A modeling approach to inform in-vitro hypoxia platforms for patient-specific analyses

Abstract and Key Terms

Purpose:

Hypoxia is a consequence of diseases such as obstructive sleep apnea (OSA) and can be detrimental to tissue function, with in-vitro platforms commonly used to study the associated impacts. However, to create a realistic experimental simulation of hypoxia, the dissolved oxygen concentration at the in-vitro cell layer must mimic in-vivo values. Considering the limitations associated with mass transfer and continuous oxygen monitoring in standard hypoxia chambers, mathematical modeling can be leveraged to inform experimental design for more accurate and clinically relevant in-vitro representations.

Methods:

A model of oxygen transfer in a standard hypoxia chamber with a cell culture set-up was calibrated and validated using experimental data. To demonstrate a direct application with clinical data, the validated model was coupled with a novel optimization approach to create a “patient-specific” gas controller protocol capable of reproducing a model-derived OSA patient profile at the in-vitro cell layer.

Results:

The validated model was successful in independently capturing the effect of endothelial cell oxygen consumption in the media. Furthermore, the predicted “patient-specific” gas controller protocol resulted in an in-vitro oxygen profile which mimicked the oxygen values and fluctuation timings seen in a model-predicted arterial oxygen pattern for an OSA patient. Combining the analyses of this study, media height, evaporation, and spatial position were determined to be important factors in experimental design.

Conclusion:

This work demonstrates the utility of applying mathematical modeling for experimental design and presents a means to bridge the gap between the clinical and experimental fields with a “bedside to bench” approach.

1. Introduction

Oxygen is vital for maintaining proper cellular function, as it plays a significant role in various biochemical processes such as ATP generation. Therefore, hypoxia, a condition of low tissue oxygen exposure, can be detrimental to the human body. It can be caused by conditions such obstructive sleep apnea (OSA) [1, 2] and chronic obstructive pulmonary disease (COPD) [3] and has been linked to health consequences including cognitive decline [4], pulmonary hypertension [1, 5], and cardiovascular dysfunction [2]. Importantly, since cells are exposed to different physiological oxygen concentrations under normal conditions in-vivo [6], the definition of hypoxia is relative and highly dependent on the cell type. Furthermore, the impact of hypoxia is also variable and determined by the state of low oxygen exposure (i.e. duration and periodicity) and the biochemical response. For example, cells can experience a short-term oxygen deficiency (acute hypoxia), long-term oxygen deficiency (chronic hypoxia), or a fluctuating pattern of oxygen levels (intermittent hypoxia) [7]. Moreover, hypoxic cellular response can differ for varying cell types [8–10]. Therefore, studying these responses and the overall impact of hypoxia on tissue remains an important research topic.

In-vitro platforms have long been used for simulating hypoxic environments to study the stress response of several cell types including endothelial, cardiac, and brain cells [11–13]. However, to accurately simulate pathophysiological conditions in an experimental setting, it is important to ensure that the dissolved oxygen concentration at the in-vitro cell layer mimics in-vivo values. For traditional hypoxia chambers which rely on oxygen diffusion from the gas phase to the cell layer in the media [14], this may be difficult to assess as only the inlet oxygen fraction is typically reported rather than cell exposure values. Moreover, it is especially challenging to reproduce in-vivo oxygen concentrations when studying intermittent hypoxia due to the additional requirement of replicating fluctuation timings in-vitro, with possibly rapid reoxygenation and desaturation in diseases such as OSA. Beyond this, there are limitations in accurately measuring the oxygen concentration in media due to possible cell disruption, potential recalibration during prolonged experiments, and the inability for simultaneous spatial recording [15]. Thus, there is a clear gap between the clinical and experimental fields that can potentially be bridged with prior knowledge of in-vitro oxygen transfer and optimization of operating protocols, which can both be informed by mathematical modeling.

There have been several instances of the use of mathematical modeling to better understand oxygen transport in experimental platforms. For standard cell culture set-ups, simple steady state models [16] and more encapsulating partial differential equation (PDE) models [17, 18] have been used to estimate the effect of culturing conditions on peri-cellular oxygen concentrations. PDE models have also been applied for complex cell culture constructs, with studies investigating the effect of parameters such as cell type and dimensions of a microfluidic device on oxygen concentration [19], predicting oxygen dynamics in 3D hydrogels [20], and predicting configurations for a muscle-on-a-chip bioreactor to allow normoxic conditions [21]. Although these studies are important for demonstrating the advantage of implementing a modeling approach for experimental design, they do not investigate scenarios such as oxygen cycling over time and lack a direct link to clinical conditions with patient-specific flexibility. Therefore, the primary aim of this work was to demonstrate the application of a mathematical model towards predicting experimental conditions which would expose cells in an in-vitro set-up to patient-specific, cycling oxygen concentrations.

To this end, first we developed a model to predict oxygen mass transfer in a gas-controlled chamber with a standard cell culture set-up. The model was successfully calibrated while accounting for measurement uncertainty and validated with direct oxygen measurements in the gas and liquid phases. Although parameter calibration in the liquid was performed in the absence of cells, the model accurately captured the effect of metabolic consumption on oxygen diffusion in the presence of endothelial cells. Using OSA as a paradigm to study fluctuating oxygen patterns, the model was then coupled with a novel optimization approach to predict a gas chamber protocol that resulted in a patient-specific target oxygen pattern at the cell layer, which was informed by polysomnography (PSG) data (Figure 1). Overall, this method can be used as a basic framework to link patient-specific clinical data and in-vitro hypoxia chamber platforms.

Fig. 1.

Generic workflow for bedside to bench approach. Respiratory function data from a polysomnography (PSG) study must first be converted to a time-dependent lung volume signal. The collected PSG signal/combination of signals used depends on which data is available, usable, and the most accurate. The lung volume signal can then be fed into a oxygen transfer model to predict the dissolved oxygen concentration at different levels of the body including the blood and deep tissue. The desired dissolved oxygen profile to be replicated in-vitro is segmented and input to our gas controller protocol prediction approach, which uses a mathematical model of the target in-vitro cell culture system. Prior to this, the mathematical model of the set-up must be calibrated and validated using experimental data and can then be used to predict oxygen levels at any location in the set-up. Once the optimization code has predicted an intermittent hypoxia protocol to replicate the desired dissolved oxygen pattern at the cell-layer, the gas controller can be programmed to run that to create a patient-specific model of hypoxia. The comprehensive analysis will then involve the patient-specific driven, model predicted dissolved oxygen profiles correlated with structural and functional in-vitro results. The red dashed region highlights the focus of the current work.

2. Materials and Methods

2.1. Experimental Set-up

A temperature-controlled chamber with a gas flow controller system (Noxygen, Elzach, Germany) was used to simulate normoxia and intermittent hypoxia conditions. Compressed gas tanks of nitrogen, carbon dioxide, and oxygen (Airgas, Radnor, PA) were connected to the controller via three input lines, and the get red-y 5 programming software (Vogtlin Instruments GmbH, Muttenz, Switzerland) was used to adjust the flow rate of each gas into the chamber. To create a cell culture set-up in the chamber (Fig. 2a), six covered 35 mm wells were placed in a tray, with an uncovered 35 mm well on top of each covered well. The purpose of the tray is to serve as a water bath for maintaining humidity in the chamber, the uncovered wells are for cell culturing, and the covered wells raise the culture wells to limit direct flow exposure and media evaporation.

