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September  2012, 17(6): 1651-1672. doi: 10.3934/dcdsb.2012.17.1651

A class of optimization problems in radiotherapy dosimetry planning

1. 

Instituto de Matemática Interdisciplinar (IMI) and Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Plaza de las Ciencias 3, Madrid 28040, Spain, Spain

2. 

Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa

3. 

Departamento de Matemática Aplicada, Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos, Sevilla 41092, Spain

4. 

Servicio de Radiofsica y Protección Radiológica, Hospital Universitario Puerta de Hierro Majadahonda, Manuel de Falla 1, 28222 Majadahonda, Madrid, Spain

Received  September 2011 Revised  January 2012 Published  May 2012

Radiotherapy is an important clinical tool to fight malignancies. To do so, a key point consists in selecting a suitable radiation dose that could achieve tumour control without inducing significant damage to surrounding healthy tissues. In spite of recent significant advances, any radiotherapy planning in use relies principally on experience-based decisions made by clinicians among several possible choices.
    In this work we consider a mathematical problem related to that decision-making process. More precisely, we assume that a well-defined target region, called planning target volume (PTV), is given. We then consider the question of determining which radiation distribution is able to achieve a maximum impact on tumour cells and a minimum one in healthy ones. Such dose distribution is defined as the solution of a multi-parameter minimization problem over the PTV and healthy tissues, subject to a number of constraints arising from clinical and technical requirements. For any choice of parameters, sufficient conditions for the existence of a unique solution of that problem are derived. Such solution is then approximated by means of a suitable numerical algorithm. Finally, some examples are considered, on which the dependence on model parameters of different clinical efficiency indexes is discussed.
Citation: Juan Carlos López Alfonso, Giuseppe Buttazzo, Bosco García-Archilla, Miguel A. Herrero, Luis Núñez. A class of optimization problems in radiotherapy dosimetry planning. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1651-1672. doi: 10.3934/dcdsb.2012.17.1651
References:
[1]

M. Abramowitz and I. A. Stegun, eds., "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,", Reprint of the 1972 edition, (1972).

[2]

G. W. Barendsen, Dose fractionation, dose rate and iso-effect relationships for normal tissue responses,, Int. J. Radiat. Oncol. Biol. Phys., 8 (1982), 1981.

[3]

B. J. Blonigen, R. D. Steinmetz, L. Levin, M. A. Lamba, R. E. Warnick and J. C. Breneman, Irradiated volume as a predictor of brain radionecrosis after linear accelerator stereotactic radiosurgery,, Int. J. Radiat. Oncol. Biol. Phys., 77 (2010), 996. doi: 10.1016/j.ijrobp.2009.06.006.

[4]

D. J. Brenner, L. R. Hlatky, P. J. Hahnfeldt, Y. Huang and R. K. Sachs, The linear-quadratic model and most other common radiobiological models result in similar predictions of time-dose relationships,, Radiat. Res., 150 (1998), 83. doi: 10.2307/3579648.

[5]

G. Buttazzo, "Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations,", Pitman Res. Notes Math. Ser., 207 (1989).

[6]

G. Buttazzo, M. Giaquinta and S. Hildebrandt, "One-Dimensional Variational Problems: An Introduction,", Oxford Lecture Series in Mathematics and its Applications, 15 (1998).

[7]

R. H. Byrd, J. Nocedal and R. A. Waltz, KNITRO: An integrated package for nonlinear optimization,, in, 83 (2006), 35.

[8]

A. Cappuccio, M. A. Herrero and L. Nuñez, Biological optimization of tumor radiosurgery,, Med. Phys., 36 (2009), 98. doi: 10.1118/1.2986141.

[9]

A. Cappuccio, M. A. Herrero and L. Nuñez, Tumour radiotherapy and its mathematical modelling,, in, 492 (2009), 77.

[10]

D. J. Carlson, R. D. Stewart, X. A. Li, K. Jennings, J. Z. Wang and M. Guerrero, Comparison of in vitro and in vivo $\alpha$/$\beta$ ratios for prostate cancer,, Phys. Med. Biol., 49 (2004), 4477. doi: 10.1088/0031-9155/49/19/003.

