Week 1: Mathematical Foundations of Pharmacokinetics

Lecture Notes

1 Course Overview

This 13–14 week program covers the first principles and fundamentals of pharmacokinetics, with an emphasis on mapping theoretical concepts to real-world applications. The course is designed to bridge the gap between understanding equations and applying them in practice.

1.1 Topics to Be Covered

Basic mathematics for pharmacokinetics
One-compartment models (IV bolus)
Volume and clearance
Constant-rate infusions
Extravascular administration and biopharmaceutical considerations
Multiple dosing concepts
Nonlinear systems
Non-compartmental analysis (NCA)
Allometric scaling

Population PK modeling will follow in a later phase, along with advanced statistics, before moving into full pharmacometric modeling.

1.2 Course Pedagogy

All lecture material is provided as structured notes (HTML/Quarto markdown documents) with accompanying Julia tutorials. Practice problems are assigned weekly and are to be completed using computational tools — all graphing, slope calculations, and analyses should be performed programmatically (e.g., in Pumas), not by hand on graph paper.

2 1. Core Terminology: Variables, Parameters, and Constants

2.1 1.1 Dependent and Independent Variables

Consider the fundamental pharmacokinetic equation for a one-compartment IV bolus:

\[C(t) = \frac{\text{Dose}}{V} \cdot e^{-K_e \cdot t}\]

In any equation or dataset:

Dependent variable (y-axis): The quantity whose value is determined by the model or system. In PK, plasma concentration $C(t)$ is the dependent variable — it changes within the individual over time.
Independent variable (x-axis): The quantity whose value is set or predefined by the investigator. Time after drug administration is the independent variable; the sampling schedule is determined by the study design.

2.2 1.2 Parameters

A parameter is a numeric value that defines the relationship between independent and dependent variables within the context of a specific model. For the equation above, $K_e$ and $V$ are parameters. Within a given model fit to a given individual, parameters are treated as fixed (constant) values — they do not change with time.

It is important to distinguish: a parameter is derived from data (it is estimated), but once estimated, it is held constant within that model context. For example, an estimated $K_e = 0.02\ \text{hr}^{-1}$ is a constant term that fully characterizes the elimination rate for that individual.

2.3 1.3 Constants

A constant is a value that is not measured or estimated within the current analysis — it is assumed known. For example, hepatic blood flow is often cited as $1.5\ \text{L/hr}$; this is treated as a physiological constant, though it was originally derived as a parameter from some prior experiment.

Key distinction for scientific communication: A parameter can also function as a constant depending on context. Misuse of these terms is common in publications and should be avoided.

2.4 1.4 Data

Data consists of measured values in relationship to one another. A list of plasma concentrations without associated time points carries limited meaning; it is the pairing of concentration with time (and other covariates) that constitutes analyzable data. This distinction becomes especially important in physiologically-based pharmacokinetic (PBPK) modeling, where individual pieces of information require context to be interpretable.

2.5 1.5 Errors

In the context of modeling, residual error is the difference between an observed value and the value predicted by the model. However, the concept of error is broader:

Independent variables such as time carry measurement error (a nominal 2-hour sample may have been collected at 2 hours 5 minutes).
Dependent variables such as concentration carry analytical error, quantified by the bioanalytical method’s standard curve precision.

The magnitude of error must always be assessed relative to its impact on inference. All data carry errors. Being cognizant of data quality — before fitting any model — is a critical skill in pharmacokinetics.

3 2. Units and Dimensionality

3.1 2.1 Why Units Matter

An equation is only valid if the units on both sides are consistent. For example:

\[\text{CL} = K_e \cdot V\]

$K_e$ has units of $\text{hr}^{-1}$
$V$ has units of $\text{L}$
Therefore $\text{CL}$ has units of $\text{L/hr}$ — consistent with the left-hand side.

3.2 2.2 Two Fundamental Rules of Dimensionality

Addition and subtraction are only valid between quantities that share the same units.
Multiplication and division of quantities with different units produce a new composite unit.

For example, $K_e + V$ is dimensionally invalid (one cannot add $\text{hr}^{-1}$ and $\text{L}$), whereas $K_e \times V$ produces $\text{L/hr}$, a physically meaningful quantity.