Fig. 2.

Experimental set-up and model geometry. a) Gas chamber with internal set-up of water bath and wells. The top layer of wells (uncovered) are for cell seeding. b) Cross-sectional view of chamber with the three layers for the gas phase model. Dashed gray lines indicate separation between layers. c) Top-down view of layers showing model approximation of geometry, with uniform elements. Blacked out elements represent flow obstructions. Solid arrows in Layer 2 indicate the total inlet and outlet from the chamber, while dashed arrows in Layers 1 and 3 indicate that the inlets and outlets are out of the plane of the respective layers (Layer 2 is the source and sink of the flow).The highlighted teal region in Layer 3 is an example of the elements used to calculate $\overline{C_{O_2,g}}$ (Equation 2a) and $\overline{P_w}$ (Equation 3c) for Well 1

2.2. Mathematical Model Formulation

Oxygen mass transfer in the chamber was modeled with a coupled system of ODEs and a PDE. To determine a spatial oxygen profile for the gas phase, the chamber was divided into three layers based on the geometry of the internal set-up. This simplification was done to create a computationally feasible workflow for estimation of the velocity profile. Each layer partitioned into uniform elements (Fig. 2b and c), and the oxygen concentration in each element was modeled using a control volume approach:

$$V_e\frac{\text{d}{C}_{O_{2,g}}}{\text{d}t}=\sum _{i=1}^n{Q}_{\left\{i\right\}}\cdot C_{O_{2,g}\left\{i\right\}}-D_{O_{2,g}}\cdot A_c\cdot \frac{\left(C_{O_{2,g}}-C_{O_{2,g}\left\{i\right\}}\right)}{\delta }-\sum _{j=1}^m{Q}_{\left\{j\right\}}\cdot C_{O_2,g}-D_{O_2,g}\cdot A_c\cdot \frac{\left(C_{O_{2,g}\left\{j\right\}}-C_{O_{2,g}}\right)}{\delta }$$ (1)

The subscript i represents an inlet, with a total of n inlets to the control volume, while subscript j represents an outlet, with a total of m outlets from the control volume. It was assumed that the total inlet and outlet flows were equivalent for each element. V_e denotes element volume, Q denotes flow rate, A_c denotes cross-sectional area, $D_{O_2,g}$ denotes oxygen diffusion coefficient through the gas phase, and δ denotes diffusion distance. Diffusion distance was approximated using the mid-points of neighboring elements. The flow profile in the chamber was estimated using results from a steady-state computational fluid dynamics (CFD) model in ANSYS (Supplementary Material, Section S2). A steady state approximation was used for the flow profile as the total inlet gas flow was expected to remain constant for each run. For modeling simplification, each layer was assumed to have laminar flow, with no convective mixing in the z direction, apart from the chamber inlet and outlet. Furthermore, diffusion in the z direction was only applied for the first and second row of elements (informed from measurements of a step change in oxygen), the chamber inlet, and chamber outlet. The gas phase model was independently validated by running the chamber with various inlet oxygen fluctuation profiles and measuring the change in oxygen partial pressure at different locations.

Oxygen diffusion in the liquid phase of each uncovered well was modeled as:

$$\frac{\partial C_{O_2,l}}{\partial t}=D_{O_2,l}\frac{{\partial }^2{C}_{O_2,l}}{\partial z^2}$$ (2)

Where the subscript l represents the liquid phase and $D_{O_2,l}$ denotes the diffusion coefficient of oxygen through the liquid. The oxygen concentration in the liquid phase was assumed to be uniform along each cross-section. The boundary conditions for Equation 2 were:

$$-{\left.D_{O_2,l}\frac{\mathrm{d}{C}_{O_2,l}}{\mathrm{d}z}\right|}_{\text{surface}}=k_{O_2}\left(\overline{C_{O_2,g}}-\frac{C_{O_2,l}{|}_{\text{surface}}}{S_F\cdot {\beta }_{O_2}\cdot R\cdot T_l}\right)$$ (2a)

$$-{\left.D_{O_2,l}\frac{\mathrm{d}{C}_{O_2,l}}{\mathrm{d}z}\right|}_{\text{bottom}}=\frac{M{R}_{O_2}}{A_s}$$ (2b)

Where $k_{O_2}$ denotes the mass transfer coefficient of oxygen on the gas phase side of the gas-liquid surface, SF denotes the solubility correction factor for oxygen in the liquid phase, ${\beta }_{O_2}$ denotes the solubility of oxygen, R denotes the ideal gas constant, T_l denotes the liquid temperature, and A_s denotes the surface area of the well. The mass transfer coefficient was represented as:

$$k_{O_2}=k_w\cdot \frac{D_{O_2,g}}{D_w}$$ (2c)

Where k_w denotes the mass transfer coefficient of water vapor on the gas phase side of the gas-liquid surface, determined by fitting with evaporation data, and D_w represents the diffusion coefficient of water through the gas phase. Expecting low variability in oxygen for the elements above a single well due to spatial proximity and for model simplification, the gas phase oxygen concentration above each well, $\overline{C_{O_2,g}}$ , was calculated as an average across the corresponding elements in Layer 3 (Fig. 2c). To account for any cellular oxygen consumption in the liquid phase, the overall metabolic rate for endothelial cells was represented as [21]:

$$M{R}_{O_2}=\frac{{\left.R_c\cdot N_c\cdot C_{O_2,l}\right|}_{\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{m}}}{{\left.C_{O_2,l}\right|}_{\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{m}}+k_c}$$ (2d)

Where R_c denotes the oxygen consumption rate per cell, N_c denotes the total number of cells seeded in the well, and k_c denotes the Michaelis-Menten constant for the cell type. This relationship can be adjusted for any cell type of interest.

Evaporation and condensation in the chamber were also modeled to account for the effect of liquid level reduction on oxygen transport to the cell layer. Assuming water vapor as the main evaporating and condensing component, its partial pressure in the gas phase was represented similarly to oxygen transport (Equation 1):

$$V_e\frac{\text{d}{P}_{w,g}}{\text{d}t}=\sum _{i=1}^n{Q}_{\left\{i\right\}}\cdot P_{w,g\left\{i\right\}}-D_w\cdot A_c\cdot \frac{\left(P_{w,g}-P_{w,g\left\{i\right\}}\right)}{\delta }-\sum _{j=1}^m{Q}_{\left\{j\right\}}\cdot P_{w,g}-D_w\cdot A_c\cdot \frac{\left(P_{w,g\left\{j\right\}}-P_{w,g}\right)}{\delta }+\left(q_E-q_C\right)\cdot R\cdot T$$ (3)

Where the transfer rate of water vapor into an element from the evaporating liquid was determined as:

$$q_E=\left\{\begin{array}{ll}{k}_w\cdot A_c\cdot \left(P_{\mathit{\text{sat}},\mathit{\text{liquid}}}-P_{w,g}\right)\cdot \frac{1}{R\cdot T},& H_l>0\\ 0,& H_l=0\end{array}\right.$$ (3a)

