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Pharm formulas, with a purpose

Choose the quantity you need, check the units, and work through the calculation one step at a time.

Based on FMK 12.2, Principles of Pharmacokinetics, and FMK 12.5, Clinical PK Calculations. Your symbols appear first; lecture equivalents stay alongside them. Slide numbers count from the first slide in each deck.

One notation throughout

Meaning	Here	Lecture / standard equivalents	Typical units
Plasma concentration	C_p	Cp; C; measured or predicted concentration	mg/L = µg/mL
Desired / target concentration	C_t	Desired Cp; target Css(avg) for maintenance	mg/L
Elimination rate	V_e	Re; rate of elimination	mg/h
Clearance	Cl	CL	L/h or mL/min
Apparent distribution volume	V_d	Vd	L or L/kg
Elimination rate constant	k_e	ke; k; kel	h⁻¹
Half-life	t_½	t1/2	h
Bioavailability	F	Fraction reaching systemic circulation	Fraction: 0.80, not 80
Dosing interval	τ	tau	h
Body amount / elapsed time	A / t	Amount present / time since reference	mg / h, respectively
Exposure	AUC	Area under concentration–time curve	mg·h/L
Dose / input rate	LD; MD; R_in	Loading dose; maintenance dose per interval; infusion rate	mg; mg; mg/h
Average steady-state concentration	C_ss(avg)	Set equal to Ct when designing maintenance	mg/L
Capacity / half-capacity concentration	V_max; K_m	Michaelis–Menten parameters	mg/h; mg/L
Organ extraction ratio	ER	Uses organ inflow Cin and outflow Cout	Unitless fraction

Keep three different quantities separate: V_e is mass removed per time; Cl is volume cleared per time; k_e is an inverse-time constant. C_p describes concentration; C_t states the goal. They are equal only when the goal is reached.

Work through a formula

What are you trying to find?

Dosing · lecture equation or rearrangement

Loading dose

LD = C_t × V_d / F

LD = desired concentration × Vd / F; lecture also uses Css for the desired concentration

When to use it

Reach a target concentration promptly rather than waiting through several half-lives of accumulation.

Why it matters

Loading is mainly about filling the apparent distribution space. The absorbed amount must equal Ct × Vd.

Worked example

Ct = 8 mg/L, Vd = 25 L and F = 0.50. What theoretical loading dose is needed?

Check before using: A loading dose does not change clearance, half-life or the kinetic time to true steady state. The calculation alone does not establish a safe administration rate or real-world regimen.

Source: FMK 12.5 slides 14–18, 31

One case connects the formulas

Imagine a hypothetical drug with linear first-order kinetics. The target is an average steady-state concentration. Change a value to see which parts of the plan move.

Target C_t: 10 mg/L Distribution V_d: 40 L Clearance Cl: 4 L/h Bioavailability F: 80% (0.80) Interval τ: 8 h

1 · Load the distribution space500 mg

LD = 10 × 40 / 0.80. A theoretical loading dose to supply Ct × Vd systemically.

2 · Replace ongoing elimination400 mg every 8 h

MD = 10 × 4 × 8 / 0.80. At the target, V_e = 10 × 4 = 40 mg/h; the administered average rate is 50 mg/h.

3 · Predict the time courset_½ ≈ 6.93 h

k_e = 4 / 40 = 0.100 h⁻¹. About five half-lives is 34.66 h.

Try these changes and explain the result

Double Vd: loading dose and half-life double; maintenance dose is unchanged if clearance stays fixed.
Halve Cl: maintenance dose halves and half-life doubles; loading dose stays the same if Vd is fixed.
Halve F: administered loading and maintenance doses double; the systemic target and its elimination rate stay the same.
Double τ: dose per interval doubles for the same average target; the average dose rate stays unchanged. Peaks and troughs may change.

Reset to the example values before comparing one change at a time.

This demonstrates relationships, not a prescription calculator. Stable linear kinetics, complete distribution within the model, constant parameters and appropriate use of average concentrations are assumed. A real regimen also requires drug-specific formulation, peak/trough, administration-rate and safety information.

All formulas at a glance

Seventeen lecture equations, rearrangements or table relationships, plus three clearly marked extensions. “Extra” does not mean the professor assigned that equation for memorization.

