Measurement and Modeling of Transport Across the Blood–Brain Barrier

Abstract

The blood–brain barrier (BBB) is a dynamic regulatory barrier at the interface of blood circulation and the brain parenchyma, which plays a critical role in protecting homeostasis in the central nervous system. However, it also significantly impedes drug delivery to the brain. Understanding the transport across BBB and brain distribution will facilitate the prediction of drug delivery efficiency and the development of new therapies. To date, various methods and models have been developed to study drug transport at the BBB interface, including in vivo brain uptake measurement methods, in vitro BBB models, and mathematic brain vascular models. Since the in vitro BBB models have been extensively reviewed elsewhere, we provide a comprehensive summary of the brain transport mechanisms and the currently available in vivo methods and mathematic models in studying the molecule delivery process at the BBB interface. In particular, we reviewed the emerging in vivo imaging techniques in observing drug transport across the BBB. We discussed the advantages and disadvantages associated with each model to serve as a guide for model selection in studying drug transport across the BBB. In summary, we envision future directions to improve the accuracy of mathematical models, establish noninvasive in vivo measurement techniques, and bridge the preclinical studies with clinical translation by taking the altered BBB physiological conditions into consideration. We believe these are critical in guiding new drug development and precise drug administration in brain disease treatment.

1 Introduction

The blood–brain barrier (BBB) is a functional interface lying between the blood flow and the brain parenchyma, controlling the transport of nutrients and preventing the entry of blood-borne toxic compounds and pathogens into the central nervous system (CNS) [1,2]. The BBB is formed by the compact brain capillary endothelial cells (ECs) that are sealed by junctional protein complexes and low levels of transcytosis of the endothelium. The endothelial cells actively interact with various cells, such as pericytes, glial cells, and neurons, to maintain barrier integrity and regulate cerebral blood flow [3,4]. While the BBB allows oxygen and nutrients such as glucose, amino acid, and vitamins to enter the brain to support normal cell metabolism, 98% of small molecules and nearly 100% of large molecules (>400–500 Da) cannot cross the BBB [5]. Therefore, the BBB is a primary obstacle in drug delivery into the brain.

It has been demonstrated that the BBB permeability is elevated in pathological situations such as brain tumors, brain injury, and neurological disorders (e.g., strokes and Alzheimer's disease) [6,7]. However, drug delivery through the impaired BBB is different from the intact BBB under normal physiological conditions because BBB alterations affect most drugs' brain accumulation and intracranial distribution [8]. Moreover, it is unknown whether this BBB disruption is uniform or whether the magnitude is sufficient to allow drug penetration in meaningful quantities. To date, approaches have been developed to overcome the BBB for drug delivery into the CNS. For example, focused ultrasound associated with circulating microbubbles can open the BBB locally and enhance drug delivery to the tumor [9]. This approach exhibits promising efficacy in clinical investigations and has evolved into a significant research domain within the field of brain tumor therapy. Recently, our group has developed a technique to modulate the BBB using ultrashort laser pulse stimulation of blood vasculature-targeted gold nanoparticles and enable the delivery of antibodies, adeno-associated virus, and cargo-laden liposomes into the brain [10]. With these powerful tools, monitoring the drug transport at the BBB interface facilitates the investigation of drug delivery efficiency and the decision on optimal treatment.

Various approaches and models have been developed to study drug delivery across the BBB, including in vivo measurements, in vitro BBB models, and mathematical models. Although highly advanced in vitro BBB models have been developed during the last decade, the full complexity of the BBB is difficult to recapitulate in vitro. Thus, drug delivery across the BBB and into the brain parenchyma must often be investigated in vivo in preclinical studies to predict drug delivery to the human brain adequately. Mathematical models can provide information that is challenging to obtain by experiments alone, especially to help describe and understand the impact of transport processes that govern drug distribution within the brain. However, to date, there is no detailed comparison of these models, which limits the selection of optimal methods for the prediction of drug delivery across the BBB. Since in vitro BBB models have been extensively reviewed [11–13], here we focused on the discussion of the mechanisms of BBB transport and summarized BBB transport studies from an engineering perspective, including measurement methods for transport across the BBB and mathematical models. We also provide insights on the future directions for developing more reliable BBB models to advance the prediction of drug delivery efficiency and spatial distribution.

2 Composition of the Blood–Brain Barrier and the Physiology of Blood–Brain Barrier Transport

The BBB operates as a part of the neurovascular unit (NVU), which defines the functional and structural relationship between the brain and the blood vessels [14]. The NVU incorporates cellular and extracellular components, including neurons, microglia, astrocytes, pericytes, ECs, and the basement membrane [15]. These NVU components closely cooperate with each other and establish a highly efficient system to maintain BBB integrity and regulate cerebral flow [16]. In particular, ECs provide the major cellular contribution to the BBB by suppressing transcytosis and limiting paracellular transport. The limited paracellular transport is largely due to the presence of junctional complexes between the ECs, namely, adherens junctions (AJs) and tight junctions (TJs). AJs are cell-cell adhesion complexes that mainly consist of classical cadherins, such as E-cadherin, maintaining the physical integrity of the epithelium [17]. TJs are the continuous barrier in the gap between endothelial cells formed by the claudin family of transmembrane proteins such as claudin-1, -3, -5, and -12, which can regulate the selectivity of molecules transport [18]. The junctional complexes prevent the passage of most molecules and ions through the paracellular space.