P_sat,liquid denotes the saturation pressure of water vapor evaluated at the liquid temperature and H_l denotes the height of the liquid. q_E was only applied to elements directly above a well. Furthermore, to account for condensation on the underside of the top chamber lid, q_C was only applied to the elements of Layer 3 as:

$$q_C=\left\{\begin{array}{ll}{k}_w\cdot A_c\cdot \left(P_{w,g}-P_{sat,lid}\right)\cdot \frac{1}{R\cdot T},& \left(P_{w,g}-P_{sat,lid}\right)>0\\ 0,& \left(P_{w,g}-P_{sat,lid}\right)<0\end{array}\right.$$ (3b)

P_sat,lid denotes the saturation pressure at the lid temperature. The conditionality in Equation 3b ensures that q_C is only activated in the case of super-saturation at the lid surface, with the lid temperature being lower than the surrounding gas, which, in turn, is lower than the liquid temperature based on experimental data. The mass transfer coefficient of water vapor, k_w, was fit using experimental data, with a different value for each well and was expected to be dependent on temperature, surrounding humidity, and gas flow velocity. In Equation 3b, k_w for elements not directly above a well was estimated as the value for the nearby well (Supplementary Material, Figure S5. The height of the liquid in each well was determined as:

$$\frac{\mathrm{d}{H}_l}{\mathrm{d}t}=-\frac{k_w\cdot M_w\cdot \left(P_{\mathit{\text{sat}},\mathit{\text{liquid}}}-\overline{P_{w,g}}\right)}{{\rho }_w\cdot R\cdot T_g}$$ (3c)

Where $\overline{P_{w,g}}$ denotes the average water vapor pressure above the well (Fig. 2c), M_w denotes the molecular weight of water, ${\rho }_w$ denotes the density of water, and T_g denotes the gas temperature above the well. All equations were solved in MATLAB 2024b using finite differences in time, with the method of lines (backward in time, centered in space) additionally applied for the PDE. All model parameters are further described in the online supplementary material (Table S1).

2.3. Experimental Data Collection

2.3.1. Gas Phase Oxygen

All gas phase oxygen experiments were performed using a NeoFox oxygen sensing system (Ocean Optics, Florida, USA). The system consisted of an adhesive patch with a FOSPOR coating and an optical sensing probe to measure fluorescence decay after LED excitation, which is a function of oxygen partial pressure. The adhesive patch was placed underneath the top lid of the chamber for all experiments. Before measurement, a 2-point probe calibration was performed at the minimum and maximum oxygen set points in the gas controller protocols (0 and 21% for all experiments). This was achieved by running the chamber with an inlet gas of 0 and 21% oxygen composition, until the fluorescence decay values (tau) for each stabilized.

To calibrate the unknown model parameters and assess output uncertainty, a time-dependent profile was measured for a step change in oxygen (0 to 21%) at two different points in Layer 3. Validation was then performed by running the chamber with two IH protocols at a total of three different points in Layer 3. Both IH protocols started with a 5-minute flush of the chamber at 4% oxygen, followed by 25 seconds at 0% oxygen. The first protocol consisted of three cycles of fluctuating oxygen (25 seconds at 21% oxygen followed by 125 seconds at 0% oxygen for each cycle). Oxygen measurements were obtained for this protocol at two points. The second IH protocol consisted of three cycles of faster fluctuations (25 seconds at 21% oxygen followed by 25 seconds at 0% oxygen for each cycle). Oxygen measurements were obtained for this protocol at one point. Flow data measured from the gas controller was recorded for each run to be used as a model input. The measurement delay (contributed by the probe itself and by time for the gas to travel from the controller to the chamber) was estimated by placing an adhesive patch in a well directly below the inlet and measuring the time taken for the probe to respond to a step increase in oxygen from an initial point of 0%. All gas phase experiments were repeated three times, with no liquid present in the water bath or cell culture wells, and with a maximum chamber inlet flow of 450 mL/min. Additionally, all oxygen profile measurements were done at 5-second intervals. Multiple adhesive patches were used for measurements to avoid potential scratching associated with movement, which may result in inaccurate readings.

2.3.2. Temperature Measurements

To collect temperature data for the gas and liquid phase, measurements were obtained for dry and wet runs in the chamber using a digital thermocouple. Temperature was recorded at 0, 0.5, 1.25, and 2 hours after starting flow, with normoxia conditions (21% oxygen) and the maximum chamber inlet of 450 mL/min. The dry runs were completed without any water in the cell culture wells and water bath. Measurement points included: the gas phase above each well (Layer 3), the gas phase directly above the water bath (at random locations in Layer 2), and the gas phase above the chamber bottom (at random locations in Layer 1). For the wet runs, each cell culture well was filled with 2.5 mL of water, and the water bath was filled with 20 mL of water. In addition to the gas phase measurement points, temperature was also recorded for the water in each well and the water in the water bath (at random locations). The dry and wet runs were repeated three times each.

In a separate set of experiments, the temperature difference between the top lid and the surrounding gas in Layer 3 was measured at random x-y positions in the chamber at 0, 0.5, 1.25, and 2 hours after starting a run. Before each run, all wells were filled with 1.5 mL of water, and the chamber inlet was set to the maximum flow rate of 450 mL/min with pure nitrogen gas. For temperature measurement during a single trial, one sensor of the thermocouple was fixed to the under-side of the lid and the other was suspended directly below it in Layer 3, at the same x-y position. Based on the measured data, an equilibrium temperature difference was reached by the 0.5-hour time point for all positions. Following estimation of the equilibrium time with the initial tests, temperature difference measurements were collected in x-y positions around each well at 0.5 hours after starting a run. Using the same set-up and experimental conditions, three replicates were obtained around each well.

2.3.3. Water Bath Impact on Evaporation

Experiments were run with and without water in the water bath to determine whether it significantly impacts evaporation from the cell culture wells. For this purpose, each cell culture well was filled with 2.5 mL of water, and the water bath was filled with either 0 or 20 mL of water. The chamber was run with normoxia (21% oxygen) and the maximum inlet flow of 450 mL/min for two hours, after which the amount of water remaining in each well and the water bath (if applicable) was measured. Each experiment was repeated three times. Considering that there were no significant differences between the groups (Supplementary Material, Fig. S4), the water bath was left empty for the liquid phase calibration/validation experiments and excluded from the evaporation model.

2.3.4. Liquid Phase Oxygen Without Cells

Dissolved oxygen concentration was recorded using a NeoFox optical sensing probe with a FOSPOR coating at the tip (Ocean Optics, Florida, USA), based on a similar principle as previously described (Section 2.3.1). Before measurement, a 2-point probe calibration was performed in water, outside the chamber, at the minimum and maximum oxygen set points in the gas controller protocols (0 and 21% for all experiments). This was achieved by bubbling a gas with a 0 and 21% oxygen composition through water, with each gas composition run until the tau value stabilized. A hot plate was used to maintain water temperature during calibration. The dissolved oxygen concentration associated with each calibration point was calculated as:

$$DO=F_{O_2}\cdot \left(P-P_w\right)\cdot {\beta }_{O_2}$$ (4)

Where DO denotes the dissolved oxygen concentration, $F_{O_2}$ denotes the oxygen fraction in the inlet, P denotes total pressure in the gas phase, and P_w and ${\beta }_{O_2}$ are the same as defined previously. The vapor pressure and solubility were determined at the specific temperature of the water.

