)}\left(T,E\right)}}{D_{tot}^{\left(i\right)}},1\le i\le N,$ where p > 0 is a characteristic power of the biological effect multiplier and $D_{tot}^{\left(i\right)}={\displaystyle .Math.{}_{T,E}{D}^{\left(i\right)}\left(T,E\right)}$ is a total dose of the i.sup.th contribution.

6. The method of claim 1, wherein p = 1 for at least one biological effect multiplier δ(T,E) of the RBE model.

7. The method of claim 6, wherein the at least one biological effect multiplier includes an α multiplier of a linear-quadratic model, such as an α.sub.0 term or $z_{1D}^{\ast }$ term of a microdosimetric-kinetic model, MKM.

8. The method of claim 1, wherein p = 2 for at least one biological effect multiplier δ(T,E) of the RBE model.

9. The method of claim 8, wherein the at least one biological effect multiplier includes a β multiplier of a linear-quadratic model.

10. The method of claim 1, wherein each contribution represents a beam or spot to be delivered in radiation therapy.

11. The method of claim 1, wherein each contribution represents a radiation treatment plan, such as a base plan or Pareto-optimal plan.

12. The method of claim 11, wherein the coefficients represent a convex combination of the contributions.

13. The method of claim 12, wherein the contributions represent base plans obtained by multi-criteria optimization, MCO, and the combination corresponds to a navigated plan.

14. The method of claim 1, wherein the combination corresponds to a scaling of a radiation treatment plan.

15. The method according to claim 1, wherein the non-photon radiation includes proton radiation, helium ions or carbon ions.

16. The method of claim 1, further comprising using the determined biological effect of the combination to support radiation treatment planning.

17. A treatment planning system configured to dynamically estimate a biological effect of a variable combination of non-photon radiation in accordance with a relative biological effectiveness, RBE, model including at least one biological effect multiplier δ(T,E) which depends on particle type T and/or particle energy E, such that its contribution to the biological effect is δ(T,E)D(T,E).sup.p, where p > 0 is a characteristic power of the biological effect multiplier, the system comprising: an interface configured to receive one or more non-photon radiation contributions D.sup.(i)(T,E), 1 ≤ i ≤ N, at least one of said contributions including multiple particle types and/or multiple particle energies, and dynamic assignments Π of the combination, the assignments being in terms of non-negative coefficients k.sub.1, k.sub.2, ..., k.sub.N ≥ 0 to be applied to the one or more contributions; a memory configured to store per-contribution dose-weighted averages δ.sup.(i), 1 ≤ i ≤ N, of said at least one biological effect multiplier with respect to each of the one or more contributions; and processing circuitry configured to determine, for each received dynamic assignment Π of the combination, a biological effect of the combination, including computing a combined dose-weighted average δ.sup.Π of said at least one biological effect multiplier on the basis of the stored per-contribution dose-weighted averages.

18. A computer program product, comprising a non-transitory storage medium containing instructions which, when executed by a computer, cause a computer to carry out the method of claim 1.

19. (canceled)

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Aspects and embodiments are now described, by way of example, with reference to the accompanying drawings, on which:

[0019] FIG. 1 illustrates schematically how the dynamic computation of biological effect –ln S.sup.Π is organized according to embodiments of the invention;

[0020] FIG. 2 is a detail of FIG. 1;

[0021] FIG. 3 is a flowchart of a method according to an embodiment;

[0022] FIG. 4 is a block diagram of a treatment planning system according to an embodiment; and

[0023] FIG. 5 shows a radiation delivery system for executing a treatment plan.

DETAILED DESCRIPTION

[0024] The aspects of the present disclosure will now be described more fully with reference to the accompanying drawings, on which certain embodiments of the invention are shown. The invention may, however, be embodied in many different forms and the embodiments should not be construed as limiting; rather, they are provided by way of example so that this disclosure will be thorough and complete, and to fully convey the scope of all aspects of invention to those skilled in the art.

[0025] FIG. 3 is a flowchart of a method 300 for dynamically estimating a biological effect of a treatment plan specifying a non-photon irradiation of a patient, wherein the treatment plan is a combination of non-photon radiation contributions. The non-photon radiation may for example be radiation by ions or protons.

