The tangible number of particles in drug delivery formulations (number concentration,

N) is of importance for quality assurance, comprehensive physicochemical characterization, and pharmacodynamics [

15]. The number of particles in a certain volume of sample, rather than merely their size, could affect their absorption, clearance and disposition [

16,

18]. The number of particles affects effective uptake of a targeted carrier system by specific cells (e.g., phagocytic cells) and the cumulative drug content of the particles determines their bioactivity and therapeutic efficacy. Knowledge of particle quantity enables the precise assessment of drug concentration in each solid (rigid, e.g., liposomes made of lipids with high phase transition temperature) or liquid (elastic) particle [

15]. Concentration of the bioactive agent or its distribution between phases determines if the system is of dissolved or dispersed type, and accordingly, it defines the drug release kinetics and mechanism. Among the techniques used for the quantification of particle number is nanoparticle tracking analysis (NTA), which is able to track and measure particles moving under Brownian motion [

17,

19]. This high-resolution method is effective in determining the size, size distribution, and concentration of colloidal and particulate drug delivery systems. It can be employed to assess particles and vesicles within the size range of 30–1000 nm [

20]. Samples are injected into the special cell of the apparatus and then it is illuminated by laser light (635 nm) that passes through a liquid layer on the optical surface [

19,

20]. Refraction occurs and the region in which the vesicles or particles are present is illuminated and visualized under microscopy. A charge-coupled device camera records a video (30 frames/sec) wherein the movement of particles under Brownian motion can be visualized. Special software identifies and tracks the center of each particle throughout the length of the video and relates it to the particle characteristics [

21]. Mathematical equations used to calculate the quantity of some examples of carrier systems are described below. Simple equations are given for both solid particles (e.g., metallic particles, polymeric microcapsules and nanocapsules) and soft vesicles (e.g., liposomes, nanoliposomes, lipospheres, solid lipid nanoparticles, niosomes and tocosomes).

#### 2.1. Quantification of Number of Metallic Particles

Particles of gold, silver, iron, copper and other metals are popular colloidal substrates employed in various sensor, imaging, and drug delivery applications. They can be synthesized and modified with several different chemical functional groups, which allow them to be conjugated with antibodies, ligands, and other bioactive agents or drugs of interest [

22]. Particle number or concentration of metallic particles determines crucial features of the formulations including stability, bioactivity and cytotoxicity. The number of metallic particles (

N_{MP}) in solution can be calculated from the ratio of the number of initial metal atoms (N

_{ma}) and the number of metal atoms per one, single metallic particle N

_{ma/smp} as described in Equation (1):

Hinterwirth and co-workers [

23] employed a similar mathematical equation to calculate number concentration of the gold nanoparticles in their formulation. Taking their study as an instance, if initially 55 mL of 1.14 mM Au(III) atoms was used in the construction of particles and the number of metal atoms per single metallic particle is calculated to be 30.89602 (see Equation (2) below), the number of gold nanoparticles in 1 L sample will be:

in which N

_{A} is Avogadro’s constant (i.e., 6.02E23). The average number of metal atoms per metallic particle of gold is calculated according to the following equation [

24,

25]:

where ρ is the density for a face-centered cubic (FCC) gold (19.3 g/cm

^{3}), D is the average size (31 ± 1.6 nm) and M stands for atomic weight of gold (197 g/mol). The average number of gold atoms per nanoparticle can also be calculated from high-resolution microscopic analysis. Rajakumar et al. [

25] reported that images of their synthesized gold nanoparticles depicted particles in the range of 23 to 46 nm, with an average size of 31 ± 1.6 nm (

D, nm) [

25]. Assuming a spherical shape and a uniform face-centered cubic (FCC) structure [

26], the average number of gold atoms for each type of nanosphere was calculated by Equation (2) [

27,

28].

The mathematical approach and the related Equation (1) explained above can be extrapolated to be used for other metallic micro- and nanoparticles including copper [

29] and silver [

30].

