Research Papers

Mathematical Model for Tissue-Level Hypoxic Response in Microfluidic Environment

[+] Author and Article Information
Adnan Morshed

School of Mechanical and Materials Engineering,
Washington State University,
Pullman, WA 99164-2920
e-mail: adnan.morshed@wsu.edu

Prashanta Dutta

Fellow ASME
School of Mechanical and Materials Engineering,
Washington State University,
Pullman, WA 99164-2920
e-mail: prashanta@wsu.edu

1Corresponding author.

Manuscript received March 2, 2017; final manuscript received August 31, 2017; published online November 9, 2017. Assoc. Editor: Nathan Sniadecki.

J Biomech Eng 140(1), 011009 (Nov 09, 2017) (10 pages) Paper No: BIO-17-1090; doi: 10.1115/1.4037915 History: Received March 02, 2017; Revised August 31, 2017

Availability of essential species like oxygen is critical in shaping the dynamics of tumor growth. When the intracellular oxygen level falls below normal, it initiates major cascades in cellular dynamics leading to tumor cell survival. In a cellular block with cells growing away from the blood vessel, the scenario can be aggravated for the cells further inside the block. In this study, the dynamics of intracellular species inside a colony of tumor cells are investigated by varying the cell-block thickness and cell types in a microfluidic cell culture device. The oxygen transport across the cell block is modeled through diffusion, while ascorbate (AS) transport from the extracellular medium is addressed by a concentration-dependent uptake model. The extracellular and intracellular descriptions were coupled through the consumption and traffic of species from the microchannel to the cell block. Our model shows that the onset of hypoxia is possible in HeLa cell within minutes depending on the cell location, although the nutrient supply inside the channel is maintained in normoxic levels. This eventually leads to total oxygen deprivation inside the cell block in the extreme case, representing the development of a necrotic core that maintains a dynamic balance with growing cells and scarce supply. The numerical model reveals that species concentration and hypoxic response are different for HeLa and HelaS3 cells. Results also indicate that the long-term hypoxic response from a microfluidic cellular block stays within 5% of the values of a tissue with the basal layer. The hybrid model can be very useful in designing microfluidic experiments to satisfactorily predict the tissue-level response in cancer research.

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Fig. 1

Schematic of the normoxic and hypoxic reaction pathways. Hypoxia-inducible factors (HIF1α) gets hydroxylated by PHDs and the FIH in parallel pathways. PHD gets activated by a number of other cofactors such as iron (Fe), DG, and AS. Hydroxylated HIF later gets degraded by ubiquitination reaction after being tagged by vHL proteins. When there is a lack of oxygen supply, both hydroxylation pathways lose their effectiveness. HIF, in this case, accumulates in cytosol and shuttles inside the nucleus where it also faces hydroxylation pathways but it can also form a dimer (HIFd) with HIF1β. This dimer can activate the hypoxia response elements and initiate a large number of transcriptional activities. Feedback mechanisms such as elevated PHD production are also activated at this stage. Uptake and consumption of extracellular nutrients (O2 and AS) have also been considered in the model.

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Fig. 2

Schematic representation of the computational domain inside a microfluidic channel. Total channel width consists of the unobstructed flow region (a) and cell-block region (b). The nondimensional parameter δ represents the ratio of a to b, which varies with the thickness of cell block. The basal layer, encountered in in vivo experiments, poses an additional impediment for free entry of different species into tissue. Extracellular species like oxygen and AS are transported in the fluid channel. Small species like oxygen can diffuse through the cell block. However, AS transport inside the block requires intervention of active transporter proteins. Depending on theavailability of oxygen and other cofactors, each cell undergoes through a cascade reactions presented in Fig. 1 and supplementary Table S2 which is available under the “Supplemental Data” tab for this paper on the ASME Digital Collection. In this study, the length of the channel is 1.5 cm and a+b=const=3000μm.

