This work deals with numerical simulation of a hyperthermia treatment of skin cancer as a state estimation problem, where uncertainties in the evolution and measurement models, as well as in the measured data, are accounted for. A reduced model is adopted, based on a coarse mesh for the solution of the partial differential equations that describe the physical problem, in order to expedite the solution of the state estimation problem with a particle filter algorithm within the Bayesian framework of statistics. The so-called approximation error model (AEM) is used in order to statistically compensate for model reduction effects. The Liu and West algorithm of the particle filter, together with the AEM, is shown to provide accurate estimates for the temperature and model parameters in a multilayered region containing a tumor loaded with nanoparticles. Simulated transient temperature measurements from one sensor are used in the analysis.
References
1.
Arulampalam
, M. S.
, Maskell
, S.
, Gordon
, N.
, and Clapp
, T.
, 2002
, “A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking
,” IEEE Trans. Signal Process.
, 50
(2
), pp. 174
–188
.2.
Ristic
, B.
, Arulampalam
, S.
, and Gordon
, N.
, 2004
, Beyond the Kalman Filter
, Artech House
, Boston, MA
.3.
Doucet
, A.
, de Freitas
, N.
, and Gordon
, N.
, 2001
, Sequential Monte Carlo Methods in Practice
, Springer
, New York
.4.
Liu
, J.
, and West
, M.
, 2001
, “Combined Parameter and State Estimation in Simulation-Based Filtering
,” Sequential Monte Carlo Methods in Practice
, A.
Doucet
, N.
de Freitas
, and N.
Gordon
, eds., Springer
, New York
.5.
Sheinson
, D. M.
, Niemi
, J.
, and Meiring
, W.
, 2014
, “Comparison of the Performance of Particle Filter Algorithms Applied to Tracking of a Disease Epidemic
,” Math. Biosci.
, 255
, pp. 21
–32
.6.
Lopes
, H. F.
, and Carvalho
, C. M.
, 2013
, “Online Bayesian Learning in Dynamic Models: An Illustrative Introduction to Particle Methods
,” Bayesian Theory and Applications
, P.
Damien
, P.
Dellaportas
, N. G.
Polson
, and D. A.
Stephens
, eds., Oxford Press University
, Oxford, UK
.7.
Smal
, I.
, Niessen
, W. J.
, Demirel
, O.
, Meijering
, E.
, and Sbalzarini
, I. F.
, 2014
, “PPF: A Parallel Particle Filtering Library
,” IET
Conference on Data Fusion and Target Tracking 2014: Algorithms and Applications
, Apr. 30.8.
Chitchian
, M.
, Simonetto
, A.
, van Amesfoort
, A. S.
, and Keviczky
, T.
, 2013
, “Distributed Computation Particle Filters on GPU Architectures for Real-Time Control Applications
,” IEEE Trans. Control Syst. Technol.
, 21
(6
), pp. 2224
–2238
.9.
Goodrum
, M. A.
, Trotter
, M. J.
, Aksel
, A.
, Acton
, S. T.
, and Skadron
, K.
, 2012
, “Parallelization of Particle Filter Algorithms
,” Computer Architecture
(Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics, Vol. 6161)), Springer-Verlag
, Berlin
, pp. 139
–149
.10.
Brown
, J. A.
, and Capson
, D. W.
, 2012
, “A Framework for 3D Model-Based Visual Tracking Using a GPU-Accelerated Particle Filter
,” IEEE Trans. Visualization Comput. Graphics
, 18
(1
), pp. 68
–80
.11.
Lamien
, B.
, Orlande
, H. R. B.
, Eliçabe
, G. E.
, and Maurente
, A. J.
, 2014
, “State Estimation Problem in Hyperthermia Treatment of Tumors Loaded With Nanoparticles
,” 15th International Heat Transfer Conference
, Kyoto, Japan.12.
Kaipio
, J. P.
, and Somersalo
, E.
, 2004
, Statistical and Computational Inverse Problems (Applied Mathematical Sciences, Vol. 160), Springer-Verlag
, New York
.13.
Kaipio
, J. P.
, and Somersalo
, E.
, 2007
, “Statistical Inverse Problems: Discretization, Model Reduction and Inverse Crimes
,” J. Comput. Appl. Math.
, 198
(2
), pp. 493
–504
.14.
Nissinen
, A.
, Heikkinen
, L. M.