For all liquid phase experiments without cells, the cell culture wells were filled with 1.5 mL of water, while the water bath was left empty. To calibrate the unknown model parameters and assess output uncertainty, a time-dependent dissolved oxygen profile was measured at 5-second intervals for a step change in oxygen (21 to 0%) at a single cell culture well for a 30-minute period. The water volume remaining in all six culture wells was also measured at the end of the experiment duration. Validation was then performed by running the chamber with an IH protocol and measuring dissolved oxygen at 5-second intervals in the same cell culture well. The IH protocol consisted of three cycles of fluctuating oxygen (125 seconds at 0% oxygen followed by 125 seconds at 21% oxygen for each cycle). Flow data measured from the gas controller was recorded to be used as a model input. All liquid phase experiments were repeated three times, with a maximum chamber inlet flow of 450 mL/min. The tip of the probe was kept as close as possible to the bottom of the well for all measurements.

2.3.5. Cell Culture Well Preparation

In preparation for experiments, 35 mm individual wells were coated with 1.5 mL of a solution composed of 1 part fibronectin (FN; Sigma Alrich, St. Louis, MO) per 20 parts of distilled water. After coating, each well was incubated for 30 minutes, before washing with PBS three times. 2 mL of phosphate-buffered saline (PBS; Thermo Fisher, Cat #20012027) were left in each well, which was wrapped in parafilm and stored at 4°C until seeding.

2.3.6. Cell Culture and Seeding

Immortalized human micro-vascular endothelial cells (HMEC) (derived from tissue biopsies) were obtained from Dr. Chandan Sen. The cells were utilized at Passage 8 and cultured in MCDB-131 medium (Thermo Fisher, Cat#10372-019) supplemented with 10% fetal bovine serum (FBS; Thermo Fisher, Cat#26140-079), 100 units/mL of penicillin, and 100 μL of streptomycin (MilliporeSigma, Cat#P4333) in a T75 flask. The flask was placed in a standard incubator with 21% oxygen and 5% carbon dioxide, at 37°C. The media was changed every two days until confluency reached 90%. At this point, the HMECs were lifted by adding 3 mL of 1 mg/mL trypsin (Sigma-Aldrich) to the T75 flask and incubating at 37°C for 3 minutes. The trypsin was the neutralized with 3 mL of MCDB-131 culture media that was supplemented with 10% FBS. The suspended HMECs were transferred to a 15 mL tube and centrifuged at 1000 rpm for 5 minutes. Following this, the liquid was aspirated to leave a cellular pellet, which was resuspended in 1 mL of the MCDB-131 culture media. 10 μL of the resuspended HMECs, 40 μL of MCDB-131 media, and 10 μL of Trypan blue were mixed in a microcentrifuge tube, and 10 μL of the resulting solution was added to a hemocytometer to estimate cellular density. Based on calculations, 750,000 cells were seeded onto each 35 mm FN-coated well, with 2 mL of added MCDB-131 culture media. The HMECs were allowed to adhere and proliferate for approximately 48 hours. Immediately before running experiments, the culture media in each well was replaced with 1.5 mL of normal Tyrode’s solution (approximate composition: 1.8 mM CaCl₂, 5 mM glucose, 5 mM HEPES, 1 mM MgCl₂, 5.4 mM KCl, 135 mM NaCl, and 0.33 mM NaH₂PO₄, with pH adjusted to ≈ 7.4 using NaOH).

2.3.7. Liquid Phase Oxygen With Cells

All cell experiments were run with normal Tyrode’s solution. The chamber set-up for each run was similar to the liquid phase experiments without cells, with the difference being that 1.5 mL of Tyrode’s solution was used to fill the wells instead of water. Oxygen measurement was performed at the same position as the previous liquid phase experiments; yet, in this case, a well seeded with cells on a FN layer was used. Furthermore, Tyrode’s solution was used for probe calibration instead of water. The dissolved oxygen concentration associated with each calibration point was calculated as:

$$DO=F_{O_2}\cdot \left(P-P_w\right)\cdot S_F\cdot {\beta }_{O_2}$$ (5)

Where all variables are the same as previously defined. The vapor pressure, solubility, and salinity correction factor were determined at the specific temperature of the Tyrode’s solution.

For the experiment, the chamber was run with the same IH protocol used for liquid phase validation: three cycles of fluctuating oxygen (125 seconds at 0% oxygen followed by 125 seconds at 21% oxygen for each cycle). Flow data measured from the gas controller was recorded to be used as a model input. All cell experiments were repeated six times, with a maximum chamber inlet flow of 450 mL/min. The tip of the probe was kept as close as possible to the bottom of the well for all measurements. Following each run, the cells in the single well used for measurement were lifted with trypsin as previously described, and the number of live cells was determined using Tyrpan blue staining and a hemocytometer.

2.4. Model Parameter Calibration and Input/Output Uncertainty

2.4.1. Bayesian History Matching and Gaussian Process Emulation

To calibrate the model and capture uncertainty in the measured data, the open-source Python package ‘chameleon’ was applied. Briefly, ‘chameleon’ implements Bayesian History Matching (BHM) [22, 23] with Gaussian Process Emulators (GPE) to narrow down the parameter space of a model through multiple iterations called waves [24]. For the first wave, a Sobol sequence was used to generate an expansive parameter space, which was then sampled to extract a small subset of parameters sets for GPE training and validation. The subset was used to run the chamber model described in Section 2.2, with 80% of the resulting data used to train the GPEs and 20% applied for independent validation. The trained GPEs were implemented to evaluate the entire parameter space, and a comparison between the GPE outputs and measured data was used to determine which parameter sets were implausible. GPEs are advantageous for this application as they allow for the evaluation of a large parameter space with a low computational cost. The non-implausible parameter sets were then used to generate the parameter space for the next wave. This process was repeated for multiple iterations, with the ultimate goal of having all model outputs approximately within two standard deviations of the measured data (95% confidence interval). It is important to note that the threshold for implausibility was lowered with each iteration. The data required to run ‘chameleon‘ includes: uncertainty ranges for each model input, mean values for model outputs, and standard deviation for model outputs.

A separate BHM/GPE process with a minimum of five waves was completed for the gas and liquid phase models using the measured step change data. 256 parameter sets were then randomly sampled from the remaining plausible parameter spaces after the final wave and used to independently run the models. From the sampled parameter sets, those resulting in the minimum, mean, and maximum value for each output were used to run with the IH protocols for model validation.

2.4.2. Estimated Input Parameters

For the gas phase model, the estimated parameters were the fraction of total inlet flow entering Layers 2 and 3 of the chamber. The uncertainty range for these inputs was estimated from trial runs of the model, with comparison of the associated results to experimental data.

For the liquid phase model, the estimated parameters were the mass transfer coefficients for evaporation from all six cell culture wells (k₁, k₂, k₃, k₄, k₅, k₆), the position of the probe tip, and the fraction of total inlet flow entering Layers 2 and 3 of the chamber. The uncertainty range for the mass transfer coefficients was estimated from trial optimization runs in MATLAB using ‘surrogateopt‘. Considering the small starting height of liquid in each cell culture well and the fact that the probe tip was kept as close as possible to the bottom of the well, the bottom half of the liquid was estimated as the uncertainty range for the probe tip position. Lastly, for the inlet flow fractions, the results of running the BHM/GPE process with the gas phase model were used to narrow down the uncertainty ranges.