[11]

D. D. Dionysiou, G. S. Stamatakos, D. Gintides, N. Uzunoglu and K. Kyriaki, Critical parameters determining standard radiotherapy treatment outcome for glioblastoma multiforme: A computer simulation,, Open Biomed. Eng. J., 2 (2008), 43.

[12]

R. E. Drzymala, R. Mohan, L. Brewster, J. Chu, M. Goitein, W. Harms and M. Urie, Dose-volume histograms,, Int. J. Radiat. Oncol. Biol. Phys., 21 (1991), 71. doi: 10.1016/0360-3016(91)90168-4.

[13]

B. Emami, J. Lyman, A. Brown, L. Coia, M. Goitein, J. E. Munzenrider, B. Shank, L. J. Solin and M. Wesson, Tolerance of normal tissue to therapeutic irradiation,, Int. J. Radiat. Oncol. Biol. Phys., 21 (1991), 109. doi: 10.1016/0360-3016(91)90171-Y.

[14]

L. Feuvret, G. Noël, J. J. Mazeron and P. Bey, Conformity index: A review,, Int. J. Radiat. Oncol. Biol. Phys., 64 (2006), 333. doi: 10.1016/j.ijrobp.2005.09.028.

[15]

A. Giese, R. Bjerkvig, M. E. Berens and M. Westphal, Cost of migration: Invasion of malignant gliomas and implications for treatment,, J. Clin. Oncol., 21 (2003), 1624. doi: 10.1200/JCO.2003.05.063.

[16]

G. G. Steel, "Basic Clinical Radiobiology,", 3rd edition, (2002).

[17]

T. S. Kehwar, Analytical approach to estimate normal tissue complication probability using best fit of normal tissue tolerance doses into the NTCP equation of the linear quadratic model,, J. Cancer Res. Ther., 1 (2005), 168.

[18]

T. S. Kehwar and S. C. Sharma, Use of normal tissue tolerance doses into linear quadratic equation to estimate normal tissue complication probability,, April 2003. Available from: \url{http://www.rooj.com/Normal%20Tissue%20Comp.htm}., (2003).

[19]

J. Nocedal and S. J. Wright, "Numerical Optimization,", 2nd edition, (2006).

[20]

J. R. Palta and T. R. Mackie, eds., "Intensity Modulated Radiation Therapy: The State of the Art,", Medical Physics Publishing, (2003).

[21]

E. Shaw, R. Kline, M. Gillin, et al., Radiation therapy oncology group: Radiosurgery quality assurance guidelines,, Int. J. Radiat. Oncol. Biol. Phys., 27 (1993), 1231. doi: 10.1016/0360-3016(93)90548-A.

[22]

R. Timmerman and L. Xing, eds., "Image Guided and Adaptive Radiation Therapy,", Published by Wolters Kluwer Lippincott Williams & Wilkins Health in Philadelphia, (2010).

[23]

C. M. West, S. E. Davidson, S. A. Roberts and R. D. Hunter, Intrinsic radiosensitivity and prediction of patient response to radiotherapy for carcinoma of the cervix,, Br. J. Cancer, 68 (1993), 819. doi: 10.1038/bjc.1993.434.

[24]

C. M. West, S. E. Davidson, S. A. Roberts and R. D. Hunter, The independence of intrinsic radiosensitivity as a prognostic factor for patient response to radiotherapy of carcinoma of the cervix,, Br. J. Cancer, 76 (1997), 1184. doi: 10.1038/bjc.1997.531.

[25]

C. M. West, Invited review: Intrinsic radiosensitivity as a predictor of patient response to radiotherapy,, Br. J. Radiol., 68 (1995), 827. doi: 10.1259/0007-1285-68-812-827.