[0026] The treatment plan may be executed by a radiation delivery system 500. As shown in FIG. 5, such system may include a gantry 510 with a radiation source, and a couch 520, which a patient rests on and is fixated to during the treatment. Rotatory and/or translational relative movement between the gantry 510 and couch 520 is possible. In particular, the gantry 510 may be rotatable with respect to one or two axes; and the couch 520 may be rotatable round a vertical axis and translatable in at least one dimension. This allows a multitude of irradiation angles and positions (or incidence directions), as may be described by a corresponding plurality of spots. The treatment plan may specify fluence and/or particle energy values for all or some of the available spots.

[0027] The biological effect is to be computed in accordance with a relative biological effectiveness (RBE) model including at least one biological effect multiplier δ(T,E) which depends on particle type T and/or particle energy E. The biological effect multiplier may further be associated with a value of a characteristic power p > 0. For particle type T and particle energy E, the biological effect multiplier corresponds to a contribution to the biological effect - In S which is equal to δ(T,E)D(T,E).sup.p. For a mixed dose, the total contribution is given by

[00002]${\displaystyle \underset{T,E}{.Math.}\delta \left(T,E\right)D{\left(T,E\right)}^p},$

where the notation .Math..sub.T,E ... is shorthand for summing over all (T, E) pairs for which the dose D(T,E) > 0. The formalism with biological effect multipliers is useful for representing macroscopic biological effects, e.g., in beam mixing, but may not be sufficient for micro- or nanodosimetric calculations.

[0028] An RBE model may be expressed as an RBE factor which is a linear combination of one or more biological effect multipliers. Within the scope of the present invention, an RBE model may be: [0029] a local effect model (LEM) (see for example the early version described in Scholz et al., “Computation of cell survival in heavy ion beams for therapy. The model and its approximation”, Radiat. Environ. Biophys. (1997), vol. 36, pp. 59-66 [doi:10.1007/soo4110050055]), [0030] a microdosimetric-kinetic model (MKM) (see for example Hawkins, “A microdosimetric-kinetic model for the effect of non-Poisson distribution of lethal lesions on the variation of RBE with LET”, Radiat. Res. (2003), vol. 160, pp. 61-69 [doi:10.1667/RR3010]).

[0031] An LEM by Krämer and Scholz (see Krämer et al., “Rapid calculation of biological effects in ion radiotherapy”, Phys. Med. Biol. (2006), vol. 51, pp. 1959-1970 [doi:10.1088/0031-9155/51/8/001]) quantifies the biological effect of dose D(T,E) as

[00003]$-\text{ln}S=\left\{\begin{array}{ll}\left(\beta \left(T,E\right)D\left(T,E\right)+\alpha \left(T,E\right)\right)D\left(T,E\right),\hfill & D\left(T,E\right)\le D_{cut}\hfill \\ \left(\beta \left(T,E\right)D_{cut}+\alpha \left(T,E\right)\right)D_{cut}+\left(D\left(T,E\right)-D_{cut}\right)s_{max},\hfill & D\left(T,E\right)>D_{cut}\hfill \end{array}\right)$

where parameters α(T,E), β(T,E), D.sub.cut and s.sub.max are independent of the macroscopic dose. Hence, the parameters can be applied without modification to any treatment plan. The biological effect - In S can be converted into equivalent dose D.sub.bio using the following relation:

[00004]$D_{bio}=\left\{\begin{array}{ll}\sqrt{-\text{ln}S/{\beta }_X+{\left({\alpha }_X/2{\beta }_X\right)}^2}-\left({\alpha }_X/2{\beta }_X\right)\hfill & -\text{}\text{ln}S\le -\text{ln}{S}_{cut}\hfill \\ \left(-\text{ln}S+\text{ln}{S}_{cut}\right)/s_{max}+D_{cut},\hfill & -\text{}\text{ln}S>-\text{ln}{S}_{cut}\hfill \end{array}\right)$

In this model, it is notable that the expression β(T,E)D(T,E) + α(T,E) includes one multiplier α, which is constant with respect to dose, and one multiplier β, which varies linearly with the dose. Accordingly, the part which is proportional to α will cause the biological effect to depend on the first power of the physical dose (p = 1, with the notation introduced above), while the part proportional to β will provide a quadratic dependence on the physical dose (p = 2). The quantity p will be referred to herein as the characteristic power of the biological effect multiplier. The present disclosure does not disclaim the special case without a modeled cutoff behavior, i.e., notionally setting D.sub.cut = ∞.

