# Volume of distribution

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In pharmacology, the volume of distribution (VD, also known as apparent volume of distribution) 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. It is defined as the distribution of a medication between plasma and the rest of the body after oral or parenteral dosing.

The VD of a drug represents the degree to which a drug is distributed in body tissue rather than the plasma. VD is directly correlated with the amount of drug distributed into tissue; a higher VD indicates a greater amount of tissue distribution. A VD 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 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 renal 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:

${V_{D}}={\frac {\mathrm {total\ amount\ of\ drug\ in\ the\ body} }{\mathrm {drug\ blood\ plasma\ concentration} }}$ Therefore, the dose required to give a certain plasma concentration can be determined if the VD for that drug is known. The VD is not a physiologic 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 liters. As body composition changes with age, VD decreases.

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

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

• VP = plasma volume
• VT = apparent tissue volume
• fu = fraction unbound in plasma
• fuT = 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 $C_{0}$ which directly corresponds to the amount of blood contained in the body $V_{blood}$ . Mathematically this would be:

$C_{0}=D/V_{blood}$ 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 $V_{D}$ 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 $V_{blood}=$ 0.08 l/kg . This gives a $C_{0}=$ 100 µg/ml if the drug stays in the blood stream only, and thus its volume of distribution is the same as $V_{blood}$ that is $V_{D}=$ 0.08 l/kg. If the drug distributes into all body water the volume of distribution would increase to approximately $V_{D}=$ 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 $V_{D}=$ 250-302 l/kg 

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

 Drug VD Comments Warfarin 8L Reflects a high degree of plasma protein binding. Theophylline, Ethanol 30L Represents distribution in total body water. Chloroquine 15000L Shows highly lipophilic molecules which sequester into total body fat. NXY-059 8L Highly charged hydrophilic molecule.

## Sample values and equations

Characteristic Description Symbol Unit Formula Worked example
value
Dose Amount of drug administered. $D$ $\mathrm {mol}$ Design parameter 500 mmol
Dosing interval Time between drug dose administrations. $\tau$ $\mathrm {s}$ Design parameter 24 h
Cmax The peak plasma concentration of a drug after administration. $C_{\text{max}}$ $\mathrm {M}$ Direct measurement 60.9 mmol/L
tmax Time to reach Cmax. $t_{\text{max}}$ $\mathrm {s}$ Direct measurement 3.9 h
Cmin The lowest (trough) concentration that a drug

reaches before the next dose is administered.

$C_{{\text{min}},{\text{ss}}}$ $\mathrm {M}$ Direct measurement 27.7 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).

$V_{\text{d}}$ $\mathrm {m} ^{3}$ ${\frac {D}{C_{0}}}$ 6.0 L
Concentration Amount of drug in a given volume of plasma. $C_{0},C_{\text{ss}}$ $\mathrm {M}$ ${\frac {D}{V_{\text{d}}}}$ 83.3 mmol/L
Elimination half-‍life The time required for the concentration

of the drug to reach half of its original value.

$t_{\frac {1}{2}}$ $\mathrm {s}$ ${\frac {\ln(2)}{k_{\text{e}}}}$ 12 h
Elimination rate constant The rate at which a drug is removed from the body. $k_{\text{e}}$ $\mathrm {s} ^{-1}$ ${\frac {\ln(2)}{t_{\frac {1}{2}}}}={\frac {CL}{V_{\text{d}}}}$ 0.0578 h−1
Infusion rate Rate of infusion required to balance elimination. $k_{\text{in}}$ $\mathrm {mol/s}$ $C_{\text{ss}}\cdot CL$ 50 mmol/h
Area under the curve The integral of the concentration-time curve

(after a single dose or in steady state).

$AUC_{0-\infty }$ $\mathrm {M} \cdot \mathrm {s}$ $\int _{0}^{\infty }C\,\operatorname {d} t$ 1,320 mmol/L·h
$AUC_{\tau ,{\text{ss}}}$ $\mathrm {M} \cdot \mathrm {s}$ $\int _{t}^{t+\tau }C\,\operatorname {d} t$ Clearance The volume of plasma cleared of the drug per unit time. $CL$ $\mathrm {m} ^{3}/\mathrm {s}$ $V_{\text{d}}\cdot k_{\text{e}}={\frac {D}{AUC}}$ 0.38 L/h
Bioavailability The systemically available fraction of a drug. $f$ Unitless ${\frac {AUC_{\text{po}}\cdot D_{\text{iv}}}{AUC_{\text{iv}}\cdot D_{\text{po}}}}$ 0.8
Fluctuation Peak trough fluctuation within one dosing interval at steady state. $\%PTF$ $\%$ ${\frac {C_{{\text{max}},{\text{ss}}}-C_{{\text{min}},{\text{ss}}}}{C_{{\text{av}},{\text{ss}}}}}\cdot 100\%$ where

$C_{{\text{av}},{\text{ss}}}={\frac {1}{\tau }}AUC_{\tau ,{\text{ss}}}$ 41.8%