*2.2. Ferrofluid*

A magnetic colloidal particle, also known as a ferrofluid, is a colloidal suspension of single-domain magnetic particles, typically about 10 nm in size, dispersed in a liquid carrier [15,16]. The liquid carrier can be polar or non-polar [14]. Since the 1960s, when ferrofluid was initially synthesized, its technical and medical applications have not stopped growing [18]. Ferrofluid differs from ordinary magnetorheological fluids used for shock absorbers, brakes and clutches, formed by micron-sized particles dispersed in oil [18]. In magnetorheological fluid, the application of a magnetic field increases the viscosity, so that for a sufficiently strong field, it can behave like a solid [16,18]. On the other hand, ferrofluid retains its fluidity even if it is exposed to a strong magnetic field. Ferrofluids are optically isotropic, but in the presence of an external magnetic field, they show induced

birefringence [29]. Wetting of certain substrates can also cause bifurcation in thin ferrofluid layers. To avoid agglomeration, the magnetic particle should be coated with a shell of a suitable material [30,31]. In relation to the coating, ferrofluid is divided into two main groups: surfactant, if the coating is a surfactant molecule, and ionic, if it is an electric shell [30,31].

Colloidal suspensions of magnetic particles in liquids that have the ability to magnetize in an external magnetic field are called ferrofluids [32]. These are magnetic materials in liquid form. The liquid can be water or an organic solution in which ferromagnetic or ferrimagnetic particles are dispersed. The particles are most often hematite, Fe2O3, or magnetite, Fe3O4, and they need to be stabilized due to high surface energy by adding a polymer or ionic component (surfactant). Usually, such stable particles are about 10 nm in diameter, and their surface energy is reduced by long-chain surfactants which, thanks to the long chains, prevent agglomeration, or the same charge on the surface of magnetic particles leads to a mutual repulsion, preventing agglomeration [30]. Ferrofluid particles do not precipitate even for a long time, they do not agglomerate and they do not separate from liquids even by applying an extremely strong magnetic field. The combination of the liquid phase and the magnetic behavior makes it possible to manipulate the fluid by changing its position using an external magnetic field [32].

To avoid agglomeration, ferrofluid particles are coated [30,31]. Depending on the coating, ferrofluids are divided into two groups: surfactant-coated ferrofluids and electrostatically stabilized ferrofluids [31]. Surfactant-coated ferrofluids contain magnetic particles coated with amphiphilic molecules such as oleate to prevent aggregation. Spherical repulsion between particles acts as a physical barrier that keeps the particles in solution and stabilizes the colloid. If the particles are dispersed in a nonpolar phase, such as oil, the polar head of the surfactant is attached to the surface of the particles, and the hydrophobic chain is in contact with the liquid (Figure 1a). If the particles are dispersed in the polar phase such as water, a two-layer coating of the particles is required to form a hydrophilic layer around them (Figure 1b).

**Figure 1.** Single layer (**a**) Two-layer. (**b**) Coated magnetic particles.

One of the fastest developed areas of research is one in which nanotechnology, biology and medicine intertwine. According to many experts, the application of nanotechnology in medicine, better known as nanomedicine, will lead to a revolution in the field of targeted drug delivery systems [33,34], disease diagnosis, bioengineering and the improvement of contrast agents in magnetic resonance imaging. The term "teragnostics" is usually used in this context, which, as the name itself implies, is a synthesis of diagnostics and therapy. The localization of ferrofluids by the applied magnetic field gives an interesting application of ferrofluids in medicine. A lot of research has been dedicated to the use of ferrofluids as a system for targeted delivery of drugs used in chemotherapy [35]. A drug is injected into tumor carcinomas and retained there for some time by a magnetic field. The amount of medicine needed is much less than the amount of medicine that would be needed to distribute the medicine throughout the body. After turning off the magnetic field, the drug will disperse in the body, but since it is a much smaller amount, there are practically no side effects.

