1 Introduction

The development of new therapies has been hampered by the limitations of traditional cell culture models and animal testing [1]. Cells grown two-dimensionally, although convenient, often lack the complexity of real organs, leading to inaccurate predictions of how drugs might work in humans. Animal testing, on the other hand, offers a more complete picture but is hampered by its high cost, time-consuming nature, and ethical concerns. These limitations translate to a high failure rate in drug development, with only a few candidate drugs progressing from pre-clinical testing to successful clinical trials [2]. In this regard, a recent technology called organ-on-a-chip (OoC) emerged and it has shown promise.

OoC devices are advanced microfluidic devices that offer a more realistic, accurate, and ethical option, with the potential to accelerate the drug discovery process at reduced costs [3, 4]. They aim to mimic the structure and function of specific human organs by incorporating living cells in a scaffold combined with microfluidic technology. These miniature platforms offer several advantages. Unlike simpler cell culture methods, OoCs provide a more realistic microenvironment for the cells, allowing for better predictions of how drugs might affect humans. Additionally, OoCs enable researchers to rapidly test numerous drug candidates, significantly accelerating the pace of drug discovery [5]. The ability to integrate micro(bio)sensors, including electrochemical and optical within the chip, further enhances OoCs by allowing researchers to monitor critical parameters in real time [6,7,8,9,10]. One example is the continuous monitoring of barrier integrity using measurements of trans-endothelial electrical resistance (TEER) without causing any disruption to the system [11]. Additionally, the integration of biosensors holds huge potential for real-time detection of specific molecules, such as drug metabolites or biomarkers released by the cells, providing a more comprehensive overview of drug effects [12]. While traditional analytical methods like flow cytometry, enzyme linked immunosorbent assay (ELISA) tests and polymerase chain reaction (PCR) remain valuable tools, sensor integration offers the significant advantages of being label-free, non-invasive, and allowing for continuous monitoring, eventually leading to a more robust and efficient drug discovery process [13]. For these reasons, OoCs have been increasingly used in biomedical research, and recently, researchers have found that they are a promising technology for understanding the complex interactions of neural cells and tissues in vitro as well as developing and/or testing new therapies [14,15,16]. This is possible by replicating the Blood–Brain Barrier (BBB) function.

The BBB is a cellular barrier that contains endothelial cells that line the walls of the brain's blood vessels surrounded by pericytes, astrocytes, neurons, and microglia [17] (Fig. 1). The BBB ensures adequate neuronal function by regulating the transport of molecules between the Central Nervous System—CNS and the blood [18,19,20,21,22]. This prevents the entrance of harmful blood-borne and large molecules while facilitating the transport of essential nutrients and oxygen to the brain. Disruptions in BBB function are linked to various debilitating neurological disorders like Alzheimer's disease, Parkinson's disease, and multiple sclerosis [23, 24].

Fig. 1
figure 1

Schematic view of the cross-section of a neurovascular unit. This consists of astrocytes, endothelial cells, pericytes, a basement membrane, and tight junctions. Astrocytes provide support and protection to blood vessels, while endothelial cells line the vessels. Pericytes surround endothelial cells and provide additional support. Tight junctions form specialized protein complexes that prevent blood and other fluids from leaking out of the vessel

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By mimicking the key structural and functional aspects of the BBB, BBB-on-a-chip devices offer a unique platform to study neurological diseases and develop new therapeutic strategies. These microfluidic systems can be employed to model BBB permeability, assess drug delivery efficiency across the barrier, and evaluate the potential neurotoxicity of drug candidates [25,26,27]. However, despite the significant advancements in OoC technology, they may lack the complexity of the in vivo environment. Mimicking structural complexity, maintaining functional barrier properties, facilitating cellular interactions, and ensuring scalability and reproducibility are some of the key aspects that BBB models on a chip must fulfill. This is where computational simulations become crucial.