[26]

M. Yoon, S. Y. Park, D. Shin, S. B. Lee, H. R. Pyo, D. Y. Kim and K. H. Cho, A new homogeneity index based on statistical analysis of the dose-volume histogram,, J. Appl. Clin. Med. Phys., 8 (2007), 9.

[27]

M. Zaider and G. N. Minerbo, Tumour control probability: A formulation applicable to any temporal protocol of dose delivery,, Phys. Med. Biol., 45 (2000), 279. doi: 10.1088/0031-9155/45/2/303.

[28]

, Radiation quantities and units,, ICRU Report 33, (1980).

[29]

, Prescribing, recording, and reporting photon-beam intensity-modulated radiation therapy (IMRT),, ICRU Report 50, (1993).

[30]

, Prescribing, recording, and reporting photon-beam intensity-modulated radiation therapy (IMRT),, ICRU Report 62, (1999).

[31]

, Prescribing, recording, and reporting photon-beam intensity-modulated radiation therapy (IMRT),, ICRU Report 83, (2010).

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, eds., "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,", Reprint of the 1972 edition, (1972).

[2]

G. W. Barendsen, Dose fractionation, dose rate and iso-effect relationships for normal tissue responses,, Int. J. Radiat. Oncol. Biol. Phys., 8 (1982), 1981.

[3]

B. J. Blonigen, R. D. Steinmetz, L. Levin, M. A. Lamba, R. E. Warnick and J. C. Breneman, Irradiated volume as a predictor of brain radionecrosis after linear accelerator stereotactic radiosurgery,, Int. J. Radiat. Oncol. Biol. Phys., 77 (2010), 996. doi: 10.1016/j.ijrobp.2009.06.006.

[4]

D. J. Brenner, L. R. Hlatky, P. J. Hahnfeldt, Y. Huang and R. K. Sachs, The linear-quadratic model and most other common radiobiological models result in similar predictions of time-dose relationships,, Radiat. Res., 150 (1998), 83. doi: 10.2307/3579648.

[5]

G. Buttazzo, "Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations,", Pitman Res. Notes Math. Ser., 207 (1989).

[6]

G. Buttazzo, M. Giaquinta and S. Hildebrandt, "One-Dimensional Variational Problems: An Introduction,", Oxford Lecture Series in Mathematics and its Applications, 15 (1998).

[7]

R. H. Byrd, J. Nocedal and R. A. Waltz, KNITRO: An integrated package for nonlinear optimization,, in, 83 (2006), 35.

[8]

A. Cappuccio, M. A. Herrero and L. Nuñez, Biological optimization of tumor radiosurgery,, Med. Phys., 36 (2009), 98. doi: 10.1118/1.2986141.

[9]

A. Cappuccio, M. A. Herrero and L. Nuñez, Tumour radiotherapy and its mathematical modelling,, in, 492 (2009), 77.

[10]

D. J. Carlson, R. D. Stewart, X. A. Li, K. Jennings, J. Z. Wang and M. Guerrero, Comparison of in vitro and in vivo $\alpha$/$\beta$ ratios for prostate cancer,, Phys. Med. Biol., 49 (2004), 4477. doi: 10.1088/0031-9155/49/19/003.

[11]

D. D. Dionysiou, G. S. Stamatakos, D. Gintides, N. Uzunoglu and K. Kyriaki, Critical parameters determining standard radiotherapy treatment outcome for glioblastoma multiforme: A computer simulation,, Open Biomed. Eng. J., 2 (2008), 43.

[12]

R. E. Drzymala, R. Mohan, L. Brewster, J. Chu, M. Goitein, W. Harms and M. Urie, Dose-volume histograms,, Int. J. Radiat. Oncol. Biol. Phys., 21 (1991), 71. doi: 10.1016/0360-3016(91)90168-4.

[13]

B. Emami, J. Lyman, A. Brown, L. Coia, M. Goitein, J. E. Munzenrider, B. Shank, L. J. Solin and M. Wesson, Tolerance of normal tissue to therapeutic irradiation,, Int. J. Radiat. Oncol. Biol. Phys., 21 (1991), 109. doi: 10.1016/0360-3016(91)90171-Y.