The ability of ferrofluid to absorb the energy of electromagnetic waves at a frequency different from the frequency at which water absorbs energy allows heating of the localized part of the tissue where the ferrofluid is injected (for example, tumors) without heating the surrounding tissues. This phenomenon is called hyperthermia. Hyperthermia is one of the

methods used in cancer therapy that is based on increasing the temperature of tumor tissue above 41 ◦C. As a result, the function of tumor cells is disturbed and they die. Magnetic hyperthermia is based on the effect of releasing heat when magnetic nanoparticles are found in a changing magnetic field. Magnetic nanoparticles can be successfully localized in tumor tissue, which allows heating only in the desired place. In therapy, these particles are most often used in colloidal form. It is possible to bind chemotherapy drugs or radionuclides to these particles and thus achieve a combined effect.

#### *2.3. State of the Art in Clinical Drug Delivery Systems Using Ferromagnetic Particles*

Drug delivery systems based on nanotechnology have improved the delivery of drugs due to their changes in pharmacokinetics, enabling a longer half-life of the drug in the bloodstream and reducing toxicity [36]. Magnetic nanoparticles play an important role in the diagnosis and treatment of diseases such as cancer, heart and neurological diseases [37]. These particles are often used in the targeted delivery of medically active substances because they deliver the drug to the desired place via tissue magnetic absorption or strong ligand–receptor interaction [38].

The drug–carrier complex can be administered intravenously or by arterial injection [39]. It can also be administered orally, but the main problem with such administration is the delivery of peptides and proteins due to their breakdown in gastric acid, low absorption and first-pass metabolism through the liver [40].

If a drug is delivered by a magnetic field, the gradient of the external magnetic field associated with the magnetic field within human tissues enables the transfer and accumulation of magnetic nanoparticles in the body [41]. However, there are a number of intracellular and extracellular barriers that can be limiting factors. One of the possible solutions is covering the surface of nanoparticles with biocompatible materials (different organic and inorganic compounds) [37]. Coating the surface of nanoparticles increases the half-life of the drug by delaying clearance [38]. Macrophages take up uncoated nanoparticles at a rate that depends on their functional surface, size and hydrophilicity, followed by clearance in the liver and spleen. Plasma proteins bind to the surface of nanoparticles, accelerating phagocytosis. Coatings enable the slowing down of detection by macrophages and thus reduce clearance. For this purpose, the most often used is polyethylene glycol (PEG), the attachment of which provides a "stealth" protective effect. PEG is suitable for this purpose because it shows low toxicity and immunogenicity and is excreted by the kidneys [42]. In addition, surface coating enables covalent binding of biomolecules such as antibodies and proteins and their transport to the target tissue. It is necessary that these coatings be sensitive to the change in pH value, which would enable the controlled release of the drug [43]. Drug release can be stimulated by chemical radiation, mechanical forces and magnetic hyperthermia [38].

Figure 2 shows the structure of a magnetic nanoparticle carrying the active molecule on the surface.

The most important characteristics of nanoparticles used for drug delivery are intrinsic magnetic properties, the shape and size of nanoparticles, non-toxicity, stability in water, surface charge and coating. Many magnetic materials with ideal magnetic properties, such as cobalt or chromium, are very toxic and cannot be used in medicine, while materials based on iron oxide (magnetite or maghemite) are safe. In addition to low toxicity, these nanoparticles show high stability against degradation [38]. The size of the particles is such that they allow entry into biological structures, and it varies from 3 to 30 nm [41].

This method can be applied to solid tumor mass, neoplasms with metastases and tumors that are in the early phase of cell growth. Treatment would involve the application of particles that specifically recognize clusters of cancer cells, carrying a medically active substance that will act [44].

**Figure 2.** Structure of a magnetic nanoparticle with active molecule on the surface (drug-carrying nanoparticle) and the influential factors on its behavior within a drug delivery system, including stimuli for drug release and cell structure that interact with such nanoparticles.

