Parameter Optimization for FEM based modeling of singlet oxygen during PDT using COMSOL

Transcription

1 Excerpt from the Proceedings of the COMSOL Conference 00 Boston Prmeter Optimiztion for FEM bsed modeling of singlet oxygen during PDT using COMSOL Xing Ling, Ken Kng-Hsin Wng, nd Timothy C. Zhu * Deprtment of Rdition Oncology, School of Medicine, University of Pennsylvni, Phildelphi, PA 904 *Corresponding uthor: Rdition Oncology, University of Pennsylvni, 400 Spruce Street/ Donner Bldg, Phildelphi, PA 904, tzhu@mil.med.upenn.edu Abstrct: Singlet oxygen ( O ) is the mjor cytotoxic gent in photodynmic therpy (PDT). The rection between O nd tumor cells defines the tretment efficcy. Bsed on previously developed model tht incorportes the diffusion eqution for the light trnsport in tissue nd the mcroscopic kinetic equtions for the genertion of the singlet oxygen, the distnce-dependent rected O ws numericlly clculted using finite-element method (FEM) on COMSOL. The formul of rected O concentrtion involves 5 photo-physiologicl prmeters which need to be determined explicitly to predict the genertion of O. We developed n lgorithm to itertively clculte the singlet oxygen concentrtion profile nd compred with the mesurements to obtin n optiml set of prmeters. The optimiztion is performed using Mtlb nd dynmiclly linked with the COMSOL for the forwrd clcultion. The optimized prmeters were then used to predict the trnsient pprent rected O concentrtion s PDT dosimetry quntity on D prostte model in COMSOL. Keywords: Singlet oxygen, PDT dosimetry, mcroscopic model, microscopic model, oxygen consumption, dynmic mechnism. mjor deficiency of current PDT technology is the lck of ccurte dosimetry to ssess PDT efficcy. We hve been focusing on developing n explicit PDT dosimetry model using pprent rected O concentrtion, [ O ] rx, s the clinicl PDT dosimetry quntity [-]. Finite element method (FEM) ws used to model the complex geometry in humn ntomy. In this pper, new optimiztion lgorithm ws implemented bsed on the previous model, in which PDT sptil-dependent light distribution ws modeled in COMSOL, nd time-dependent [ O ] rx ws modeled in Mtlb dynmiclly linked with COMSOL. Once the sptil light distribution ws determined, the optimiztion lgorithm compred the model with experimentl necrosis rdius from mice studies, nd determined the optiml photophysiologicl prmeters. PDT dosimetry prediction using [ O ] rx ws then obtined using these prmeters, nd bsed on prostte ultrsound imges. The flow chrt of this work is shown Fig., in which the red rrows represent processes implemented by COMSOL, while blue rrows represent processes implemented by MATLAB dynmiclly linked with COMSOL.. Introduction Photodynmic therpy (PDT) is n importnt tretment modlity for cncer nd other loclized diseses. During the tretment, photosensitizers excited by light rect with ground stte oxygen O, which leds to genertion of the mjor cytotoxic gent - singlet oxygen O - to kill the surrounding tissues nd cells. Compred with other tretments, PDT hs its unique dvntges. First, light cn be delivered only to region of interest, to increse specificity. Second, PDT is non-ionizing modlity thus voiding potentil cumultive rdition effects. Finlly, PDT usully combines fster post opertive recovery nd better cosmetic outcome. However, one Figure. Flow chrt for PDT photophysiologicl prmeter optimiztion nd dosimetry prediction. Red rrows denote COMSOL process, while blue rrows denote Mtlb + COMSOL process.