In biomedical research, numerical simulations have emerged as a valuable and cost-effective tool for studying complex biological systems, analyzing physiological processes, and developing innovative devices suitable for testing new drugs and therapies. In the field of OoC, numerical simulations have been mainly used to assess fluid flow behavior and physical phenomena, such as, oxygen transport [28, 29]. In particular, in the field of BBB-on-a-chip, these have been a powerful tool to estimate the shear stress (SS) as well as, evaluate the fluid flow given the complexity of performing this in vivo [30, 31] and also to improve the device accordingly. By integrating computational models with BBB-on-a-chip devices, researchers can bridge the gap between in vitro and in vivo models, potentially originating a new era of discovery in neurological drug development. Nevertheless, this integration is still lacking in most of the current published studies.

Despite there are several literature reviews on the scope of engineered Brain-on-a-chip models [10, 32,33,34,35,36,37,38], there is a notable gap in these reviews regarding the advancements made in numerical simulations within this field of research. This literature review aims to address this gap by discussing diverse approaches and techniques used to model BBB function, including fluid dynamics, mass transport, and electrical properties as well as the challenges and future directions. To the best of the authors' knowledge, this is the first literature review that provides an overview of the numerical models developed so far in the scope of in vitro BBB-on-a-chip devices.

2 Brief Overview of BBB Replication and its Integration on Microfluidic Devices

Many attempts have been made to mimic the BBB in vitro and these started in the early 1970s by isolating brain microvessels [39, 40]. Since then, several in vitro BBB models have been developed, from simple 2D models [41,42,43,44] to single [45,46,47] or multi-organ [28, 48, 49] advanced microfluidic devices. In particular, membrane-based models emerged as a valuable tool for studying the BBB and its function. Different polymeric membranes have been used, namely polycarbonate [50,51,52,53,54,55,56,57,58,59,60,61], teflon [62], polyethylene terephthalate [46, 63,64,65,66,67,68,69,70], polydimethylsiloxane (PDMS) coated membranes [71,72,73], polycarbonate track-etched (PCTE) embossed with topographic patterns from a master mold [74], and also commercial ultrathin nanoporous silicon nitride membranes [75,76,77]. Despite being different materials, all of them exhibit biocompatibility, and transparency, and often require coatings to enhance cell adhesion and growth. However, while materials like polycarbonate, Teflon, and polycarbonate track-etched are typically rigid, potentially offering stability for cell growth, PDMS is characterized by its flexibility. Thus, when selecting the appropriate membrane factors such as the type of device/assembly, cell types, assay aims, and working methods should be considered.

Although the use of membranes is the most common, other approaches have been considered to mimic the BBB function, including hollow tubular microvessels, where cells are seeded aiming to better mimic the in vivo conditions [78,79,80,81,82,83,84,85,86,87,88,89]. Other authors cultured brain cells and endothelial cells in two independent vascular channels separated by an interface with 3 μm pores which replace the need for membranes [90, 91], while other researchers included microhole structures [92], or pillars/posts [93] on the microfluidic device for trapping the cells. Alternatively, the use of hydrogels has also increased for this type of application, such as collagen gels developed on the chip at 37ºC for posterior cell seeding [94], or through the injection of cell-hydrogel solutions [95,96,97,98,99]. In other studies, cells are seeded in previously coated channels [100,101,102,103,104,105].

In addition to the variety of biomaterials used to recreate the BBB, the geometries and dimensions of the developed chips also significantly differ from author to author as can be observed in Fig. 2. Some devices have a two-part composition, consisting of lower and upper compartments (Fig. 2a–e). Meanwhile, others are designed with a single layer but include adjacent channels in parallel (Fig. 2f–i).