Volume of distribution

V_d = A / C_p

Vd = amount of drug in body / plasma concentration

FMK 12.2 slide 26; FMK 12.5 slide 13

Plasma concentration from body amount

C_p = A / V_d

Initial IV concentration = loading dose / Vd (ideal one-compartment model)

FMK 12.2 slides 26–28; FMK 12.5 slides 17, 31

Loading dose

LD = C_t × V_d / F

LD = desired concentration × Vd / F; lecture also uses Css for the desired concentration

FMK 12.5 slides 14–18, 31

Maintenance dose per interval

MD = C_t × Cl × τ / F

MD = Css(avg) × Cl × τ / F

FMK 12.5 slides 16, 19, 31

Average steady-state concentration

C_ss(avg) = F × MD / (τ × Cl)

Lecture uses Css(avg); set Css(avg) = Ct when solving for a target regimen

FMK 12.5 slides 16, 31

Continuous IV infusion rate

R_in = C_t × Cl

At steady state, rate of infusion = rate of elimination; IV F = 1

FMK 12.2 slide 37; FMK 12.5 slides 14, 20, 31

Elimination rate

V_e = Cl × C_p

Lecture: Re (rate of elimination). Your Ve = Re, in mass/time.

FMK 12.2 slide 37; FMK 12.5 slides 20–21 (rearranged)

Clearance from elimination rate

Cl = V_e / C_p

Cl = Re / concentration

FMK 12.2 slide 37; FMK 12.5 slides 20–21

Clearance from dose and AUC

Cl = F × Dose / AUC

IV: Cl = DoseIV / AUCIV; extravascular: Cl = F × Dose / AUC

FMK 12.2 slide 38; FMK 12.5 slides 22–23

Absolute bioavailability

F = (AUC_EV / AUC_IV) × (Dose_IV / Dose_EV)

Lecture F(%) = 100 × AUCev/AUCiv × Doseiv/Doseev; EV includes PO

FMK 12.2 slide 19; FMK 12.5 slides 9–10

Equivalent dose after a route change

Dose_new = Dose_old × F_old / F_new

Lecture: new dose = amount absorbed from current dosage form / Fnew

FMK 12.2 slide 21; FMK 12.5 slides 11–12

Elimination rate constant

k_e = Cl / V_d

Lecture: ke; standard texts may use k or kel

FMK 12.2 slide 40; FMK 12.5 slides 24–25

First-order half-life

t_½ = 0.693 / k_e = 0.693 × V_d / Cl

Lecture: t1/2 = 0.693/ke

FMK 12.2 slide 41; FMK 12.5 slides 26–30

Amount remaining after half-lives

n = t / t_½; fraction remaining = (½)ⁿ

Lecture table: 50%, 25%, 12.5%, 6.25%, 3.13% remaining; eliminated = 1 − fraction remaining

FMK 12.2 slides 42–43; FMK 12.5 slides 27–28

Time to approach steady state

t ≈ 5 × t_½; fraction of steady state ≈ 1 − (½)ⁿ

Lecture: about five half-lives; 96.9% of steady state at five

FMK 12.2 slides 42–43; FMK 12.5 slides 15, 27–29

Organ extraction ratio

ER = (C_in − C_out) / C_in

Lecture: ER, Cin and Cout; concentrations entering and leaving the organ

FMK 12.2 slide 39

Michaelis–Menten elimination

V_e = V_max × C_p / (K_m + C_p)

Lecture introduces Vmax and Km. This is the standard equation for that saturation model.

FMK 12.5 slides 6–8 (concept and parameters); standard model equation

Extra model / derived practice

Constant-amount elimination

A(t) = A₀ − V_e × t

Lecture zero-order table: constant mg removed per unit time

FMK 12.5 slides 4, 8; equation restates the lecture table

Concentration at a later time

C_p(t) = C_p0 × e^−kₑt = C_p0 × (½)^t/t½

Derived helper: continuous version of the lecture half-life table

Derived from FMK 12.5 slides 26–28

Extra model / derived practice

Time to a desired residual concentration

t = ln(C_p0 / C_t) / k_e

Equivalent: t = t½ × log2(Cp0/Ct); Ct is the chosen residual threshold here

Derived from FMK 12.5 slides 26–28

Extra model / derived practice

Before you calculate

Convert 80% to F = 0.80. IV F = 1.
1 µg/mL = 1 mg/L; 1 L = 1,000 mL. Convert mL/min to L/h by multiplying by 0.06.
If Vd is supplied in L/kg, multiply by body mass in kg before using it as L.
Dose is mg; dose rate is mg/h. The dosing interval belongs in the dose-per-interval equation.
AUC has concentration × time units. It is not a concentration.
Do not apply fixed-clearance, constant-half-life shortcuts across saturable kinetics.

Scope: supplied FMK PK materials. FMK 9.5 and the available 13.2 pharmacodynamics transcript were checked for additional equations. The separately scheduled dose-response lecture is still missing, so its future formulas are not presented as verified lecture requirements.