Volume of distribution

In pharmacology, the volume of distribution (V_D, also known as apparent volume of distribution, literally, volume of dilution) is the theoretical volume that would be necessary to contain the total amount of an administered drug at the same concentration that it is observed in the blood plasma. In other words, it is the ratio of amount of drug in a body (dose) to concentration of the drug that is measured in blood, plasma, and un-bound in interstitial fluid.

The V_D of a drug represents the degree to which a drug is distributed in body tissue rather than the plasma. V_D is directly proportional with the amount of drug distributed into tissue; a higher V_D indicates a greater amount of tissue distribution. A V_D greater than the total volume of body water (approximately 42 liters in humans) is possible, and would indicate that the drug is highly distributed into tissue. In other words, the volume of distribution is smaller in the drug staying in the plasma than that of a drug that is widely distributed in tissues.

In rough terms, drugs with a high lipid solubility (non-polar drugs), low rates of ionization, or low plasma protein binding capabilities have higher volumes of distribution than drugs which are more polar, more highly ionized or exhibit high plasma protein binding in the body's environment. Volume of distribution may be increased by kidney failure (due to fluid retention) and liver failure (due to altered body fluid and plasma protein binding). Conversely it may be decreased in dehydration.

The initial volume of distribution describes blood concentrations prior to attaining the apparent volume of distribution and uses the same formula.

Equations

The volume of distribution is given by the following equation:

${\displaystyle {V_{D}}={\frac {\mathrm {total\ amount\ of\ drug\ in\ the\ body} }{\mathrm {drug\ blood\ plasma\ concentration} }}}$

Therefore, the dose required to give a certain plasma concentration can be determined if the V_D for that drug is known. The V_D is not a physiological value; it is more a reflection of how a drug will distribute throughout the body depending on several physicochemical properties, e.g. solubility, charge, size, etc.

The unit for Volume of Distribution is typically reported in litres. As body composition changes with age, V_D decreases.

The V_D may also be used to determine how readily a drug will displace into the body tissue compartments relative to the blood:

${\displaystyle {V_{D}}={V_{P}}+{V_{T}}\left({\frac {fu}{fu_{t}}}\right)}$

Where:

• V_P = plasma volume
• V_T = apparent tissue volume
• fu = fraction unbound in plasma
• fu_T = fraction unbound in tissue

Examples

If you administer a dose D of a drug intravenously in one go (IV-bolus), you would naturally expect it to have an immediate blood concentration ${\displaystyle C_{0}}$ which directly corresponds to the amount of blood contained in the body ${\displaystyle V_{blood}}$. Mathematically this would be:

${\displaystyle C_{0}=D/V_{blood}}$

But this is generally not what happens. Instead you observe that the drug has distributed out into some other volume (read organs/tissue). So probably the first question you want to ask is: how much of the drug is no longer in the blood stream? The volume of distribution ${\displaystyle V_{D}}$ quantifies just that by specifying how big a volume you would need in order to observe the blood concentration actually measured.

An example for a simple case (mono-compartmental) would be to administer D=8 mg/kg to a human. A human has a blood volume of around ${\displaystyle V_{blood}=}$0.08 L/kg . This gives a ${\displaystyle C_{0}=}$100 µg/mL if the drug stays in the blood stream only, and thus its volume of distribution is the same as ${\displaystyle V_{blood}}$ that is ${\displaystyle V_{D}=}$ 0.08 L/kg. If the drug distributes into all body water the volume of distribution would increase to approximately ${\displaystyle V_{D}=}$0.57 L/kg

If the drug readily diffuses into the body fat the volume of distribution may increase dramatically, an example is chloroquine which has a ${\displaystyle V_{D}=}$250-302 L/kg

In the simple mono-compartmental case the volume of distribution is defined as: ${\displaystyle V_{D}=D/C_{0}}$, where the ${\displaystyle C_{0}}$ in practice is an extrapolated concentration at time = 0 from the first early plasma concentrations after an IV-bolus administration (generally taken around 5 min - 30 min after giving the drug).