#### *2.4. Motion of Particles in an External Field and as a Random Variable*

Nanoparticles move after collision with smaller water molecules. Smaller water molecules come across the nanoparticle from all directions and collide with it, thus imparting their momentum to it. When at some point after such a collision a particle receives a momentum in that direction, it starts to move in that direction, until another water molecule collides with it and gives it its momentum by collision. If in the field of classical mechanics, this mode of motion is, in principle, deterministic and there is a corresponding Hamiltonian [45]. In reality, due to the large number of water molecules that interact with each other and with the nanoparticle, and due to the unknown initial conditions, such a system cannot be described using the Hamiltonian in practice, i.e., it is impossible to say with certainty how much and in which direction the particle will move in at some point. On the contrary, a shift in any direction is equally probable (this can be seen intuitively from the symmetry of the system). That is why the tool of statistical physics is used to study such a system. Due to the large number of water molecules that make up the system, instead of observing the microscopic effect on the nanoparticle of each of them, one can observe their "total" macroscopic effect after some time. After repeating such an experiment several times with the same initial conditions (as much as it can be controlled, e.g., placing a nanoparticle with the same physical values in the same place), a different value is obtained for the total macroscopic displacement of the nanoparticle after a certain time. This gives an empirical distribution of the probability of the total movement of a nanoparticle, and therefore, instead of looking at exactly how much a nanoparticle will move at a given moment, one can look at the probability that the particle will move by a certain value at a given moment.

A fluid containing a large number of particles (Brownian particles) and moving in one dimension is observed. The density is given by *n(x, t),* where *n* is the particle density, *x* is the position coordinate and *t* is the time. Brownian motion over time makes the density uniform. The flow of particles during diffusion *jd* is defined by:

$$j\_d(\mathbf{x}, t) = -D \frac{\partial n(\mathbf{x}, t)}{\partial \mathbf{x}} \tag{1}$$

The flow of particles causes a change in density in time according to the continuity equation:

$$\frac{\partial n(\mathbf{x},t)}{\partial t} = -\frac{\partial j\_d(\mathbf{x},t)}{\partial \mathbf{x}} = D \frac{\partial^2 n(\mathbf{x},t)}{\partial \mathbf{x}^2} \tag{2}$$

In the equation, *D* [m2/s] is the diffusion constant (it depends on the type of particle material), and the given equation is the diffusion equation. The flow of particles is opposite to the direction of the density gradient and the flow direction is from the area of higher density to the area of lower density.

A uniform force field (gravitational field) acts on the Brownian particles. The stated field accelerates the particles until the velocity reaches a certain limiting velocity, *ug*, and where the sum of the forces acting on the particle, the force field, *F*, and the friction force is equal to zero, *F=mγug*, where *m* is the mass of the particle and *γ* [1/s] is the velocity gradient. Now, the particle moves with a velocity *ug*, which is determined by the force *F* and the friction that the fluid acts on the particles. Now, the flux of particles is *jF* in the gravitational field:

$$j\_F(\mathbf{x}, \mathbf{t}) = n(\mathbf{x}, \mathbf{t}) u\_{\S} = \frac{nF}{m\gamma} \tag{3}$$

where *m* is the mass of the particle and *mγ* is the mass flux [kg/s] and *n* is the density, while *F* is the strength of the force field. The total flow of particles is written:

> *j*

$$
\dot{\lambda} = \dot{f}\_d + \dot{f}\_F \tag{4}
$$

Now, the diffusion equation is:

$$\frac{\partial n(\mathbf{x},t)}{\partial t} = -\frac{\partial j(\mathbf{x},t)}{\partial \mathbf{x}} = D \frac{\partial^2 n(\mathbf{x},t)}{\partial \mathbf{x}^2} - \frac{F}{m\gamma} \frac{\partial n(\mathbf{x},t)}{\partial \mathbf{x}} \tag{5}$$