Fig. 2
figure 2

Overview of different geometries developed by several authors: a A dissected view of the chip reveals two distinct components: a top plate and a bottom plate, interconnected by a porous PDMS membrane. Reproduced with permission [73]. Copyright 2020, Royal Society of Chemistry. OpenAccess; b Photo of the fabricated device with the blue color representing the upper channel, and red color, the lower channels. Reproduced with permission [34]. Copyright 2020, Springer Nature. Open Access; c 3D configuration of a single BBB unit of the chip. The upper channel represents the luminal channel, and the lower channel represents the abluminal channel. Reproduced with permission [106]. Copyright 2021, Elsevier; d Picture of the fabricated BBB microfluidic system. It contains a bottom and top channel separated by a polyester membrane. Reproduced with permission [107]. Copyright 2016, Wiley; e Schematic view of the microfluidic platform. This comprises a lower perfusion layer containing microchannels and lower electrodes, an intermediate layer with two reservoirs and the neuronal chamber, and an upper lid layer with upper electrodes that covers the channel and reduces evaporation. Reproduced with permission [108]. Copyright 2016, Wiley; f Schematic representation of the device comprising five parallel PDMS channels. Reproduced with permission [47]. Copyright 2019, Wiley. Open Access; g Exploded view of the microdevice. This consists of three layers, a bottom layer with the electrode, a middle layer with fluidic channels, and a top layer containing the media reservoir. Reproduced with permission [109]. Copyright 2020, Elsevier; h Diagram illustrating the placement of inlet/outlet and access points for filling the hydrogel reservoir within the microfluidic device. Reproduced with permission [110]. Copyright 2017, Elsevier; i Top view representation of the microfluidic chip designed to simultaneously handle multiple porous tubes. Reproduced with permission [111]. Copyright 2017, Wiley

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While the previous models were originally developed by researchers, commercial BBB models are also being adopted in research. Examples include SynBBB from SynVivo [103, 112], brain-chip from Emulate [113], and OrganoPlate from Mimetas [114, 115].

Many authors have focused their research using static models, which lack the presence of SS [100,101,102, 104, 116,117,118,119,120,121]. This is a key factor for maintaining the structure and function of endothelial cells [122,123,124,125,126] and it is reported to vary from 4–30 \({dyn}/{{cm}}^{2}\) in the arterial circulation and 1–4 \({dyn}/{{cm}}^{2}\) in the venous circulation [127,128,129,130]. In general, it has been shown that under physiological SS conditions, the differentiation of vascular endothelial cells is potentiated, increased protein levels with the upregulation of tight and adherent junctions proteins, genes, and enzymes [118, 131,132,133,134], as well as optimal TEER [135], and increased expression of multidrug resistance transporters [136]. Nonetheless, it is interesting to notice the wide range of flow rates found in the literature to attain the physiological SS in brain tissues. This varies from 0.01–10 µL/min [55, 90, 103], 10 µL/min – 100 µL/min [14, 113, 137], and 1–6 mL/min [54, 133, 135, 138, 139], while some authors just mention that physiological SS is obtained without saying the flow rate applied in the experiments [28, 46, 74, 96]. Although usually the same flow rate is applied in all channels of the device, in order to better recreate the in vivo conditions, different flow rates may be applied [70]. Moreover, generally, the flow rate is induced through the use of peristaltic pumps, however, some authors used pump-free systems to generate a flow rate similar to the one found in vivo driven by gravity [84, 108]. On the other hand, other authors explored both steady and pulsatile SS through an orbital rotator, and CellMax artificial capillary system, respectively [131].

It is also interesting to notice that SS is commonly mentioned, but the formula to calculate it varies from author to author. For rectangular cross sections, the most common is \(6{\mu Q}/({{h}}^{2}{w})\) [94, 97, 140, 141], where µ represents the fluid viscosity, \(Q\) denotes the flow rate, \(h\) represents the channel height, and \(w\) stands for the channel width. However, there are variations among authors regarding the value used, with some considering 4 [131] or 12 [29]. Consequently, the quantification of SS may not be accurate, and standardization is necessary. For circular channels, the SS can be calculated through \(32{\mu Q}/(\uppi {{d}}^{3})\), where d is the diameter of the channel cross section [142].

Table 1 summarizes the advantages and disadvantages of using in vivo and in vitro models to study the BBB.