Figure 1 and Table 1 show the main pathways of BBB transport and their characteristics [26,27]. A number of small lipophilic molecules (<400 Da) and gases can passively diffuse across the BBB [28] (Fig. 1(a)) such as ethanol and oxygen. Paracellular transport (Fig. 1(b)) is severely limited by junctional complexes [29,30]. Only small hydrophilic molecules will pass through TJs with low-level diffusions, such as peptide drugs octreotide [31]. For other substances, including essential nutrients and proteins, transport across BBB relies on transport proteins, specific receptors, or vesicles. Receptor-mediated transport (Fig. 1(c)) provides a route for large molecular-weight substances, such as proteins, to enter the brain [32]. It relies on endocytosis by the cell membrane, where a coat protein on the cell membrane begins to polymerize a coat that draws the membrane with it into a vesicle. An example of receptor-mediated endocytosis is the import of iron by transferrin receptors. The transferrin binds to the transferrin receptor, is transported by receptor-mediated endocytosis via an endocytic vesicle, and then releases the iron by decreasing pH [33]. Carrier-mediated transport (Fig. 1(d)) is one major active transport mechanism for some small molecular substances such as hexose, amino acid, and glucose [34]. Table 2 summarizes transporters and transport systems that are commonly expressed at the BBB [35]. The corresponding transporters are expressed on the brain capillary endothelial cells, carrying nutrients and resulting in the transport from the blood to the brain or from the brain to the blood. Adenosine triphosphate is involved in this process to provide energy for transport against the concentration gradient. For nonreceptor-bound materials, such as albumin and histone, adsorptive-mediated transcytosis (Fig. 1(e)) is the main route of transport related to electrostatic interaction. When a positively charged substance (such as modified albumin) contacts negatively charged cell membranes, the transcytosis process will be triggered, and the substance will cross the cell membrane with endocytic clathrin-coated vesicles [25].

Fig. 1.

Physiology of blood-brain barrier transport. There are mainly five transport pathways: (a) Passive diffusion. Blue dots represent small lipophilic molecules. (b) Paracellular transport. Brown dots represent small hydrophilic molecules. (c) Receptor-mediated transport. The large molecules bind to receptors and transport by endocytosis via vesicles (d) Carrier-mediated transport. The molecules are carried by transporter proteins and pass through the BBB. (e) Adsorptive mediated transcytosis. Substance crosses the cell membrane with endocytic vesicles.

Table 1.

Characteristics of five BBB transport pathways

	Passive diffusion	Paracellular transport	Receptor-mediated transport	Carrier-mediated transport	Adsorptive mediated transcytosis
Molecular size	<400 Da	<400 Da	Large molecules	Small molecules	Large molecules
Solubility	Lipophilic	Hydrophilic	Both hydrophilic and lipophilic	Hydrophilic	Both hydrophilic and lipophilic
Required medium	None	None	Receptors and vesicles	Transport proteins	Vesicles
Adenosine triphosphate requirement	No	No	Yes	Yes	Yes
Examples	O2, CO2, sucrose, ethanol	Octreotide	Transferrin, insulin	Glucose, amino acids	Albumin, histone
References	[19,20]	[21,22]	[23]	[24]	[25]

Table 2.

Transporters and transport systems expressed at the BBB

Transporters	Substrates
Energy transport system
GLUT1	Glucose
MCT1	l-Lactate, monocarboxylates
CRT	Creatine
Amino acid transport system
LAT1	Large neutral amino acids
CAT1	Cationic amino acids
EAAT1, 2, 3	Anionic amino acids
Neurotransmitter transport system
GAT2/BGT1	γ-Aminobutyric acid
SERT	Serotonin
ABC transporters
ABCA1	Cholesterol
ABCB1/MDR1	Vincristine, cyclosporin A
ABCC4/MRP4	Topotecan

3 The Basic Concept of Blood–Brain Barrier Transport

Here we review several basic concepts for the transport across the BBB, including the Renkin–Crone equation, the relationship between BBB permeability and molecule lipophilicity, the Krogh cylinder, and the Starling equation. They are the foundations of modeling the BBB transport, which takes blood flow, BBB permeability, and molecule transport rate into consideration. With these basic concepts, the BBB transport can be described mathematically and quantitatively, which will further guide the drug delivery strategy in clinical studies.

3.1 Renkin–Crone Equation.

Blood–brain barrier permeability is one of the most important factors that affect the molecule transport rate in the brain. The most commonly used parameter for modeling BBB permeability is the permeability surface area products (PS), where P is the permeability and S is the surface area [36]. According to Fick's law, tissue uptake $j$ (mol/s/g) of a substance equals the blood flow $F$ (volume/s/g) multiplied by the difference between arterial inlet $C_a$ and venous outlet $C_v$ concentrations (mol/volume) of the substance in a steady-state

$$j=F\times (C_a-C_v)$$ (1)

The permeability $P$ (cm/s) of the brain capillary to a given substance is defined as the amount of substance that passes a unit area in unit time for unit concentration difference across the membrane

$$P=\frac{j}{S\times \Delta C}$$ (2)

where $S$ (cm²/g) is the surface of the capillaries per gram of brain tissue, and $\Delta C$ (mol/volume) is the average concentration difference across the capillary wall. Assume that the substance passes through the capillary wall passively so that the concentration falls exponentially from the arterial to the venous end of the capillary. The log-mean concentration difference is given by

$$\Delta C=\frac{C_a-C_v}{\ln \left(C_a\right)-\ln \left(C_v\right)}$$ (3)

Therefore, the extraction fraction of a specific substance from blood to the brain (E), defined as the fractional reduction of the arterial concentration of the substance during the conversion of arterial to venous blood, is derived as the following equation

$$E=\frac{C_a-C_v}{C_a}=1-e^{-PS/F}$$ (4)

This equation is called Renkin–Crone equation [37,38], indicating that the transport of compounds with passive diffusion is not only related to the permeability but also the supplied blood flow. The blood flowrate in the mouse brain ( $F$) is about 8 × 10⁻³ml/s/g, [39] and the capillary surface area of the mouse brain ( $S$) is 240 cm²/g [37]. Figure 2(a) shows if PS is much lower than the blood flowrate, few molecules will pass through the BBB, known as extraction-limited transport. Sucrose and mannitol are examples of such molecules, which have low permeability values of 10⁻⁶ml/s [41]. On the contrary, when PS is much larger than the blood flowrate, most of the molecules will be extracted from the blood, indicating a high transport rate across the BBB or flow-limited transport. Such molecules include ethanol and caffeine with high permeability values of 10⁻⁴ml/s.