[0032] The biological effect according to the MKM may be expressed as follows:

[00005]$-\text{ln}S=\left({\alpha }_0\left(T,E\right)+\beta z_{1D}^{\ast }\left(T,E\right)\right)D\left(T,E\right)+\beta {\left(D\left(T,E\right)\right)}^2,$

The characteristic power is p = 1 for both α.sub.0 (T, E) and

[00006]$z_{1D}^{*}\left(T,E\right).$

Since β is constant with respect to different radiation types according to a current version of MKM, it may be applied directly to the total dose.

[0033] Further, the RBE factors may be in accordance with one or more phenomenologically based parameterizations of a linear energy transfer (LET) model, such as: [0034] a Carabe model (see for example Carabe-Fernandez et al., “The incorporation of the concept of minimum RBE (RBEmin) into the linear-quadratic model and the potential for improved radiobiological analysis of high-LET treatments”, Int. J. Radiat. Biol. (2007), vol. 83, pp. 27-39 [doi:10.1080/09553000601087176]), [0035] a Chen & Ahmad model (see for example Chen et al., “Empirical model estimation of relative biological effectiveness for proton beam therapy”, Radiat. Prot. Dosim. (2012), vol. 149, pp. 116-123 [doi:10.1093/rpd/ncr218]), [0036] a McNamara model (see for example McNamara et al., “A phenomenological relative biological effectiveness (RBE) model for proton therapy based on all published in vitro cell survival data”, Phys. Med. Biol. (2015), vol. 60, pp. 8399-8416 [doi:10.1088/0031-9155/60/21/8399]), [0037] a Wedenberg model (see for example Wedenberg et al., “A model for the relative biological effectiveness of protons: The tissue specific parameter α/β of photons is a predictor for the sensitivity to LET changes”, Acta Oncologica (2013), vol. 52, pp. 580-588 [doi:10.3109/0284186X.2012.705892]). In this disclosure, a named RBE model includes not only the cited disclosure by the named author but also further developments by same or other authors, as well as quantitative and qualitative variations of the disclosed model.

[0038] A still further option is to use external software which inputs a dose of specified particle type T and particle energy E and outputs a value of a biological effect multiplier, an RBE factor, an equivalent dose or a biological effect. The software may be provided as source code which is caused to be executed by the method 300. Alternatively, repeated calls to a local software library are made during execution of the method 300. Further alternatively, and especially if low latency can be ensured, calls are made to a web application programming interface (API). The software is external in the sense of being opaque to the treatment planner, i.e., it returns an output (biological effect) for every admissible input (physical dose) but the treatment planner need not be aware of the RBE model that it implements or other considerations underlying the software.

[0039] The method 300 may be implemented in a treatment planning system 400 of the type illustrated in FIG. 4. The treatment planning system may include an interface 410, a memory 420 and processing circuitry 430. The interface 410 is configured to receive, via a text or graphical user interface, a script or by data transfer, one or more non-photon radiation contributions D.sup.(i)(T,E), 1 ≤ i ≤ N, dynamic assignments Π of the combination in terms of non-negative coefficients k.sub.1, k.sub.2, ..., k.sub.N ≥ 0 and possible further data. The interface 410 may connect the treatment planning system 400 to a data network (not shown), so as to enable communication with users, clinicians, researchers, treatment planning personnel, radiation delivery systems etc. The memory 420 may be configured to store a computer program 421 with instructions for causing the treatment planning system 400 to carry out the method 300. The processing circuitry 430 may execute the instructions of the computer program 421, in particular to carry out the steps to be described in the next paragraphs.

[0040] In a first step 310 of the method 300, one or more non-photon radiation contributions D.sup.(i)(T, E), 1 ≤ i ≤ N, are obtained. An i.sup.th one of the contributions may be represented as a list, table or matrix, which provides a value of the dose D.sup.(i)(T,E) at a location x for a pair of a particle type T and particle energy E. The location x may refer to a point, voxel or other region and will be implicit in the notation herein. The representation of the dose may be discrete or continuous with respect to the particle energy E. Each contribution may correspond to a beam or spot to be delivered in radiation therapy. Alternatively, each contribution may correspond to a preliminary treatment plan, such as a base plan or Pareto-optimal plan. It may not be explicit from a particular treatment plan how large physical dose will be absorbed in a particular volume of the patient when the treatment plan is carried out. If the treatment plan is not expressed in terms of physical dose, but rather in terms of, say, machine-level instructions, relatively complex computations may be required to determine or estimate the physical dose.