3.3 2.3 Dimensionless Quantities

Certain quantities in pharmacokinetics are dimensionless — they carry no units:

Bioavailability ($F$): a fraction representing the ratio of absorbed drug to administered dose.
Fraction unbound ($f_u$): similarly a ratio.
Percentages: when a percentage represents a fraction, it remains dimensionless (multiplying a dimensionless fraction by 100 does not introduce a dimension). Note, however, that percentages can sometimes be applied to quantities with dimensions (e.g., percent weight/volume concentrations used in dispensing pharmacy).
Logarithms: $\log(C)$ is dimensionless, regardless of the units of $C$. This is because a logarithm is fundamentally the logarithm of a ratio (discussed further in Section 4). Although it is conventional in PK to label log-concentration axes with concentration units as a communication aid, the mathematical quantity $\ln(C)$ itself is unitless. Publications that assign units to log-transformed values are technically imprecise.

4 3. Unit Conversions

Correct unit conversions are essential before any pharmacokinetic analysis. The general strategy is to handle each dimension (mass, volume, time) separately.

4.1 3.1 Single-Dimension Conversions

Example — dose conversion: \[100\ \text{mg} \times 10^3\ \frac{\mu\text{g}}{\text{mg}} = 10^5\ \mu\text{g}\]

Example — concentration conversion (mass only):

To convert from $\text{ng/mL}$ to $\text{mg/mL}$, only the mass unit changes: \[C\ [\text{ng/mL}] \times 10^{-6}\ \frac{\text{mg}}{\text{ng}} = C\ [\text{mg/mL}]\]

4.2 3.2 Multi-Dimensional Conversions

Example — clearance:

To convert $15\ \text{L/hr}$ to $\text{mL/min}$: \[15\ \frac{\text{L}}{\text{hr}} \times \frac{1000\ \text{mL}}{1\ \text{L}} \times \frac{1\ \text{hr}}{60\ \text{min}} = 250\ \text{mL/min}\]

Each dimension is scaled independently.

4.3 3.3 Molar Conversions

Converting from mass-based to molar concentration units (e.g., $\text{ng/mL}$ to $\text{nM}$) requires knowledge of the drug’s molecular weight (in $\text{g/mol}$). The steps are:

Convert nanograms to milligrams (mass unit).
Convert milligrams to grams.
Convert grams to moles using molecular weight.
Convert moles to nanomoles.
Convert mL to liters (volume unit).

Real-world datasets frequently present dose in milligrams and concentration in nanomolar, requiring multi-step conversions. Fluency in these conversions is a practical necessity.

4.4 3.4 Machine Precision and Floating-Point Arithmetic

When performing conversions computationally, results may appear with a large number of decimal places (e.g., $250.00000000000003$ instead of $250$). This is an artifact of floating-point machine precision — the way a computer’s hardware represents real numbers internally. The result is mathematically correct to hardware precision but carries more decimal digits than are meaningful.

This introduces the practical need to understand significant figures and rounding.

5 4. Significant Figures and Rounding

5.1 4.1 Definitions

Significant digits (significant figures): The total number of meaningful digits in a number. Leading zeros are not significant; trailing zeros after a decimal point are significant.
- Example: $123.4$ has 4 significant digits.
- Example: $0.00234$ has 3 significant digits (the leading zeros are not counted).
Decimal places (digits): The number of digits to the right of the decimal point.
- Example: $12.345$ has 3 decimal places.

5.2 4.2 Rounding Rules

When rounding, if the digit immediately following the last retained digit is $\geq 5$, the last retained digit is incremented by one.

5.3 4.3 When to Use Each

Decimal places are preferred when communicating precision in a specific numerical range — e.g., reporting a PK parameter to 2 decimal places conveys the precision of the estimate.
Significant figures are preferred when values span a wide range of magnitudes — e.g., setting all entries in a results table to 3 significant figures ensures consistency regardless of whether the value is $0.023$ or $980$.

Practical example — bioequivalence: The 90% confidence interval limits of $80.00$–$125.00\%$ must not be rounded. A lower bound of $79.9357\%$ cannot be rounded up to $80\%$ to claim a passing result. Regulatory guidelines (e.g., FDA bioequivalence guidance) specify exactly how decimal places are to be handled in such assessments.

5.4 4.4 Printing vs. Rounding in Code

In literate programming environments (e.g., Quarto/R Markdown documents), it is often preferable to display a number with reduced precision without altering the underlying stored value. Formatting functions (e.g., Printf.@sprintf in Julia) print a formatted string while retaining the full-precision value in memory. By contrast, round(x; digits=2) creates a new variable with the rounded value. Both approaches are valid, but their distinction matters when the rounded value is subsequently used in further calculations.

6 5. Exponential and Logarithmic Functions

6.1 5.1 Definitions

The logarithm of a number is the power to which a base must be raised to produce that number.

\[\log_b(x) = y \iff b^y = x\]

Two bases are used in pharmacokinetics:

Common logarithm ($\log_{10}$): logarithm to base 10.
Natural logarithm ($\ln$): logarithm to base $e$, where $e \approx 2.718$.