For the diffusion coefficient, Einstein finds that the Brownian particle is related to its mobility *μ* [s/kg] via the equation:

$$D = \mu kT \tag{6}$$

where *k* is the Boltzmann constant and *T* is the temperature. This equation is derived as follows. For an arbitrary distribution of the density of Brownian particles after a long time, the flow of particles will equalize and an equilibrium state is obtained (it does not change with time), *n(x, t) = n(x)*. This is the sedimentation equilibrium, which is independent of time *t*0, and then the particle density is where the position coordinate is *x*0:

$$m(\mathbf{x}) = n(\mathbf{x}\_0)e^{\frac{F(\mathbf{x} - \mathbf{x}\_0)}{kT}} \tag{7}$$

Now, the first and second derivatives of the particle flow are performed and the total flow is zero and is obtained as:

$$\frac{D}{kT} = \frac{1}{m\gamma} \tag{8}$$

Accordingly, Einstein's relation is obtained as:

$$D = \frac{kT}{m\gamma} = \mu kT\_\prime \tag{9}$$

where *μ = 1/mγ* is the mobility of the particles, which is equal to the ratio of the force *F* and the limiting velocity *ug*. The force field can also be of electric potential if the Brownian particle is charged.

If the density is high, then the interaction of Brownian particles is ignored and the density is defined as follows. The probability that a Brownian particle is located at the position coordinate *x* at time *t* is defined if it was at time *t*0 at *x*0 with *V*(*<sup>x</sup>*, *tIx*0, *t*0). Now, the density is *n(x, t)*, the integral of the product *<sup>n</sup>*(*<sup>x</sup>*0,*<sup>t</sup>*0) and the probability of passing over all possible values of *x*0. Moreover, since there are many Brownian particles of density at some *x*, there is a certain probability that the Brownian particles arrived at a certain *x* in some time *t* − *t*0 from all other particles of the liquid as:

$$n(\mathbf{x}, t) = \int n(\mathbf{x}\_0, t\_0) V(\mathbf{x}, t / \mathbf{x}\_0, t\_0) d\mathbf{x}\_0 \tag{10}$$

where *V*(*<sup>x</sup>*,*t/x*0,*<sup>t</sup>*0) is conditional probability.

> The given equation is included in the diffusion equation as:

$$\int n(\mathbf{x}\_{0}, t\_{0}) \frac{\partial V(\mathbf{x}, t/\mathbf{x}\_{0}, t\_{0})}{\partial t} d\mathbf{x}\_{0} = D \int n(\mathbf{x}\_{0}, t\_{0}) \frac{\partial^{2} V(\mathbf{x}, t/\mathbf{x}\_{0}, t\_{0})}{\partial \mathbf{x}^{2}} d\mathbf{x}\_{0} \tag{11}$$

where *V* is the probability of the position *x* of the particle at time *t*.

Both equations are shifted to one side and equalized to zero:

$$\int \left[ n(\mathbf{x}\_0, t\_0) \left( \frac{\partial V(\mathbf{x}, t/\mathbf{x}\_0, t\_0)}{\partial t} - D \frac{\partial^2 V(\mathbf{x}, t/\mathbf{x}\_0, t\_0)}{\partial \mathbf{x}^2} \right) \right] d\mathbf{x}\_0 = 0 \tag{12}$$

The equation will be correct regardless of the value of *dx*0, which is the derivative of the position coordinate. Accordingly, the probability *V* (*<sup>x</sup>*, *tIx*0, *t*0) fulfills the diffusion equation as:

$$\frac{\partial V(\mathbf{x},t/\mathbf{x}\_0, t\_0)}{\partial t} = D \frac{\partial^2 V(\mathbf{x}, t/\mathbf{x}\_0, t\_0)}{\partial \mathbf{x}^2} \tag{13}$$

Since the equation is valid regardless of the selected initial position coordinate *x*0 and initial time *t*0, it is written further as:

$$\frac{\partial V(\mathbf{x},t)}{\partial t} = D \frac{\partial^2 V(\mathbf{x},t)}{\partial \mathbf{x}^2} \tag{14}$$

It is assumed that the initial condition is that at the initial time *t*0, all particles are at the same coordinate position *x*0, and the following can be stated:

$$V(\mathbf{x}\_0, \mathbf{t}\_0) = \delta(\mathbf{x} - \mathbf{x}\_0),\tag{15}$$

where the right side of the equation is the Dirac delta function.

