A model of tumour growth under chemotherapy from time t = 0 tot = t₁> 0 is-C logwhere, at time t, C(t) is the size of the tumour, Cmax > 0 is the maximumsize of the tumour, a constant, so that 0 0 is therate at which the drug is administered.dCdtequation (*) becomesdxdt=-XCCmaxrt₁S[D] = fª dt D(t).S[x] = -D1+ D'wherewhere ·∞ < x≤ 0.It is required to reduce the size of the tumour from Co at t = 0 to C₁ att = t₁. Let x = log(Co/Cmax) and x₁ = log(C1₁/Cmax). For the health of thepatient, it is desired to minimise the total amount of drug administered,which is given by the functionalrt₁["dα=dtDC1+ D2x + 3x + x = -1x + x1 + x + xx = log(C/Cmax),Euler-Lagrange equation for S[x] is given bywhere x = d²x/dt².Solve (**) with the boundary conditions x(0) = x₁ and x(t₁) = x1, toshow that the stationary path of S[x] is given byx(t) = aet+ (xo +1-a)e-t/2 - 1,(x + 1)e−t₁/² − (x₁ +1)e-t₁/2 - e-ti(*)(**)

(.) Euler - Lagrange equation for is, where. .Now we will solve Euler - Lagrange…

A model of tumour growth under chemotherapy from time t = 0 to t = t₁ > 0 is dC dt = -Clog ( C Cmax equation (*) becomes dx dt where, at time t, C(t) is the size of the tumour, Cmax > 0 is the maximum size of the tumour, a constant, so that 0 < C(t) ≤ Cmax, and D(t) ≥ 0 is the rate at which the drug is administered. x = log(C/Cmax), =-x- where D 1+ D' rt₁ S[x] = - " where - < x≤0. t = t₁. Let co = It is required to reduce the size of the tumour from Co at t = 0 to C₁ at log(Co/Cmax) and x₁ log(C₁/Cmax). For the health of the patient, it is desired to minimise the total amount of drug administered, which is given by the functional S[D] = [ª dt D(t). α= DC 1+D - dt x + x 1+x+x Euler-Lagrange equation for S[x] is given by -t x(t) = = ae + (xo + 1 − a)e¯t/2 – 1, (*) 2x + 3x + x = −1 where = d²x/dt². Solve (**) with the boundary conditions x(0) = xo and x(t₁) = x₁, to show that the stationary path of S[x] is given by (xo + 1)e−t₁/² − (x₁ + 1) e-t₁/2 - e-ti (**)

A model of tumour growth under chemotherapy from time t = 0 to t = t₁ > 0 is dC dt = -Clog ( C Cmax equation () becomes dx dt where, at time t, C(t) is the size of the tumour, Cmax > 0 is the maximum size of the tumour, a constant, so that 0 < C(t) ≤ Cmax, and D(t) ≥ 0 is the rate at which the drug is administered. x = log(C/Cmax), =-x- where D 1+ D' rt₁ S[x] = - " where - < x≤0. t = t₁. Let co = It is required to reduce the size of the tumour from Co at t = 0 to C₁ at log(Co/Cmax) and x₁ log(C₁/Cmax). For the health of the patient, it is desired to minimise the total amount of drug administered, which is given by the functional S[D] = [ª dt D(t). α= DC 1+D - dt x + x 1+x+x Euler-Lagrange equation for S[x] is given by -t x(t) = = ae + (xo + 1 − a)e¯t/2 – 1, () 2x + 3x + x = −1 where = d²x/dt². Solve () with the boundary conditions x(0) = xo and x(t₁) = x₁, to show that the stationary path of S[x] is given by (xo + 1)e−t₁/² − (x₁ + 1) e-t₁/2 - e-ti ()

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter6: Exponential And Logarithmic Functions

Section6.8: Fitting Exponential Models To Data

Problem 1TI: Table 2 shows a recent graduate’s credit card balance each month after graduation. a. Use...

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