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Optimal control for bio-heat equation due to induced microwave

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Abstract

A distributed optimal control problem for a system described by a bio-heat equation for a homogeneous plane slab of tissue is analytically investigated. The required tissue temperature at a particular location of the tumour in hyperthermia can be attained within the total operation time of the process due to induced microwave radiation which is taken as control. The tissue temperature against the tissue length at different operation time of the process is considered to attain the desired temperature of the tumor.

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Abbreviations

c :

specific heat of tissue, J/(kg·°C)

h :

heat transfer coefficient between skin and ambient air, W/(m2·°C)

k :

thermal conductivity of tissue, W/(m·°C)

L :

lengthofplate, m

x 1 :

desired position of heating (center of tumor), m

χ :

temperature, °C

χ a :

arterial temperature, °C

χ 0 :

initial temperature, °C

u(t):

temperature of surrounding medium, °C

χ*:

desired temperature to be attained, °C

T :

total time of process, s

t 1 :

switching time, s

Q(t):

heat generation rate due to volumetric heating, W/m3

ρ :

density of tissue, kg/m3

δ :

dirac-delta function

ω :

product of flow and heat capacity of blood, W/(m3·°C)

Q m :

rate of metabolic heat generation, W/m3

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Authors and Affiliations

  1. Department of Mathematics, Heritage Academy, Chowbagha Road, Kolkata, 700107, India

    Piyanka Dhar

  2. Office of the Registrar, Vidyasagar University, Midnapore, 721102, West Bengal, India

    Ranajit Dhar

Authors
  1. Piyanka Dhar
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  2. Ranajit Dhar
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Corresponding author

Correspondence to Piyanka Dhar.

Additional information

Communicated by Zhe-wei ZHOU

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Dhar, P., Dhar, R. Optimal control for bio-heat equation due to induced microwave. Appl. Math. Mech.-Engl. Ed. 31, 529–534 (2010). https://doi.org/10.1007/s10483-010-0413-x

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  • DOI: https://doi.org/10.1007/s10483-010-0413-x

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2000 Mathematics Subject Classification

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