Fig. 2.

Basic concepts of BBB transport. (a) The relation between BBB permeability and blood flowrate. If the BBB permeability is much larger than the blood flowrate, most of the drug will be extracted from the blood, called flow-limited transport. If the BBB permeability is much lower than the blood flowrate, few drugs will pass through the BBB, which is extraction-limited transport (adapted from Sprowls et al. [40]). (b) The relationship between BBB permeability and lipophilicity. The permeability of compounds that pass through BBB passively is a function of the lipophilicity of these compounds, represented as a solid line. (c) Starling forces on a capillary. $P_c$ and $P_i$ are hydrostatic pressure in the capillary and interstitial region, respectively; ${\pi }_p$ and ${\pi }_i$ are osmotic pressure of plasma and interstitial region, respectively. (d)Schematic of the Krogh cylinder. The single capillary is represented as a red cylinder in the center, and the brain tissues are represented as a sublayer. The drug is diffused unidirectionally from the capillary layer to the brain tissue layer.

3.2 Blood–Brain Barrier Permeability and Molecule Lipophilicity.

The permeability of a substance is the key property that affects brain uptake. It can be measured with the brain perfusion technique. In this technique, a cannula is inserted into the afferent vessel of the brain to inject the solution, and another cannula fitted is introduced into the efferent vessel. The solution containing the test substance is injected from the afferent vessel to the efferent vessel. The solute concentration of the inlet and outlet can be measured to obtain the perfusion rate. Therefore, the permeability can be calculated by Renkin–Crone equation. The lipophilicity of a substance, also known as lipid solubility, can be characterized by the octanol–water partition coefficient, which is defined as the ratio of the concentration of a solute in two phases, i.e., the water-saturated octanoic phase and the octanol-saturated aqueous phase [42]. Table 3 summarizes the permeability and octanol-water partition coefficient values of different substances. It has been found that the permeability of compounds that pass BBB passively, such as sucrose and mannitol, has a linear relationship with their lipophilicity (Fig. 2(b)) [28,44]. The compounds with permeability and lipophilicity above that line, such as levodopa and phenylalanine, are actively taken up by the brain. Those compounds with permeability and lipophilicity below that line, such as colchicine and quinidine, are actively efflux substances.

Table 3.

Permeability and octanol–water partition coefficient values of different substances

Transport pathway	Substance	Permeability (cm/s)	Octanol-water partition coefficient	References
Passive diffusion	Sucrose	3.5 × 10−8	2.0 × 10−4	[43]
	Mannitol	9.1 × 10−8	7.9 × 10−4	[43]
	Glycerol	2.1 × 10−7	3.2 × 10−3	[43]
	Urea	3.0 × 10−7	2.0 × 10−3	[43]
	Ethylene glycol	1.7 × 10−6	4.0 × 10−2	[43]
	Trimethylene glycol	2.6 × 10−6	6.3 × 10−2	[43]
Active efflux	Colchicine	2.1 × 10−6	1.0 × 101	[28]
	Quinidine	1.7 × 10−6	5.0 × 101	[28]
Active uptake	Levodopa	6.6 × 10−5	3.2 × 10−3	[28]
	Phenylalanine	2.1 × 10−4	6.3 × 10−2	[28]

3.3 Starling Equation.

Trans-endothelial fluid exchange mainly occurs in the capillaries. It's important to describe the driving forces of fluid exchange for modeling BBB transport. Taking the BBB as a semipermeable membrane, the Starling equation describes the passive exchange of water between the capillary microcirculation and the interstitial fluid (Fig. 2(c))

$$J_v=L_{p}S\left(\right[P_c-P_i]-\sigma [{\pi }_p-{\pi }_i\left]\right)$$ (5)

where $J_v$ is the trans endothelial solvent transport rate (m³/s); $L_p$ is the hydraulic conductivity of the membrane (m²·s/kg); $S$ is the surface area of fluid exchange (m²); $P_c$ and $P_i$ are hydrostatic pressure in the capillary and interstitial region, respectively; $\sigma$ is Staverman's reflection coefficient; and ${\pi }_p$ and ${\pi }_i$ are osmotic pressure of plasma and interstitial region, respectively. $\left[P_c-P_i\right]-\sigma [{\pi }_p-{\pi }_i]$ is the net driving force (kg/m·s²); Hydrostatic pressure is the primary force driving fluid transport between the capillaries and tissues exerted by the blood confined within blood vessels or the interstitial fluid. The osmotic pressure is created by the concentration of colloidal proteins in the blood and interstitial fluid. As a result, blood has a higher colloidal concentration and lower water concentration than tissue fluid. Therefore, it attracts water. Table 4 lists these values for brain capillary transport. Starling equation indicates that the rate at which fluid is filtered across vascular endothelium is determined by hydrostatic and osmotic pressure drops, widely used in mathematical BBB transport models.