[0041] In an optional second step 312, a total dose

[00007]$D_{tot}^{\left(i\right)}={\Sigma }_{T,E}{D}^{\left(i\right)}\left(T,E\right)$

of each contribution is stored for later use in the method 300. The notation .Math..sub.T,E ... is shorthand for summing over all (T, E) pairs for which the dose D.sup.(i)(T,E) > 0. If step 312 is not performed separately, the total dose of the contribution can be computed at a later stage.

[0042] In a likewise optional third step 314, a per-contribution dose-weighted average at least one biological effect multiplier is computed. The computation may be in accordance with the following equation:

[00008]${\left({\overline{\delta }}^{\left(i\right)}\right)}^{1/p}=\frac{{\displaystyle {.Math.}_{T,E}{\left(\delta \left(T,E\right)\right)}^{1/p}{D}^{\left(i\right)}\left(T,E\right)}}{D_{tot}^{\left(i\right)}},1\le i\le N,$

where p is the characteristic power of the biological effect multiplier and

[00009]$D_{tot}^{\left(i\right)}={\displaystyle \underset{T,E}{.Math.}{D}^{\left(i\right)}}\left(T,E\right)$

is a total dose of the i.sup.th contribution. The above expression can be classified as a power mean with exponent p. For the Krämer & Scholz model discussed above, and bearing in mind the respective characteristic powers of the multipliers, this step 314 would include computing:

[00010]${\overline{\alpha }}^{\left(i\right)}=\frac{{\displaystyle {.Math.}_{T,E}\alpha \left(T,E\right)D^{\left(i\right)}\left(T,E\right)}}{D_{tot}^{\left(i\right)}},1\le i\le N,$

[00011]${\overline{\beta }}^{\left(i\right)}={\left(\frac{{\displaystyle {.Math.}_{T,E}\sqrt{\beta \left(T,E\right)}{D}^{\left(i\right)}\left(T,E\right)}}{D_{tot}^{\left(i\right)}}\right)}^2,1\le i\le N.$

Since this operation may be at least partly performed by a different entity, e.g. by having δ(T, E) or δ(T, E).sup.1/pD.sup.(i)(T,E) computed by external software in the manner explained above, step 314 is optional in the method 300.

[0043] In a next step 316, the per-contribution dose-weighted averages δ.sup.(i), 1 ≤ i ≤ N, of the at least one biological effect multiplier with respect to each of the N contributions are stored. The storing may for example be performed in the memory 420 of the treatment planning system 400.

[0044] In a subsequent step 318 of the method 300, an assignment Π of the combination of the N contributions is obtained. The assignment Π may be in terms of non-negative coefficients k.sub.1, k.sub.2, ..., k.sub.N ≥ 0 to be applied to the N contributions. The assignment Π may be obtained by setting the interface 410 in a mode where it is ready to accept input of the coefficients k.sub.1, k.sub.2, ..., k.sub.N from a user or another processor. Alternatively, the assignment Π may be obtained by polling a memory space where they are to be found. In FIG. 3, the block 318 has one right-hand exit, representing no input of coefficients, in which case the execution of the method 300 loops back. The downward exit from block 318, representing that coefficients have been obtained, continues into the subsequent step 320 of determining a biological effect of the combination according to this assignment II, denoted - In S.sup.Π. After completion of step 320, the execution of the method 300 may loop back to block 318 to receive a new assignment Π′.