The relationship between the two:

\[\ln(x) = 2.303 \cdot \log_{10}(x)\]

Important note for Julia: The built-in log() function computes the natural logarithm. To compute a base-10 logarithm, use log10().

6.2 5.2 Why Logarithmic Transformation is Used in PK

Two principal reasons:

Visualization: Concentration data spanning several orders of magnitude are compressed onto a manageable scale, separating data points that would otherwise overlap.
Linearization: An exponential decay curve becomes a straight line on a semi-logarithmic plot. Linear relationships are far easier to analyze, parameterize, and interpret than nonlinear ones. This is the foundation for graphical estimation of PK parameters (e.g., terminal slope estimation).

Example: The log-linear PK equation

\[\ln[C(t)] = \ln\left(\frac{\text{Dose}}{V}\right) - K_e \cdot t\]

has the form $y = \text{intercept} + \text{slope} \times t$. The slope equals $-K_e$, and if $K_e = 0.38\ \text{hr}^{-1}$, the interpretation is that approximately 38% of the drug present in plasma is eliminated per hour (under first-order kinetics).

6.3 5.3 Why Logarithms Are Dimensionless

A logarithm is the logarithm of a ratio — the argument of a logarithm must be dimensionless. When we write $\ln[C(t)]$, we are implicitly computing $\ln[C(t)/C_{\text{ref}}]$ where $C_{\text{ref}}$ is a reference concentration of 1 (in whatever units $C$ is expressed). The result carries no units. This is why plotting log-concentration versus time yields a dimensionless y-axis, even though the original concentration had units.

6.4 5.4 Fundamental Rules of Logarithms

Rule	Expression
Product rule	$\log(xy) = \log(x) + \log(y)$
Quotient rule	$\log(x/y) = \log(x) - \log(y)$
Power rule	$\log(x^n) = n \cdot \log(x)$
Log of 1	$\log(1) = 0$

PK application of the power rule: Allometric scaling relationships take the form:

\[\log(\text{CL}) = \log(a) + b \cdot \log(\text{BW})\]

where body weight (BW) is the predictor. This is a log-log linear relationship arising directly from the power rule.

6.5 5.5 Rules of Exponents

Exponents are shorthand for repeated multiplication. The rules of exponents are used extensively in PK — particularly in model-building and parameter transformations:

Rule	Expression
Product rule	$a^m \cdot a^n = a^{m+n}$
Power of a power	$(a^m)^n = a^{mn}$
Zero exponent	$a^0 = 1$ ($a \neq 0$)
Negative exponent	$a^{-n} = 1/a^n$

7 6. Rates and Orders of Process

7.1 6.1 Zero-Order Processes

For a zero-order process, the rate of change of concentration is constant and independent of the current concentration:

\[-\frac{dC}{dt} = K_0\]

Integrating yields:

\[C(t) = C_0 - K_0 \cdot t\]

This is a linear function of time, yielding a straight line on a standard (linear) concentration–time plot.

Units of $K_0$: Since concentration has units of (e.g.) $\text{ng/mL}$ and time has units of $\text{min}$, $K_0$ has units of $\text{ng/mL/min}$ (concentration per unit time).

Intuition: The rate of elimination does not depend on how much drug remains. A canonical example from physiology is alcohol metabolism at saturating concentrations — the enzymatic machinery is fully occupied and processes a fixed amount of substrate per unit time regardless of how much substrate is present.

7.2 6.2 First-Order Processes

For a first-order process, the rate of change is proportional to the current concentration:

\[-\frac{dC}{dt} = K_1 \cdot C\]

The rate decreases as concentration decreases — a larger concentration drives a faster rate of elimination.

Units of $K_1$: $\text{hr}^{-1}$ (or $\text{min}^{-1}$, etc.) — a pure rate constant with units of reciprocal time.

Intuition: The fraction of drug eliminated per unit time is constant. If $K_1 = 0.38\ \text{hr}^{-1}$, then 38% of the drug present at any given moment is eliminated each hour.

Most pharmacokinetic absorption and elimination processes are assumed to be first-order. Understanding the mechanistic basis for this assumption — rather than simply accepting it as convention — is essential for building correct PK models.