Table 4.

Brain capillary parameters in Starling equation

Parameters	Value	References
Capillary hydrostatic pressure	37 mm Hg	[45]
Interstitial hydrostatic pressure	6 mm Hg	[45]
Capillary osmotic pressure	25 mm Hg	[46]
Interstitial osmotic pressure	5 mm Hg	[46]
Hydraulic conductivity	18.4 × 10−4 cm·s−1·mm Hg−1	[47]
Reflection coefficient	1	[47]

3.4 Krogh Cylinder.

Krogh cylinder is a basic mathematical model which is widely used for modeling molecule transport across the capillary and predicting drug distribution in the blood and tissue. It is originally proposed for oxygen transport analysis [48]. Krogh cylinder model includes a cylinder capillary geometry and the equation which describes molecule diffusion in the blood and brain tissue. Figure 2(d) shows that the single capillary is represented as a red cylinder in the center [49], and the brain tissue is represented as a sublayer. The molecule diffuses unidirectionally from the capillary layer to the brain tissue layer. Considering Fick's law and mass conservation, we will get

$$\frac{\partial C}{\partial t}-D{\nabla }^{2}C+u·\nabla C=0$$ (6)

where $C$ is molecule concentration in the blood or tissue (mol/m³), $D$ is the diffusivity of the molecule (m²/s), and $u$ is the blood or interstitial flow velocity (m/s). Given certain boundary conditions, the molecule concentration distribution can be solved. The BBB surface where $r=r_c$ is assumed to have a permeability $P$ to the solute. The radial diffusive flux in the tissue at the surface equals the rate of solute permeation through the surface [50]

$$-D{\frac{\partial C}{\partial r}}_{r=r_c}=-P\left[C\right(r_c^{+})-C(r_c^{-}\left)\right]$$ (7)

where $C\left(r_c^{+}\right)$ and $C\left(r_c^{-}\right)$ are the molecule concentration at the interstitial side and blood side of the BBB surface, respectively. It should be noted that this model is based on passive diffusion, and the diffusivity is constant, indicating constant BBB permeability. Therefore, this model can only be used to predict passive drug transport across the BBB during a unidirectional blood flow and passive drug diffusion in the brain tissue.

4 Mathematical Modeling of Drug Transport

Based on the above basic concepts, mathematical models are developed to describe the drug delivery in the entire brain vascular network, including transport in the blood flow, across the BBB, and in the interstitial region. These models provide a quantitative understanding and prediction of spatial drug distribution within the brain. Furthermore, mathematical modeling is also a cost-effect approach for drug transport prediction to serve as a guide for clinical drug administration.

4.1 Cylinder Model.

Based on the concept of the Krogh cylinder, Mohsen et al. proposed a cylinder model in which the brain capillary is surrounded by multiple layers of brain tissue (Fig. 3(a)) [51], and the thickness of each layer is selected close to a real neurovascular unit. They used this model to investigate brain drug delivery by targeting transferrin receptors using aptamers. In this study, a computational fluid dynamics (CFD) model has been developed to calculate the spatial distribution of aptamer concentration. The results show that the pericyte layer accounts for a 6.7% reduction in the concentration of the aptamer. Based on the assumption that BBB layers are porous, they found the drug delivery efficacy increased from 10.9% to 13.8% when the porosity of all layers increased from 0.1 to 0.9 to account for tissue damage. However, the cylinder model is based on the assumption of constant BBB permeability and only considers one single capillary. In real conditions, the drug will be distributed in a complex capillary network, and the BBB permeability depends on the blood rate according to the Renkin–Crone equation.

Fig. 3.

Mathematical model of BBB transport. (a) Schematic of a neovascular unit, which includes the EC layer, BM, pericyte, and astrocyte cell layers surrounded by the brain parenchyma domain (adapted from Sarafraz et al. [51]). (b) A two-dimensional (2D) percolation vascular network generated through a stochastic process. The algorithm begins from one corner of the domain, which serves as the inlet, and an invasive process is simulated by adding points adjacent to the previously chosen points (adapted from Stylianopoulos et al. [52]). (c) Left: a model for drug transport across the BBB. Drug delivery is considered active transport during the unidirectional flow from left to right. Right: a cubic geometry model of the brain vascular network. A cubic lattice represents a piece of brain tissue with a volume of 1 cm³, and the blood vessel network is represented as lines (adapted from Langhoff et al. [53]). (d) Full view of the microvascular network with cerebral arterioles, draining veins, and capillaries, acquired by two-photon microscopy (adapted from Gould et al. [54]).

4.2 Capillary Network Model.

Large-scale anatomical models of brain vascular networks have been developed to better predict drug distribution. With the help of advanced algorithms and imaging techniques, the geometry of the brain blood vascular system can be more precisely recapitulated to further improve the reliability of these models.