, and Kaipio
, J. P.
, 2008
, “The Bayesian Approximation Error Approach for Electrical Impedance Tomography—Experimental Results
,” Meas. Sci. Technol.
, 19
(1
), p. 015501
.15.
Nissinen
, A.
, Heikkinen
, L. M.
, Kolehmainen
, V.
, and Kaipio
, J. P.
, 2009
, “Compensation of Errors Due to Discretization, Domain Truncation and Unknown Contact Impedances in Electrical Impedance Tomography
,” Meas. Sci. Technol.
, 20
(10
), p. 105504
.16.
Nissinen
, A.
, 2011
, “Modelling Errors in Electrical Impedance Tomography
,” Ph.D. thesis
, University of Eastern Finland, Kuopio, Finland.http://epublications.uef.fi/pub/urn_isbn_978-952-61-0428-7/urn_isbn_978-952-61-0428-7.pdf17.
Nissinen
, A.
, Kolehmainen
, V. P.
, and Kaipio
, J. P.
, 2011
, “Compensation of Modelling Errors Due to Unknown Domain Boundary in Electrical Impedance Tomography
,” IEEE Trans. Med. Imaging
, 30
(2
), pp. 231
–242
.18.
Mozumder
, M.
, Tarvainen
, T.
, Kaipio
, J. P.
, Arridge
, S. R.
, and Kolehmainen
, V.
, 2014
, “Compensation of Modeling Errors Due to Unknown Domain Boundary in Diffuse Optical Tomography
,” J. Opt. Soc. Am. A
, 31
(8
), pp. 1847
–1855
.19.
Tarvainen
, T.
, Kolehmainen
, V.
, Pulkkinen
, A.
, Vauhkonen
, M.
, Schweiger
, M.
, Arridge
, S. R.
, and Kaipio
, J. P.
, 2010
, “An Approximation Error Approach for Compensating for Modelling Errors Between the Radiative Transfer Equation and the Diffusion Approximation in Diffuse Optical Tomography
,” Inverse Probl.
, 26
(1
), p. 015005
.20.
Kolehmainen
, V.
, Tarvainen
, T.
, Arridge
, S. R.
, and Kaipio
, J. P.
, 2011
, “Marginalization of Uninteresting Distributed Parameters in Inverse Problems—Application to Diffuse Optical Tomography
,” Int. J. Uncertainty Quantif.
, 1
(1
), pp. 1
–17
.21.
Huttunen
, J. M. J.
, and Kaipio
, J. P.
, 2007
, “Approximation Error Analysis in Nonlinear State Estimation With an Application to State-Space Identification
,” Inverse Probl.
, 23
(5
), pp. 2141
–2157
.22.
Huttunen
, J. M. J.
, Lehikoinen
, A.
, Hämäläinen
, J.
, and Kaipio
, J. P.
, 2010
, “Importance Sampling Approach for the Nonstationary Approximation Error Method
,” Inverse Probl.
, 26
(12
), p. 125003
.23.
Huttunen
, J. M. J.
, and Pikkarainen
, H. K.
, 2007
, “Discretization Error in Dynamical Inverse Problems: One-Dimensional Model Case
,” J. Inverse Ill-Posed Probl.
, 15
(4
), pp. 365
–386
24.
Huttunen
, J.
, and Kaipio
, J. P.
, 2007
, “Approximation Errors in Nonstationary Inverse Problems
,” Inverse Probl. Imaging
, 1
(1
), pp. 77
–93
.25.
Huttunen
, J. M. J.
, and Kaipio
, J. P.
, 2009
, “Model Reduction in State Identification Problems With an Application to Determination of Thermal Parameters
,” Appl. Numer. Math.
, 59
(5
), pp. 877
–890
.26.
Lamien
, B.
, Orlande
, H. R. B.
, and Eliçabe
, G. E.
, 2015
, “Comparison of Particle Filter Algorithms Applied to the Temperature Filed Estimation in Hyperthermia Phantoms
,” 1st Thermal and Fluid Engineering Summer Conference
, ASTFE, New York, Paper No. TFESC–13764.27.
Lamien
, B.
, Orlande
, H. R. B.
, and Eliçabe
, G. E.
, 2015
, “Inverse Problem in the Hyperthermia Therapy of Cancer With Laser Heating and Plasmonic Nanoparticles
,” Inverse Probl. Sci. Eng.
(in press).28.