2.4.3. Measured Input Parameters

The experimentally measured input parameters were Well 1 gas temperature and oxygen measurement time delay. The initial gas temperature above Well 1 (Fig. 2c) was used as an input parameter to the model, while the gas temperatures above the cell culture wells and for Layers 1 and 2 were determined using the measured time-dependent profiles (Supplementary Material, Fig. S5a). Corresponding water temperatures for each cell culture well were determined by using the calculated gas temperatures and a linear regression between measured gas and water temperatures for each cell culture well from the wet runs (Supplementary Material, Fig. S5c). Layer 3 gas temperatures and the lid-gas temperature difference for elements around the wells were based on the values above the wells (Supplementary Material, Fig. S5b). Uncertainty ranges for the input temperature and measurement delay were estimated based on the maximum and minimum over three trials.

2.5. Gas Controller Protocol Prediction Using MATLAB Optimization

2.5.1. Patient-Specific Target Oxygen Pattern

A patient-specific dissolved oxygen pattern created in our recent work [25] was applied to show the applicability of our optimization approach. Briefly, our previous mass-transfer compartment model used nasal pressure PSG recordings from OSA patients in a controlled clinical environment to estimate lung volume, which was then input to predict time-varying oxygen profiles in the alveoli, pulmonary capillaries, and systemic arterial and venous circulation [25]. A segment of the model-predicted systemic arterial oxygen concentration associated with multiple breathing events was selected for the current demonstration. A zero-phase filter was first applied to smooth the signal and eliminate higher-frequency fluctuations caused by normal breathing or recording noise. Next, the sequence was segmented using the local extrema in the filtered signal as boundary points. The target pattern was then created by utilizing fluctuation timings from the filtered signal and local extrema values for each segment from the original signal.

2.5.2. General Optimization Process

An optimization code (Supplementary Material, Fig. S8) was implemented to output a gas controller protocol (consisting of inlet oxygen fractions and run durations) which can mimic target oxygen fluctuations at the cell layer of a specific well. Inputs to the code include: a parameter set from the liquid phase calibration process, training data for GPEs to be used during the optimization, bounds for the inlet oxygen fraction (different for increasing and decreasing segments) and run duration, threshold values for optimization, and a .mat file with the input sequence, which is segmented and analyzed to extract local extrema values as previously described (Section 2.5.1). With these local extrema serving as boundary points for the segments, the code runs through an iterative loop to output an inlet oxygen fraction and its corresponding run duration for each target oxygen segment. Starting with normoxia as the initial condition of the chamber, the output values for the first iteration were chosen based on a trial and error process of running the oxygen transfer model. For the remaining iterations, a Pareto front solver (gamultiobj) in the Global Optimization Toolbox of MATLAB was used to find multiple possible solutions for the optimization variable(s). This type of solver was selected as it has the capability of (1) using multiple objective functions during variable optimization and (2) outputting solutions resulting in a range of values for the defined objective functions, which allows an optimal solution to be chosen based on user preference. In the current study, achieving the correct oxygen concentration values was prioritized over fluctuation times in the target sequence. Furthermore, to improve computational efficiency, the code was set up so that the chamber initial condition of a particular iteration was defined as the ending point of the previous iteration.

2.5.3. Primary Optimization Method

For iterations 3 and above of the main code loop (Section 2.5.2), the optimization variables were the run duration for the inlet oxygen fraction of the current iteration (inlet oxygen fraction determined during the previous iteration) and the inlet oxygen fraction for the next iteration, with the following objective functions:

$$f(1)=\left|t_{\left\{\text{target}\right\}}-t_{\left\{\text{interval}\right\}}\right|$$ (6a)

$$f(2)=\left|e_{\left\{\text{end}\right\}}^{\mathrm{*}}-a_{\left\{\text{end}\right\}}^{\mathrm{*}}\right|$$ (6b)

$$f(3)=\left|e_{\left\{\text{start}\right\}}^{\mathrm{*}}-a_{\left\{\text{start}\right\}}^{\mathrm{*}}\right|$$ (6C)

Where t_{target} and t_{interval} denote the target segment duration and the actual duration between the segment-bounding local extrema from the objective function run, respectively. Furthermore, $e_{\left\{\text{start}\right\}}^{\mathrm{*}}$ denotes the local extremum at the start of the target segment, $e_{\left\{\text{end}\right\}}^{\mathrm{*}}$ denotes the local extremum at the end of the target segment, $a_{\left\{\text{start}\right\}}^{\mathrm{*}}$ denotes the predicted local extremum at the start of the segment, and $a_{\left\{\text{end}\right\}}^{\mathrm{*}}$ denotes the predicted local extremum at the end of the segment. During the optimization, the model was run 50 seconds beyond the time for the current iteration to capture any remnant increase or decrease in oxygen concentration (Supplementary Material, Fig. S9). $a_{\left\{\text{start}\right\}}^{\mathrm{*}}$ and $a_{\left\{\text{end}\right\}}^{\mathrm{*}}$ were determined by identifying inflection points in the oxygen concentration profile during the run. If less than two inflection points were detected, the objective functions were assigned a placeholder value to disregard that specific solution set. The possible solution sets were first narrowed down by applying threshold values to the 1st and 2nd objective functions (Equations 6a and 6b, respectively), and the optimal solution set was then selected as the filtered solution set with the lowest value of the 1st objective function (Equation 6a).

2.5.4. Secondary Optimization Method

For iteration 2 of the main code loop (Section 2.5.2) or in the case where the primary optimization method (Section 2.5.3) failed, an alternative method was used. For this, the optimization variables were the inlet oxygen fraction and run duration (iteration 2 of the code described in Section 2.5.2) or just the run duration (iteration number > 2 of the code described in Section 2.5.2), with the following objective functions:

$$f(1)=\left|t_{\left\{\text{target}\right\}}-t_{\left\{\text{run}\right\}}\right|$$ (7a)

$$f(2)=\left|e_{\left\{\text{end}\right\}}^{\mathrm{*}}-\mathrm{e}\mathrm{n}\mathrm{d}\left(c_{O_2\left\{\text{sol}\right\}}\right)\right|$$ (7b)

$$f(3)=\left\{\begin{array}{ll}\left|e_{\left\{\text{start}\right\}}^{\mathrm{*}}-\mathrm{m}\mathrm{i}\mathrm{n}\left(c_{O_2\left\{\text{sol}\right\}}\right)\right|& e_{\left\{\text{start}\right\}}^{\mathrm{*}}<e_{\left\{\text{end}\right\}}^{\mathrm{*}}\\ \left|e_{\left\{\text{start}\right\}}^{\mathrm{*}}-\mathrm{m}\mathrm{a}\mathrm{x}\left(c_{O_2\left\{\text{sol}\right\}}\right)\right|& e_{\left\{\text{start}\right\}}^{\mathrm{*}}>e_{\left\{\text{end}\right\}}^{\mathrm{*}}\end{array}\right.$$ (7C)

Where $t_{\text{{run}}}$ denotes the inlet oxygen fraction run time for the current iteration and $c_{O_2\left\{\text{sol}\right\}}$ denotes the oxygen concentration profile at the cell layer during the solution run time for the current iteration. $t_{\left\{\text{target}\right\}}$, $e_{\left\{\text{start}\right\}}^{\mathrm{*}}$, and $e_{\left\{\text{end}\right\}}^{\mathrm{*}}$ are the same as previously defined.