Table 1 Advantages and disadvantages of using in vivo and in vitro models to study the BBB
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3 Numerical Models Developed to Recreate the BBB In Vitro

As aforementioned, there is a vast variety of research papers in the field of in vitro BBB models, however, few combine numerical tools to obtain insights about the physical phenomena within the device.

Some authors have used numerical simulations to better understand and evaluate the fluid flow behavior [73, 111], SS [106, 108, 146], mass transfer [28, 34, 47, 77, 109, 110, 137], including the transport of nanoparticles (NPs) across the BBB [107, 147, 148], sound field simulation [149], voltage [60] and zeta potential [150], and also cellular alignment [151]. In addition, to Computational Fluid Dynamics (CFD) simulations, recently some researchers have implemented molecular simulations [152, 153] to gain more insights into the transport mechanisms of molecules across the BBB. Other authors have resorted to 4D simulations (both spatial and temporal dimensions) to understand the sensitivity and challenges associated with characterizing BBB permeability, providing a non-invasive testing option that can be good for optimizing imaging protocols for future studies [154].

3.1 Fluid Flow Patterns in BBB-on-a-Chip Devices

In terms of fluid flow behavior, computational simulations have been used to analyze the flow patterns within the devices. For instance, Marino and co-workers [111] developed a BBB-on-a-chip through two-photon lithography, a high-resolution 3D printing technique to fabricate models at a real-scale size. This approach allowed for precise control over the microstructure of the BBB model. The model's performance was validated by its ability to produce results consistent with those observed in vivo in terms of microcapillary diameter size and fluid flow. In this case, the authors used numerical simulations to analyze the flow patterns within the 3D BBB model, considering the pores of the fabricated model (Fig. 3a). Similarly, Zakharova and co-workers [73] evaluated the flow distribution in the channels of the device as well as the velocity fields (Fig. 3b). The authors developed a multiplexed BBB-on-a-chip platform capable of simultaneously analyzing multiple experimental conditions. This innovative device overcomes the limitations of traditional BBB-on-a-chip models, enabling researchers to conduct a wider range of tests in a more efficient and reproducible manner, namely evaluating the effects of different drugs on BBB permeability in a parallel manner.

Fig. 3
figure 3

Copyright 2017, Wiley; b simulated flow profile in the chip using finite element modeling. Reproduced with permission [73]. Copyright 2020, Royal Society of Chemistry. Open Access; c) Serum transport simulation through the microfluidic channels in the device without considering the effect of cells. Reproduced with permission [34]. Copyright 2020, Springer Nature. Open Access; d finite element analysis of concentration distribution of a chemokine in the abluminal compartment fluid without (i) and with (ii) oscillating laminar flow. Reproduced with permission [77]. Copyright 2023, Elsevier; e Dextran transport under pulsating flow conditions at the inlet boundary (mg/mL). Reproduced with permission [110]. Copyright 2017, Elsevier; f simulation results of the liquid/air volume fraction after 10 min. Reproduced with permission [107]. Copyright 2016, Wiley

Computational simulations outputs: a visualization of fluid flow patterns derived from numerical calculations for a simplified two-dimensional geometry. Reproduced with permission [111].

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As previously mentioned, SS is a key mechanical stimulus experienced by endothelial cells in blood vessels, which consequently, helps to replicate the physiological conditions and functional characteristics of the BBB. However, this is difficult to measure experimentally. For this reason, several researchers have used computational simulations to estimate it. Some efforts have been made to attain uniform SS patterns where cells are cultured. This can be obtained, for instance, by creating a step chamber in the device [108]. With this, SS can be minimized while providing the same desired flow rate. Booth et al. [146] also showed that their device provides a uniform SS distribution. In this case, a parallel multichannel device was developed for studying the effects of SS on vascular cells. The channels have different dimensions which allows subjecting cells to different levels of SS, the smaller the channel the bigger the SS. Different responses in terms of cell morphology, alignment, migration, and gene expression were observed, stressing the importance of characterizing the full range of SS effects. By increasing SS, permeabilities of Fluorescein isothiocyanate (FITC) conjugated dextran and propidium iodide decreased, while the TEER, protein expression of the tight junction component, and the alignment of cells along the flow direction increased when compared to static control. In addition to varying the channel dimensions, other authors [106] investigated the SS levels by modifying the flow rates, and porosities of the polycarbonate membrane of the BBB model. Similarly, they found that the SS exhibited an increasing trend with the reduction of the microfluidic channel dimension and porosity. With these results, it was shown that the properties of the membrane affect the SS undergone by cells.

