Biomedical Microdevices

, Volume 12, Issue 6, pp 1061–1072

Numerical studies of continuous nutrient delivery for tumour spheroid culture in a microchannel by electrokinetically-induced pressure-driven flow

Authors

  • Saeid Movahed
    • Department of Mechanical and Mechatronics EngineeringUniversity of Waterloo
    • Department of Mechanical and Mechatronics EngineeringUniversity of Waterloo
Article

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

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 flow

1 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, Coxygen = 0.023 mM and Cglucose = 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

In this section, a novel method is put forward to generate electrokinetically-induced pressure-driven flow in an H-shape microchannel. A schematic diagram of the proposed method is shown in Fig. 1. The system is placed horizontally on a lab table surface. The main objective is to generate continuous pressure-driven flow and nutrient delivery in the cell culture channel. The channel C-D is used as an electroosmotic pump. All four reservoirs, A, B, C and D, are open to air and have atmospheric pressure. By keeping a potential at point O equal to zero and applying a negative electric potential at points C and D, electroosmotic flow is generated in the channel C-D from point O to reservoirs C and D. Because of the flow continuity, point O has negative pressure that induces a flow from reservoirs A and B to point O and hence generates a continuous pressure-driven flow in the cell culture channel from reservoirs A and B. It should be noticed that, in the channel C-D, the electroosmotic flow is in the directions from the point O to reservoirs C and D (because of the driving electric field), respectively, the negative pressure at point O also generates a reverse flow in the middle of the channel rom the ends towards point O in the upper channel. However, the net flow is from point O to the ends of the channel C-D. In this system, it is clear that without any moving parts, pressure gradient is created to generate flow in the channel P-O. Thus there is no electric field in the vertical cell-culture channel; by putting a MTS tumour cell in this channel, the effects of nutrients’ concentration, flow velocity and the channel size on tumour growth can be studied.
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9460-z/MediaObjects/10544_2010_9460_Fig1_HTML.gif
Fig. 1

Schematic diagram of the proposed method to generate electrokinetically-induced pressure-driven flow. The s-s is the central plane of the spheroid. Nutrients are in reservoir A and reservoir B. Reservoirs C and D contains initially the buffer solution and later are used for holding waste materials. Voltage at point O is fixed to be zero, and negative voltages applied at C and D

2.2 Velocity field

The Navier-Stokes and the continuity equations, Eqs. 1 and 2, must be solved in order to find steady state flow field in the system. To find the electric potential, the Laplace equation must be solved.
$$ \rho \overrightarrow u \bullet \nabla \overrightarrow u = - \nabla p + \mu {\nabla^2}\overrightarrow u $$
(1)
$$ \nabla \bullet \overrightarrow u = 0 $$
(2)
$$ {\nabla^2}V = 0 $$
(3)

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.

The electrical and velocity boundary conditions are summarized below:
$$ {u_{B.C.}} = \left\{ {\begin{array}{*{20}{c}} { - \frac{{\varsigma {\varepsilon_T}{\varepsilon_0}E}}{\mu }\quad {\hbox{upper}}\;{\hbox{horizontal}}\;{\hbox{channel}}} \hfill \\{\quad \;\;0\quad \quad \,\;\;{\hbox{lower}}\;{\hbox{horizontal}}\;{\hbox{channel}}} \hfill \\\end{array} } \right. $$
(4)
$$ \nabla \overrightarrow u = {0:}\;{\hbox{at boundaries}}\;{\hbox{A,}}\;{\hbox{B,}}\;{\hbox{C}}\;{\hbox{and}}\;{\hbox{D}} $$
(5)
$$ \overrightarrow n \bullet \overrightarrow J = 0\;:\;{\hbox{at}}\;{\hbox{walls}} $$
(6)
$$ V = \left\{ {\begin{array}{*{20}{c}} {0\;\;V\quad {\hbox{at}}\;{\hbox{O}}} \hfill \\{{V_e}\quad {\hbox{at}}\;{\hbox{C}}\;{\hbox{and}}\;{\hbox{D}}} \hfill \\\end{array} } \right. $$
(7)