[0045] It is noted that a first set of coefficients k.sub.1, k.sub.2, ..., k.sub.N may sum to one,

[00012]${\displaystyle {.Math.}_{i=1}^N{k}_i}=1,$

and thereby define a convex combination

[00013]$.Math.={\displaystyle {.Math.}_{i=1}^N{k}_i{P}_i}$

of base plans P.sub.1, P.sub.2, ..., P.sub.N. The convex combination may be referred to as a navigated plan Π. A user may select the coefficients and inspect the resulting properties of the navigated plan using a navigation interface of the type described in the applicant’s disclosure EP3581241A1. The navigation interface may include display means for displaying a list of clinical goals and an associated value range for each clinical goal, and a user input means enabling a user to input navigation weights. For each clinical goal, there is also preferably an indicator of whether the clinical goal is fulfilled. A set of altered coefficients

[00014]$k_1^{\prime },k_2^{\prime },\mathrm{...},k_N^{\prime }$

may be obtained as a result of the user’s continued navigation. The present way of computing the biological effect allows the user to receive responsive feedback with minimal latency when biological effect is one of the clinical goals.

[0046] Step 320 more precisely includes computing a combined dose-weighted average δ.sup.Π of said at least one biological effect multiplier on the basis of the stored per-contribution dose-weighted averages δ.sup.(i). The combined dose-weighted average is given by

[00015]$\begin{array}{l}{\left({\overline{\delta }}^{.Math.}\right)}^{1/p}=\frac{1}{D_{.Math.,tot}}{\displaystyle \underset{T,E}{.Math.}{\left(\delta \left(T,E\right)\right)}^{1/p}{D}_{.Math.}\left(T,E\right)}=\\ \frac{1}{D_{.Math.,tot}}{\displaystyle \underset{T,E}{.Math.}{\displaystyle \underset{i=1}{\overset{N}{.Math.}}{\left(\delta \left(T,E\right)\right)}^{1/p}{k}_i{D}^{\left(i\right)}\left(T,E\right)}}\\ =\frac{1}{D_{.Math.,tot}}{\displaystyle \underset{i=1}{\overset{N}{.Math.}}{k}_i}{\displaystyle \underset{T,E}{.Math.}{\left(\delta \left(T,E\right)\right)}^{1/p}{D}^{\left(i\right)}}\left(T,E\right)=\\ \frac{1}{D_{.Math.,tot}}{\displaystyle \underset{i=1}{\overset{N}{.Math.}}{k}_i{\left({\overline{\delta }}^{\left(i\right)}\right)}^{1/p}{D}_{tot}^{\left(i\right)}.}\end{array}$

If the RBE model comprises no other biological effect multiplier than δ(T, E), the biological effect is given as the product of δ.sup.Π and the p.sup.th power of the total dose D.sub.Π,tot of the combination Π, as follows:

[00016]$-\text{ln}{S}^{.Math.}={\overline{\delta }}^{.Math.}{D}_{.Math.,tot}^2={\left({\displaystyle \underset{i=1}{\overset{N}{.Math.}}{k}_i{\left({\overline{\delta }}^{\left(i\right)}\right)}^{1/p}{D}_{tot}^{\left(i\right)}}\right)}^p.$

See Zaider and Rossi, “The synergistic effects of different radiations”, Radiat. Res. (1980), vol. 83, pp. 732-739 [doi:10.2307/3575352]. As the inventors have realized, the per-contribution dose-weighted averages δ.sup.(i) and the total per-contribution doses

[00017]$D_{tot}^{\left(i\right)}$

are independent of the coefficients k.sub.1, k.sub.2, ..., k.sub.N. Therefore, when a new assignment Π′ of the combination is obtained in an iteration of step 318 (e.g., by obtaining new coefficient values

[00018]$k_1^{\prime },k_2^{\prime },\mathrm{...}{k}_N^{\prime }),$

its biological effect - In S.sup.Π′ can be computed by substituting

[00019]$k_i\mapsto {k^{\prime }}_i$

in the above expression. There is no need to recompute δ.sup.(i) or

[00020]$D_{tot}^{\left(i\right)}.$

The biological effect - In S.sup.Π or new biological effect - In S.sup.Π′ may be used to support radiation treatment planning.