[14]

L. Feuvret, G. Noël, J. J. Mazeron and P. Bey, Conformity index: A review,, Int. J. Radiat. Oncol. Biol. Phys., 64 (2006), 333. doi: 10.1016/j.ijrobp.2005.09.028.

[15]

A. Giese, R. Bjerkvig, M. E. Berens and M. Westphal, Cost of migration: Invasion of malignant gliomas and implications for treatment,, J. Clin. Oncol., 21 (2003), 1624. doi: 10.1200/JCO.2003.05.063.

[16]

G. G. Steel, "Basic Clinical Radiobiology,", 3rd edition, (2002).

[17]

T. S. Kehwar, Analytical approach to estimate normal tissue complication probability using best fit of normal tissue tolerance doses into the NTCP equation of the linear quadratic model,, J. Cancer Res. Ther., 1 (2005), 168.

[18]

T. S. Kehwar and S. C. Sharma, Use of normal tissue tolerance doses into linear quadratic equation to estimate normal tissue complication probability,, April 2003. Available from: \url{http://www.rooj.com/Normal%20Tissue%20Comp.htm}., (2003).

[19]

J. Nocedal and S. J. Wright, "Numerical Optimization,", 2nd edition, (2006).

[20]

J. R. Palta and T. R. Mackie, eds., "Intensity Modulated Radiation Therapy: The State of the Art,", Medical Physics Publishing, (2003).

[21]

E. Shaw, R. Kline, M. Gillin, et al., Radiation therapy oncology group: Radiosurgery quality assurance guidelines,, Int. J. Radiat. Oncol. Biol. Phys., 27 (1993), 1231. doi: 10.1016/0360-3016(93)90548-A.

[22]

R. Timmerman and L. Xing, eds., "Image Guided and Adaptive Radiation Therapy,", Published by Wolters Kluwer Lippincott Williams & Wilkins Health in Philadelphia, (2010).

[23]

C. M. West, S. E. Davidson, S. A. Roberts and R. D. Hunter, Intrinsic radiosensitivity and prediction of patient response to radiotherapy for carcinoma of the cervix,, Br. J. Cancer, 68 (1993), 819. doi: 10.1038/bjc.1993.434.

[24]

C. M. West, S. E. Davidson, S. A. Roberts and R. D. Hunter, The independence of intrinsic radiosensitivity as a prognostic factor for patient response to radiotherapy of carcinoma of the cervix,, Br. J. Cancer, 76 (1997), 1184. doi: 10.1038/bjc.1997.531.

[25]

C. M. West, Invited review: Intrinsic radiosensitivity as a predictor of patient response to radiotherapy,, Br. J. Radiol., 68 (1995), 827. doi: 10.1259/0007-1285-68-812-827.

[26]

M. Yoon, S. Y. Park, D. Shin, S. B. Lee, H. R. Pyo, D. Y. Kim and K. H. Cho, A new homogeneity index based on statistical analysis of the dose-volume histogram,, J. Appl. Clin. Med. Phys., 8 (2007), 9.

[27]

M. Zaider and G. N. Minerbo, Tumour control probability: A formulation applicable to any temporal protocol of dose delivery,, Phys. Med. Biol., 45 (2000), 279. doi: 10.1088/0031-9155/45/2/303.

[28]

, Radiation quantities and units,, ICRU Report 33, (1980).

[29]

, Prescribing, recording, and reporting photon-beam intensity-modulated radiation therapy (IMRT),, ICRU Report 50, (1993).

[30]

, Prescribing, recording, and reporting photon-beam intensity-modulated radiation therapy (IMRT),, ICRU Report 62, (1999).

[31]

, Prescribing, recording, and reporting photon-beam intensity-modulated radiation therapy (IMRT),, ICRU Report 83, (2010).

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