Numerical studies of continuous nutrient delivery for tumour spheroid culture in a microchannel by electrokinetically-induced pressure-driven flow
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DOI: 10.1007/s10544-010-9460-z
- Cite this article as:
- Movahed, S. & Li, D. Biomed Microdevices (2010) 12: 1061. doi:10.1007/s10544-010-9460-z
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Abstract
Continuous nutrient delivery to cells by pressure-driven flow is desirable for cell culture in lab-on-a-chip devices. An innovative method is proposed to generate an induced pressure-driven flow by using an electrokinetically-driven pump in a H-shape microchannel. A three-dimensional numerical model is developed to study the effectiveness of the proposed mechanism. It is shown that the average velocity of the generated pressure-driven flow is linearly dependent on the applied voltage. Considering the culture of a multicellular tumour spheroid (MTS) in such a microfluidic system, numerical simulations based on EMT6/Ro tumour cells is performed to find the effects of the nutrient distribution (oxygen and glucose), bulk velocity and channel size on the cell growth. Using an empirical formula, the growth of the tumour cell is studied. For low nutrient concentrations and low speed flows, it is found that the MTS grows faster in larger channels. It is also shown that, for low nutrient concentrations, a higher bulk liquid velocity provide better environment for MTS to grow. For lower velocities, it is found that the local MTS growth along the flow direction deviates from the average growth.
Keywords
Multicellular tumour spheroidCell cultureH-shaped microchannelElectrokinetically-induced flow1 Introduction
In order to develop better treatments and new curing methods for cancers, studying the growth kinetics of malignant tumours is essential. During the avascular growth stage, there are no blood vessels within the tumours; therefore only surface contact exists between blood and tumour and consequently tumours can access to the nutrients only through their outer surface. Many factors such as the supply, distribution and uptake of nutrients (e.g., oxygen, glucose and hydrogen as well as the growth modulating chemical substances) are parameters that affect the growth of tumour cells. The growth process of tumour cells in the avascular growth stage strongly depends on the nutrient transport to the cells and the nutrient uptake by the cells at the cell surface. These are the fundamentals of predictive mathematical tumour models (Araujo and McElwain 2004; Byrne et al. 2006).
Many theoretical models have been proposed for the tumour growth (Hu and Li 2007; Bartha and Rieger 2006; Rejniak 2007). Multicellular tumour spheroid (MTS) is typically used as the simplified model to investigate the tumour growth kinetics though cancers are extremely diverse and heterogeneous (Casciari et al. 1992; Delsanto et al. 2004; Freyer 1998; Freyer and Sutherland 1985; Kelm et al. 2003; Landry et al. 1982; Sutherland 1986, 1988). Sutherland experimentally showed the spheroids growth until they reached a critical size (Sutherland 1988). This critical size is determined by the balance between cell proliferation and cell death (apoptosis) depending on the supply of nutrients from outside environment as well as the transport and consumption of nutrients within the tumour. Based on energy conservation and scaling argument and also considering EMT6/Ro as the sample cells, in 1992 Casciari et al. developed a model to describe the effects of oxygen concentration, glucose concentration and extracellular pH on MTS growth (Casciari et al. 1992). Using this model, Hu and Li numerically studied the effects of oxygen and glucose as nutrients on the growths of MTS tumours in rectangular microchannels (Hu and Li 2007). Their work provided valuable findings on the impacts of flow velocity and channel size on the growth of the single and multiple MTSs. They showed that as long as there is a bulk flow, the growth of single tumour spheroid at early stages is insensitive to the flow velocity and channel size (Hu and Li 2007). However, their findings are based on sufficiently high concentration of nutrients so that the tumour growth is independent of the nutrient concentration. By considering this fact that the EMT6/Ro cells are able to grow under poor condition (for example, the cells can grow with a doubling time of 18 h even at a pH = 6.67, C_{oxygen} = 0.023 mM and C_{glucose} = 0.4 mM (Casciari et al. 1992)), it is desirable to find the effects of the flow velocity and channel size on MTS growth under low concentration of nutrients. When the concentration is low, the growth rate should be affected by concentration distribution around the tumour which is the function of the microchannel size and the flow velocity. Furthermore, previous studies consider tumour as a sphere that grows uniformly. When the nutrient concentration is sufficiently high, uniform growth can be a valid assumption. However, for low concentrations, the concentration distribution on tumour surface can influence the growth rate and therefore may lead to non-uniform local growth.
