irtpre
As mentioned in Introduction, in radiation therapy the dose absorbed from particle field ψ= (ψ1, ψ2, ψ3) is defined by
D(x) = (Dψ)(x) :=
3 j=1
S×Iςj(x, E)ψj(x, ω, E)dωdE, where ςj ∈L∞(G×I), ςj ≥0 are the so-called (total) stopping powers.
Also, it is worth recalling that the component fields of ψ, relevant to photon and electron radiation therapy, are ψ1= photons,ψ2= electrons andψ3= positrons. Clearly, nothing we have said above depends on the number, or designation (to certain particle species) of fields treated (or even the dimensionality of the spaces G, S or I in fact), with the exception that typically only charged particle fields (are assumed to) obey CSDA version of the transport equation (cf. (5)), while non-charged particles obey the standardintro5-2 linear BTE (cf. (intro5-14)). Thus with very minor modifications, and in particular if one is interested in radiation therapy, what will and has been said works, in principle, equally well in proton (and ion) therapy framework as well.
We find thatD:L2(G×S×I)3→L2(G) is a bounded linear operator and its adjoint operatorD∗:L2(G)→L2(G×S×I)3 is simply a multiplication type operator,
D∗d= (ς1, ς2, ς3)d, ford∈L2(G).
We describe shortly an optimization problem related to inverse radiation treatment plan-ning. We restrict ourselves toexternal radiation therapyin which the particles are inflowing through the patch(es) of patient surface. This means that in the transport problemf= 0 (i.e. the internal particle source vanishes) andg (the inflow particle flux) is the variable to be controlled. Conversely, for the internal radiation therapy problems one setsg= 0, and f would be the variable to be controlled. We refer to[29] (section 8) and to the referencestervo17-up therein for a more detail exposition ofinverse problem (optimization) in this setting.
Let g ∈T2(Γ−)3 and let ψ=ψ(g)∈ HP(G×S×I◦) be the solution of the problem (74)–(csda1a77) guaranteed by Theoremcsda3 4.3. The deposited dose is thenm-d-j-co1
D(x) = (D(ψ(g)))(x), x∈G.
We shall also denoteD(g) :=D(ψ(g)).
Denote the target region by T ⊂ G, the critical organ region by C ⊂ G and the normal tissue region by N ⊂ G. Then G = T∪C∪N where the union is mutually disjoint. Suppose thatdT∈L2(T),dC∈L2(C),dN∈L2(N) are given dose distributions in the respective regions (for example, they may be constants). We define a strictly convex object (cost) function J:X→Rby (see[29])tervo17-up
J(g) = cT
2 dT− D(g)2L2(T)+cC
2 dC− D(g)2L2(C)
+cN
2 dN− D(g)2L2(N)+ c 2g2X,
for some constants (weights)cT, cC, cN, c >0, whereX :=T2(Γ−)×H1(I, T2(Γ−))2and it is equipped with the inner product
g, hX :=g1, h1T2(Γ−)+ 3 j=2
I
∂gj
∂E,∂hj
∂E
T2(Γ−)
dE.
Let Y be a closed subspace ofX defined by (here we denoteg(y, ω, E) := (g(E))(y, ω)) Y :={g∈X |gj(·,·, Em) = 0, j = 2,3}.
A relevantadmissible set (of controls) is
Uad={g∈Y | g≥0 a.e.onG×S×I}, which is a closed convex subset ofX.
Suppose that the assumptions of Theorem 4.3 are valid. Furthermore, suppose thatm-d-j-co1 ςj j= 1,2,3 are regular enough. Then the minimum ming∈UadJ(g) exists and the optimal controlgis obtained from the system of variational equations (cf. [29, the proof of Theoremtervo17-up 8.7])
− γ−(ψ∗), wT2(Γ−)3+cg, wX≥0 ∀w∈Uad, (93) irtp16
− γ−(ψ∗), gT2(Γ−)3+cg2X = 0, (94) irtp16a B˜0(ψ, v) =g, vT2(Γ−)3 ∀v∈H˜, (95) irtp18 B˜∗0(ψ∗, v) +cTDψ, DvL2(T)+cCDψ, DvL2(C)+cNDψ, DvL2(N)
=cTdT, DvL2(T)+cCdC, DvL2(C)+cNdN, DvL2(N) ∀v∈H˜. (96) irtp19 Here ˜B0(·,·) is the bilinear form corresponding to the problem (74)–(csda1a77) and ˜csda3 B∗0(·,·) is
the bilinear form corresponding to the adjoint problemof (74)–(csda1a77); seecsda3 [28] sections 3.2,tervo16-up 5, and 7. Likewise, the Hilbert space ˜H is defined in Eq. (371) in tervo16-up[28]. We notice that if we used Z := {(g1,0,0)| g1 ∈ T2(Γ−)} as a control space instead of Y, the relations (93), (irtp16irtp16a94) imply (as in[29], Theorem 8.7) thattervo17-up g1= 1c(γ−(ψ∗))+. Here (h)+ denotes the positive part of a functionh.