So, the solution for conditional probability, *V*, is:

$$V(\mathbf{x}\_0, t\_0/\mathbf{x}, t) = \frac{1}{\sqrt{4\pi D(t - t\_0)}} e^{-\frac{(\mathbf{x} - \mathbf{x}\_0)^2}{4D(t - t\_0)}}\tag{16}$$

The resulting solution is a Gaussian distribution with expected value *x*0 and variance *4D*(*t* − *t*0), and the resulting equation is the Green's function [46].

Now, the time interval within which Brownian motion is observed can be divided into *t*0, ... ,*ti*, ... ,*tN*. *ΔX(ti)* is the displacement of the Brownian particle between time *ti* − 1 and *ti*. Next, *X(ti)* is the position of the particle at time *ti*, and we put *X(t*0 *= 0) = 0*. The Brownian particle is surrounded on all sides by an average equal number of particles (which is shown by the symmetry of the system), and since there is no total flow of water in which the particles are located (the average total velocity of water molecules is zero), the expected probability of the Brownian particle motion is zero because it is equally likely that it can be hit by a particle from any direction. Accordingly, it can be stated:

$$
\langle \Delta X(t\_i) \rangle = 0 \tag{17}
$$

Now, the position of the particle is:

$$X(t\_N) = \sum\_{i=0}^{N} \Delta X(t\_i) \tag{18}$$

The probability of the position is zero:

$$
\langle X(t\_N) \rangle = \langle \sum\_{i=0}^N \Delta X(t\_i) \rangle = \sum\_{i=0}^N \langle \Delta X(t\_i) \rangle = 0 \tag{19}
$$

The autocovariance of the displacement of the Brownian motion is:

$$
\langle \Delta X(t\_i) \Delta X(t\_j) \rangle = 0, i \neq j \tag{20}
$$

So, the covariance between two variables is observed. Autocovariance is a measure of the covariance between the value of a stochastic variable at some time t and its value at some other time. Correlation is the covariance divided by the product of the variances of both variables (normalized to the interval from −1 to 1). The correlation between two variables measures how much one variable changes as the other variable changes, and it is a measure of their mutual linear dependence. Two variables with a correlation of −1 change exactly the opposite (when one increases, the other always decreases); when the correlation is 0, there is no linear dependence of one variable on the other; and when the correlation is 1, when one increases, the other always increases. Furthermore, it is valid for the variance if there is no autocorrelation between the shifts:

$$
\langle X^2 \rangle = \sum\_{i=0}^{N} \langle \Delta X(t\_i)^2 \rangle \tag{21}
$$

It is assumed that they are all *Δ X*(*ti*) 2 equal and their value is *Δ X*2, so:

$$
\langle X^2 \rangle = N \langle \Delta X^2 \rangle = t \frac{\Delta X^2}{\Delta t} \tag{22}
$$

The time *Δt* is the time between collisions between water molecules and Brownian particles, where a water molecule hits a Brownian particle and then it moves by ± *Δ X*. When the time *Δt* has passed, the Brownian particle collides with the water molecule again and moves by ± *Δ X*, and so on. The speed of all particles has been replaced with the average speed, which is the most probable in thermal equilibrium. The displacement of the Brownian particle *Δx = x(t)* − *x(0)* is related to the diffusion coefficient, as:

$$D = \frac{1}{2\Delta t'} \tag{23}$$

and in the following way:

$$
\langle X^2 \rangle = 2Dt \tag{24}
$$

Based on the obtained equation and Einstein's relation [2], it is concluded that the collision time between two molecules is inversely proportional to the friction constant and temperature. A higher friction constant (viscosity coefficient) actually means that collisions with a Brownian particle are more frequent, and a higher speed (temperature) of the particle leads to more frequent collisions.