A 2-tiered approach was taken to select the solution. As an initial step, the possible solution sets were narrowed down by applying a threshold to the value of the 3rd objective function (Equation 7c). For the second step, a GPE was first trained to predict the remnant increase or decrease associated with running a possible solution set (Supplementary Material, Fig. S9). The inputs to the GPE were the initial oxygen concentration at the cell layer for the specific iteration, the inlet oxygen fraction and run duration for the iteration (solution set), and the inlet oxygen fraction for the next iteration, which was estimated as the user-defined upper and lower bound for the inlet oxygen fraction. Starting with the filtered solution set resulting in the smallest value of the 1st objective function (Equation 7a), a possible set was run through the GPE until the value of the 2nd objective function (Equation 7b) fell within the prediction of the two GPE runs (associated with assuming the value of the inlet oxygen fraction for the next iteration), with a user-specified value of uncertainty. At this point, the GPE was evaluated with the selected solution set assuming that the inlet oxygen fraction for the next iteration was at the mid-point of the user-defined limits. The three GPE predictions using the selected solution set and the assumed values for the inlet oxygen fraction of the next iteration were then applied to interpolate the actual value of the next inlet oxygen fraction with the value of the 2nd objective function (Equation 7b) corresponding to the selected solution set. If the GPE predictions were not unique or were within a certain threshold of each other, the upper limit for the inlet oxygen fraction was selected as the value for the next iteration if $e_{\left\{\text{start}\right\}}^{\mathrm{*}}>e_{\left\{\text{end}\right\}}^{\mathrm{*}}$ and the lower limit was selected if $e_{\left\{\text{start}\right\}}^{\mathrm{*}}$ $<e_{\left\{\text{end}\right\}}^{\mathrm{*}}$. Considering the possibility of error in the trained GPE, after the inlet oxygen value for the next iteration was selected, the run duration for the inlet oxygen fraction in the current iteration solution set was adjusted by running the oxygen model until the value of the following function fell within the user-defined threshold and an inflection point was detected during the run:

$$h=\left\{\begin{array}{ll}{a}_{\left\{\text{end}\right\}}^{\mathrm{*}}-e_{\left\{\text{end}\right\}}^{\mathrm{*}}& e_{\left\{\text{start}\right\}}^{\mathrm{*}}>e_{\left\{\text{end}\right\}}^{\mathrm{*}},{\mathrm{N}}_{\text{inflection}}>1\\ \mathrm{m}\mathrm{i}\mathrm{n}\left(c_{O_2\left\{\text{after}\right\}}\right)-e_{\left\{\text{end}\right\}}^{\mathrm{*}}& e_{\left\{\text{start}\right\}}^{\mathrm{*}}>e_{\left\{\text{end}\right\}}^{\mathrm{*}},{\mathrm{N}}_{\text{inflection}}<2\\ e_{\left\{\text{end}\right\}}^{\mathrm{*}}-a_{\left\{\text{end}\right\}}^{\mathrm{*}}& e_{\left\{\text{start}\right\}}^{\mathrm{*}}<e_{\left\{\text{end}\right\}}^{\mathrm{*}},{\mathrm{N}}_{\text{inflection}}^{\mathrm{*}}>1\\ e_{\left\{\text{end}\right\}}^{\mathrm{*}}-\mathrm{m}\mathrm{a}\mathrm{x}\left(c_{O_2\left\{\text{after}\right\}}\right)& e_{\left\{\text{start}\right\}}^{\mathrm{*}}<e_{\left\{\text{end}\right\}}^{\mathrm{*}},{\mathrm{N}}_{\text{inflection}}^{\mathrm{*}}<2\end{array}\right.$$ (8)

Where $e_{\left\{\text{start}\right\}}^{\mathrm{*}}$ and $e_{\left\{\text{end}\right\}}^{\mathrm{*}}$ are the same as previously defined and $c_{O_2\left\{\text{after}\right\}}$ denotes the oxygen concentration profile at the cell layer 200 seconds beyond the solution run time for the current iteration (determined oxygen fraction for the next iteration was used as the chamber inlet beyond the solution run time). N_inflection represents the number of inflection points detected during the total run time (equivalent to the sum of the current and next inlet oxygen fraction run times). Furthermore, if a solution set could not be found by comparing the value of Equation 7b and the GPE predictions, each initially filtered solution set was run through the GPE twice as previously described, and a squared function was calculated as:

$$g_{\left\{\mathrm{i}\right\}}={(f(2{)}_{\left\{\mathrm{i}\right\}}-GP{E}_{\left\{\text{ll}\right\}})}^2+{(f(2{)}_{\left\{\mathrm{i}\right\}}-GP{E}_{\left\{\mathrm{u}\mathrm{l}\right\}})}^2$$ (9)

Where the subscript i denotes a filtered solution set, and the subscripts ll and ul indicate that the GPE was run assuming the lower and upper limit for the inlet oxygen fraction of the next iteration, respectively. The solution set with the lowest function value was selected, and the interpolation and time adjustment methods were performed as previously described.

Once a solution set was identified, the oxygen profile segment time associated with the adjusted run time of the current iteration was compared to the target segment time as:

$$p=t_{\left\{\text{target}\right\}}-t_{\left\{\text{interval}\right\}}$$ (10)

Where t_{target} and t_{interval} are the same as previously defined. t_{interval} was determined after running the model for 200 seconds beyond the adjusted run time for the current iteration. If the value of Equation 10 was outside the defined threshold, the inlet oxygen fraction for the current iteration was gradually adjusted (Supplementary Material, Fig. S8) and the whole secondary optimization process repeated until either the time threshold was satisfied or the inlet oxygen fraction reached the defined bounds.

3. Results

3.1. Gas Phase Data

The gas phase portion of the model was calibrated using measured oxygen data at two different positions in the chamber (P1 and P2, Fig. 3a). For a step increase in inlet oxygen fraction, the model output resembled the experimental data closely enough that it could be fit with four parameters (Fig. 3b). After successful model calibration (Fig. 3b), validation was performed by comparing simulated results to experimental data at the calibration positions and one additional location (P3, Fig. 3a) in response to IH protocols (Fig. 3c). Notably, the model captures the overall gas-phase behavior in the chamber by successfully predicting the relative oxygen changes during IH at each position (Fig. 3c). However, small discrepancies exist at some of the locations. Although there appears to be an approximate 3% oxygen fraction difference in the peak values at P2 (Fig. 3cii), this is likely the result of temporal aliasing since measurements were taken at 5-second intervals (Supplementary Material, Fig. S6). In addition, there is a slight phase difference between the simulated and experimental data at P3 (Fig. 3ciii), which is possibly a consequence of an unrepresentative estimation of the measurement time delay due to the variable response of different adhesive patches used for recording. Overall, the model successfully recapitulates the significant features of the measured data. This validated model was then applied towards predicting oxygen profiles at the approximate well locations (Fig. 3d) in response to IH, with the observed differences further highlighting spatial variations in the gas phase (Fig. 3d) which may impact the dissolved oxygen values at the cell layer.