Omitting details we mention that equivalently, this variational system (irtp1895), (96) canirtp19 be written as a coupled system
(P∗(x, ω, E, D) + Σ∗−K∗)ψ∗+cTD∗eTDψ+cCD∗eCDψ+cND∗eNDψ
=cTD∗eTdT+cCD∗eCdC+cND∗eNdN, (P(x, ω, E, D) + Σ−K)ψ= 0
ψ|Γ− =g, ψj(·,·, Em) = 0, j= 2,3 ψ∗|Γ+ = 0, ψ∗j(·,·,0) = 0, j= 2,3,
where, moreover,g∈Uadsatisfies (93), (irtp1694). The operatorirtp16a P(x, ω, E, D) isPC(x, ω, E, D) whenC = 0 (see (eq:def_P_C87)), and the operatorP∗(x, ω, E, D) is defined by (seetervo16-up[28], section 5)
P1∗(x, ω, E, D)ψ∗1:= −ω· ∇xψ1∗, Pj∗(x, ω, E, D)ψ∗j :=Sj
∂ψj∗
∂E −ω· ∇xψj∗, j= 2,3, P∗(x, ω, E, D)ψ∗:=
P1∗(x, ω, E, D)ψ1∗, P2∗(x, ω, E, D)ψ2∗, P3∗(x, ω, E, D)ψ∗3 . Finally, Σ∗ = Σ and K∗ is the adjoint on K. In addition, for any subset A of G and function hdefined onA, we wroteeAh:G→Rfor the extension by zero of hontoG.
We emphasize that here the described solutiong of the optimal control problem can be used only as aninitial point for the actual treatment planning. The more realistic object function is given in[29], section 8 and it has been found to be very multi-extremal. Hencetervo17-up the actual optimization requiresglobal optimization (see e.g. [23]). The determination ofpinter a carefully chosen initial point for a large dimensional global optimization scheme is very essential for achieving (time savings and) satisfactory results.
We also notice that if we contented ourselves with so-called mild solutions ([22], p.pazy83 146), then the existence of an optimal control g (the admissible set being a subset of T2(Γ−)3), together with the explicit formulag= 1c(γ−(ψ∗))+for it, could be proven under quite weak assumptions. In any case, the validity of estimates such as (diss-co-es92) is essential for guaranteeing that D be a bounded linear operator in appropriate spaces. The use of mild solutions, however, has the drawback that the inflow boundary conditions are not necessarily satisfied by the solutions (and thus the solutions might be non-physical).
We remark that, for example with respect to x-variable the solution of the transport problem is generally at most inH2,(s,0,0)(G×S×I◦) withs < 32whereH2,(s,0,0)(G×S×I◦) is the mixed-norm Sobolev-Slobodeckij space with the Lebesgue index 2 and with the fractional indexs(with respect tox-variable). This must be kept in mind when solving the problem e.g. by the finite element methods (FEM). In[28] we derived the correspondingtervo16-up variational formulation of the problem together with the pertinent estimates for the due bilinear form. These estimates imply (by the Cea’s estimate) that the FEM-scheme in principle convergences (in relevant spaces), when one uses appropriate basis functions.
We omit in this paper further discussion of the inverse radiation treatment problem which was outlined above and refer to e.g. frank10[11] for related treatments.