2 . Method The governing equtions describe the dynmic process tht determines the bsorption of light, φ, the consumption of photosensitizing drug, [S], oxygen [ O ], nd the production of singlet oxygen, [ O ]. For simplicity, ll the clcultions were crried out either in onedimension for sphericl geometry or in twodimension for semi-infinite geometry.. Singlet oxygen model The following re the governing equtions of our mcroscopic singlet oxygen model. ( µ + ε u u u ) = S µ s ' du κ u( u + δ ) u + S γη u dt + α u = du α uu u + S ( 0) γη u g dt + α u u t = 4 α uu u fs γη dt + α u 0 () () () du (4) where u, u, u, nd u 4 represent light fluence rte φ, the concentrtions of sensitizer [S 0 ], oxygen [ O ], nd rected singlet oxygen [ O ], respectively. In the cses we considered here µ, µ s, ε, nd S re the bsorption, reduced scttering coefficients t tretment wvelength, extinction coefficient for photosensitizer t the tretment wvelength, nd source strength in mw/cm for liner source, respectively. α, β, γ, κ, η nd S re the photophysicl prmeters for photosensitizer. The detil of the origin nd definition cn be found in []. g is the mximum oxygen supply rte, nd f is the frction of [ O ] rx efficiently inducing cell deth. In this work, we ssume f equl to one. [ O ] rx is the [ O ] recting with cell trgets, which is the dosimetry quntity fundmentlly determining the tretment efficcy, nd cn be described s = T [ O ] rx [ O ] dt (5) 0 Compred with the model used in our previous works [-4], new prmeter δ is introduced in the current model, which is low photosensitizer concentrtion correction term [5-8]. In most clinicl environments, the sensitizer O concentrtion is low enough tht the molecule is more likely to rect with its prent sensitizer thn would be predicted bsed solely on the species reltive concentrtions. Therefore, the introduction of the δ term improves the theoreticl description of the bleching mechnism in low photosensitizer concentrtion environment. In the optimiztion process, Eq. () is considered s stedy stte eqution, nd being solved independently from the other three time dependent Eqs. (-4). The complex photochemicl prmeters cn be further lumped into independent prmeters, where ξ = S γηα/(+α), σ = κ/α, nd β. The finl modeling equtions cn be described s µ u u = S µ s' du u( u + δ ) u + ξσ u dt u = du uu u + ξ u g ( 0) dt u u t = 4 uu u ξ dt u 0 (6) (7) (8) du (9) In Eq. 6, ε u is dropped, becuse in the study, the sensitizer extinction coefficient ε is set to be (cm - um - ) [9] nd the initil in vivo Photofrin concentrtion is ssumed to be 7 µμ, 4 hr fter 5 mg/kg i.v. injection [4]. The bsorption coefficient for Photofrin is 0.05 cm - which is 8 folder lower thn the bsorption coefficient of tumor tissue such s 0.7 (cm - ) mesured interstitilly (dt not shown). Therefore, the 4 independent prmeters g, ξ, σ, nd β needs to be determined precisely using the optimiztion lgorithm.. Prmeter optimiztion In the optimiztion process, the min progrm ws coded in MATLAB environment which executes clcultion procedure by clling COMSOL model. First, the stedy stte light diffusion eqution (Eq. (6)) for liner source in D cylindricl geometry ws built in COMSOL environment. Due to the symmetricl feture of the cylinder, we re only interested in the light distribution long D rdil xis. This D light distribution profile will be pssed to

3 next step, the clcultion of PDT kinetics eqution (Eqs. (7-9)). Agin, due to the cylindriclly-symmetricl feture, we cn consider the [ O ] rx production in D rdil xis. A fitting quntity [ O ] rx,sd ws introduced in the lgorithm to represent pprent singlet oxygen threshold concentrtion. During the process, our fitting routine vries the 4 independent prmeters globlly so tht the [ O ] rx t the necrotic rdius for ech niml is close to the [ O ] rx,sd. In the current fitting routine, n initil guess of [ O ] rx,sd is ssigned rndomly within the rnge of 0.4 to. mm nd the vlue is fixed throughout the subsequent fitting procedure. The differentil evolution lgorithm developed by Storn et l. [0] nd modified by us re used s the optimiztion routine. It is n efficient directserch lgorithm for nonliner problem. Our gol is to minimize the mximum devition MAX ( [ O ] rx /[ O ] rx,sd ) between dt nd fit. The lgorithm ws implemented in MATLAB using function fminserch, nd certin number of itertions were clculted to give the optimized prmeters. The necrosis distnces from different tretment conditions were obtined from in vivo mice experiments. Tble lists the boundry, initil conditions, nd other necessry prmeters used in this model []. Tble. Boundry, initil conditions, nd prmeters for singlet oxygen model Boundry conditions u i=,,4 u i u (µm) 7 Initil conditions u (µm) 8 u 4 (µm) 0 Light ssocited µ (cm - ) 0.7 prmeters µ s (cm - ) 9.4 Physiologicl δ(µm) prmeter g (mm) 0.7 Photochemicl ξ (cm /mws).7 x 0 - Prmeter (for σ (/µm) 7.6 x 0-5 Photofrin) β (µm).9 The vlues of the initil conditions of u nd u re extrcted from in vivo experiment. The vlues of µ nd µ s re determined by interstitil point source technique [] within RIF mice tumor. The result of this optimiztion fitting is single set of the prmeters g, ξ, σ, nd β.. Rected singlet oxygen concentrtion prediction Once the optiml photophysiologicl prmeters were determined, they were used to clculte the [ O ] rx for ny geometry s PDT dosimetry quntity. COMSOL Multiphysics ws used to model the PDT rected singlet oxygen concentrtion. The light distribution (Eq. 6) ws modeled using PDE mode in COMSOL, while PDT kinetics eqution (Eqs. (7-9)) were modeled using time-dependent PDE mode. The geometry nd opticl properties of the PDT dosimetry utilized the humn prostte reconstruction previously introduced []. Briefly, in COMSOL meshes were generted in the D prostte geometry, reconstructed using trnsrectl ultrsound imges of treted prostte. Opticl properties were ssumed to be homogeneous in the prostte in the clcultion, including µ. cm - nd µ s = 4 cm -. The refrctive indices were ssumed to be.4 nd.0 for the prostte nd the cylindricl surrounding lyer, respectively. liner sources were modeled for tretment, nd intensity of 50 mw/cm ws used for ech source. A -D geometry of the ptient s prostte ws reconstructed nd shown in Fig., s well s the mesh plot of the geometry. Photosensitizer used in the prediction model is Photofrin.. Results nd Discussions. Numericl results simulting in vivo Photofrin-PDT tretment From the optimiztion process described in., n optimized set of prmeters g, ξ, σ, nd β re obtined, s shown in tble. These photochemicl prmeters re lso compred with our previous fitting results nd reported vlues for PDT ppliction of Photofrin t 60 nm from litertures. It is obvious tht the photochemicl oxygen consumption rte per light fluence rte ξ hs lower vlue compred with previous results. The probbility rtio of O molecule recting with ground-stte