[0047] For an RBE model with two or more biological effect multipliers, the equivalent-dose contributions are summed. In the particular case of the Krämer & Scholtz model discussed above, such summing yields:

[00021]$\begin{array}{l}-\text{ln}{S}^{.Math.}={\displaystyle \underset{T,E}{.Math.}\left(\beta \left(T,E\right)D{\left(T,E\right)}^2+\alpha \left(T,E\right)D\left(T,E\right)\right)=}\\ {\overline{\beta }}_{.Math.}{D}_{.Math.,tot}^2+{\overline{\alpha }}_{.Math.}{D}_{.Math.,tot}\\ ={\left({\displaystyle \underset{i=1}{\overset{N}{.Math.}}{k}_i\sqrt{{\overline{\beta }}^{\left(i\right)}}}{D}_{tot}^{\left(i\right)}\right)}^2+{\displaystyle \underset{i=1}{\overset{N}{.Math.}}{k}_i{\overline{\alpha }}^{\left(i\right)}}{D}_{tot}^{\left(i\right)}.\end{array}$

[0048] As seen above, the combination provides, for each assignment Π, a mixed radiation field whose total biological effect is obtained by summing the contributions to –ln S.sup.Π over all (T, E) pairs for which there is a non-zero dose. The calculations are structured in the manner presented above to enable a computationally efficient refresh when the coefficients k.sub.1, k.sub.2, ..., k.sub.N are altered.

[0049] The computational structure is visualized in FIG. 1, where input quantities

[00022]$\delta \left(T,E\right),{\left(D^{\left(i\right)}\left(T,E\right)\right)}_1^N\text{and}{\left(k_i\right)}_1^N$

and are shown at the left-hand size. The notation shall be understood in the sense that

[00023]${\left(f\left(i\right)\right)}_1^N=\left\{f\left(1\right),f\left(2\right),.Math.,f\left(N\right)\right\}.$

The intermediate quantities δ.sup.(i) and

[00024]$D_{tot}^{\left(i\right)},$

which can be calculated without knowledge of

[00025]${\left(k_i\right)}_1^N$

are shown outside the area 101. The intermediate quantity

[00026]$D_{.Math.,tot}={\displaystyle {.Math.}_{i=1}^N{k}_i{D}_{tot}^{\left(i\right)}}$

is shown inside this area 101. The total biological effect - In S.sup.Π is calculated by a function illustrated by the block 102 on the basis of the three intermediate quantities and the coefficients

[00027]${\left(k_i\right)}_1^N.$

[0050] FIG. 2 shows the inner workings of the block 102, which reveals that the total biological effect - In S.sup.Π is calculated in two steps. Initially the combined dose-weighted average δ.sup.Π of each biological effect multiplier is computed. Then, the combined dose-weighted average δ.sup.Π is multiplied by the p.sup.th power of the total dose D.sub.Π,tot and output as - In S.sup.Π.

[0051] An advantage of embodiments of disclosed herein is that only the computations inside the area 101, which represent a relatively limited effort, need to be repeated when a new set of coefficients

[00028]${k^{\prime }}_1,{k^{\prime }}_2,.Math.,{k^{\prime }}_N$

is received.

[0052] The computational structure of FIG. 1 may provide benefit also in the case of a scaling of a treatment plan. The treatment plan to be scaled can then be regarded as a base plan P.sub.1 constituting the sole contribution N = 1, and its coefficient k.sub.1 will be the scaling factor. Accordingly, the total dose is

[00029]$D_{.Math.,tot}=k_1{D}_{tot}^{\left(1\right)},\text{so that}{\overline{\alpha }}_{.Math.}={\overline{\alpha }}^{\left(1\right)}$

and β.sub.Π = β.sup.(1). The biological effect is given by:

[00030]$\begin{array}{l}-\text{ln}S={\displaystyle \underset{T,E}{.Math.}\left(\beta \left(T,E\right)D{\left(T,E\right)}^2+\alpha \left(T,E\right)D\left(T,E\right)\right)}={\overline{\beta }}_{.Math.}{D}_{.Math.,tot}^2\\ +{\overline{\alpha }}_{.Math.}{D}_{.Math.,tot}\\ ={\overline{\beta }}^{\left(1\right)}{D}_{.Math.,tot}^2+{\overline{\alpha }}^{\left(1\right)}{D}_{.Math.,tot},\end{array}$

where α.sup.(1), β.sup.(1), the dose-weighted averages with respect to the base plan P.sub.1, can be precalculated. The biological effect is a second-order polynomial in k.sub.1.

[0053] The aspects of the present disclosure have mainly been described above with reference to a few embodiments. However, as is readily appreciated by a person skilled in the art, other embodiments than the ones disclosed above are equally possible within the scope of the invention, as defined by the appended patent claims.