6 Discussion
In this paper we studied a linear Boltzmann transport equations primarily intended for dose calculation. It has been reported in literature that the non-linear effects are negligible in dose calculation for the radiation therapy needs. The coupled system consisted of one original transport equation and of two approximations of transport equations, so-called Continuous Slowing Down Approximations (CSDA). The CSDA-type modelling is neces-sary due to hyper-singularities in the differential cross-sections related to certaincharged particle interactions. These singularities produce pseudo-differential-like terms. Under certain relevant assumptions these terms can be, at least formally, approximated with terms containing pure partial derivatives with respect to energy. These approximations are essentially founded on Taylor’s formulas and the mono-directionality of (new) primary particles. In the dose calculation the scattering events containing hyper–singularities are primary Møller scattering and Bremstrahlung.
We gave a systematic study of existence and uniqueness of solutions (that is, well-posedness of the problem) in appropriateL2-based spaces. The well-posedness can be con-sidered among others applying the Lions-Lax-Milgram Theorem (variational approach),
the theory ofm-dissipative operators and the theory of non-autonomous evolution opera-tors (semigroup approach) ([28]). Our analysis based on thetervo16-up m-dissipativity and partially on semigroup approach. The main technique consisted of that the partial differential part of the operator was m-dissipative, and the partial integral operator part was considered as a dissipative perturbation. The pertinent a priori estimates have been derived as well.
The existence theory containing exact hyper-singular partial integral-type operators (or alternatively pseudo-differential-like operators) remains open.
We remark that the initial inflow boundary value problems related to transport prob-lems are so calledvariable multiplicitythat is, the dimension of the kernel of the boundary operator is not constant (e.g. nishitani98
[21]). Generally, these problems require additional assump-tions concerning for the ”transition with a non-zero derivative” over the manifold Γ0. It is known that for the general first order partial differential (initial) boundary value problems, the mentioned transition assumption is needed even to guarantee the unique existence of solutions, that is, to guarantee that the problem is well-posed. For transport problems, such as the one considered in this paper, the transition assumption is not necessarily required for well-posedness as we have verified in this paper.
In the last section 5 outlined an inverse problem occurring in radiation therapy. Itsirtpre refined analysis and numerical solving remain open. In addition, the analysis for Sobolev regularity of solutions (needed e.g. in various approximation and convergence treatments) remains to our knowledge open. It is known that the transport problems have a lim-ited Sobolev regularity but higher order weighted (co-normal) Sobolev regularity can be achieved.
Acknowledgments
This work has been supported by DFG HE5386/13-15,1 and DAAD MIUR 2016–2017. The authors like to thank an anonymous referee for suggestions to improve the manuscript.
References
asadzadeh [1] Asadzadeh M. and T. Geb¨ack. Analytical solutions for the pencil beam equation with energy loss and straggling. Preprint 2011:23, Univ. of Gothenburg.
boman [2] Boman, E.,Radiotherapy Forward and Inverse Problem Applying Boltzmann Transport Equation, Doctoral Dissertation, University of Kuopio, 2007.
borgers [3] B¨orgers C. and E.W. Larsen.On the accuracy of the Fokker-Planck and Fermi pencil beam equations for charged particle transport. Med. Phys. 23 (10): 1749-1759, 1996 borgers99 [4] B¨orgers, C.Radiation Therapy Planning Problem, Computational Radiology and
Imag-ing Therapy and Diagnostic, The IMA Volumes in Mathematics and Applications, Volume 110, pp. 1-16, 1999
dautraylionsv6 [5] Dautray, R., Lions, J-L. Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 6. Evolution Problems II, Springer, 1999.
egger2014 [6] Egger, H. and M. Schlottbom. An Lp theory for stationary radiative transfer. Appl.
Anal. 93 (6), pp. 1283-1296, 2014.
schlottbom2014 [7] Egger, H. and Schlottbom, M.,Stationary radiative transfer with vanishing absorption, Math. Mod. Meth. Appl. Sci., 24 (2014) 973–990.
engelnagel [8] Engel, K.-J., Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000.
nselKlar2007aa [9] M. Frank, H. Hensel, and A. Klar,A fast and accurate moment method for dose calculation in electron radiotherapy, SIAM J. Appl. Math., 67 (2007), pp. 582–603.
frank08 [10] Frank, M., M. Herty, M. Sch¨afer. Optimal Treatment planning in radiotherapy based on Boltzmann transport calculations, Math. Mod. Meth. Appl. Sci. 18 (4), 2008, pp.