2.3 Nutrient distribution

For nutrient component i, the concentration field is governed by the mass balance equation in the microchannels:
$$ \frac{{\partial {C_i}}}{{\partial t}} + \overrightarrow u \bullet \nabla {C_i} = {D_i}{\nabla^2}{C_i} $$
(8)
where Ci is the concentration of nutrient component i, Di is the diffusion coefficient. The associated boundary conditions of the concentration field include the boundary conditions at the four reservoirs and at the tumour cell surface.
Michaelis-Menten (MM) kinetics is used to find the volumetric uptake rate for nutrient component i by MTS (Ri) (Hu and Li 2007):
$$ {R_i} = \frac{{{V_{\max, i}}{C_i}}}{{{K_{m,i}} + {C_i}}} $$
(9)
where Vmax,i and Km,i are the MM parameters and can be found in Table 1. If the value of Kmi is much lower than the local nutrient concentration Ci, Eq. 9 can be further simplified to zero-order kinetics (Hu and Li 2007):
$$ {R_i} = {V_{\max, i}} $$
(10)
Table 1

Model parameters (Hu and Li 2007)

Parameter

value/range

Unit

Reference

ζ

−40

mV

Li 2004

εT

80

Li 2004

ε0

8.85 × 10−8

F/m

Li 2004

ρ

1,000

Kg/m3

a

ρs

2.01 × 108

Cell/cm3

Casciari et al. 1992

μ

1 × 10−3

Pa.s

a

Do

2 × 10−5

cm2/s

Perry 1984

Dg

9.25 × 10−6

cm2/s

Li 1982

Vmax,o

−2.95 to 2.48 × 10−8

mol/cm3/s

Freyer and Sutherland 1985

Kmax,o

4.64 × 10−3

mM

Casciari et al. 1992

Vmax,g

−4.73 to 4.02 × 10−8

mol/cm3/s

Freyer and Sutherland 1985

Kmax,g

0.04

mM

Li 1982

Ko

0.0533

1/h

Landry et al. 1982

G1

0.0138

-

Casciari et al. 1992

G2

7.29 × 10−3

mM

Casciari et al. 1992

G3

1.76 × 10−2

mM

Casciari et al. 1992

n

0.4566

mM

Casciari et al. 1992

aAssumed to be the same as those in water at 37°C. Subscripts o and g stand for oxygen and glucose, respectively.

The mass transfer flux (Fi) for nutrient component i at the outer surface of tumour spheroid is given by:
$$ - \overrightarrow n \bullet ({D_i}\nabla {C_i}) = {F_i} $$
(11)
where ni is the unit normal vector to the surface. The flux Fi at the tumour surface can be obtained by multiplying the volumetric uptake rate (Ri) by the tumour spheroid volume (Vo) and then divided by the surface area of the tumour spheroid (S):
$$ {F_i} = {R_i}\frac{{{V_o}}}{S} $$
(12)
Other boundary conditions for the concentration field are as follows:
$$ {C_i} = {C_{in,i}}\quad {\hbox{at the boundaries}}\;A\;{\hbox{and}}\;B $$
(13)
$$ \overrightarrow n \bullet ({D_i}\nabla {C_i}) = 0\;{\hbox{at the boundaries}}\;C\;{\hbox{and}}\;D $$
(14)

2.4 MTS growth

The concentration of nutrients will affect the MTS growth and the tumour growth in turn will affect the distribution of nutrient. The change of MTS volume (Vo) is given by (Casciari et al. 1992):
$$ \frac{{d{V_o}}}{{dt}} = K{V_o} - \frac{{2{N_S}}}{{{\rho_c}}} $$
(15)
In the Eq. 15ρc is the constant that can be found in Table 1. Ns is the rate of cell shedding and can be found as:
$$ {N_s}\left( { \frac{{cells}}{h} } \right) = \left\{ {\begin{array}{*{20}{c}} {\quad \quad \left( {2.18 \times {{10}^4}\; \frac{{cells}}{{c{m^2}h}} } \right){A_s} - 41.5\quad \quad \quad d > 400\mu m} \hfill \\{\left( {1.66 \times {{10}^6}\frac{{cell{s^2}}}{{c{m^4}{h^2}}}} \right)A_s^2 + \left( {520\frac{{cells}}{{c{m^2}h}}} \right){A_s}\quad \quad d \leqslant 400\mu m} \hfill \\\end{array} } \right. $$
(16)
K in Eq. 15 is the growth coefficient that can be modeled as:
$$ K = {K_0}f\left( {{C_0},{C_g},{C_h}} \right) $$
(17)
where K0 is the growth constant for the tumour cells, and f is an empirical function of the concentrations of oxygen (Co), glucose (Cg) and hydrogen (Ch). This function can be written as:
$$ f = {G_1}\left( {\frac{{{C_o}}}{{{C_o} + {G_2}}}} \right)\left( {\frac{{{C_g}}}{{{C_g} + {G_3}}}} \right){\left( {\frac{1}{{{C_h}}}} \right)^n} $$
(18)
where G1, G2, G3 and n are empirical values given in Table 1.
Equation 15 can be rewritten in terms of required time for a spherical tumour to growth from radius R0 to radius R as:
$$ \frac{{dt}}{{dr}} = {\left( {\frac{{Kr}}{3} - \frac{{2{N_s}}}{{4\pi {r^2}{\rho_c}}}} \right)^{ - 1}} $$
(19)
If the MTS growth is spherical and symmetric, the required time for tumour to growth from radius R0 to radius R can be modified as:
$$ t = \int_{{R_0}}^R {\frac{{4\pi {r^2}dr}}{{\frac{4}{3}K\pi {r^3} - \frac{{2{N_S}}}{{{\rho_S}}} }}} dr $$
(20)
If d ≤ 400 μm, the integral of Eq. 20 becomes:
$$ t = \left. {\frac{{2ArcTan\left( {\frac{{b + 2ar}}{{\sqrt {{ - {b^2} + 4ac}} }}} \right)}}{{\sqrt {{ - {b^2} + 4ac}} }}} \right|_{{R_0}}^R $$
(21)
where:
$$ a = \frac{k}{3} $$
$$ b = - \frac{{\left( {2 \times 2.18 \times {{10}^4}} \right)}}{{{\rho_c}}} $$
$$ c = \frac{{83}}{{{\rho_c}}} $$