3.2 Transport Phenomena Across the BBB

Computational simulations have also been useful to understand the transport phenomena across the BBB membrane. Serum transport is one of the examples [34] (Fig. 3c). In this case, the porosity of the membrane was considered, but without cellular components. On the other hand, other authors [77] presented a BBB-on-a-chip that integrates digital immunosensors for in situ and real-time monitoring of cytokine secretion. Similarly, the authors simulated the transport and distribution of cytokines within the chip (Fig. 3 d)). Computational simulations provided insights into the spatiotemporal distribution of cytokine concentrations at the brain endothelial barrier, considering the lateral diffusion of analyte molecules traversing the membrane pores. Similarly, Shin and co-workers [47] used numerical simulations to simulate the concentration profiles of amyloid-beta peptides within the microfluidic system which are known to accumulate in the brain of Alzheimer's patients. The authors developed a three-dimensional in vitro model of Alzheimer's disease that successfully recapitulates key features of BBB dysfunction, such as increased permeability to small molecules, altered expression of tight junction proteins, and enhanced secretion of inflammatory cytokines compared to control conditions. The authors also demonstrated the possibility of restoring BBB integrity and reducing permeability by testing different compounds. A more complex model was presented by Partyka and co-workers [110]. They incorporated a compliant three-dimensional model of the BBB, that can mimic the physiological response to mechanical stress. By subjecting the model to different flow rates and durations, the researchers could evaluate the impact of mechanical stress on the transport mechanisms. The findings revealed that higher levels of mechanical stress led to increased permeability and enhanced diffusion of solutes across the BBB. A key advantage of compliant models is their ability to investigate fluid transport mechanisms induced by mechanical stretch along the walls of brain vessels. The authors numerically simulated the dextran transport within the compliant 3D model of the BBB considering a pulsatile inlet boundary condition, the geometry, and material properties of the compliant BBB model (Fig. 3e). On the other hand, Wong et al. [109] designed and fabricated a 3D hydrogel-based microfluidic vascular device with an integrated electrochemical system for in situ measurement of endothelial permeability. The researchers employed electrochemical impedance spectroscopy to monitor changes in endothelial permeability within the microfluidic vascular model. This integrated electrochemical measurement approach provided real-time and continuous monitoring of the barrier function. This method was validated by comparing the measured permeability with the traditional, fluorescent-based method. The authors numerically simulated the time-dependent transport of the electroactive tracer within the device and they used their previous numerical works [155, 156] to prove that diffusive transport was the prevailing transport mode across a broad range of experimental setups, including cell-free and cell-laden gel matrices.

Oxygen transport is another topic of interest for OoC, due to its critical role in cellular function and tissue health. For instance, Maoz et al. [137] simulated the transport of oxygen, but also the cascade-blue within the microfluidic device. For the applied flow rate, physiologically relevant oxygen levels were observed throughout the model. The researchers further showed that the cultured endothelium preserved its barrier function by simulating the cascade blue diffusion. The authors developed a linked OoC model consisting of two microfluidic chambers representing the vascular and neural compartments interconnected by a porous membrane. This study demonstrates that hypoxia and ischemia-like conditions in the vascular compartment led to altered metabolic and functional responses in both endothelial and neuronal cells.

