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ABSTRACT
Particle filters are general methods for the solution of state estimation problems, which can be applied to nonlinear models with non-Gaussian uncertainties. In this paper, an algorithm of the particle filter is used for the simultaneous estimation of model parameters and state variables in a bioheat transfer problem associated with the radio frequency (RF) hyperthermia treatment of cancer. Results obtained with simulated measurements indicate an excellent agreement between the estimated and the exact quantities, even for cases with large uncertainties in the measurements, as well as in the evolution and measurement models.
Nomenclature
| cp | = | specific heat |
| D | = | number of measurements |
| E | = | electric field strength |
| f | = | frequency |
| f, h | = | general functions for the evolution and observation models, respectively |
| H | = | intensity of the magnetic field |
| hf | = | heat transfer coefficient |
| k | = | thermal conductivity |
| Lx, Ly, Lz | = | domain dimensions in the x, y, and z directions, respectively |
| n | = | measurement noise vector |
| n | = | number of nanoparticles |
| N | = | number of particles for the particle filter |
| m | = | Gaussian kernel center |
| M | = | total number of elements |
| Q | = | volumetric heat source |
| r | = | mean radius of nanoparticles |
| R | = | radius of the tumor |
| s | = | interface between the tumor and the surrounding tissue |
| T | = | temperature |
| Tb | = | blood temperature |
| Tf | = | temperature of the surrounding medium |
| t | = | time |
| U | = | voltage |
| w | = | weights of the particles |
| x | = | state vector |
| x,y,z | = | Cartesian coordinates |
| v | = | state noise vector |
| V | = | Monte Carlo covariance matrix of the posterior distribution |
| V | = | volume |
| z | = | vector of measurements |
| Greeks | = | |
| δ | = | discount factor for Liu & West's algorithm |
| ε | = | permittivity |
| π(a|b) | = | conditional probability of a when b is given |
| ρ | = | density |
| Ω | = | surface of the domain |
| Ω′1, Ω′2 | = | boundary patches with electrodes set to voltages U and ground, respectively |
| ωb | = | blood perfusion rate |
| φ | = | electric potential |
| θ | = | parameter vector |
| Θ | = | volumetric concentration of nanoparticles |
| σ | = | electric conductivity |
| χ | = | susceptibility of the magnetic nanoparticles |
| μ0 | = | dielectric permeability constant |
| ν | = | constant standard deviation |
| ξ | = | Gaussian random vector with zero mean and constant standard deviation |
| Superscripts | = | |
| i | = | particle index |
| meas | = | measurements |
| Subscripts | = | |
| 1 | = | health tissue |
| 2 | = | tumor |
| 3 | = | nanoparticles |
| b | = | blood |
| e | = | electrical |
| est | = | estimated |
| exa | = | exact |
| k | = | index to time step |
| m | = | metabolism |
Nomenclature
| cp | = | specific heat |
| D | = | number of measurements |
| E | = | electric field strength |
| f | = | frequency |
| f, h | = | general functions for the evolution and observation models, respectively |
| H | = | intensity of the magnetic field |
| hf | = | heat transfer coefficient |
| k | = | thermal conductivity |
| Lx, Ly, Lz | = | domain dimensions in the x, y, and z directions, respectively |
| n | = | measurement noise vector |
| n | = | number of nanoparticles |
| N | = | number of particles for the particle filter |
| m | = | Gaussian kernel center |
| M | = | total number of elements |
| Q | = | volumetric heat source |
| r | = | mean radius of nanoparticles |
| R | = | radius of the tumor |
| s | = | interface between the tumor and the surrounding tissue |
| T | = | temperature |
| Tb | = | blood temperature |
| Tf | = | temperature of the surrounding medium |
| t | = | time |
| U | = | voltage |
| w | = | weights of the particles |
| x | = | state vector |
| x,y,z | = | Cartesian coordinates |
| v | = | state noise vector |
| V | = | Monte Carlo covariance matrix of the posterior distribution |
| V | = | volume |
| z | = | vector of measurements |
| Greeks | = | |
| δ | = | discount factor for Liu & West's algorithm |
| ε | = | permittivity |
| π(a|b) | = | conditional probability of a when b is given |
| ρ | = | density |
| Ω | = | surface of the domain |
| Ω′1, Ω′2 | = | boundary patches with electrodes set to voltages U and ground, respectively |
| ωb | = | blood perfusion rate |
| φ | = | electric potential |
| θ | = | parameter vector |
| Θ | = | volumetric concentration of nanoparticles |
| σ | = | electric conductivity |
| χ | = | susceptibility of the magnetic nanoparticles |
| μ0 | = | dielectric permeability constant |
| ν | = | constant standard deviation |
| ξ | = | Gaussian random vector with zero mean and constant standard deviation |
| Superscripts | = | |
| i | = | particle index |
| meas | = | measurements |
| Subscripts | = | |
| 1 | = | health tissue |
| 2 | = | tumor |
| 3 | = | nanoparticles |
| b | = | blood |
| e | = | electrical |
| est | = | estimated |
| exa | = | exact |
| k | = | index to time step |
| m | = | metabolism |