4.2.1 Stochastic Capillary Model.

The Krogh cylinder model is widely used in normal capillaries, which are relatively straight and regularly spaced. In contrast, tumor vascular networks are tortuous, and a wide range of avascular spaces are found in tumors. The fractal dimensions of tumor vascular networks suggest that the tortuous vessels are better represented by invasion percolation, a well-known statistical growth process governed by local substrate properties [55]. Invasive percolation is introduced in a stochastic capillary model, which considers vascular growth with random local substrate heterogeneity [56]. Since tumor vasculature grows in response to local heterogeneity, it is better to model tumor vascular networks using the stochastic capillary model. Figure 3(b) shows a vascular network model with one inlet and one outlet, which is generated through a stochastic process. This model considers the coupling of blood flow, transvascular flow, and interstitial fluid flow. The blood flow in a vessel ( $Q_{\mathrm{vascular}}$) is considered as laminar flow, and the flowrate follows Poiseuille's equation

$$Q_{\mathrm{vascular}}=-\frac{\pi d^4}{128\mu }\frac{\Delta p_v}{\Delta x}$$ (8)

where $d$ is the vessel diameter, $\Delta p_v$ is the vascular pressure difference that corresponds to a vascular length $\Delta x$, and $\mu$ is the blood viscosity. Volumetric fluid flowrate across the vessel wall ( $Q_{\mathrm{transvascular}}$) follows the Starling equation

$$Q_{\mathrm{transvascular}}=L_{p}S(p_v-p_i)$$ (9)

where $L_p$ is the hydraulic conductivity of the vessel wall shown in Table 3. $S$ is the surface area of the vessel. Interstitial volumetric fluid flowrate ( $Q_{\mathrm{tissue}}$) follows Darcy's law [57]

$$Q_{\mathrm{tissue}}=-K_t{A}_C\frac{\Delta p_i}{\Delta x}$$ (10)

where $p_i$ is the pressure of the blood and the brain tissue. $K_t$ is the hydraulic conductivity of the interstitial space, which is set as 1 × 10⁻⁷ cm²/mm Hg/s [52]. $A_C$ is the tissue cross-sectional area. The mass transport equations based on Fick's law are as follows:

$$\frac{d{c}_v}{dt}=-\nu \frac{\Delta c_v}{\Delta x}$$ (11)

$$\frac{d{c}_i}{dt}+v_i\nabla c_i=D{\nabla }^2{c}_i$$ (12)

where $c_v$ and $c_i$ are the drug concentration in the blood flow and the brain tissue, respectively. $\nu$ is the blood velocity. $D$ is the diffusion coefficient of the free drug. The geometry of this model with high vascular density and high irregularity of capillary distribution can mimic tumor vasculature very well. Using this model, Stylianopoulos et al. [57] studied how poor blood perfusion affects drug delivery in tumors. The leakiness is defined by the size of the pores of the vessel wall. They analyzed different pore sizes of blood vessels from 50 nm (poorly permeable tumor vessels) to 400 nm (hyperpermeable vessels). Vascular decompression by alleviating physical forces inside tumors is a therapeutic strategy to improve perfusion for better drug delivery to tumor cells. The simulation results show that vessel decompression improves blood perfusion in tumors for vessels with a pore size of less than 200 nm, providing an optimal perfusion region when vessels are uncompressed. Within this region, drug distribution inside the tumor is optimized.

4.2.2 Cubic Mesh Model.

The 2D geometry model lacks spatial information on drug distribution. To model the three-dimensional drug transport and drug distribution, a cubic mesh model is developed, where the capillary network is assumed as a constant topology and geometry with a cubic lattice for simplicity. This model is applied in the drug delivery prediction in the three-dimensional (3D) capillary space. Figure 3(c) shows that the brain tissues are represented as constant pieces of cubes with blood capillaries in them. Drug delivery is considered active transport for every capillary during the blood flow from the inlet to the outlet. For capillary $i=1,\hspace{0.17em}2,\dots$, the drug concentration variation in the blood results from drug loss to the downstream flow, drug transport to the brain tissue, and inflow from upstream vessels. The drug concentration variation in the brain tissue is due to the drug transport from the blood and the drug clearance from the brain. To sum up, the mathematical transport equation is shown as follows:

$$\frac{d{v}_i}{dt}=-f_i{v}_i\left(t\right)-\frac{k_1{v}_i\left(t\right)}{K+v_i\left(t\right)}+f_i\frac{\sum V_j{v}_j\left(t\right)}{\sum V_j}$$ (13)

$$B\frac{d{w}_i}{dt}=\frac{k_1{v}_i\left(t\right)}{K+v_i\left(t\right)}-k_2{w}_i\left(t\right)$$ (14)

where $v_i$ and $w_i$ are the drug concentration in the blood flow and the brain tissue, respectively. $f=Q/V$ represents the reciprocal of the transit time of the blood, which is the volumetric flowrate $Q$ divided by the vessel volume $V$. $B$ is the ratio of brain tissue volume to capillary volume for a given capillary. $k_1$ is the maximal transport rate. $K$ is the drug concentration at which the transport occurs at half the maximal rate. $k_2$ is the rate of clearance of the drug from the brain. This model is applicable for carrier-mediated transport in the brain capillary mesh. Langhoff et al. used this model to predict the transport of l-Dopa, an amino acid for treating PD, across the BBB via LAT1 transporter [53]. The simulation results show that as the blood permeates the vascular network, the drug concentration in the brain tissue decreases due to the brain clearance of the drug. It is found that brain concentration of l-Dopa decreases by 10% along a 1 cm-long vessel. Vendel et al. [59] used a similar 3D cubic model to predict the transport of drugs whose parameters, such as permeability and diffusivity, are set within the physiological ranges. The brain capillary blood flow and active transport across the BBB have been considered so that the impact of the interaction of cerebral blood flow, BBB characteristics, and brain diffusion can be realistically predicted.