8 7. Differential Equations in Pharmacokinetics

8.1 7.1 The Noyes–Whitney Equation

The Noyes–Whitney equation describes the rate of dissolution of a solid drug in a medium:

\[\frac{dA}{dt} = D \cdot S_A \cdot \frac{C_s - C}{h}\]

where $D$ is the diffusion coefficient, $S_A$ is the effective surface area, $h$ is the boundary layer thickness, $C_s$ is the saturation concentration, and $C$ is the bulk concentration. This equation is the physical basis for understanding drug dissolution and permeability, which underlies all formulation design and in vitro–in vivo correlations.

8.2 7.2 Rules for Writing Differential Equations for PK Compartmental Models

Four rules govern the construction of differential equations for compartmental models:

Number of equations equals number of compartments. Each compartment requires exactly one differential equation.
Number of terms on the right-hand side equals the number of arrows (transfer processes) connected to that compartment.
Sign of each term is determined by the direction of the arrow:
- Arrow pointing into the compartment → positive term.
- Arrow pointing out of the compartment → negative term.
Coefficient of each term is the associated rate constant (flow constant).

Example — one-compartment model with IV bolus:

For a single compartment containing drug amount $X$, with elimination proceeding by first-order kinetics:

\[\frac{dX}{dt} = -K_{10} \cdot X\]

Units check: $[X] = \text{mg}$; $[K_{10}] = \text{hr}^{-1}$; therefore $\frac{dX}{dt}$ has units of $\text{mg/hr}$. Both sides are consistent.
There is one arrow (elimination out of the compartment) → one term.
The arrow points outward → negative sign.
The subscript notation $K_{10}$ denotes transfer from compartment 1 to the outside (compartment 0).

Two-compartment extension: Transfer between compartments 1 and 2 is denoted $K_{12}$ (from 1 to 2) and $K_{21}$ (from 2 to 1). Terminal elimination from compartment 1 is $K_{10}$.

9 8. Integration and the Area Under the Curve (AUC)

Integration is the mathematical inverse of differentiation. In PK, integration is used to compute the area under the concentration–time curve (AUC), which represents total drug exposure.

\[\text{AUC}_{0}^{t} = \int_0^t C(t')\ dt'\]

The most common numerical method for computing AUC from discrete data is the linear trapezoidal rule:

\[\text{AUC}_{t_i \to t_{i+1}} = \frac{C_i + C_{i+1}}{2} \cdot (t_{i+1} - t_i)\]

The total AUC is the sum of all such trapezoids. This forms the computational basis of non-compartmental analysis (NCA). Extrapolation of AUC from the last observed concentration to infinity requires additional rules (covered in the NCA section).

Relationship to the analytical equation: For a one-compartment IV bolus:

\[\text{AUC}_{0}^{\infty} = \frac{\text{Dose}}{\text{CL}} = \frac{F \cdot \text{Dose}}{\text{CL}}\]

where $F$ is bioavailability (dimensionless) and CL has units of $\text{L/hr}$.

10 9. Common PK Parameters: Symbols and Units

Parameter	Symbol	Typical Units
Plasma concentration	$C$	ng/mL, µg/L, nM
Dose	$D$	mg, µg
Volume of distribution	$V$	L, L/kg
Clearance	CL	L/hr, mL/min
Elimination rate constant	$K_e$	hr⁻¹
Absorption rate constant	$K_a$	hr⁻¹
Half-life	$t_{1/2}$	hr, min
Bioavailability	$F$	dimensionless (fraction)
AUC	AUC	ng·hr/mL, µg·hr/L
Zero-order rate constant	$K_0$	ng/mL/hr

11 Summary

This lecture establishes the mathematical and conceptual foundation required for all subsequent pharmacokinetic work:

Precise use of the terms variable, parameter, and constant is essential for clear scientific communication.
Dimensionality governs the validity of all PK equations; units must balance, and one must always verify unit consistency when deriving or interpreting equations.
Unit conversions must be performed correctly before any analysis; computational tools assist but the analyst must understand the logic.
Significant figures and rounding must be applied appropriately, especially in regulatory contexts such as bioequivalence.
Logarithms linearize exponential PK data and are dimensionless by nature; the natural log is the default in Julia (log()).
Zero-order processes proceed at a rate independent of concentration; first-order processes proceed at a rate proportional to the remaining concentration. This distinction defines the structure of all PK differential equations.
Differential equations for compartmental models follow four simple rules; understanding these rules enables one to translate any pharmacokinetic diagram into a system of equations.
Integration (and specifically the trapezoidal rule) underlies NCA and AUC calculation.