Fig. 3.

Gas phase calibration and validation results. a) Top-down view of chamber showing oxygen measurement points in Layer 3 and approximate positions of cell culture wells below (shaded gray regions). b) Calibration data and results. The red arrows indicate the time points used to run the BHM/GPE process. c) Validation results. Measured flow data (mean values for each position) was used as an input to the model. b and c) The shown outputs are the result of running the model with selected parameter sets from the 256 randomly sampled sets remaining in the plausible parameter space after the final wave of the BHM/GPE process. A single parameter set to run the model consists of: {T₁ , Inlet_L2, Inlet_L3, t_d}.The measured data is shown as mean ± 2×SD, with n=3. d) Model output for sample IH run, with average oxygen concentrations in gas phase for each well location

3.2. Liquid Phase Data

Once the gas phase fitting was finalized, the whole dual-phase model without cells was calibrated to experimental data of a step decrease in inlet oxygen fraction with eleven parameters (Fig. 4). As expected, the model accurately predicts dissolved oxygen change in the liquid, while fully capturing uncertainty in the measured data for the time associated with most of the decrease (Fig. 4a). However, there is a slight discrepancy in the asymptote, with the model output approaching the ideal 0 μM (Fig. 4a), while the experimental asymptote is slightly higher and likely caused by the chamber not being completely air-tight during liquid phase measurement. The same model was also successfully calibrated to capture a complete representation of experimental data for water volume remaining in the cell culture wells (Fig. 4b).

Fig. 4.

Liquid phase calibration results. a) Model and experimental results for dissolved oxygen in Well 3 after a step change in gas phase oxygen from 21% to 0%. Top right figure shows results for running the model with selected parameter sets from the 256 randomly sampled sets remaining in the plausible parameter space after the final wave of the BHM/GPE process. The model output for three of those parameter sets is highlighted, which result in the approximate mean, maximum, and minimum volume of water remaining in the cell culture wells. Dashed lines in the top right figure represent mean ± 2×SD of the measured data. The main figure on the left shows an averaged result of all parameter set runs in the top right. A single parameter set to run the model consists of: {T₁ , k₁ , k₂, k₃, k₄, k₅, k₆, Inlet_L2, Inlet_L3, position, t_d}. The measured data is shown as mean ± 2×SD, with n=3. The red arrows indicate time points used to run the BHM/GPE process. b) Model and experimental results for volume of water remaining in Well 3 at the end of step change run. The shown data points are the result of running the model with the same parameter sets used for the top right figure in (a). The measured data is shown as mean ± 2×SD, with n=3

Following calibration, the model prediction for liquid phase oxygen transfer was initially validated by running the chamber with an IH protocol in the absence of cells (Fig. 5a). The results indicate a clear oxygen gradient, with a distinct variability in profiles for the middle and bottom of the liquid level (Fig. 5a). A parameter sensitivity analysis reinforces the link between vertical position in the liquid and oxygen cycling, with more rapid changes closer to the surface as expected (Supplementary Material, Fig. S7). Based on the model prediction, it appears that the probe was within approximately the bottom 35% of the liquid level for the IH runs (Fig. 5a) which is difficult to visualize during measurement owing to the extremely small level of liquid in the cell culture well. Accordingly, the model reasonably predicts dissolved oxygen change in response to IH (Fig. 5a). As a final validation step, HMECs were utilized (Fig. 5ci) to assess the model’s incorporation of cellular metabolism. Although any cell type can be used in the model by adjusting the metabolic parameters in Equation 2d, HMECs are relevant for this study as they provide a platform to investigate endothelial dysfunction in OSA [26], which can be achieved by exposing them to intermittent hypoxia. The viability of the cells after IH was confirmed with Trypan blue staining (Fig. 5cii). Using the estimated live cell count, the model was able to as accurately capture oxygen fluctuations in the liquid as in the case without cells, with the probe position estimated to be in the same location (Fig. 5b). Considering that the data with HMECs was not used for calibration, the dissolved oxygen results validate the model’s ability to successfully account for the effects of cellular consumption on oxygen transport. The validated model was then applied towards confirming the importance of gas-phase variations (Fig. 3d) on cell layer exposure to oxygen (Fig. 5d). Notably, there is an increase in amplitude with time for each well profile (Fig. 5d), which highlights the impact of evaporation and reduced diffusion limitation.

Fig. 5.

Liquid phase validation. Model and experimental results for dissolved oxygen in Well 3 after running IH protocol in the gas phase. a) Case without cells. b) Case with seeded HMECs. a and b) Top right figure shows results for running the model with the same selected parameter sets used for the top right of Fig. 4a. The model output for two of those parameter sets is highlighted, which assume probe position at the middle and bottom of the liquid level. Dashed lines in the top right figure represent mean ± 2×SD of the measured data. The main figure on the left shows an averaged result of all parameter set runs in the top right and those parameter set runs in the top right which assume probe position in the bottom 25% and 35% of the liquid. A single parameter set to run the model consists of: {T₁ , k₁ , k₂, k₃, k₄ , k₅, k₆, Inlet_L2, Inlet_L3, position, t_d}. The measured data is shown as mean ± 2×SD, with n=3 for a) and n=6 for b). ci) Cell image taken at 10X before experiment. cii) Images taken at different locations of the hemocytometer at 10X magnification, with Trypan blue-stained dead cells after running the chamber. Live and dead cells are marked with colored arrows. All scale bars represent 250 microns. d) Model output associated with IH protocol in Fig. 3d, with HMECs modeled in each well

3.3. Model Application: A Bedside to Bench Approach

Oxygen data derived from the breathing pattern (Fig. 6a) of an OSA patient (55 year-old male with an apnea-hypopnea index of 11) [25] was used as an input to the optimization approach (Section 2.5), which provided a gas controller protocol (Fig. 6b) that resulted in an oxygen concentration profile at the cell layer of Well 3 which not only closely matched the model-predicted patient oxygen values (Fig. 6c) but also mimicked the fluctuation timings seen in the profile (Fig. 6c). Despite previously identified mass transfer delays in the gas and liquid phases, the success of our approach in reproducing the model-predicted [25] patient oxygen pattern in-vitro highlights the power and importance of mathematical modeling in accounting for effects that are difficult to estimate experimentally.

Fig. 6.