3.3 NPs Transport Across the BBB

Other authors have investigated the NPs' transport across the BBB. Falanga et al. [107] investigated the transport of NPs across the BBB, by using an in vitro model of brain endothelium. The NPs were engineered with a shuttle molecule that enhances their transport across the endothelial barrier. Their uptake and transport were monitored by evaluating their distribution and concentration within the endothelial cell layer. The findings revealed that the shuttle-mediated transport of NPs across the brain endothelium under flow conditions is significantly enhanced, and also showed the potential of shuttle-mediated strategies for enhancing drug delivery to the brain, particularly when combined with controlled flow conditions. To validate the direction of fluid flow through the permeable layer, numerical simulations were conducted. The model considered two opposing forces, namely gravity pulling the liquid downward and the pressure in the bottom channel pushing in the opposite direction. Despite these opposing forces, the simulations revealed a consistent downward movement of the liquid/air interface in the reservoir (Fig. 3 f)). The authors concluded that particle transfer from the bottom microchannel to the upper chamber does not occur based on these findings. Similarly, Gkountas et al. [147] used a simplified numerical model of the BBB and simulated the transport of magnetic nanoparticles (MNPs). They investigated the interaction of MNPs with the BBB and their subsequent crossing into the brain region. The factors that can affect NP crossing efficiency and distribution within the brain region were investigated such as the impact of NP size (10 and 100 nm), magnetic field strength (0, 0.5, 0.8, and 1 T), and the blood flow (10−3 m/s) in the vessel. The researchers also assessed the available space between endothelial cells relative to the proportion of MNPs size. The findings indicated that MNP size and available area, as well as blood flow, had a minimal impact on BBB permeability, contrary to the external magnetic force, which can cause an increase of the NPs passing through the BBB up to 45%. Another interesting work was presented by Hassanzadeganroudsari et al. [148]. The authors presented a 2D and 3D numerical study on the molecular mass transfer across the BBB in brain capillaries. The mathematical model considers the permeability of the endothelial cell layer, the diffusion coefficient of the molecules, and the concentration gradients across the barrier. The authors evaluated the effect of the number of red blood cells, as well as the blood flow velocity, and the neuron distance from the capillary wall on NPs’ concentration. These were found to increase the mass transfer resistance, and thus reduce the NPs' concentration. Furthermore, the capillary diameter effect on the concentration and diffusion flux of NPs was investigated. By increasing the capillary diameter, a reduction both in the concentration and diffusion flux of NPs is observed.

3.4 Other Computational Models Developed for Investigating the BBB

Other researchers [149] have focused their investigation on developing a non-invasive method to open the BBB using focused ultrasound. In this case, simulation techniques were applied to predict and optimize the sound field parameters required for effective BBB opening. By modeling the acoustic propagation through the rat skull and brain, the authors were able to determine the optimal ultrasound frequency, intensity, and beam shape to achieve targeted BBB opening. The effects of BBB opening were evaluated through the penetration and accumulation of contrast agents within the brain tissue. The optimized sound field parameters that effectively opened the BBB in rats and the corresponding changes observed in the permeability of the BBB are also presented. In general, the results showed the potential of focused ultrasound as a non-invasive technique for targeted drug delivery and therapeutic interventions in the brain by temporarily opening the BBB. Taking a different approach, Ugolini and colleagues [60] employed numerical simulations to identify the optimal placement of voltage-sensing electrodes, aiming to achieve uniform current density within the chamber, while simultaneously increasing the voltage difference between the culture medium and cells, and consequently improving the assessment of TEER. On the other hand, Kincses and the research team [150] created a dynamic BBB-on-a-chip system that can measure both transcellular electrical resistance and streaming potential along the surface of cell layers by incorporating "zeta electrodes." In parallel with experimental assays, the authors conducted numerical simulations both in static and dynamic conditions. Streaming potential data obtained from the device, were compared with zeta potential measurements obtained using the conventional laser-Doppler velocimetry method. With this setup, new insights into the relationship between surface charge and barrier function under both normal and abnormal conditions.

