CalcTMax determines the time (Tmax) at which a pharmacokinetic model achieves its maximum concentration. Because Tmax cannot be directly calculated using peak() statement for models with closed-form solutions, this function employs a numerical method to find it (see “CalcTMax numerical method”).
lambertw: the Lambert W-function is the inverse of function x given by x(y)=y*exp(y).
lgamm: logarithm of the gamma function (e.g., lgamm(x)).
factorial: factorial function (e.g., factorial(x) returns x factorial, and is the same as x!=x*(x – 1)*…*1).
erfunc: error function (e.g., erfunc(x) returns the value of the error function at x).
abs: absolute (e.g., abs(x) return the absolute value of x).
min: minimum (e.g., min(x, y))
max: maximum (e.g., max(x, y))
log10: log base 10 (e.g., log10(x))
ln: natural log (e.g., ln(x))
vfwt: observation variance function with its syntax given by vfwt(f, p), which returns the value of max(0, f^(p/2).
The CalcTMax function supports pharmacokinetic models represented by combinations of exponential terms. The most general form is the 3-compartment model with first-order absorption:
f(x) = A * exp(-a * x) + B * exp(-b * x) + C * exp(-c * x) - (A + B + C) * exp(-Ka * x)
where:
x typically represents time.
A, B, C: Coefficients representing the contribution of each compartment.
a, b, c: Elimination rate constants for each compartment.
Ka: Absorption rate constant.
The function CalcTMax can be called with different numbers of arguments depending on the complexity of the model.
3-Compartment Model with First-Order Absorption:
Tmax = CalcTMax(A, a, B, b, C, c, Ka)
A, B, C: Coefficients of the three compartments.
a, b, c: Elimination rate constants for the compartments.
Ka: Absorption rate constant.
2-Compartment Model with First-Order Absorption:
Tmax = CalcTMax(A, a, B, b, Ka)
1-Compartment Model with First-Order Absorption:
Tmax = CalcTMax(A, a, Ka)
Deprecated Interface (2-Compartment Model):
Tmax = CalcTMax(A, a, B, b, C, c)
In this deprecated form:
C must be equal to -(A + B).
c represents the absorption rate constant (equivalent to Ka in the preferred interface).
Notes:
The function assumes that a, b, c, and Ka are positive values representing elimination and absorption rates.
The function also assumes the absorption rate Ka > max(a, b, c). If this is not the case the function will return an inaccurate result.
The numerical method used relies on finding the root of the first derivative of the model function.
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