Varon
, L. A. B.
, Orlande
, H. R. B.
, and Elicabe
, G.
, 2016
, “Combined Parameter and State in the Radiofrequency Hyperthermia Treatment of Cancer
,” Numer. Heat Transfer, Part A
(in press).29.
Lamien
, B.
, Varon
, L. A. B.
, Orlande
, H. R. B.
, and Eliçabe
, G. E.
, 2016
, “State Estimation in Bioheat Transfer: A Comparison of Particle Filter Algorithms
,” Int. J. Numer. Methods Heat Fluid Flow
(in press).30.
Varon
, L. A. B.
, Orlande
, H. R. B.
, and Elicabe
, G.
, 2015
, “Estimation of State Variables in the Hyperthermia Therapy of Cancer With Heating Imposed by Radiofrequency Electromagnetic Waves
,” Int. J. Therm. Sci.
, 98
, pp. 228
–236
.31.
dos Santos
, I.
, Haemmerich
, D.
, Schutt
, D.
, da Rocha
, A. F.
, and Menezes
, L. R.
, 2009
, “Probabilistic Finite Element Analysis of Radiofrequency Liver Ablation Using the Unscented Transform
,” Phys. Med. Biol.
, 54
(3
), pp. 627
–640
.32.
de Greef
, M.
, Kok
, H. P.
, Correia
, D.
, Borsboom
, P. P.
, Bel
, A.
, and Crezee
, J.
, 2011
, “Uncertainty in Hyperthermia Treatment Planning: The Need for Robust System Design
,” Phys. Med. Biol.
, 56
(11
), pp. 3233
–3250
.33.
Liu
, J.
, 2001
, “Uncertainty Analysis for Temperature Prediction of Biological Bodies Subject to Randomly Spatial Heating
,” J. Biomech.
, 34
(12
), pp. 1637
–1642
.34.
Liu
, J.
, 2014
, “Ways Toward Targeted Freezing or Heating Ablation of Malignant Tumor: Precisely Managing the Heat
,” 15th International Heat Transfer Conference
, IHTC-15, Kyoto, Japan.35.
Xu
, X.
, Meade
, A.
, and Bayazitoglu
, Y.
, 2011
, “Numerical Investigation of Nanoparticle-Assisted Laser-Induced Interstitial Thermotherapy Toward Tumor and Cancer Treatments
,” Lasers Med. Sci.
, 26
(2
), pp. 213
–222
.36.
Tjahjono
, I. K.
, and Bayazitoglu
, Y.
, 2008
, “Near-Infrared Light Heating of a Slab by Embedded Nanoparticles
,” Int. J. Heat Mass Transfer
, 51
(7–8), pp. 1505
–1515
.37.
Vera
, J.
, and Bayazitoglu
, Y.
, 2009
, “A Note on Laser Penetration in Nanoshell Deposited Tissue
,” Int. J. Heat Mass Transfer
, 52
(13–14), pp. 3402
–3406
.38.
Vera
, J.
, and Bayazitoglu
, Y.
, 2009
, “Gold Nanoshell Density Variation With Laser Power for Induced Hyperthermia
,” Int. J. Heat Mass Transfer
, 52
(3–4), pp. 564
–573
.39.
Bayazitoglu
, Y.
, Kheradmand
, S.
, and Tullius
, T. K.
, 2013
, “An Overview of Nanoparticle Assisted Laser Therapy
,” Int. J. Heat Mass Transfer
, 67
, pp. 469
–486
.40.
Dombrovsky
, L. A.
, Timchenko
, V.
, and Jackson
, M.
, 2012
, “Indirect Heating Strategy for Laser Induced Hyperthermia: An Advanced Thermal Model
,” Int. J. Heat Mass Transfer
, 55
(17–18), pp. 4688
–4700
.41.
Dombrovsky
, L. A.
, Timchenko
, V.
, Jackson
, M.
, and Yeoh
, G. H.
, 2011
, “A Combined Transient Thermal Model for Laser Hyperthermia of Tumors With Embedded Gold Nanoshells
,” Int. J. Heat Mass Transfer
, 54
(25–26), pp. 5459
–5469
.42.
Chatterjee
, D.
, and Krishnan
, S.
, 2013
, “Gold Nanoparticle—Mediated Hyperthermia in Cancer Therapy
,” Cancer Nanotechnology: Principles and Applications in Radiation Oncology
, S.
