# Non Compartmental Analysis

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### Student Objectives for this Chapter

• To understand and use the non compartmental approach to parameter estimation
• be able to define, use, and calculate the parameters:
• AUMC (area under the first moment curve)
• MRT (mean residence time)
• MAT (mean absorption time)
• MDT (mean dissolution time)

## Non Compartmental Analysis

Non compartmental methods can be used to determine certain pharmacokinetic parameters without deciding on a particular compartmental model. The basic calculations are based on the area under the plasma concentration versus times curve (zero moment) and the first moment curve (AUMC). The AUC can be calculated as before using the trapezoidal rule. The first moment is calculated as concentration times time (Cp • t). The AUMC is the area under the concentration times time versus time curve. Maybe best covered with an example. Consider a drug given both by IV and oral administration. Both the AUC and AUMC were calculated using the trapezoidal rule without making any assumption concerning the number of compartments. The final segment of the AUC curve is calculated as Cp(last)/k', where k' is the last exponential term (the slowest) calculated from the Cp versus time graph. The last segment for the AUMC curve is:

Equation 20.1.1 Equation of the last segment AUMC

Table 20.1.1 Typical Cp versus Time Data after 100 mg IV Bolus Administration
Time (hr) Cp (mg/L) Cp • t (mg.hr/L) AUC (mg.hr/L) AUMC (mg.hr2/L)
0 8 0 0 0
1 7.09 7.09 7.55 3.55
2 6.29 12.58 14.24 13.39
3 5.58 16.74 20.18 28.05
4 4.95 19.80 25.45 46.32
6 3.89 23.34 34.29 89.46
9 2.71 24.39 44.19 161.06
12 1.89 22.68 51.09 231.67
18 0.92 16.56 59.52 349.39
24 0.44 10.56 63.60 430.75
67.27 549.31

These data, Cp versus time and Cp x time versus time, can be plotted on linear graph.

Figure 20.1.1 Plot of Cp versus Time (IV)

Figure 20.1.2 Plot of Cp x Time versus Time (IV)

and various parameters can be calculated as:

Equation 20.1.2 Equation for Mean Residence Time (MRT)

Equation 20.1.3 Equation for Apparent Elimination Rate constant (kel')

Equation 20.1.4 Equation for Total Body Clearance (TBC)

Equation 20.1.5 Equation for Apparent Volume of Distribution, Steady State (Vss)

From the AUC and AUMC values we can calculate the mean residence time, MRT. This is the average time that the drug stays in the body (or plasma as measured here). It can be related to the average elimination rate constant as 1/MRT. The values from the above data are

MRT = 549.31/67.27 = 8.17 hr

and

kel' = 1/8.17 = 0.12 hr-1

Remember we can also calculate the clearance,

CL = Dose/AUC = 100/67.27 = 1.49 L.hr-1

Finally a steady state volume can be calculated as

Vss = CL • MRT = 1.49 x 8.17 = 12.14 L

Oral data can be analyzed by these methods as well.

Table 20.1.2 Typical Cp versus Time Data after 250 mg Oral Administration
Time (hr) Cp (mg/L) Cp • t (mg.hr/L) AUC (mg.hr/L) AUMC (mg.hr2/L)
0 0 0 0 0
1 12.18 12.2 6.09 6.09
2 14.12 28.24 19.24 26.30
3 13.43 40.29 33.02 60.57
4 12.16 48.64 45.82 105.04
6 9.64 57.84 67.62 211.52
9 6.73 60.57 92.18 389.14
12 4.69 56.24 109.31 564.42
18 2.28 41.22 130.25 856.92
24 1.11 26.64 140.45 1060.50
149.70 1359.58

These data, Cp versus time and Cp x time versus time, can be plotted on linear graph.

Figure 20.1.3 Plot of Cp versus Time (PO)

Figure 20.1.4 Plot of Cp x Time versus Time (PO)

and additional parameters can be calculated (NOTE: We don't calculate Clearance or Vss using oral data).

Equation 20.1.6 Equation for Mean Absorption Time (MAT)

Equation 20.1.7 Equation for Apparent Absorption Rate Constant (ka')

Equation 20.1.8 Equation for Oral Bioavailability (F)

The data were calculated after a 250 mg oral dose of the same drug. From these data a MRT was calculated as

MRT = AUMC/AUC = 1361/149.8 = 9.08 hr

We can subtract from this MRT(PO) the MRT(IV) to get an idea of the absorption process, the mean absorption time (MAT). That is

MAT = MRT(PO) - MRT(IV) = 9.08 - 8.17 = 0.92 hr

From this we can calculate an average absorption rate constant

ka' = 1/MAT = 1/0.92 = 1.09 hr-1

Of course we can calculate the bioavailability of the oral dosage form using the dose adjusted AUC ratio. Thus

F = (149.70/67.27) x (100/250) = 0.89

## Another Example

### using Data from Bevill et al. 1977

Pharmacokinetics of Sulfamethazine in Cattle following IV and Three Oral Dosage Forms Dosage Forms

• IV Bolus 107 mg/Kg
• Oral Solution 107 mg/Kg
• Oral Rapid Release Tablet 105 mg/Kg
• Oral Sustained Release Tablet 249 mg/Kg

Figure 20.1.5 The Average Data from Each Dosage Form

Table 20.1.6 Non Compartmental Analysis Results
AUC AUMC MRT (hr) MAT (hr) MDT (hr)
IV 437 6393 14.6 - -
Solution 431 9454 21.9 7.3 -
Tablet 450 11303 25.1 10.5 3.2
Slow Tablet 765 45484 59.5 44.9 37.6

Table 20.1.7 Comparison with Non Linear Regression Analysis (SAAM)
Rate Constant (hr-1) Non Linear Regression Analysis Non Compartmental Analysis
kel' 0.077 0.068
ka' 0.11 0.14
kd' (fast) 0.41 0.31
kd' (slow) 0.026 0.027

Non compartmental analysis:

kel' = 1/MRT

ka' = 1/MAT

kd' = 1/MDT

where MDT is the mean dissolution time

Want more practice with this type of problem!
References

• Bevill, R.F., Dittert, L.W. and Bourne, D.W.A. 1977 Pharmacokinetics of Sulfamethazine in Cattle following IV and Three Oral Dosage Forms, J. Pharm. Sci., 66, 619-23