On the other hand, Pellicciotta and co-workers [151] investigated the impact of cilia density and flow velocity on the alignment of motile cilia found in brain cells using numerical simulations. For this purpose, they considered an alignment parameter. They found that increasing the cilia density resulted in enhanced alignment. Similarly, when the flow velocity was increased, ciliary alignment also improved. Table 2 provides a concise overview of papers that have conducted numerical simulations in in vitro BBB models. It focuses on key aspects, including flow rates, device and membrane material, the goal of the simulations, software used, and whether model validation was included or not.

Table 2 Summary of the papers that performed numerical simulations to better understand the BBB comprising the goal of the numerical simulations, the software used, the flow rates the inclusion of the model validation, the device and membrane material and the cells types and configuration of the advanced microfluidic device.
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Regarding the software, COMSOL® seems to be preferred by the researchers for simulating this type of device. Although ANSYS® is less common, both are powerful simulation tools, but it is important to note that they differ in their underlying numerical methods. COMSOL® primarily employs the Finite Element Method (FEM), while ANSYS® Fluent/Open Foam applies the Finite Volume Method (FVM). The selection of a particular method has a direct impact on the strengths and weaknesses of each software for specific applications. FEM has the ability to handle complex geometries [157]. It also allows for higher-order approximations, leading to potentially more accurate results. However, FEM can be computationally demanding. On the other hand, FVM offers a simpler implementation approach. This makes it particularly well-suited for problems governed by conservation laws, such as those frequently encountered in CFD [158, 159]. While FVM can achieve good accuracy by refining the mesh, it typically relies on lower-order approximations compared to FEM.

The choice between the software depends on the specific simulation needs. If complex geometry and high accuracy are paramount, FEM might be the better option. However, if working with conservation law-based problems, FVM approach could be more suitable. Choosing the right tool depends on the scale and nature of the simulation.

3.5 Validation Techniques for Numerical Models in BBB-on-a-Chip Research

Despite the variety of numerical models developed, their validation is important to ensure the reliability and accuracy of the outputs. This can be achieved in different ways, and some of the previously discussed papers have conducted experimental validation. For instance, Booth and co-workers [146] compared the simulated SS values with volumetric measurements and also flow sensor measurements for different channel widths. The results revealed to be in close agreement with a low error percentage (\(\approx\) 10%) for two of the cases tested. Whereas, for the sensor measurement, bigger errors were obtained for the smallest and largest channel. Jeong et al. [106] also validate the numerical model by SS measurements for different flow rates and considering a permeability of 0.01%. The error percentage observed was lower than in the previous study (2.17%) (Fig. 4a). These results demonstrate the accuracy and reliability of the computational model developed by the authors. On the other hand, Hassanzadeganroudsari et al. [148] verified the accuracy of the numerical model by analyzing the numerical and experimental findings of NP mass transfer resistance across a range of red blood cell distances within the capillary. The results presented good agreement with an average deviation of 6% (Fig. 4b). Marino and the research team [111] validated the numerical simulations by comparing the numerical results for two different outlet conditions with the corresponding analytical solutions. For both cases, the analytical solution suitably matched the numerical outputs (Fig. 4c).

Fig. 4
figure 4

Copyright 2021, Elsevier; b model validation and comparison of experimental data in various red blood cell distances in the capillary. Reproduced with permission [148]. Copyright 2020, Elsevier; c analytical and numerical solutions comparison. Reproduced with permission [111]. Copyright 2017, Wiley; d fluorescence intensity distribution profiles obtained by analyzing a transverse cut line across the channel and compared to the results obtained from the finite element model. Reproduced with permission [110]. Copyright 2017, Elsevier; e validation through food dye visualization, and experimental result validating high mixing efficiency. Reproduced with permission [77]. Copyright 2023, Elsevier; f percentage of the total MNPs inserted in the blood flow obtained experimentally and numerically. Reproduced with permission [147]. Copyright 2021, Elsevier; g) Pressure distribution patterns with and without the presence of skull bone compared against simulated results. Reproduced with permission [149]. Copyright 2022, Elsevier

Numerical validation approaches: a validation of SS values between numerical and experimental results. Reproduced with permission [106].