”Patient-specific” gas controller protocol prediction. a) OSA patient lung volume segment previously derived from PSG data [25], selected from section of PSG with multiple breathing events. The model-dervied arterial oxygen concentration associated with the selected lung volume segment [25] was used to create the target oxygen protocol for the optimization code. Abbreviations: RERA; respiratory effort-related arousal. b) Gas controller protocol predicted by optimization code to achieve target oxygen concentration at cell layer. The shown protocol is after the onset of oxygen cycling; the chamber is initially run at an inlet oxygen fraction of 0.1 for 600 seconds to bring the cellular oxygen concentration down from the initial condition of normoxia (approximately 250 μM). c) Comparison of model-derived patient arterial oxygen pattern associated with a) and the oxygen concentration profile at the cell layer of Well 3 resulting from running the chamber with b). The extended dashed lines indicate boundaries of the step changes from b)

4. Discussion

4.1. Importance of Experimental Conditions for Targeted Cellular Oxygen Exposure

A realistic representation of any physiological system in-vitro requires a replication of the conditions that the cells are exposed to in-vivo. For hypoxia-related experimental studies, the target variable is dissolved oxygen concentration at the in-vitro cell layer. When using traditional hypoxia chamber set-ups for this purpose, diffusion through culture media delays oxygen equilibration between the gas phase and cell layer [27]. This idea is recapitulated by our model results, which show faster oxygen cycling in response to IH closer to the liquid surface (Fig. 5a and b). Furthermore, the equilibration time is dependent on multiple factors such as media height (Fig. 5d), covering of cell culture wells [27], convective mixing [28], temperature (Supplementary Material, Fig. S7b), cell seeding density, cellular oxygen consumption rate, and media composition. Although some variables may have a more significant impact than others, optimizing each condition through an experimental trial-and-error process is difficult to achieve. Furthermore, limitations in oxygen measurement can be associated with long-term recording or physical restrictions with the actual system. As an example, avoiding probe contact with the cell layer and maintenance of temperature due to probe sensitivity were specific problems encountered in the current study.

Conveniently, in recent years, the power of mathematical modeling has been leveraged to better understand and improve experimental design, with several instances related to cell culture [21, 29]. Similarly, in the current work, our model was applied to inform experimental design. As an example, our calibration results for accurately modeling evaporation show that a run resulting in the minimum volume of water remaining for one cell culture well will not necessarily lead to the minimum water volume for all cell culture wells (Fig. 4b). Furthermore, evaporation causes an increasing amplitude of dissolved oxygen fluctuations in response to a regular gas-phase pattern over time (Fig. 5d). Since liquid phase transfer is impacted by diffusion distance, these results highlight the importance of considering evaporation rate and all its contributing factors when designing experiments for multiple cell culture wells with gas flow. Moreover, well position is another important factor owing to spatial variations in temperature and flow velocity, both of which impact evaporation rate and gas phase oxygen.

During the testing/refinement of the protocol prediction process, media height was identified as a critical factor for mass transfer to the cell layer, with 0.8 mm (0.75 mL in a 35 mm well) chosen as the fluid level capable of fully capturing the behavior of the patient profile. Although higher fluid levels were also able to mimic the local extrema of the patient oxygen values, the required cycling times were significantly longer. Moreover, considering the effects of evaporation as previously mentioned, since the required media height is very low, there is a possibility of the wells drying out over time. To overcome these limitations in the current system and traditional hypoxia set-ups in general, a mechanism must be put in place to either periodically replenish the evaporated media during the chamber run or to humidify the air entering the chamber to lower the driving force for evaporation. Furthermore, it must be ensured that any evaporation-limiting method does not interfere with oxygen transport from the gas phase, in which case the media may need to be pre-treated or the oxygen may need to enter the bottom of the well through a semi-permeable membrane [15]. Whatever the remedial approach, as long as it is realistically incorporated into the system model, our optimization framework can be effectively deployed for ”patient-specific” analyses.

4.2. A Stepping Stone for Clinically Driven Research

The most significant achievement of this work is the development of a novel optimization approach that incorporates the validated chamber model towards achieving desired experimental oxygen fluctuations, with a demonstrated application using a model-derived, patient-specific pattern (Fig. 6). Although the current study presents a single patient case, the optimization approach can be implemented for any subject given that a corresponding dissolved oxygen profile is provided. While this work focuses primarily on the methods used to reproduce a given oxygen pattern in cell culture, it is important to highlight that a realistic experimental simulation of IH requires physiological accuracy of the input. Considering this, a directly measured profile of deep tissue oxygen [30] would be the ideal pattern to replicate in-vitro. However, the methods/technologies required to obtain this have not been sufficiently tested or applied in humans and may be invasive, especially for tissue in proximity to the vital organs. As an alternative, previous experimental studies of fluctuating oxygen have utilized pulse oximetry recordings to inform in-vitro IH protocols [13]. Yet, in addition to the many limitations of pulse oximetry [31, 32], research has suggested that this may not provide an accurate representation of tissue-level oxygenation [30]. In recent years, mathematical mass transfer models of the human body have been developed [25, 33, 34] to predict oxygen profiles in the blood and various organ tissues. Although this provides a more feasible approach to generate the input to our optimization protocol, the majority of current literature lacks a direct incorporation of clinical data. Therefore, to demonstrate a feasible patient-specific application of our method, a previously predicted oxygen profile [25] based on a PSG nasal pressure signal was used. In practice, any combination of clinical signals and physiological oxygen models can be applied to generate the dissolved oxygen input, as long as chosen method is sufficiently validated. This is an essential aspect of our presented framework (Figure 1) to ensure that there is a true representation of physiological conditions in cell culture. Although the dissolved oxygen input used to demonstrate our optimization approach has not yet been validated with measured data, this study addresses a vital component of the overall pipeline (red dashed region in Figure 1) and provides an important stepping stone for more reverse translational research in the future.

4.3. Limitations

As with any approach, there are some limitations to the current study. To avoid model complication, we have not accounted for uncertainty and possible variations in the velocity profile, along with potential turbulence and z-convective. One limitation of the model calibration approach is that it must be run every time a parameter, such as flow distribution, is changed. However, the primary limitation of this study is computational efficiency. Although the code is capable of running a multi-hour simulation in several minutes, the time for calibration was several hours, likely due to the initial model runs required to train/validate the GPEs (approximate run-time for a single parameter set was 1 minute in MATLAB 2024b on a machine with a AMD Ryzen 7 5700U Processor and 16 GB of RAM). In addition, the gas controller protocol prediction for the target sequence (Fig. 6b) also required over an hour of optimization time, likely due to multiple iterations of the optimization solvers. Reducing run-time with the use of methods such as surrogate models will significantly improve the power of our approach.

4.4. Future Directions

Indeed, our method can be adapted for several future applications. Firstly, in addition to a monolayer of cells, our model can be adjusted to study oxygen availability to cells embedded in hydrogels or other engineered substrates. Such set-ups better mimic the physiological environment and are important for applications such as tissue engineering research. Next, our approach can be implemented for the analysis of various cell types. If metabolic parameters are not available for certain cells, our model can be used to fit their values using measured oxygen data. Further experimental optimization is also a possibility, such as predicting the minimum starting volume of culture media in each well to prevent drying out from evaporation. This will be important to isolate the effect of hypoxia-related consequences on the cells, as absence of media can add to cellular stress. Arguably, the most compelling direction is a bedside to bench approach, with the use of multi-hour clinical data from patients suffering from diseases such as OSA. Such applications of our method will be important for bridging the gap between the clinical and experimental fields, especially considering that current commercially available systems do not provide measures of dissolved oxygen as a function of time and space and lack a feedback mechanism to adjust operating conditions for maintaining a physiologically relevant environment.

5.2. Data Availability and Online Supplementary Materials

Supplemental figures and materials are available with this manuscript as a single online file. All MATLAB codes and Jupyter Notebooks used in this study are available at: https://github.com/Cardiovascular-Modeling-Laboratory/Hypoxia_Chamber_Model_Codes.git. Experimental data has been made publicly available on the Dryad repository at: https://doi.org/10.5061/dryad.t1g1jwtg9