3 Results and discussion

3.1 Numerical simulation

3-D solver of the finite element package COMSOL Mutliphysics 3.5a is used in order to simulate the problem. Equation 3 subjected to boundaries (6) and (7) is solved to find the electric potential in the system. It is assumed that the walls of the system are completely insulated. At point O, a plane with the dimension 200 μm × 200 μm is assumed to have a zero potential. In Y-direction, the side of this plane is equal to the height of the channel; in X-direction, the orientation of the plane is completely symmetry with respect to the line S-S, see Fig. 1. It is also assumed that the boundaries C and D have the potential value Ve.
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9460-z/MediaObjects/10544_2010_9460_Fig2_HTML.gif
Fig. 2

Pressure gradient and velocity vectors. Color bar is for pressure, Ve = −50 V

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

By applying the electric potential of −50 V at the two ends of the channel C-D, electroosmotic flow is generated in the channel C-D and consequently a pressure gradient is produced in the system. Figure 2 shows the pressure gradient in the middle plane (half of the depth) of the channels. It can be seen that there is a negative pressure at the top of channel P-O. This generates a flow field from reservoirs A and B to point O as shown in Fig. 2. In the channel C-D, on one hand the liquid near the channel walls flows from point O to the ends of the channel due to electroosmosis; on the other hand there is a backward flow in the middle of the channel because of the pressure gradient between the reservoirs (C and D) and the point O. Therefore, we have some circulations near the intersection of the channels C-D and P-O. However, the net flow is from the point O to the ends of the channel. To have a better visualization, for different XZ-cross sections, Figs. 3 and 4 illustrate the velocity vectors in vicinity of point O and boundary D, respectively. From these figures, one can clearly see that the dominant flow is the electroosmotic flow near the walls and hence the net flow is from the point O to the ends of channel C-D. To have a better clarification, in channel OC (OD) the velocity has been integrated across the channel to find the flow rate. For different values of applied electric field, Table 2 shows the flow rate in channels OC and OD. Positive sign of the flow rate means that the overall flow is from point O to the outlets of the system (points C and D).
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9460-z/MediaObjects/10544_2010_9460_Fig3_HTML.gif
Fig. 3

Velocity Vectors at the intersection of P-O and C-D channels and for different cross sections. Color bar is for the magnitude of the velocity. Ve = −50 V. (a) Y = 100 μm, (b) Y = 150 μm (c) Y = 170 μm (d) bottom of the cannel OC (e) middle of the cannel OC

https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9460-z/MediaObjects/10544_2010_9460_Fig4_HTML.gif
Fig. 4

Velocity Vectors at the boundary D at different XZ-cross sections. Color bar is for the magnitude of the velocity (a) Y = 100 μm, (b) Y = 150 μm (c) Y = 170 μm

Table 2

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 (m3/s)

large channel (m3/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

For the two different cases, Fig. 5 presents the variation of the average velocity in the channel P-O with respect to the applied electric potential at C and D (in the rest of the paper, V refers to the electric potential at the two points C and D). The electric potential at the point O is kept at zero. From this figure, it is evident that the average velocity of the channel P-O is linearly dependent on the applied voltage (Ve). By increasing the negative value of the applied electric potential, the average velocity of channel P-O increases.
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9460-z/MediaObjects/10544_2010_9460_Fig5_HTML.gif
Fig. 5