4.2.3 Imaging-Based Vascular Model.

The cubic mesh model lacks the complexity and tortuosity of the brain vascular network. Therefore, there is a need for a highly realistic model. The vascular geometry model based on 3D images has emerged with the development of imaging technologies. The vascular anatomical network model shown in Fig. 3(d) is acquired by two-photon microscopy. The flow and distribution of the drug are calculated by bringing these realistic vascular networks into a mathematical model. Physiological parameters of the vascular network, such as geometry information, are measured with microscopic approaches to obtain analytical results closer to reality. This model has been used to study tracer distribution in the 3D tumor vasculature using mathematical simulation [60]. Results from the simulations demonstrate that drug delivery to the tumor is affected by flow, permeability, and diffusion in the interstitial space. To be specific, a higher flowrate results in higher values of average tissue concentrations; the rise of blood vessel permeability leads to a rapid increase of tissue concentration; when the diffusion through the interstitial space is slow, the drug accumulates rapidly within the capillary network. With the help of highly realistic computational models, simulation results become convincing and enable more precise prediction of drug permeability at the BBB.

5 In Vivo Blood–Brain Barrier Transport Measurement

5.1 In Vivo Techniques for Measuring Brain Uptake.

In vivo methods are the most direct way to measure brain drug uptake as it represents entirely physiological conditions. As the field related to brain research has expanded exponentially over the decades, in vivo methods for measuring the brain uptake and permeability of small molecules have also evolved.

5.1.1 In Situ Brain Perfusion.

In situ brain perfusion technique is one of the techniques to assess the brain uptake of the molecules, which involves the injection into the carotid artery and has the advantage of quantitative analysis. In situ brain perfusion technique is developed by Takasato et al. [43], which consists of the perfusion of the drug-containing buffer using a catheter in the external carotid artery (Fig. 4(a)). After the perfusion, the brain is extracted, and the solute concentration in the ipsilateral hemisphere can be measured by high-performance liquid chromatography (HPLC) or scintillation counting if the solute is radiolabeled [63]. Then, the drug delivery rate, $K_{in}$ (ml/s), can be calculated with the following equation:

$$\frac{Q_{br}}{C_{pf}}=K_{in}t+V_0$$ (15)

Fig. 4.

In vivo techniques for measuring brain uptake. (a) Schematic of in situ brain perfusion. A catheter in the external carotid artery is used for the perfusion of the drug-containing buffer (adapted from Alata et al. [61]). (b) Single time point analysis method. Blood samples are collected at different time points during the transcranial perfusion, as well as the final brain sample. The drug concentration of these samples and be measured, and a curve representing the relation can be drawn. (c) Multiple time point analysis method. Taking brain samples at different time points and assuming a linear range of uptake for a given tracer, a linear relationship between the ratio of brain/blood concentration and the exposure time will be displayed. The slope represents the drug transport rate ( $K_{in}$) (adapted from Sommariva et al. [62]).

where $Q_{br}$ is the final quantity of tracer in the brain (mol); $C_{pf}$ is the tracer concentration in the perfusion buffer (mol/ml); $t$ is the perfusion time (s); and $V_0$ is the volume of distribution of vascular tracer (ml). Cannon et al. [64] applied this technique to measure brain accumulation of the P-glycoprotein substrate. They perfused the rats with oxygenated Ringer's solution (isotonic solution relative to the animal body fluids containing NaCl, KCl, CaCl₂, and NaHCO₃) and infused three different P-glycoprotein substrates, ³H-verapamil, ³H-loperamide, and ³H-paclitaxel. They found that the application of sphingosine-1-phosphate reduced P-glycoprotein activity and increased brain uptake of these three substrates for five-folds, which would otherwise be pumped out of the brain. This technique has the advantages of high sensitivity, complete control of perfusion, and avoiding the effect of metabolism.

5.1.2 Single Time Point Analysis.

Single time point analysis is applied for measuring brain uptake of drugs after intravenous injections (Fig. 4(b)) [65]. In this technique, blood samples are collected at different time points during the transcardial perfusion, and the terminal brain sample is also collected. The drug concentration of these samples can be measured, and a curve representing the relation can be drawn, as shown in Fig. 4(b). Then, the brain uptake for the compound $K_{\mathit{in}}$ (ml/s), defined as flux from blood to the brain, can be calculated as

$$K_{\mathit{in}}=\frac{C_{\mathrm{br}}}{\mathrm{AU}{\mathrm{C}}_0^T}$$ (16)

where $C_{\mathit{br}}$ is terminal brain concentration (mol) and $\mathrm{AU}{\mathrm{C}}_0^T$ is the area under the curve for plasma concentration (mol·s/ml) from time 0 to time $T$. Miah et al. [66] used this technique to measure BBB permeability associated with [¹⁴C] sucrose as a tracer. The method proved a linear concentration relationship between the diluted blood and brain homogenate, demonstrating that [¹⁴C] sucrose is a valid marker for studying BBB permeability. Chowdhury et al. measured BBB permeability in mice using [¹³C₁₂] sucrose as the vascular marker. The $K_{in}$ value acquired by single time point analysis is 0.501 ± 0.064 μl/min. This technique will be more accurate when the drug transport occurs only from the blood to the brain, but it can only be applied to molecules with linear pharmacokinetics, such as sucrose.

5.1.3 Multiple Time Point Analysis.

The single time point analysis method only collects the brain sample at the terminal time point and can be subjected to errors. Based on this technique, Patlak et al. developed an equation by taking brain samples at different time points (Fig. 4(c)) [67]. Then, assuming a linear range of uptake for a given tracer, a linear relationship between the ratio of brain/blood concentration and the exposure time will be displayed, where the slope $K_{\mathit{in}}$ represents the brain uptake flux (ml/s), following the equation below:

$$\frac{C_{\mathit{br}}}{C_{\mathit{pl}}}=K_{\mathit{in}}\times \frac{C_{\mathit{pl}}}{\mathrm{AU}{\mathrm{C}}_0^T}+(V_p+V_0)$$ (17)

where $C_{\mathit{br}}$ (mol) and $C_{\mathit{pl}}$ (mol/ml) are brain and blood concentration, respectively. $\frac{C_{\mathit{pl}}}{\mathrm{AU}{\mathrm{C}}_0^T}$ represents the exposure time (s). $(V_p+V_0)$ (ml) is the interception, representing the volume of plasma in the tissue sample. Due to the large sample data sets, this technique has high accuracy and reliability. Still, it has the same limitation as the single time point analysis method, which only applies to unidirectional transport.