--- title: "Week 1: Mathematical Foundations of Pharmacokinetics" subtitle: "Lecture Notes" format: html: toc: true toc-depth: 3 --- ## Course Overview This 13–14 week program covers the first principles and fundamentals of pharmacokinetics, with an emphasis on mapping theoretical concepts to real-world applications. The course is designed to bridge the gap between understanding equations and applying them in practice. ### Topics to Be Covered 1. Basic mathematics for pharmacokinetics 2. One-compartment models (IV bolus) 3. Volume and clearance 4. Constant-rate infusions 5. Extravascular administration and biopharmaceutical considerations 6. Multiple dosing concepts 7. Nonlinear systems 8. Non-compartmental analysis (NCA) 9. Allometric scaling Population PK modeling will follow in a later phase, along with advanced statistics, before moving into full pharmacometric modeling. ### Course Pedagogy All lecture material is provided as structured notes (HTML/Quarto markdown documents) with accompanying Julia tutorials. Practice problems are assigned weekly and are to be completed using computational tools — all graphing, slope calculations, and analyses should be performed programmatically (e.g., in Pumas), not by hand on graph paper. --- ## 1. Core Terminology: Variables, Parameters, and Constants ### 1.1 Dependent and Independent Variables Consider the fundamental pharmacokinetic equation for a one-compartment IV bolus: $$C(t) = \frac{\text{Dose}}{V} \cdot e^{-K_e \cdot t}$$ In any equation or dataset: - **Dependent variable** (y-axis): The quantity whose value is determined by the model or system. In PK, plasma concentration $C(t)$ is the dependent variable — it changes within the individual over time. - **Independent variable** (x-axis): The quantity whose value is set or predefined by the investigator. Time after drug administration is the independent variable; the sampling schedule is determined by the study design. ### 1.2 Parameters A parameter is a numeric value that defines the relationship between independent and dependent variables within the context of a specific model. For the equation above, $K_e$ and $V$ are parameters. Within a given model fit to a given individual, parameters are treated as fixed (constant) values — they do not change with time. It is important to distinguish: a parameter is *derived from data* (it is estimated), but once estimated, it is held constant within that model context. For example, an estimated $K_e = 0.02\ \text{hr}^{-1}$ is a constant term that fully characterizes the elimination rate for that individual. ### 1.3 Constants A constant is a value that is not measured or estimated within the current analysis — it is assumed known. For example, hepatic blood flow is often cited as $1.5\ \text{L/hr}$; this is treated as a physiological constant, though it was originally derived as a parameter from some prior experiment. **Key distinction for scientific communication:** A parameter *can* also function as a constant depending on context. Misuse of these terms is common in publications and should be avoided. ### 1.4 Data Data consists of measured values in relationship to one another. A list of plasma concentrations without associated time points carries limited meaning; it is the pairing of concentration with time (and other covariates) that constitutes analyzable data. This distinction becomes especially important in physiologically-based pharmacokinetic (PBPK) modeling, where individual pieces of information require context to be interpretable. ### 1.5 Errors In the context of modeling, **residual error** is the difference between an observed value and the value predicted by the model. However, the concept of error is broader: - Independent variables such as time carry measurement error (a nominal 2-hour sample may have been collected at 2 hours 5 minutes). - Dependent variables such as concentration carry analytical error, quantified by the bioanalytical method's standard curve precision. The magnitude of error must always be assessed relative to its impact on inference. **All data carry errors.** Being cognizant of data quality — before fitting any model — is a critical skill in pharmacokinetics. --- ## 2. Units and Dimensionality ### 2.1 Why Units Matter An equation is only valid if the units on both sides are consistent. For example: $$\text{CL} = K_e \cdot V$$ - $K_e$ has units of $\text{hr}^{-1}$ - $V$ has units of $\text{L}$ - Therefore $\text{CL}$ has units of $\text{L/hr}$ — consistent with the left-hand side. ### 2.2 Two Fundamental Rules of Dimensionality 1. **Addition and subtraction** are only valid between quantities that share the same units. 