573-592
frank10 [11] Frank, M., M. Herty and A.N. Sandjo. Optimal radiotherapy treatment planning governed by kinetic equations, Math. Mod. Meth. Appl. Sci. 20 (4), 2010, pp. 661-678 friedman [12] Friedman A.Partial Differential Equations, Robert, E. Krieger Publishing Co., 1976 friedrich44 [13] Friedrichs, K. O.,The identity of weak and strong extensions of differential operators,
Trans. Amer. Math. Soc. 55, pp. 132-151, 1944.
friedrichs [14] Friedrichs, K. O., Symmetric Positive Definite Systems of Differential Operators, Comm. Pure. Math. 7, 1954.
halmos [15] P.R. Halmos and V.S. Sunder. Bounded Integral Operators on L2 Spaces, Springer 1978
anSiedow2006aa [16] H. Hensel, R. Iza-Teran, and N. Siedow,Deterministic model for dose calcula-tion in photon radiotherapy, Phys. Med. Biol., 51 (2006), pp. 675–693.
hsiao [17] Hsiao, G.C. and Wendland, W.L.Boundary Integral Equations, Springer, 2008.
lee [18] Lee, J.,Introduction to Smooth Manifolds, Springer, 2002.
Lewis-Miller [19] Lewis, E.E., Miller Jr., W.F. Computational Methods in Neutron Transport, John Wiley and Sons, 1984.
mayles07 [20] Mayles, P., Nahum A., Rosenwald, J. C. Handbook of Radiotherapy Physics: Theory and Practice, CRC Press, 2007
nishitani98 [21] T. Nishitani and M. Takayama. Characteristic initial boundary value problems for symmetric hyperbolic systems. Osaka J. Math. 35, pp. 629-657, 1998
pazy83 [22] Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983.
pinter [23] Pinter, J.D.Global Optimization in Action, Kluwer, 1996
rauch85 [24] Rauch, J. Symmetric positive systems with boundary characteristic of constant mul-tiplicity. Trans. Amer. Math. Soc. 291 (1), pp. 167-187
schepard [25] Schepard, D.M., M.C. Ferris, G.H. Olivera and T.R. Mackie.Optimization the delivery of radiation therapy to cancer patients. SIAM Reviews 41, 1999, pp. 721-744
tanabe [26] Tanabe, H. Equations of Evolution, Pitman, 1979
tervo07 [27] Tervo, J., On coupled Boltzmann transport equation related to radiation therapy, J.
Math. Anal. Appl. 335:2, 819-840, 2007.
tervo16-up [28] Tervo, J., Kokkonen, P., Frank, M., Herty, M.On Existence ofL2-solutions of Coupled Boltzmann Continuous Slowing Down Transport Equation System, arXiv:1603.05534v2 [ math.OC ] 13 Dec 2016
tervo17-up [29] Tervo, J. and Kokkonen, P.On existence ofL1-solutions for coupled Boltzmann trans-port equation and radiation therapy treatment planning, ArXiv: 1406.3228v1 [ math.OC ] 12 Jun 2014
tillikainen08 [30] Tillikainen, L., Helminen, H., Torsti, T., Siljam¨aki, S., Alakuijala, J., Pyyry, J., Ulmer, W.A 3D pencil-beam-based superposition algorithm for photon dose calculation in heterogeneous media, Phys Med Biol. 2008 Jul 21;53(14):3821-39
treves [31] Treves, F.Basic Linear Partial Differential Equations, Academic Press, 1975 ulmer [32] Ulmer, W., Pyyry, J. and Kaissl, W. A 3D-photon superposition/convolution
algo-rithm and its foundation on results of Monte Carlo calculations, Phys. Med. Biol. 50, pp. 1767-1790, 2005
vassiliev [33] Vassiliev, O.N., Wareing, T.A., McGhee, J., Failla, G., Salehpour, M.R. and Mour-tada, F.Validation of a new grid-based Boltzmann equation solver for dose calculation in radiotherapy with photon beams, Phys. Med. Biol. 55 (2010) 581-598.
webb [34] Webb, S. The Physics of Three-Dimensional Radiation Therapy, IOP Publishing, 1993