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Fig. 3

Flowchart for the hybrid algorithm. Each cell in the domain must be evaluated individually at each time for temporal evolution of species, which is also connected to the spatial distribution of the species presented in the extracellular domain that freely diffuses through the cellular block or, transported in actively. The spatial solution must go through iterations in order to converge at every time step.

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Fig. 4

Spatiotemporal oxygen concentration distribution with HeLaS3 cells. (a) The contour of oxygen levels in the whole computational domain after 30 min showing a jump in concentration across the basal layer of the cell block. Concentration distribution inside the block approaches dynamic equilibrium within minutes. (b) Oxygen levels across the domain at different times for δ = 1.0 and (c) δ = 3.0, where the equilibrium level inside the cell block is compared with results from Ref. [30] for model validation. At the inlet boundary, supply for oxygen was maintained at 207 μM, while the initial intracellular concentrations are reported in supplementary Table S4 which is available under the “Supplemental Data” tab for this paper on the ASME Digital Collection. The side walls of the channel are taken as impervious and the outlet is fully developed.

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Fig. 5

Effect of cell-block thickness on oxygen concentration. Results are obtained at the midlength of the channel. Four δ values (a) 11.0, (b) 3.0, (c) 1.0, and (d) 0.33 were considered which correspond to 10, 30, 60, and 90 HeLa cells in the y-direction, respectively. For δ value as low as 1.0, although the cells are in hypoxia, they maintain a steady dynamic equilibrium. However, for δ less than 1.0, we see almost negligible oxygen further inside the block. For all cases, entry of extracellular medium was maintained at 207 μM for oxygen, and the initial intracellular concentrations are given in supplementary Table S3 which is available under the “Supplemental Data” tab for this paper on the ASME Digital Collection.

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Fig. 6

Effect of cell-block thickness on the transcriptional activity. The variation of tissue thickness and initial conditions are kept exactly same as those in Fig. 5. Although oxygen concentration reaches equilibrium fairly quickly, the extent of hypoxic activity increases steadily once the cells are kept in hypoxic conditions. The highest level of activity was observed for the cells furthest away from the channel. The level found at 1 h for δ = 0.33 was used to normalize transcriptional activity for all four cases considered.

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Fig. 7

Effect of cell type variation through a change in the OCR. Different OCR levels for HeLa and HeLaS3 were used from experimental measurements [21]. In all cases, absolute values of the difference were normalized by corresponding levels observed for HeLa cells. (a) Numerically predicted difference in oxygen concentration at different time and location. Different OCR values for HeLa and heLaS3 result in a significant variation in oxygen levels inside the block initially. However, it reaches steady levels quickly and maintains a fixed level after a distance of about 15 cell diameters. (b) Difference in transcriptional activities at different locations (along the y-direction) inside the tissue calculated at midlength. Initial difference is high until a dynamic equilibria is reached over the domain.

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Fig. 8

Difference in (a) oxygen levels and (b) transcriptional activity with and without the artificial basal layer with HeLa cells at midlength of the channel. In both cases, absolute values of the difference were normalized by corresponding levels observed for the cell block with the base layer. The sharp difference in oxygen at the channel-block interface is due to the absence of the basal layer which has an insulator like effect on gradient driven mass transport. Initially, the cells deeper inside the block have similar levels of oxygen; hence, the percentage difference is almost zero. But within minutes the diffusion front reaches cells at the distal end in the case without the basal layer resulting in the rise in percentage difference in oxygen concentration. Again, the change becomes same after 10 min and constant after ∼15 cells from the start of the block. Due to the comparatively large difference in concentration just at the interface for the two cases, transcriptional activity is most prominent near this area. However, as we move inside the block, it reaches a rather small (∼5%) and consistent level in most of the cell block. All initial concentrations are presented in supplementary Table S4 which is available under the “Supplemental Data” tab for this paper on the ASME Digital Collection.



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