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Unformatted text preview: xture of these m formulations. We let a(k) ∈ Rm denote the amounts of
+
the m formulations in treatment k , for k = 1, . . . , K .
(k ) Each formulation i has a time proﬁle pi (t) ∈ R+ , for t = 1, 2, . . .. If an amount ai of formulation
i from treatment k is administered at time t0 , the drug concentration in the bloodstream (due to
(k )
this formulation) is given by ai pi (t − t0 ) for t > t0 , and 0 for t ≤ t0 . To simplify notation, we will
deﬁne pi (t) to be zero for t = 0, −1, −2, . . .. We assume the eﬀects of the diﬀerent formulations and
diﬀerent treatments are additive, so the total bloodstream drug concentration is given by
K m c ( t) =
k=1 i=1 (k ) pi ( t − τk ) a i , t = 1, . . . , T. (This is just a vector convolution.) Recall that pi (t − τk ) = 0 for t ≤ τk , which means that the
eﬀect of treatment k does not show up until time τk + 1.
We require that c(t) ≤ cmax for t = 1, . . . , T , where cmax is a given maximum permissible concentration. We deﬁne the therapeutic t...
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 Fall '13
 F.Borrelli
 The Aeneid

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