As the fingerprint method is an extension of the general partial fraction method, it can be applied only to fractions of one function in "s" over another. Furthermore, the highest exponent of "s" that occurs when the denominator function is fully expanded (called the "degree of s") must be higher than that of the expression in the numerator. Finally, there can be no repeating terms in the simplified version of the denominator, which would yield repeating roots. For example, a function with s^2 in the denominator cannot be inverse-transformed with the fingerprint method, because the term "s" is repeated (s^2 = s * s), which would yield two roots of zero. The same is also true of a denominator of (s + 1)(s + 1).
Once the function is in the correct format and found to be in the proper format for the fingerprint method, the next step is to determine the roots of the denominator. This is accomplished by setting each of the factoring terms in it to zero, one at a time, and recording the value of "s" in each case. For example, if the denominator consists of the expression "s(s + 3)(s + 4)," then the three roots in order from left to right would be 0, -3 and -4. In pharmacokinetic equations and functions, these roots will almost always be either zero or negative numbers.
With the roots known, the entire function as a whole is now examined, both numerator and denominator. One of the factors in the denominator is ignored or "covered up" (hence the name "fingerprint"), and all other instances of "s" are set equal to the root corresponding to this omitted factor. The resulting fraction is then multiplied by Euler's number ("e") raised to the power of the product of the root and "t," the time variable.
This step is repeated for each term and root pair in the denominator, then each result is summed to generate the final inverse transform.
Suppose a pharmacokinetic expression for the rate of absorption of a drug is (s + 1) / [ s(s + 2)(s + 3) ]. The degree of the denominator, three, is higher than the first-degree numerator, and no repeating factor terms are present, so the fingerprint method can be used to approximate an inverse Laplace transform.
From observation of the denominator, the roots are determined to be 0, -2 and -3. By covering up the "s," and setting all other "s" to zero, the function becomes "[1 / (2)(3)] * e^(0*t)," or "(1/6)."
Covering up the second term, (s + 2), and setting "s" in the remaining expression to -2 gives "{(-2 + 1)/[-2(-2 +3)] * e^(-2t)}," which can be simplified as "(1/2)*e^(-2t)."
Repeating this process for the third and last root gives "{(-3 +1)/[-3(-3+2)]} * e^(-3t)," or "-(2/3)*e^(-3t)."
Since the coefficients of all three have a common factor of 1/6, the sum of all three iterations of the fingerprint method can be simplified to produce a final answer of:
(1/6) * [1 + 3e^(-2t) - 4e^(-3t)].