#### 2.2. Particle Number of Vesicular Carriers

Vesicular drug-delivery carriers comprise liposomes, nanoliposomes, micelles, tocosomes, niosomes, solid lipid nanoparticles and archaeosomes to name a few [

16,

31]. Among the vesicular carriers, liposomes (and nanoliposomes) are the most applied encapsulation techniques with the highest number of products approved for Human use on the market. Also known as a bilayer phospholipid vesicle, liposome is a mesomorphic structure mainly composed of lipid, phospholipid and water molecules [

32]. The main chemical components of liposomes and nanoliposomes are amphiphilic lipid/phospholipid molecules (

Figure 1). They improve the efficacy of pharmaceutical, nutraceutical and other bioactive compounds by entrapment and release of water-soluble, lipid-soluble and amphipathic materials, as well as targeting the encapsulated drug molecules to particular cells or tissues [

33,

34]. Vesicular drug carriers, including liposomes, can be prepared in different forms with respect of their number of lamella (phospholipid bilayers) as depicted in

Figure 2.

Currently, there are no validated experimental approaches for the determination of the particle number-concentration of liposomal and nanoliposomal formulations [

15]. Here we present a simple mathematical approach to calculate the number of phospholipid vesicles, in the form of unilamellar vesicle, in any certain volume of sample. Once the total concentration of phospholipids, lipids and other ingredients of our vesicles (such as cholesterol, phytosterols, vitamin E, etc.) in the suspending media are known, then the total number of particles per ml can easily be calculated using Equation (3).

where:

N_{Ves} is the number of drug delivery vesicles per milliliter; M

_{ing} is the molar concentration of ingredients of the vesicles; N

_{A} is the Avogadro Number (6.02E23); and N

_{tot} is the total number of ingredients per vesicle. Equation (3) is the main equation by which particle number of vesicular bioactive carrier systems can be easily calculated once we know N

_{tot}.

N

_{tot} can be calculated using the following equation:

in which: [4π(d/2)

^{2}] is the surface area of vesicle’s monolayer;

d is the diameter of the vesicle;

h is the thickness of the phospholipid bilayer (i.e., ~5 nm);

a is the phospholipid head group area and E2 is exponent two (to the power 2). The headgroup area of phosphatidylcholine (a generally used ingredient in the manufacture of lipid vesicles, niosomes, tocosomes, etc.) is about 0.71 nm square, as depicted in

Figure 1 [

35,

36,

37]. Accordingly, Equation (4) can be simplified to:

in which 17.69 is 4π/

a.

As an example the total number of ingredients for a unilamellar formulation with 400 nm mean particle size is:

and using this number as N

_{tot} in Equation (3), we will find out the number of vesicles in a milliliter of the prepared sample, assuming molar concentration (M

_{ing}) of 1 micromole:

Exceptional cases for the use of Equation (3) for quantification of particle number of vesicular drug carriers would be multilamellar vesicles (MLV) or multivesicular vesicles (MVV) (see

Figure 2). The mathematical equations described above are straightforward means for calculation of particle number. Other approaches mentioned in the literature are complicated and involve multistep calculations. For instance, Pidgeon and Hunt [

38] have presented formulas based on the volume of the entrapped water by liposomal vesicles, which involve solving several equations in order to find the estimated particle number.

#### 2.3. Particle Number of Polymeric Carriers

A method used to assess the number concentration of polymeric carriers is scanning mobility particle sizer (SMPS) [

39,

40]. In this method, the formulations are atomized to aerosol droplets that are then dried in a silica-gel diffusion drier. The dry particles flow into the SMPS apparatus, which is composed of the differential mobility analyzer (DMA) unit. DMA provides size information according to the size-dependent electric mobility of the particles. The particles then move into the condensation particle counter (CPC) section, which counts the number of particles in each size group. The combined measurements result in a highly resolved particle count for the full-size distribution range. Integrating the counts over the full-size range yields the total particle number concentration

N_{c} using Equation (5):

in which N’

_{c} is the number of particles per cm

^{3} measured by SMPS, CF is the calibration factor for the particle size, F

_{g} is the flow rate (cm

^{3} min

^{−1}) of the carrier gas (e.g., nitrogen). R

_{ev} is the measured evaporation rate of the sample solution (ml min

^{−1}) and is calculated by measuring the rate of solution loss from the atomizer compartment for the calibrated gas flow through the atomizer. The calibration factor (CF) is employed to account for the loss of particles in the system.