A model for the mass of dye in the heart (mg) at any time from $t=2$ seconds until the end of the procedure: $H(t)=35e^{-0.916t}$
How was this derived?
The following information was used:
- 60% of the dye in the patient’s heart is ejected into her circulatory system each second (this meant an exponential model was an appropriate choice)
- A total of $8$ mg of dye is injected into her heart at a constant rate for a period of $2$ seconds
- Assume that the heart beats once per second during the procedure
- $S=-4$ where $S(t)$ is the mass of dye in the syringe (mg)
- $H'=4-0.6H$ where $H(t)$ is the mass of dye in the heart (mg)
- $C'=0.6H$ where $C(t)$ is the mass of dye that has entered general circulation (mg)
- When $t=2$, $S=0$, $H=5.6$ and $C=2.4$
Assumptions:
- Heart beats once per second
- No dye that has entered circulation has time to return to the heart
- All flows occur continuously
The description is not for a continuous model, it reads more like a discretized model based on one-second intervals or heart beats. Then in the interpretation "heart beat, then wait one second, measure" to get the evolution $$ H_1=4+0.4H_0\\ H_2=4+0.4H_1\\ H_{k+1}=0.4H_k $$ with values $H_0=0$, $H_1=4$, $H_2=5.6$, and for $k>2$ $$ H_k=5.6⋅0.4^{k-2}=35⋅0.4^k=35⋅e^{-0.916290731874155⋅k} $$ These are the concentrations directly before a heart beat.