These in vivo measurement methods have been proven practical in numerous studies, but the limitations cannot be ignored. First, we can only quantify the drug delivery into the brain from in vivo measurements without understanding the transport mechanisms, such as the transport pathways. Therefore, it is challenging to model and predict the transport of new drugs. Second, brain uptake cannot be monitored in real-time or continuously due to the perfusion process. Therefore, it is necessary to develop noninvasive measurement methods.

5.2 Imaging-Based Techniques to Measure Blood–Brain Barrier Transport.

The techniques described above involve the animal perfusion process, which lacks continuous monitoring of drug transport. Recently, in vivo imaging methods have been developed to observe drug transport at the BBB. Table 5 lists some of the general characteristics of representative imaging techniques, including Positron emission tomography (PET), Single-photon emission computerized tomography (SPECT), Magnetic resonance imaging (MRI), and two-photon imaging.

Table 5.

Characteristics of representative imaging techniques

Imaging technique	Spatial resolution	Temporal resolution	Mechanism	Advantage	Disadvantage	References
Positron emission tomography (PET)	0.1–1 mm	Seconds–minutes	Positron emitting radio isotope	High contrast; high tissue penetration depth;	High cost	[68,69]
Single-photon emission computerized tomography (SPECT)	4–15 mm	Minutes	Gamma emitting radio isotope	High tissue penetration depth; low cost	Poor spatial resolution	[68,69]
Magnetic resonance imaging (MRI)	1–2 mm	Minutes–hours	Excitation by a magnetic field	No radiation; high tissue penetration depth	High cost; long scanning time	[68,69]
Two-photon imaging	250 nm	Seconds–minutes	Fluorescence	High spatial resolution; no ionizing radiation	Poor penetration depth (<2 mm)	[70]

Positron emission tomography is a noninvasive imaging technique to measure brain uptake of drugs by using radioactive substances [71]. In this technique, a PET tracer (molecular probe), a positron-emitting radiopharmaceutical such as fludeoxyglucose (F-18), is introduced into the live body and interacts with specific target molecules. Then the annihilation radiation occurs, and two gamma photons with the same energy but opposite directions are generated and detected by a PET scanner. It is widely used in clinical study and basic research in neurology and neuro-oncology due to its advantages of high sensitivity and high tissue penetration depth. Tournier et al. [72] used PET imaging to acquire brain distribution of P-glycoprotein and breast cancer resistance protein substrate [¹¹C] erlotinib in mice to improve brain delivery of this substrate. Tariquidar is an effective agent for transporter inhibition, which facilitates erlotinib entering the brain. After the infusion of erlotinib and tariquidar into the mice, they found that the brain distribution volume of [¹¹C] erlotinib increased by three folds (Fig. 5(a)). This finding may enhance brain delivery of molecularly targeted anticancer drugs for more effective treatment of brain tumors. Therefore, PET imaging is a powerful technique for investigating the brain uptake of drugs in vivo. However, PET needs to employ a huge and high-cost cyclotron to produce most of the isotopes [76].

Fig. 5.

Imaging-based techniques to measure BBB transport. (a) Left: Representative coronal, axial, and sagittal [¹¹C] erlotinib PET summation images obtained in macaques. Right: Metabolite-corrected arterial input function of the parent [¹¹C] erlotinib in the brain of macaques under the different tested conditions (adapted from Tournier et al. [72]). (b) Coronal and sagittal SPECT images of representative tg-ArcSwe mice at day 14 after injection of RmAb158-scFv8D3 or RmAb158 (adapted from Gustavsson et al. [73]). (c) Left: a representative example with BBB disruption achieved by 0.6-MPa focused ultrasound exposure. The BBB opening was easily monitored by leakage of the MR contrast agent into the brain parenchyma (arrows). The location of the BBB opening was confirmed by trypan blue staining of the affected area. Right: magnitude of BBB disruption monitored by the MR-intensity change. Absolute values of the MR intensity are plotted for repeated image acquisitions after sonication (adapted from Kinoshita et al. [74]). (d) 3D reconstruction of the signal from nanoparticles associated with endothelium 2 h postinjection in vivo. It is based on two-photon in vivo images of RI7-L-A550 nanoparticles (adapted from Kucharz et al. [75]).

Single-photon emission computerized tomography is very similar to PET in its use of a radioactive tracer and detection of gamma rays. In contrast with PET, the tracers used in SPECT emit gamma radiation that is measured directly. Compared with PET, SPECT also has the advantage of high sensitivity and high tissue penetration depth but with lower spatial resolution. Moreover, SPECT is significantly less expensive than PET because the radionuclides used in SPECT have a longer half-life and are relatively easily obtained than PET. It can also be used in preclinical neurology, oncology, and drug development studies, as well as clinical applications such as the diagnosis or monitoring of brain disorders. Gustavsson et al. [73] studied the long-term brain distribution of two radiolabeled monoclonal amyloid-β (Aβ) antibody variants, RmAb158 and RmAb158-scFv8D3, which are used in AD immunotherapy. in vivo SPECT was used to investigate brain retention and brain distribution of the antibodies. The results showed that RmAb158-scFv8D3 has higher brain uptake and uniform brain distribution than RmAb158 (Fig. 5(b)), indicating that the application of the bispecific antibody contributes to improving immunotherapy efficacy.