4 photosensitizer compred to the O molecule recting with cellulr trget σ hs much lrger vlue. The rtio of the monomoleculr decy rte of the triplet stte photosensitizer to the bimoleculr rte of the triplet photosensitizer quenching by O β remin the sme, while mximum oxygen supply rte g nd pprent singlet oxygen threshold concentrtion [ O ] rx,sd were clculted s 0.8 mm/s nd 0.4mM/s, respectively. At the optiml prmeters, the result of fitting the PDT-induced necrotic rdius with the clculted photo sensitizer concentrtion, triplet oxygen concentrtion nd pprent rected singlet oxygen concentrtion re shown in Fig.. The symbols represent the necrotic rdius for different mice. One cn notice in Fig. tht the pprent rected singlet oxygen concentrtion decrese with the rdius increse, minly due to the light distribution decrese over distnce in tissue. b b Figure. Reconstructed prostte geometry. () Prostte Geometry surfce. (b) Mesh plot of the prostte with surrounding cylindricl lyer. c Tble. Prmeters from optimiztion process Prmeters Finl fit Previous fit [] Published vlues ξ (cm /s/mw) σ (/mm) β (mm) g (mm/s) [ O ] rx,sh (mm) [6] [6] [] _ _ Figure : Results for optimiztion clcultion. () Sensitizer, (b) oxygen, nd (c) rected singlet oxygen concentrtions s functions of necrosis rdius from mice.

5 . Results of PDT tretment simultion using optiml prmeters Using the clculted prmeters in Tble, we re ble to predict PDT quntities in rther complex geometry, which ws described in.. The prediction resutls re shown in Fig. 4. Fig. 4 demonstrtes pprent rected singlet oxygen concentrtion distribution in the prostte nd surrounding cylinder model t 00 s of tretment. Slices in x-y nd y-z plnes re used to better visulize the results. The sme result is lso shown in Fig. 4b, in which the isosurfce of 0.4 mm (pprent singlet oxygen threshold concentrtion) ws plotted for the purpose of visulizing tretment efficcy. Fig. 4c shows the light fluence rte distribution t 00 s of tretment. It is obvious tht from our prediction model, the pprent rected singlet oxygen bove threshold is not covering the whole prostte t 00 s, which mens the tretment is not completed t this time point. However, if one uses light fluence distribution in tretment prediction (s in []), it is not cler how well the tretment hs been pplied. To better understnd the prediction model, top views of pprent rected singlet oxygen concentrtion distribution nd light fluence distribution re shown in Fig. 5, in which Fig. 5 nd d re results t tretment time of 00 s, Fig. 5b nd e re results t tretment time of 500 s, while Fig. 5c nd f re results t tretment time of 000 s. The pprent rected singlet oxygen concentrtion distribution strts from close to the liner sources, nd lmost cover the whole prostte model t 500 s, but still with some gps for tretment. At 000 s of tretment, the whole prostte is treted completely. From this model, the time threshold for complete PDT tretment on the prostte model is ~560 s. This prediction model precisely estimte the clinicl PDT dosimetry quntity [ O ] rx, which my help ssess PDT tretment efficcy ccurtely. On the other hnd, it my be difficult to predict PDT dosimetry from light fluence distribution, s it cn hrdly predict the tretment efficcy over time. 4. Conclusions Finite element method from COMSOL ws pplied to model light fluence rte distribution in PDT, combined with optimiztion lgorithm implemented in MATLB to clculte the mcroscopic kinetic equtions for the genertion of the singlet oxygen nd photochemicl prmeters. Four photochemicl prmeters g, ξ, σ, nd β were obtined from the optimiztion process. Using the optiml prmeters, pprent singlet oxygen concentrtion cn be clculted nd used s n improved dosimetry quntity. Compred with previous PDT dose (light distribution), [ O ] rx is dosimetry quntity with b c Figure 4: Results of PDT dosimetry quntities for tretment up to 00 s in homogeneous prostte with µ. cm - nd µ s = 4 cm - nd uniform source strength loding ssumption. () Slides demonstrtion of pprent rected singlet oxygen concentrtion distribution. Unit for color br is mm. (b) Isosurfce demonstrtion of pprent rected singlet oxygen concentrtion distribution for tretment up to [ O ] rx,sd.4 mm. (c) Isosurfce demonstrtion of light fluence rte distribution t 00 mw/cm.