2. **Multiplication and division** of quantities with different units produce a new composite unit. For example, $K_e + V$ is dimensionally invalid (one cannot add $\text{hr}^{-1}$ and $\text{L}$), whereas $K_e \times V$ produces $\text{L/hr}$, a physically meaningful quantity. ### 2.3 Dimensionless Quantities Certain quantities in pharmacokinetics are dimensionless — they carry no units: - **Bioavailability ($F$)**: a fraction representing the ratio of absorbed drug to administered dose. - **Fraction unbound ($f_u$)**: similarly a ratio. - **Percentages**: when a percentage represents a fraction, it remains dimensionless (multiplying a dimensionless fraction by 100 does not introduce a dimension). Note, however, that percentages can sometimes be applied to quantities *with* dimensions (e.g., percent weight/volume concentrations used in dispensing pharmacy). - **Logarithms**: $\log(C)$ is dimensionless, regardless of the units of $C$. This is because a logarithm is fundamentally the logarithm of a ratio (discussed further in Section 4). Although it is conventional in PK to label log-concentration axes with concentration units as a communication aid, the mathematical quantity $\ln(C)$ itself is unitless. Publications that assign units to log-transformed values are technically imprecise. --- ## 3. Unit Conversions Correct unit conversions are essential before any pharmacokinetic analysis. The general strategy is to handle each dimension (mass, volume, time) separately. ### 3.1 Single-Dimension Conversions **Example — dose conversion:** $$100\ \text{mg} \times 10^3\ \frac{\mu\text{g}}{\text{mg}} = 10^5\ \mu\text{g}$$ **Example — concentration conversion (mass only):** To convert from $\text{ng/mL}$ to $\text{mg/mL}$, only the mass unit changes: $$C\ [\text{ng/mL}] \times 10^{-6}\ \frac{\text{mg}}{\text{ng}} = C\ [\text{mg/mL}]$$ ### 3.2 Multi-Dimensional Conversions **Example — clearance:** To convert $15\ \text{L/hr}$ to $\text{mL/min}$: $$15\ \frac{\text{L}}{\text{hr}} \times \frac{1000\ \text{mL}}{1\ \text{L}} \times \frac{1\ \text{hr}}{60\ \text{min}} = 250\ \text{mL/min}$$ Each dimension is scaled independently. ### 3.3 Molar Conversions Converting from mass-based to molar concentration units (e.g., $\text{ng/mL}$ to $\text{nM}$) requires knowledge of the drug's molecular weight (in $\text{g/mol}$). The steps are: 1. Convert nanograms to milligrams (mass unit). 2. Convert milligrams to grams. 3. Convert grams to moles using molecular weight. 4. Convert moles to nanomoles. 5. Convert mL to liters (volume unit). Real-world datasets frequently present dose in milligrams and concentration in nanomolar, requiring multi-step conversions. Fluency in these conversions is a practical necessity. ### 3.4 Machine Precision and Floating-Point Arithmetic When performing conversions computationally, results may appear with a large number of decimal places (e.g., $250.00000000000003$ instead of $250$). This is an artifact of **floating-point machine precision** — the way a computer's hardware represents real numbers internally. The result is mathematically correct to hardware precision but carries more decimal digits than are meaningful. This introduces the practical need to understand significant figures and rounding. --- ## 4. Significant Figures and Rounding ### 4.1 Definitions - **Significant digits (significant figures):** The total number of meaningful digits in a number. Leading zeros are not significant; trailing zeros after a decimal point are significant. - Example: $123.4$ has 4 significant digits. - Example: $0.00234$ has 3 significant digits (the leading zeros are not counted). - **Decimal places (digits):** The number of digits to the right of the decimal point. - Example: $12.345$ has 3 decimal places. ### 4.2 Rounding Rules When rounding, if the digit immediately following the last retained digit is $\geq 5$, the last retained digit is incremented by one. ### 4.3 When to Use Each - **Decimal places** are preferred when communicating precision in a specific numerical range — e.g., reporting a PK parameter to 2 decimal places conveys the precision of the estimate. - **Significant figures** are preferred when values span a wide range of magnitudes — e.g., setting all entries in a results table to 3 significant figures ensures consistency regardless of whether the value is $0.023$ or $980$. **Practical example — bioequivalence:** The 90% confidence interval limits of $80.00$–$125.00\%$ must not be rounded. A lower bound of $79.9357\%$ cannot be rounded up to $80\%$ to claim a passing result. Regulatory guidelines (e.g., FDA bioequivalence guidance) specify exactly how decimal places are to be handled in such assessments. ### 4.4 Printing vs. Rounding in Code In literate programming environments (e.g., Quarto/R Markdown documents), it is often preferable to *display* a number with reduced precision without *altering* the underlying stored value. Formatting functions (e.g., `Printf.