The recent development of microfluidic devices for cell culture and cellular scale studies has attracted more and more research efforts. Microfluidics based experiments and theoretical models have been developed to investigate the cell culture and therapy based on individual cells or cell monolayers (Gu et al. 2004; Zeng et al. 2006). Microfluidic devices offer many advantages such as significant reduction in the consumption of the expensive matrix gel and the culture medium. Furthermore, the continuous flow can deliver the nutrients to the cells and remove the waste produced by the cells in a manner similar to the real situation in a living biological system. However, in order to generate flow it usually requires using external syringe pumps, tubing and valves, etc. This will increase the difficulty, complexity and the cost of making the microfluidic cell culture devices, and reduce the reliability of the operation due the failure of these external liquid handling components. Using electroosmotic flow requires only simple electrodes inserted in wells at the ends of the microchannel and can avoid these problems. However, the applied electric field may affect the cell growth and should not be present in the cell culture channel. Therefore, in the present study, a novel method is proposed to use an electroosmotic flow to generate a pressure-driven flow in the cell culture channel. Using the proposed mechanism, we can generated the pressure-driven flow and deliver the nutrients continuously for cell culture in lab-on-a-chip devices without using external mechanical liquid handling devices such a syringe pumps, tubing and valves. Numerical simulations are carried out to demonstrate the feasibility of the proposed mechanism. Furthermore, three-dimensional numerical simulations were performed to study the growth of the tumour spheroid (volumetric increase) in the low concentration medium in a pseudo-steady state.
2 Modeling
2.1 System description
2.2 Velocity field
Slip velocity boundary condition is applied at the walls of the channel C-D in order to consider the electroosmosis effect. The no-slip boundary condition is applied in the other channels where the flow is pressure driven. Open boundary condition is assumed at boundaries A, B, C and D.
2.3 Nutrient distribution
Model parameters (Hu and Li 2007)
Parameter | value/range | Unit | Reference |
---|---|---|---|
ζ | −40 | mV | Li 2004 |
ε_{T} | 80 | – | Li 2004 |
ε_{0} | 8.85 × 10^{−8} | F/m | Li 2004 |
ρ | 1,000 | _{Kg/m}^{3} | ^{a} |
ρ_{s} | 2.01 × 10^{8} | Cell/cm^{3} | Casciari et al. 1992 |
μ | 1 × 10^{−3} | Pa.s | ^{a} |
D_{o} | 2 × 10^{−5} | cm^{2}/s | Perry 1984 |
D_{g} | 9.25 × 10^{−6} | cm^{2}/s | Li 1982 |
V_{max,o} | −2.95 to 2.48 × 10^{−8} | mol/cm^{3}/s | Freyer and Sutherland 1985 |
K_{max,o} | 4.64 × 10^{−3} | mM | Casciari et al. 1992 |
Vmax,g | −4.73 to 4.02 × 10^{−8} | mol/cm^{3}/s | Freyer and Sutherland 1985 |
K_{max,g} | 0.04 | mM | Li 1982 |
K_{o} | 0.0533 | 1/h | Landry et al. 1982 |
G_{1} | 0.0138 | - | Casciari et al. 1992 |
G_{2} | 7.29 × 10^{−3} | mM | Casciari et al. 1992 |
G_{3} | 1.76 × 10^{−2} | mM | Casciari et al. 1992 |
n | 0.4566 | mM | Casciari et al. 1992 |
2.4 MTS growth
3 Results and discussion
3.1 Numerical simulation
In order to find velocity field, Eqs. 1 and 2 subjected to boundary conditions (4, 5) must be solved. The open boundary condition, Eq. 5, is assumed for boundaries A, B, C and D. The electroosmotic slip velocity is considered on the walls of channel C-D while the walls in the rest of the system have no-slip velocity condition.
For each nutrient component, Eq. 8 subjected to boundary conditions (11, 13 and 14) must be solved to find the concentration in the system. Since the transport of nutrients does not affect the flow field, Navier-Stokes equation, (1), only needs to be solved once for the given flow rate. The mass balance equation, (8), can be decoupled from the MTS growth equation, (19), because the time to reach a steady state of concentration distribution is much shorter than the cell growth time (Hu and Li 2007). Therefore, Eq. 8 is solved separately for the steady state of concentration field for a given MTS size. Once the concentrations of oxygen and glucose at the spheroid surface are determined, the growth coefficient can be estimated to calculate the MTS growth.