6 b c d e f Figure 5: Isosurfce results of PDT dose quntities t different tretment time. () [ O ] rx t 0.4 mm t 00 s. (b) [ O ] rx t 0.4 mm t 500 s. (c) [ O ] rx t 0.4 mm t 000 s. (d) Light fluence distribution for 0 J/cm t 00 s. (e) Light fluence distribution for 75 J/cm t 500 s. (f) Light fluence rte distribution t 50 J/cm t 000 s. better ccurcy, especilly in the wy to demonstrte time-dependnt PDT tretment efficcy. These optiml prmeters were lso pplied to D prostte geometry, to compre the PDT dosimetry quntities in time-dependent simultion implemented in COMSOL. 5. Acknowledgements This work is supported by grnts from Ntionl Institute of Helth (NIH) R0 CA nd P0 CA References. S. B. Brown, E. A. Brown, nd I. Wlker, "The present nd future role of photodynmic therpy in cncer tretment," Lncet Oncol. 5, (004).. K.-H. K. Wng, M. T. Busch, J. C. Finly, nd T. C. Zhu, "Optimiztion of physiologicl prmeter for mcroscopic modeling of rected singlet oxygen concentrtion in n invivo model," Proc. of SPIE 764, 7640O (009).. T. C. Zhu nd X. Zhou, "Finite-element modeling of singlet oxygen during photodynmic therpy," Proc. COMSOL Multiphysics, (007). 4. T. C. Zhu, J. C. Finly, X. Zhou, nd J. Li, "Mcroscopic modeling of the singlet oxygen

7 prodution during PDT," Proc. SPIE 647, (007). 5. K. K. Wng, J. C. Finly, T. M. Busch, S. M. Hhn, nd T. C, Zhu, Explicit dosimetry for photodynmic therpy: mcroscopic singlet oxygen modeling, J. Biophoton., -5, K. K. Wng, S. Mitr, nd T. H. Foster, A comprehensive mthemticl model of microscopic dose deposition in photodynmic therpy, Med. Phys. 4, 8 9 (007). 7. J. S. Dysrt, G. Singh, nd M. S. Ptterson, Clcultion of singlet oxygen dose from photosensitizer fluorescence nd photobleching during mthpc Photodynmic Therpy of MLL Cells, Photochem. Photobiol. 8, (005). 8. K. K. Wng, nd T. C. Zhu, " Explicit dosimetry for photodynmic therpy; singlet xxygen modeling bsed on finite-element method" Proc. COMSOL Multiphysics, (009). 9. S. Mitr, nd T. H. Foster, "Photophysicl prmeters, photosensitizer retention nd tissue opticl properties completely ccount for the higher photodynmic efficcy of meso-tetr-hydroxyphenyl-chlorin vs Photofrin," Photochem. Photobiol. 8, (005). 0. R. Storn nd K. Price, "Differentil evolution-a simple nd efficient heuristic for globl optimiztion over continuous spces," J. Globl Optimiztion, 4-59 (996).. A. Dimofte, J. C. Finly, nd T. C. Zhu, "A method for determintion of the bsorption nd scttering properties interstitilly in turbid medi," Phys. Med. Biol. 50, 9- (005).. J. Li, T. C. Zhu, nd J. C. Finly, Study of light fluence rte distribution in photodynmic therpy using finite-element method, Proc. SPIE 69, 690M M-8 (006).. I. Georgkoudi, M. G. Nichols, nd T. H. Foster, The mechnism of photofrin photobleching nd its consequences for photodynmic dosimetry, Photochem. Photobiol. 65, 5 44 (997).