@sprintf` in Julia) print a formatted string while retaining the full-precision value in memory. By contrast, `round(x; digits=2)` creates a new variable with the rounded value. Both approaches are valid, but their distinction matters when the rounded value is subsequently used in further calculations. --- ## 5. Exponential and Logarithmic Functions ### 5.1 Definitions The **logarithm** of a number is the power to which a base must be raised to produce that number. $$\log_b(x) = y \iff b^y = x$$ Two bases are used in pharmacokinetics: - **Common logarithm** ($\log_{10}$): logarithm to base 10. - **Natural logarithm** ($\ln$): logarithm to base $e$, where $e \approx 2.718$. The relationship between the two: $$\ln(x) = 2.303 \cdot \log_{10}(x)$$ **Important note for Julia:** The built-in `log()` function computes the natural logarithm. To compute a base-10 logarithm, use `log10()`. ### 5.2 Why Logarithmic Transformation is Used in PK Two principal reasons: 1. **Visualization:** Concentration data spanning several orders of magnitude are compressed onto a manageable scale, separating data points that would otherwise overlap. 2. **Linearization:** An exponential decay curve becomes a straight line on a semi-logarithmic plot. Linear relationships are far easier to analyze, parameterize, and interpret than nonlinear ones. This is the foundation for graphical estimation of PK parameters (e.g., terminal slope estimation). **Example:** The log-linear PK equation $$\ln[C(t)] = \ln\left(\frac{\text{Dose}}{V}\right) - K_e \cdot t$$ has the form $y = \text{intercept} + \text{slope} \times t$. The slope equals $-K_e$, and if $K_e = 0.38\ \text{hr}^{-1}$, the interpretation is that approximately 38% of the drug present in plasma is eliminated per hour (under first-order kinetics). ### 5.3 Why Logarithms Are Dimensionless A logarithm is the logarithm of a ratio — the argument of a logarithm must be dimensionless. When we write $\ln[C(t)]$, we are implicitly computing $\ln[C(t)/C_{\text{ref}}]$ where $C_{\text{ref}}$ is a reference concentration of 1 (in whatever units $C$ is expressed). The result carries no units. This is why plotting log-concentration versus time yields a dimensionless y-axis, even though the original concentration had units. ### 5.4 Fundamental Rules of Logarithms | Rule | Expression | |------|------------| | Product rule | $\log(xy) = \log(x) + \log(y)$ | | Quotient rule | $\log(x/y) = \log(x) - \log(y)$ | | Power rule | $\log(x^n) = n \cdot \log(x)$ | | Log of 1 | $\log(1) = 0$ | **PK application of the power rule:** Allometric scaling relationships take the form: $$\log(\text{CL}) = \log(a) + b \cdot \log(\text{BW})$$ where body weight (BW) is the predictor. This is a log-log linear relationship arising directly from the power rule. ### 5.5 Rules of Exponents Exponents are shorthand for repeated multiplication. The rules of exponents are used extensively in PK — particularly in model-building and parameter transformations: | Rule | Expression | |------|------------| | Product rule | $a^m \cdot a^n = a^{m+n}$ | | Power of a power | $(a^m)^n = a^{mn}$ | | Zero exponent | $a^0 = 1$ ($a \neq 0$) | | Negative exponent | $a^{-n} = 1/a^n$ | --- ## 6. Rates and Orders of Process ### 6.1 Zero-Order Processes For a zero-order process, the rate of change of concentration is **constant** and **independent of the current concentration**: $$-\frac{dC}{dt} = K_0$$ Integrating yields: $$C(t) = C_0 - K_0 \cdot t$$ This is a linear function of time, yielding a straight line on a standard (linear) concentration–time plot. **Units of $K_0$:** Since concentration has units of (e.g.) $\text{ng/mL}$ and time has units of $\text{min}$, $K_0$ has units of $\text{ng/mL/min}$ (concentration per unit time). **Intuition:** The rate of elimination does not depend on how much drug remains. A canonical example from physiology is alcohol metabolism at saturating concentrations — the enzymatic machinery is fully occupied and processes a fixed amount of substrate per unit time regardless of how much substrate is present. ### 6.2 First-Order Processes For a first-order process, the rate of change is **proportional to the current concentration**: $$-\frac{dC}{dt} = K_1 \cdot C$$ The rate decreases as concentration decreases — a larger concentration drives a faster rate of elimination. **Units of $K_1$:** $\text{hr}^{-1}$ (or $\text{min}^{-1}$, etc.) — a pure rate constant with units of reciprocal time. **Intuition:** The fraction of drug eliminated per unit time is constant. If $K_1 = 0.38\ \text{hr}^{-1}$, then 38% of the drug present at any given moment is eliminated each hour. Most pharmacokinetic absorption and elimination processes are assumed to be first-order. Understanding the mechanistic basis for this assumption — rather than simply accepting it as convention — is essential for building correct PK models. --- ## 