# Presentation: 333884

## Proposal for Kinetic Analysis of Metabolism Data

- William P. Eckel
- USEPA/OPP/EFED
- October 2006
- Revised January 2007

## The problem...

- Metabolism data sets often do not follow first-order kinetics, but...
- Models require first-order rate constants, and...
- First-order data reduction techniques misrepresent non-1st-order data sets

## Definitions

- Half-life = ln(2)/k, where k is the rate constant from data that obey first-order rate law
- DT50: time to 50% decay (empirical or fitted)
- DT90: time to 90% decay (empirical or fitted)

## Definitions

- Log-linear:  take natural log, do linear regression
- Exponential decay/SFO-2/”Non-linear”: regress data without transformation according to Ct = C0e-kt

## First-order Data Reduction Techniques: Problems

- Log-linear overestimates both DT50 and DT90, often by a large margin
- Log-linear: Calculated half-life is a function of the length of the experiment
- Exponential Decay (Single First-Order 2-parameter) fits DT50 reasonably well, but underestimates DT90
- Other methods given in FOCUS report share the problem of applying 1st-order methods to non-1st-order data

## Draft Data Quality Objectives

- _Descriptive_:  find the simplest kinetic equation that accurately reproduces the concentration-time data, paying particular attention to C0, DT50 and DT90.  The kinetic equation should have the minimum number of adjustable parameters needed to accurately reproduce the data, and need not be first-order.  “Accurately reproduce” means that it gives the smallest error (vs. alternatives) when compared to the actual data, particularly at the points C0, DT50, DT90.

## Draft Data Quality Objectives

- _Model-Input_:  derive a metabolism input parameter (half-life or first-order rate constant) that conservatively represents the data at least up to DT90 and minimizes overestimation of concentrations before DT90.  “Conservatively represents” means that the equation gives a concentration equal to or greater than the observed average concentration.

## Degradation of a Single Reactant

- dC/dt = -kCn
- dC/Cn = -kdt
- If n = 1,
- ln(C0/C) = -kt   (“first-order”)
- If n ≠ 1,
- (1/(n-1))*((1/Cn-1) – (1/C0n-1)) = -kt
- Reference: Moore & Pearson, Kinetics and Mechanism, 3rd Ed., 1981, pp. 19-20

## Integrals

- ∫ dx/x = log x
- ∫ xn dx = xn+1 / n+1  (except n=1)

## “First Order Multi-Compartment” vs. IORE

- C = C0/(t /β + 1)α
- C = C0(bt + 1)a
- a = -α (FOMC), -1/n-1 (IORE)
- b = 1/β (FOMC), [k(n-1)]*C0n-1
- FOMC and IORE are therefore equivalent, *non-first-order* equations

## Indeterminate Order Rate Equation 

- (1/n-1)*(1/Cn-1  -  1/C0n-1) = kt        n ≠ 1
- Zero:    C0 – C = kt
- Half:    2(C0)1/2 – 2(C)1/2 = kt
- Three-halves:  2/C1/2 – 2/C01/2 =kt
- Two:  1/C – 1/C0 = kt
- 2-Parameter fit: k and n

## Expected DT90/DT50 ratios

| Reaction Order, n | Expected
DT90/DT50 | Expected
DT75/DT50 |
| --- | --- | --- |
| 0 | 1.8 | 1.5 |
| 0.5 | 2.33 | 1.71 |
| 1 | 3.32 | 2 |
| 1.5 | 5.22 | 2.41 |
| 2 | 9 | 3 |

- Expected DT90/DT50 ratios

## Slide 13

## Comparison of 3 Methods to a 2nd-order Data set with DT50 = 1, DT90 = 9 

| Procedure | DT50 | DT90 |
| --- | --- | --- |
| Second-order
(example data) | 1 | 9 |
| Log-Linear | 12.2 | 40.5 |
| SFO/2
“non-linear” | 1.35 | 4.48 |
| Proposed method | 2.71 | 9 |

**Notes:**
Data set goes through 6 halvings to reach 1/64 (1.5625%) at t = 63.  DT50 and DT90 for log-linear increase as data are included beyond DT90.  If only followed to DT90, log-linear gives 2.9 and 9.67.  The second-order and SFO/2 methods do not change 

## Example 2: DT50 = 40, DT90 = 360

| Procedure | DT50 | DT90 |
| --- | --- | --- |
| 2nd-order (example data) | 40 | 360 |
| Log-Linear | 165
(4.1x) | 548 |
| Proposed method | 108
(2.7x) | 360 |

**Notes:**
Data set extends to 600 days, conc = 6.25% (4 halvings)

## Proposed Methods

- “Divide and Conquer”: Descriptive and Model-Input methods are separated
- _Descriptive_: Indeterminate order rate  equation if n ≠ 1
- _Model-Input_: from best descriptive fit, input parameter = DT90/3.32 (from expected ratio in first-order kinetics, if n > 1)

## Benefits

- Data sets described more simply & accurately by I.O.R.E.
- Input parameter does not over- or under-estimate DT90: conservatively represents data to this point
- Input parameter over-estimate of DT50 is limited to a knowable factor (2.7x for 2nd-order data): conservative for acute
- Robust to data set length (time)
- May be used to make estimates for data sets with 4 or more data points
- Provides data on commonly observed kinetics for future model modifications

## Slide 18