After injection of a dose D of insulin, the concentration of insulin in a patien

Question

After injection of a dose D of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as De^(-at) where t respresents time in hours and a is a positive constant.

a) If a dose D is injected every T hours, write an expresssion of the sum of the residual concentrations just before the (n+1)st injection.

b) Determine the limiting pre-injection concentration.

c) If the concentration of insulin must always remain at or above a critical value C, determine the minimal dosage D in terms of C,a, and T.

Explanation / Answer

interesting computations...

note : f_1(t) = D e^(-a t) for t < T

f_2 (t) = [ f_1(T) ] e^ (-a [t- T] ) + D e^(-a{ t - T ] ) , for t in [T, 2T)

f_2(t) = [ D e^(-a T ) + D ] e^( -a [ t - T ])

f_3 (t) = { [ f_2( 2T) ]e^(-a T) + D }e^(-a [ t - 2T ) , for t in [2T , 3T)

f_3(t) = [ D e^ ( - 2 a T ) + D e^( - a T) + D ] e^( - a [ t - 2T ] ) ;

f_4 (t) = { [ f_3(3T) ] e^(-a T) + D } e^( - a [ t - 3T ] ) , t in [ 3T , 4 T) ;

f_4(t) = [ D e^( -3 a T) + D e^(- 2 a T) + D e^(-1 a T) + D] e^( -a [ t - 3T] ) ;

f_4(t) = D [ 1 +e^(-1aT) + e^(-2aT) + e^(- 3aT) ] e^( - a [ t - 3T] ) ;

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