7. Differential Equations in Pharmacokinetics ### 7.1 The Noyes–Whitney Equation The Noyes–Whitney equation describes the rate of dissolution of a solid drug in a medium: $$\frac{dA}{dt} = D \cdot S_A \cdot \frac{C_s - C}{h}$$ where $D$ is the diffusion coefficient, $S_A$ is the effective surface area, $h$ is the boundary layer thickness, $C_s$ is the saturation concentration, and $C$ is the bulk concentration. This equation is the physical basis for understanding drug dissolution and permeability, which underlies all formulation design and in vitro–in vivo correlations. ### 7.2 Rules for Writing Differential Equations for PK Compartmental Models Four rules govern the construction of differential equations for compartmental models: 1. **Number of equations equals number of compartments.** Each compartment requires exactly one differential equation. 2. **Number of terms on the right-hand side equals the number of arrows (transfer processes) connected to that compartment.** 3. **Sign of each term is determined by the direction of the arrow:** - Arrow pointing *into* the compartment → positive term. - Arrow pointing *out of* the compartment → negative term. 4. **Coefficient of each term is the associated rate constant** (flow constant). **Example — one-compartment model with IV bolus:** For a single compartment containing drug amount $X$, with elimination proceeding by first-order kinetics: $$\frac{dX}{dt} = -K_{10} \cdot X$$ - Units check: $[X] = \text{mg}$; $[K_{10}] = \text{hr}^{-1}$; therefore $\frac{dX}{dt}$ has units of $\text{mg/hr}$. Both sides are consistent. - There is one arrow (elimination out of the compartment) → one term. - The arrow points outward → negative sign. - The subscript notation $K_{10}$ denotes transfer from compartment 1 to the outside (compartment 0). **Two-compartment extension:** Transfer between compartments 1 and 2 is denoted $K_{12}$ (from 1 to 2) and $K_{21}$ (from 2 to 1). Terminal elimination from compartment 1 is $K_{10}$. --- ## 8. Integration and the Area Under the Curve (AUC) Integration is the mathematical inverse of differentiation. In PK, integration is used to compute the area under the concentration–time curve (AUC), which represents total drug exposure. $$\text{AUC}_{0}^{t} = \int_0^t C(t')\ dt'$$ The most common numerical method for computing AUC from discrete data is the **linear trapezoidal rule**: $$\text{AUC}_{t_i \to t_{i+1}} = \frac{C_i + C_{i+1}}{2} \cdot (t_{i+1} - t_i)$$ The total AUC is the sum of all such trapezoids. This forms the computational basis of non-compartmental analysis (NCA). Extrapolation of AUC from the last observed concentration to infinity requires additional rules (covered in the NCA section). **Relationship to the analytical equation:** For a one-compartment IV bolus: $$\text{AUC}_{0}^{\infty} = \frac{\text{Dose}}{\text{CL}} = \frac{F \cdot \text{Dose}}{\text{CL}}$$ where $F$ is bioavailability (dimensionless) and CL has units of $\text{L/hr}$. --- ## 9. Common PK Parameters: Symbols and Units | Parameter | Symbol | Typical Units | |-----------|--------|---------------| | Plasma concentration | $C$ | ng/mL, µg/L, nM | | Dose | $D$ | mg, µg | | Volume of distribution | $V$ | L, L/kg | | Clearance | CL | L/hr, mL/min | | Elimination rate constant | $K_e$ | hr⁻¹ | | Absorption rate constant | $K_a$ | hr⁻¹ | | Half-life | $t_{1/2}$ | hr, min | | Bioavailability | $F$ | dimensionless (fraction) | | AUC | AUC | ng·hr/mL, µg·hr/L | | Zero-order rate constant | $K_0$ | ng/mL/hr | --- ## Summary This lecture establishes the mathematical and conceptual foundation required for all subsequent pharmacokinetic work: - Precise use of the terms **variable**, **parameter**, and **constant** is essential for clear scientific communication. - **Dimensionality** governs the validity of all PK equations; units must balance, and one must always verify unit consistency when deriving or interpreting equations. - **Unit conversions** must be performed correctly before any analysis; computational tools assist but the analyst must understand the logic. - **Significant figures** and rounding must be applied appropriately, especially in regulatory contexts such as bioequivalence. - **Logarithms** linearize exponential PK data and are dimensionless by nature; the natural log is the default in Julia (`log()`). - **Zero-order** processes proceed at a rate independent of concentration; **first-order** processes proceed at a rate proportional to the remaining concentration. This distinction defines the structure of all PK differential equations. - **Differential equations** for compartmental models follow four simple rules; understanding these rules enables one to translate any pharmacokinetic diagram into a system of equations. - **Integration** (and specifically the trapezoidal rule) underlies NCA and AUC calculation.