Cho
, and S.
Krishnan
, eds., CRC Press
, Boca Raton, FL
.43.
van der Zee
, J.
, 2002
, “Heating the Patient: A Promising Approach?
,” Ann. Oncol.
, 13
(8
), pp. 1173
–1184
.44.
Tamarov
, K. P.
, Osminkina
, L. A.
, Zinovyev
, S. V.
, Maximova
, K. A.
, Kargina
, J. V.
, Gongalsky
, M. B.
, Ryabchikov
, Y.
, Al-Kattan
, A.
, Sviridov
, A. P.
, Sentis
, M.
, Ivanov
, A. V.
, Nikiforov
, V. N.
, Kabashin
, A. V.
, and Timoshenko
, V. Y.
, 2014
, “Radio Frequency Radiation-Induced Hyperthermia Using Si Nanoparticle-Based Sensitizers for Mild Cancer Therapy
,” Sci. Rep.
, 4
, p. 7034
.45.
Wang
, Q.
, Xie
, L.
, He
, Z.
, Di
, D.
, and Liu
, J.
, 2012
, “Biodegradable Magnesium Nanoparticle-Enhanced Laser Hyperthermia Therapy
,” Int. J. Nanomed.
, 7
, pp. 4715
–4725
.46.
Rengan
, A. K.
, Bukhari
, A. B.
, Pradhan
, A.
, Malhotra
, R.
, Banerjee
, R.
, Srivastava
, R.
, and De
, A.
, 2015
, “In Vivo Analysis of Biodegradable Liposome Gold Nanoparticles as Efficient Agents for Photothermal Therapy of Cancer
,” Nano Lett.
, 15
(2
), pp. 842
–848
.47.
Pearce
, J. A.
, 2013
, “Comparative Analysis of Mathematical Models of Cell Death and Thermal Damage Processes
,” Int. J. Hyperthermia
, 29
(4
), pp. 262
–280
.48.
van Rhoon
, G. C.
, 2016
, “Is CEM43 Still a Relevant Thermal Dose Parameter for Hyperthermia Treatment Monitoring?
,” Int. J. Hyperthermia
, 32
(1
), pp. 50
–62
.49.
Clinicaltrials
, 2010
, “Pilot Study of AuroLase™ Therapy in Refractory and/or Recurrent Tumors of the Head and Neck
,” National Institutes of Health, Bethesda, MD.50.
Çetingül
, M. P.
, and Herman
, C.
, 2010
, “A Heat Transfer Model of Skin Tissue for the Detection of Lesions: Sensitivity Analysis
,” Phys. Med. Biol.
, 55
(19
), pp. 5933
–5951
.51.
Çetingül
, M. P.
, and Herman
, C.
, 2011
, “Quantification of the Thermal Signature of a Melanoma Lesion
,” Int. J. Therm. Sci.
, 50
(4
), pp. 421
–431
.52.
Star
, W. M.
, 2011
, “Diffusion Theory of Light Transport
,” Optical-Thermal Response of Laser-Irradiated Tissue
, A. J.
Welch
, and M. J. C.
van Gemert
, eds., Springer
, New York
.53.
Carp
, S. A.
, Prahl
, S. A.
, and Venugopalan
, V.
, 2004
, “Radiative Transport in the Delta-P1 Approximation: Accuracy of Fluence Rate and Optical Penetration Depth Predictions in Turbid Semi-Infinite Media
,” J. Biomed. Opt.
, 9
(3
), pp. 632
–647
.54.
Elliott
, A. M.
, Stafford
, R. J.
, Schwartz
, J.
, Wang
, J.
, Shetty
, A. M.
, Bourgoyne
, C.
, O'Neal
, P.
, and Hazle
, J. D.
, 2007
, “Laser-Induced Thermal Response and Characterization of Nanoparticles for Cancer Treatment Using Magnetic Thermal Imaging
,” Med. Phys. J.
, 34
(7
), pp. 3102
–3108
.55.
Dombrovsky
, L. A.
, Randrianalisoa
, J. H.
, Lipinski
, W.
, and Timchenko
, V.
, 2013
, “Simplified Approaches to Radiative Transfer Simulations in Laser-Induced Hyperthermia of Superficial Tumors
,” Comput. Therm. Sci.
, 5
(6
), pp. 521
–530
.56.
Pennes
, H. H.
