A process of simplification is used in solving a variety of problems which can eliminate the need for numerical solutions. In this paper, the method of quasi-steady approximation to study the effect of metabolic heat generation on one-dimensional ice-crystallization during cryosurgery has been used. Temperature profiles selleck compound and motion of freezing interface are obtained for different values of metabolic heat generation.2. Mathematical ModelIn the present study, one-dimensional ice-crystallization in biological tissue of length L has been considered as shown in Figure 1. Cryoprobe with temperature T0 = ?196��C is applied at x = 0, while at the other end x = L an adiabatic condition is used. In the frozen region, blood perfusion and metabolic heat generation are zero [4, 5, 10, 11, 13].

Figure 1Schematic representation of one-dimensional model.The governing equations for one-dimensional ice-crystallization in biological tissue are as follows.In frozen for??0��x��xi.(1)In unfrozen?region:��fcf?Tf?t=kf?2Tf?x2 for??xi��x��L.(2)Initial?region:��ucu?Tu?t=ku?2Tu?x2+qb+qm conditions:Tu(x,0)=TI=37C��,Tf(x,0)=TI=37C��.(3)Boundary conditions:Tf(0,t)=T0=?196C��,?Tu(L,t)?x=0.(4)Conditions at phase change interface:Tf(xi,t)=Tph=Tu(xi,t),kf?Tf(xi,t)?x?ku?Tu(xi,t)?x=��uldxidt,(5)where �� is the density of tissue; c the specific heat; k the thermal conductivity; xi the interface position of freezing front; T the temperature; x the space coordinate; t the time; qb the blood perfusion term; l the latent heat of fusion; and qm the metabolic heat generation in the tissue.

Subscripts u and f are for unfrozen and frozen state, respectively, and ph and I are for phase change and initial states, respectively. Assuming the negligible effect of blood perfusion and using the following dimensionless variables and Tu?=Tu?T0Tph?T0,qm?=qmL2ku(Tph?T0),(6)where??K?=kukf,Tf?=Tf?T0Tph?T0,??t?=��utL2Ste,??��?=��u��f,x?=xL,??��u=ku��ucu,??constants:��f=kf��fcf, Ste is the Stefan number defined as Ste = cu(Tph ? T0)/l.Equations (1) and (2) for??xi?��x?��1.(7)The?for??0��x?��xi?,Ste???Tu??t?=?2Tu??x?2+qm??becomeSte?��??Tf??t?=?2Tf??x?2 initial and boundary conditions (3)-(4) become(8)(9)(10)Condition at phase change interface equations (5) is transformed toTf?(xi?,t?)=1=Tu?(xi?,t?),(11)1K??Tf?(xi?,t?)?x???Tu?(xi?,t?)?x?=dxi?dt?.(12)3.

Quasi-Steady ApproximationDue to nonlinearity of the interface energy equation, there are few exact solutions to the problems with phase Anacetrapib change. The condition at phase change equation is nonlinear because the interface velocity dxi/dt depends on the temperature gradients. In this model, the Stefan number is taken small compared to the unity. A small Stefan number corresponds to the sensible heat which is small compared to the latent heat. The interface moves slowly for a small Stefan number, and the temperature distribution at each instant corresponds to that of steady-state. Quasi-steady approximation is justified for Ste < 0.1 [35, 36].