# Assume that r is constant and obtain an equahon for the quanaty of therapeutics in the heart as a function of time yp(I) [mg]. You will need to write and solve an ordinary differential equation

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# Model 1: Drug Transport into the Heart 40%]

Figure 1. Schematic of blood flow in the heart.

### Nutrients and therapeutics are transported throughout our body and into organs via the circulation of blood. Suppose that blood flowing in and out of the heart with rate r [cm3 min- ] carries a drug at a concentraaon c [mg cm-3]. The volume of blood in the heart is also given by Pg [cm3]. The average human heart can pump 4 litres of blood per minute and has an average volume of 280 cm*.

a) [30 marks] Assume that r is constant and obtain an equahon for the quanaty of therapeutics in the heart as a function of time yp(I) [mg]. You will  need  to write and   solve   an   ordinary   differential   equation   that   models   the   mass   rate  of {mg min-`] at any time the heart. At I = 0 there is no drug in the heart, but the concentration in the bloodstream is r.

### Although r was assumed to be constant in queshon la, the eoncentrahon of drugs injected in the bloodstream is known to decay as an exponential function of time given by:

In this expression, D [mgJ is the dose of drug injected, d = 1.3 R !8"* [ “1 is

constantforthe drugand tisthetñnein {mi"]

TheerxpBieenB1 •  `=• ol d()[ â   ]* 8 the   * ” an•‹ied,o,Nspapw;nE«a(xL)andMATLA8(magfonnatusethammm

urruznu4auoninihehean,oGeoneobtaIneJing«rsHo› JswiGacomtarab1oo•J

# Table 1. Experimental measurements for drug concentration in blood plasma.

 I [otin] Meaauzezneat 1 Meaauzezneot 2 hteaauzement 3 0 0.044899 o os5775 o.0445zt 0.2 o.os9223 0.040378 o.04 0.4 0.034452 0.033888 0.034524 0.6 0.029701 0.029814 0.02P149 0.8 0.026727 0.02711 0.026908 1 0.023718 0.02256 0.022417 12 0.019134 0.020123 0.019274 1.4 0.016724 0.018039 o.m6481 1.6 0.015413 0.015993 0.014639 1.8 0.012824 0.013373 0.012533 2 0.012316 0.010604 0.011679 2.2 o.o1084 0.tO8988 0.tD`9tH7 2.4 0.008908 0.008857 0.008179 16 0.00822 o ooz64 0.007076 2.8 0.006342 0.00745 0.007288 0.005951 0.tXifi177 0.O3SS8 3.2 0.005914 0.005275 0.fDS46 3.4 0.00443 0.tXi4359 0.fD4S17 3.6 0.004612 0.003279 0.004315 3.8 0.003704 o.O341as o.a2S05 4 0.002814 0.002925 0.003792

Model 2: Conservation of Endangered Species [io• ]