Magnetic resonance imaging (MRI) is a diagnostic procedure that generates 2D images of the organs in the body. MRI uses a strong magnetic field to force the protons in the body to align with this field. When pulses of radio frequency current pass through the body, the protons are stimulated and spin out of balance, resisting the pull of the magnetic field. When the radio frequency field is turned off, the MRI sensor is able to detect the energy released when the protons realign with the magnetic field. MRI takes the advantages of high spatial resolution, no penetrating tissue limit, and no radiation. MRI has a wide range of applications in medical diagnosis, including neurological cancers and cardiovascular diseases. Moreover, BBB integrity can be monitored by MRI. For example, focused ultrasound (FUS) combined with the infusion of contrast agent microbubbles (MBs) is a noninvasive technique that can cause BBB disruption in targeted brain regions [77]. FUS is usually applied in combination with MRI to monitor the effects of the BBB opening and consequent substance administration with high precision. Kinoshita et al. [74] investigated the passage of Herceptin, a therapeutic antibody for breast cancer treatment, using MRI-guided FUS. They found that Herceptin can be delivered locally into the mouse CNS after BBB disruption, and the degree to which the BBB is open can be estimated with MRI by measuring the MR-intensity change caused by leakage of the MR contrast agent into the brain (Fig. 5(c)). MRI may represent a powerful technique for monitoring the delivery of macromolecular agents, such as antibodies, to treat patients with brain diseases.

Two-photon excitation microscopy is a fluorescence imaging technique with very high spatial resolution. It allows the visualization of living tissue at depths unachievable with conventional microscopy using simultaneous excitation by two photons. The basic principle of two-photon excitation is that in the case of high photon density, fluorescent molecules can simultaneously absorb two long-wavelength photons and emit a shorter-wavelength photon that can be detected by the camera. Two-photon microscopy has been involved in numerous fields, including physiology, neurobiology, embryology, and tissue engineering. Kucharz et al. [75] applied two-photon microscopy to characterize the receptor-mediated transcytosis of nanoparticles that deliver to the brain in vivo. Using this in vivo imaging method, they monitored the entire process of transferrin-mediated transcytosis of nanoparticles across the endothelial cells as well as interstitial transport. They found that transferrin-targeted nanoparticles bind to brain endothelial cells at venules and capillaries but not at arterioles, and the trafficking of nanoparticles to the brain occurs almost exclusively at postcapillary venules (Fig. 5(d)). These observation results prove that postcapillary venules are the key site for transcytosis-mediated brain entry of nanoparticles and provide a potential strategy of therapeutic nanoparticle delivery to the brain, which is targeting venules.

Unlike brain perfusion techniques, these imaging techniques are noninvasive methods for measuring drug transport, and they provide information on drug distribution or the drug transport mechanism. These studies will shed light on future investigations of drug delivery. To better understand whether a drug can pass through the BBB and the rate, high throughput in vivo screening and measurements are essential for developing new drugs for brain disease treatment.

6 Conclusion and Future Perspective

Drug delivery at effective concentrations to the brain remains challenging for CNS disease treatment, largely due to the presence of the BBB. Therefore, advancing our understanding of BBB transport physiology is necessary to evaluate the drug efficacy and develop novel therapies. This review summarizes BBB transport studies including basic concepts, mathematical models, and in vivo measurement methods. In particular, basic concepts of BBB transport are cornerstones of mathematical models. Understanding the geometry of the Krogh cylinder and transport equations is important for building an integrated mathematical model of the brain vascular network. The mathematical models provide a quantitative understanding and prediction of spatial drug distribution within the brain, serving as a guide for clinical drug administration. in vivo measurement methods are the most direct way to measure brain drug uptake. Therein, in vivo brain perfusion techniques provide an intuitive understanding of drug kinetics at BBB. in vivo imaging techniques are noninvasive methods for measuring drug transport, and they provide information on drug distribution or the drug transport mechanism. These in vivo methods are essential for new drug development for brain disease treatment.

Future development will mainly focus on improving the accuracy of current models and bridging the gap between theoretical studies and clinical translation. First, the in vivo measurements of drug transport kinetics cannot satisfy the requirements of noninvasiveness and accuracy at the same time. Recent developments in genetically encoded, implantable, and portable biosensors can be integrated with drug delivery to monitor drug transport in real-time [78,79]. Second, current mathematical models do not take BBB transport pathways into consideration, which limits the accuracy when determining the drug transport rate. For example, carrier-mediated transport and receptor-mediated transport have different mechanisms, but both of them are considered active transport in mathematical models. The specific model of each transport pathway should be developed. Moreover, abnormal BBB integrity and function during CNS disease progression may be associated with altered drug transport pathways. Therefore, a comprehensive study on drug transport at altered BBB physiological conditions will facilitate the development of potential therapeutics against CNS disorders. In summary, improving mathematical models and in vivo measurement techniques will build a solid foundation for new drug development and precise drug administration in brain disease treatment.

Funding Data

• Cancer Prevention and Research Institute of Texas (Award No. RP190278 and RP190278; Funder ID: 10.13039/100004917).

• National Institute of General Medical Sciences (Award No. R35GM133653; Funder ID: 10.13039/100000057).

• National Institutes of Health (Award No. 2123971; Funder ID: 10.13039/100000002).

Data Availability Statement

No data, models, or code were generated or used for this paper.