Sample values and equations

• v
• t
• e

Pharmacokinetic metrics

Characteristic	Description	Symbol	Unit	Formula	Worked example value
Dose	Amount of drug administered.	${\displaystyle D}$	${\displaystyle \mathrm {mol} }$	Design parameter	500 mmol
Dosing interval	Time between drug dose administrations.	${\displaystyle \tau }$	${\displaystyle \mathrm {h} }$	Design parameter	24 h
Cmax	The peak plasma concentration of a drug after administration.	${\displaystyle C_{\text{max}}}$	${\displaystyle \mathrm {mmol/L} }$	Direct measurement	60.9 mmol/L
tmax	Time to reach Cmax.	${\displaystyle t_{\text{max}}}$	${\displaystyle \mathrm {h} }$	Direct measurement	3.9 h
Cmin	The lowest (trough) concentration that a drug reaches before the next dose is administered.	${\displaystyle C_{{\text{min}},{\text{ss}}}}$	${\displaystyle \mathrm {mmol/L} }$	Direct measurement	27.7 mmol/L
Cavg	The average plasma concentration of a drug over the dosing interval in steady state.	${\displaystyle C_{{\text{av}},{\text{ss}}}}$	${\displaystyle \mathrm {h\times mmol/L} }$	${\displaystyle {\frac {AUC_{\tau ,{\text{ss}}}}{\tau }}}$	55.0 h×mmol/L
Volume of distribution	The apparent volume in which a drug is distributed (i.e., the parameter relating drug concentration in plasma to drug amount in the body).	${\displaystyle V_{\text{d}}}$	${\displaystyle \mathrm {L} }$	${\displaystyle {\frac {D}{C_{0}}}}$	6.0 L
Concentration	Amount of drug in a given volume of plasma.	${\displaystyle C_{0},C_{\text{ss}}}$	${\displaystyle \mathrm {mmol/L} }$	${\displaystyle {\frac {D}{V_{\text{d}}}}}$	83.3 mmol/L
Absorption half-life	The time required for 50% of a given dose of drug to be absorbed into the systemic circulation.[1]	${\displaystyle t_{{\frac {1}{2}}a}}$	${\displaystyle \mathrm {h} }$	${\displaystyle {\frac {\ln(2)}{k_{\text{a}}}}}$	1.0 h
Absorption rate constant	The rate at which a drug enters into the body for oral and other extravascular routes.	${\displaystyle k_{\text{a}}}$	${\displaystyle \mathrm {h} ^{-1}}$	${\displaystyle {\frac {\ln(2)}{t_{{\frac {1}{2}}a}}}}$	0.693 h−1
Elimination half-‍life	The time required for the concentration of the drug to reach half of its original value.	${\displaystyle t_{{\frac {1}{2}}b}}$	${\displaystyle \mathrm {h} }$	${\displaystyle {\frac {\ln(2)}{k_{\text{e}}}}}$	12 h
Elimination rate constant	The rate at which a drug is removed from the body.	${\displaystyle k_{\text{e}}}$	${\displaystyle \mathrm {h} ^{-1}}$	${\displaystyle {\frac {\ln(2)}{t_{{\frac {1}{2}}b}}}={\frac {CL}{V_{\text{d}}}}}$	0.0578 h−1
Infusion rate	Rate of infusion required to balance elimination.	${\displaystyle k_{\text{in}}}$	${\displaystyle \mathrm {mol/h} }$	${\displaystyle C_{\text{ss}}\cdot CL}$	50 mmol/h
Area under the curve	The integral of the concentration-time curve (after a single dose or in steady state).	${\displaystyle AUC_{0-\infty }}$	${\displaystyle \mathrm {M} \cdot \mathrm {s} }$	${\displaystyle \int _{0}^{\infty }C\,\mathrm {d} t}$	1,320 h×mmol/L
		${\displaystyle AUC_{\tau ,{\text{ss}}}}$	${\displaystyle \mathrm {M} \cdot \mathrm {s} }$	${\displaystyle \int _{t}^{t+\tau }C\,\mathrm {d} t}$
Clearance	The volume of plasma cleared of the drug per unit time.	${\displaystyle CL}$	${\displaystyle \mathrm {m} ^{3}/\mathrm {s} }$	${\displaystyle V_{\text{d}}\cdot k_{\text{e}}={\frac {D}{AUC}}}$	0.38 L/h
Bioavailability	The systemically available fraction of a drug.	${\displaystyle f}$	Unitless	${\displaystyle {\frac {AUC_{\text{po}}\cdot D_{\text{iv}}}{AUC_{\text{iv}}\cdot D_{\text{po}}}}}$	0.8
Fluctuation	Peak–trough fluctuation within one dosing interval at steady state.	${\displaystyle \%PTF}$	${\displaystyle \%}$	${\displaystyle 100{\frac {C_{{\text{max}},{\text{ss}}}-C_{{\text{min}},{\text{ss}}}}{C_{{\text{av}},{\text{ss}}}}}}$ where ${\displaystyle C_{{\text{av}},{\text{ss}}}={\frac {AUC_{\tau ,{\text{ss}}}}{\tau }}}$	41.8%

External links

• Tutorial on volume of distribution
• Overview at icp.org.nz
• Overview at cornell.edu
• Overview at stanford.edu
• Overview at boomer.org