, 1948
, “Analysis of Tissue and Arterial Blood Temperatures in the Resting Human Forearm
,” J. Appl. Physiol.
, 1
(2
), pp. 93
–122
.http://jap.physiology.org/content/jap/1/2/93.full.pdf57.
Singh
, R.
, Das
, K.
, Okajima
, J.
, Maruyama
, S.
, and Mishra
, S. C.
, 2015
, “Modeling Skin Cooling Using Optical Windows and Cryogens During Laser Induced Hyperthermia in a Multilayer Vascularized Tissue
,” Appl. Therm. Eng.
, 89
, pp. 28
–35
.58.
Andrieu
, C.
, Doucet
, A.
, Singh
, S. S.
, and Tadic
, V. B.
, 2004
, “Particle Methods for Change Detection, System Identification, and Control
,” Proc. IEEE
, 92
(3
), pp. 423
–438
.59.
Carpenter
, J.
, Clifford
, P.
, and Fearnhead
, P.
, 1999
, “An Improved Particle Filter for Non-Linear Problems
,” IEEE Proc. Radar Sonar Navig.
, 146
(1
), pp. 2
–7
.60.
Del Moral
, P.
, and Jasra
, A.
, 2007
, “Sequential Monte Carlo for Bayesian Computation
,” Bayesian Statistics
, J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith and M. West, eds., Oxford University Press, Oxford, UK, pp. 1–34.61.
Candy
, J. V.
, 2008
, Bayesian Signal Processing: Classical, Modern, and Particle Filtering Methods
, Wiley
, Hoboken, NJ
.62.
Kalman
, R. E.
, 1960
, “A New Approach to Linear Filtering and Prediction Problems
,” ASME J. Basic Eng.
, 82
(1
), pp. 35
–45
.63.
West
, M.
, 1993
, “Approximating Posterior Distributions by Mixture
,” J. R. Stat. Soc.: Ser. B
, 55
(2
), pp. 409
–422
.http://www.jstor.org/stable/234620264.
Lehikoinen
, A.
, Finsterle
, S.
, Voutilainen
, A.
, Kowalsky
, M. B.
, and Kaipio
, J. P.
, 2009
, “Dynamical Inversion of Geophysical ERT Data: State Estimation in the Vadose Zone
,” Inverse Probl. Sci. Eng.
, 17
(6
), pp. 715
–736
.65.
Lipponen
, A.
, Seppänen
, A.
, and Kaipio
, J. P.
, 2011
, “Nonstationary Approximation Error Approach to Imaging of Three-Dimensional Pipe Flow: Experimental Evaluation
,” Meas. Sci. Technol.
, 22
(10
), p. 104013
.66.
Pacheco
, C. C.
, Orlande
, H. R. B.
, Colaço
, M. J.
, and Dulikravich
, G. S.
, 2015
, “Estimation of a Location-and-Time Dependent High Magnitude Heat Flux in a Heat Conduction Problem Using the Kalman Filter and the Approximation Error Model
,” Numer. Heat Transfer, Part A
, 68
(11
), pp. 1198
–1219
.67.
Hasgall
, P. A.
, Di Gennaro
, F.
, Baumgarther
, C.
, Neufeld
, E.
, Gosselin
, M. C.
, Payne
, D.
, Klingenbock
, A.
, and Kuster
, N.
, “IT'IS Database for Thermal and Electromagnetic Parameters of Biological Tissues, Version 3.0
,” IT'IS Foundation, Zurich.68.
Bashkatov
, A. N.
, Genina
, E. A.
, and Tuchin
, V. V.
, 2011
, “Optical Properties of Skin, Subcutaneous, and Muscle Tissues: A Review
,” J. Innovative Opt. Health Sci.
, 4
(1
), pp. 9
–38
.69.
Jain
, P. K.
, Lee
, K. S.
, El-Sayed
, I. H.
, and El-Sayed
, M. A.
, 2006
, “Calculated Absorption and Scattering Properties of Gold Nanoparticles of Different Size, Shape, and Composition: Applications in Biological Imaging and Biomedicine
,” J. Phys. Chem. B
, 110
(14
), pp. 7238
–7248
.70.
Prahl
, S. A.
, 1988
, “Light Transport in Tissue
,” Ph.D. thesis
, University of Texas at Austin, Austin, TX.http://elibrary.ru/item.asp?id=5928423Copyright © 2017 by ASME
You do not currently have access to this content.