The assumed parameters for the above equations have been listed in Table 1.
Our numerical studies consist of two main parts. In the first step, it is desired to investigate the effectiveness of the proposed method of generating electrokinetically-induced pressure-driven flow. To do this, two different configurations of the H-shape channel are considered. In both cases, the width, the depth and the length of the channels A-B and C-D are 200 μm, 200 μm and 15 mm, respectively; the length of the channel P-O is 10 mm. The channel P-O’s cross section in these two different cases are 200 μm × 200 μm and 400 μm × 400 μm, respectively. The applied voltages at points C and D are −50 V and the electric potential of point O is kept to zero.
In the next step, by assuming that the input nutrients have low concentrations, the effects of bulk velocity and the channel size on the tumour growth in the channel P-O are studied. Also, how local growth may deviate from the average growth in the case of low nutrients concentrations is investigated. We choose EMT6/Ro mouse mammary tumour cells as a sample in the channel P-O. Basal Medium Eagle (BME) at pH of 7.2 is assumed as the culture medium and the transport of two different nutrients, oxygen and glucose, are investigated. Values of required constants and coefficients are given in Table 1.
3.2 Velocity field
Flow rate in channels OC (OD). This table shows that for various values of the applied electric potential, flow rate has positive value indicating that the flow directs from point O to points C and D
Flow Rate in Channel OC (OD) | ||
---|---|---|
Applied Voltage (V) | small channel (m^{3}/s) | large channel (m^{3}/s) |
−2 | 8.29 × 10^{−14} | 2.53 × 10^{−13} |
−10 | 4.15 × 10^{−13} | 1.26 × 10^{−12} |
−25 | 1.04 × 10^{−12} | 3.16 × 10^{−12} |
−40 | 1.66 × 10^{−12} | 5.05 × 10^{−12} |
−50 | 2.07 × 10^{−12} | 6.31 × 10^{−12} |
3.3 Nutrient concentrations
Two cases with different inflow concentrations are numerically solved and the results were compared with Hu’s results (the boundary conditions of inlets A and B are: C_{in,g} = 0.4 mM, C_{in,o} = 0.28 mM, and, C_{in,g} = 5.5 mM, C_{in,o} = 0.28 mM (subscript o and g means oxygen and glucose, respectively). The tumour growth rates are approximately identical with those predicted in Hu’s work, which indicates that the nutrient concentrations around the tumour are identical and consequently present numerical method is valid. For example, if V_{e} = −5 V, C_{in,g} = 0.4 mM, C_{in,o} = 0.28 mM, the required time for the radius to increase from r_{0} = 50 μm to r = 100 μm is 45 h.
3.4 Effect of V_{e} (the applied voltage) on tumour growth
3.5 Effect of channel size on tumour growth
3.6 MTS growth
In a low concentration medium the tumour growth is much more sensitive to the nutrient’s concentration, and the concentration distribution around the tumour cell may cause the deviation between the local growth and the average growth. To study the MTS growth rate, the larger cross section for channel P-O (400 μm × 400 μm) is selected. The nutrient concentrations at reservoirs A and B are C_{o,in} = 0.175 mM and C_{g,in} = 0.4 mM. The MTS tumour with the initial radius r_{0} = 50 μm is located at the center of the channel P-O. To have a better understanding on the local growth, the average growth of the front-half and the back-half of the MTS tumour are studied separately.
4 Conclusion
In this study, a new method is proposed to generate electrokinetically-induced pressure-driven flow. This method is applied to an H-shape microchannel for cell culture. Three-dimensional numerical simulations are conducted to examine the pressure gradient, velocity field and nutrient distributions in the system. It was shown that the average velocity of the generated pressure-driven flow is linearly dependent on the applied voltage. Furthermore, the growth of a multicellular tumour spheroid (MTS) in such a microfluidic system is studied under low nutrient concentration condition. The MTS grows faster in larger channels and with higher flow velocities. The front-face and the back-face MTS growth deviating from the average growth is also predicted for low nutrient concentrations.
Acknowledgement
The authors wish to thank the financial support of the Natural Sciences and Engineering Research Council through a research grant to D. Li.