# Chapter 6

# Intravenous Infusion

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## Continuous IV Infusion - Time to Reach Steady State

Another important factor is the time to reach the steady state concentration. The time to reach half the steady state can be derived:
Since

and

then

or

taking the **ln** of both sides

Thus

Thus the approach to Cp^{ss} is exponential in nature and is controlled by the elimination process **NOT** the infusion process. NOTE however that the value of Cp^{ss} **IS** controlled by k0.

Halfway - 50% | to steady state | in | one | half-life |

75% | to steady state | in | 2 | half-life |

87.5% | to steady state | in | 3 | half-life |

94% | to steady state | in | 4 | half-life |

For theophylline with a t_{1/2} equal to 4 hours the time to reach 94% of steady state will be 16 hours. We could calculate how long it might take to reach a therapeutic concentration. For theophylline this might be 10 mg/L

Thus

Using the values from before

k0 = 60 mg/hr; kel = 0.17 hr^{- 1}; V = 25 L; and C_{required} = 10 mg/L

thus

and taking the ln of both sides gives -0.17 * t = -1.231 or t = 7.24 hr

**Figure 6.3.1 Plot of Cp ***versus* Time showing Approach to Steady State
Thus if we started an infusion to achieve a steady state plasma concentration of approximately 15 mg/L (actually 14.1 mg/L) it would take 7.25 hours to reach a therapeutic level of 10 mg/L. This is probably too long so another strategy should be explored.

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