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A different way of validating the numerical model was conducted by Partyka et al. [110]. Cellular channels were used to observe the dextran distribution at the outlet by using a fluorescent microscope. The simulated and experimental results present a similar distribution (Fig. 4d). Likewise, Wong et al. [109] introduced a fluorescent tracer into the microfluidic device, enabling the in situ observation of tracer distribution over time within the cell culture channel and collagen gel. The numerically simulated tracer concentration over time in the gel channel, matches the experimental results, showing the potential of the numerical model. Su et al. [77] also validated their numerical model by comparing the simulated cytokine profiles with the experimental measurements obtained from the integrated digital immunosensors (Fig. 4e).

On the other hand, Kincses et al. [150] compared streaming potential through both experimental measurements and model simulations, and a similar trend between the numerical and the experimental cases was observed. Gkountas and co-workers [147] validated the numerical model by comparing the results with the literature. The MNPs’ concentration that crosses the BBB with and without magnetic field was compared as a function of the MNPs inserted in the blood flow, and similar results were obtained (Fig. 4f). Grudzenski et al. [149] validated their numerical model, by assessing the pressure distribution through experimental data and numerical simulations considering two cases, three and eight elements. A strong correlation was observed between the results for both cases (Fig. 4g).

4 Conclusions and Future Perspectives

OoCs are a breakthrough technology that is developing rapidly and revolutionalizing the biomedical engineering field, namely in the understanding of neurological disorders and the development of effective therapies and personalized medicine, by providing a controlled and physiologically relevant environment capable of reproducing the key physiological and biochemical properties of BBB. BBB-on-a-chip models also hold great promise for clinical translation. Furthermore, by combining the experimental assays with numerical computational simulations, researchers can gain a more comprehensive understanding and complementary insights, enhance data interpretation, and optimize both the devices and experiments. Nevertheless, the model validation should be performed thoroughly to ensure the accuracy and reliability of the combined approach.

In the present review, an overview of the numerical models developed for studying neurological disorders was presented, by discussing the objectives and outcomes of the numerical simulations as well as the different ways that researchers used to validate their numerical models. In brief, it was found that the majority of studies resort to computational simulations to evaluate the fluid flow behavior, SS, and convection–diffusion phenomena within the BBB-on-a-chip devices. However, given the versatility of the computational software, its application for understanding the transport of NPs across the BBB, sound field, voltage, and zeta potential distributions was also observed. Regarding the models’ validation, the diffusion visualization and concentration of dyes as well as NPs transport, SS measurements, zeta potential, and pressure distribution were the methods identified. However, current papers often lack detailed information regarding the numerical models developed, including the specifics of the mesh/mesh study, equations used, and other computational details, which hampers reproducibility and validation.

Despite the progress made so far, several challenges must be addressed in order to optimize the functionality and potential translational of BBB-on-a-chip models. Efforts are still needed to increase the physiological relevance of the current BBB-on-a-chip models, namely assure long-term stability and functionality in terms of cell viability, and barrier integrity for reliable investigation over extended periods. Additionally, the integration of sensors, and imaging modalities into BBB-on-a-chip models to provide real-time monitoring and analysis of barrier function, drug transport, and cellular responses is needed, as well as improving scaling-up and incorporating patient-specific and disease-specific features, while assuring the reproducibility and standardization. The latter is the major challenge of BBB-on-a-chip. To mitigate this, standardized protocols, quality control measures, and benchmarking criteria should be established. From a computational perspective, improving the accuracy of numerical models is imperative and requires incorporating multi-scale simulations that integrate molecular, cellular, and tissue-level data. Future directions should also focus on enhancing computational efficiency to handle large-scale simulations and employing machine learning techniques to predict BBB behavior under various conditions. Additionally, addressing the variability in experimental data and standardizing model parameters will be essential for achieving reliable and reproducible simulations. Collaboration between experimental and computational approaches will be crucial in overcoming these challenges and advancing the development of BBB-on-chip technologies.