The average velocity in the channel P-O versus the applied voltage in the channel C-D

3.3 Nutrient concentrations

3D experimental data or analytical solutions for nutrient concentration distributions around the MTS are rare. Therefore, to prove the accuracy of our numerical simulation, we compare the results with Hu’s numerical simulations (Hu and Li 2007). Their results have a good agreement with the simplified 2D analytical solutions. Figure 6 shows the comparison for one case (nutrient = glucose, Cin,g = 11 mM, U = 10 μm/s, cross-section: 200 μm × 200 μm, OD = 100 μm). This figure illustrates that there is a good agreement between the result of this study and Hu’s result.
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9460-z/MediaObjects/10544_2010_9460_Fig6_HTML.gif
Fig. 6

Comparison between the results of this study and Hu’s results (Hu and Li 2007). Nutrient: glucose, U = 10 μm/s, Cin,g = 11 mM, cross-section: 200 μm × 200 μm, OD = 100 μm

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: Cin,g = 0.4 mM, Cin,o = 0.28 mM, and, Cin,g = 5.5 mM, Cin,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 Ve = −5 V, Cin,g = 0.4 mM, Cin,o = 0.28 mM, the required time for the radius to increase from r0 = 50 μm to r = 100 μm is 45 h.

Figure 7 is the example of concentration distribution of glucose in the system (Cin,g = 0.4 mol/m3 and Ve = −10 V). Figure 8 shows the distributions of glucose at the central plane (plane s-s, see Fig. 1) and the MTS outer surface for three different values of the applied voltage (the boundary conditions of inlets A and B are: Cin,g = 0.4 mM and Cin,o = 0.175 mM). From this figure, it is evident that there is a concentration difference between the upstream side and the downstream side of the spheroid. This is because the tumour consumes the nutrients (Ri in Eq. 9). The nutrients transfer to the surface of the tumour cell by diffusion and by convection. By increasing the value of the applied voltage (and consequently the flow velocity), this concentration difference decreases. Similar figures can be plotted for oxygen and other nutrient components.
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9460-z/MediaObjects/10544_2010_9460_Fig7_HTML.gif
Fig. 7

Example of concentration distribution of glucose in the system, Cin,g = 0.4 mol/m3 and Ve = −10 V

https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9460-z/MediaObjects/10544_2010_9460_Fig8_HTML.gif
Fig. 8

Concentration distributions of glucose at the central plane (plane s-s, see Fig. 1) and the MTS outer surface. Cin,g = 0.4 mM. The color bar is for concentration. (a) Ve = −2 V (b) Ve = −5 V and (c) Ve = −10 V

3.4 Effect of Ve (the applied voltage) on tumour growth

In order to study the effects of Ve on tumour growth, the smaller H-shaped microchannel is selected (channel P-O’s cross section: 200 μm × 200 μm). The nutrient concentrations at reservoirs A and B are Cin,g= 0.4 mM and Cin, o= 0.175 mM. For three voltages (−2 V, −5 V and −10 V), the growth of the EMT6/Ro tumour spheroid with the initial radius r0 = 50 μm is calculated under these conditions. Figure 9 compares the increase in radius of the spheroid with time under these conditions. From this figure, it can be seen that by increasing Ve, the growth rate increases too. It means that for low nutrient concentrations, even in the early stages the growth of the tumour spheroid is sensitive to the flow velocity. For example, if Ve = −2 V, the required time for MTS to grow from r0 = 50 μm to r = 70 μm is about 30 h, while this time is less than 25 h for Ve = −10 V. However, there is no appreciable difference in the growth rate when the applied voltage is sufficiently high (e.g., above −5 V), as the flow carries sufficient nutrients to the spheroid.
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9460-z/MediaObjects/10544_2010_9460_Fig9_HTML.gif
Fig. 9

Increase in radius of the EMT6/Ro spheroid. r0 = 50 μm. Concentrations in Reservoirs A and B: Cin,o = 0.175 mM and Cin,g = 0.4 mM

3.5 Effect of channel size on tumour growth

In this section, we are interested in studying the effect of channel size on tumour growth. Two cross sections of the channel P-O: 200 μm × 200 μm and 400 μm × 400 μm, are considered here. Similar to the previous section, the nutrient concentrations at reservoirs A and B are Co,in = 0.175 mM and Cg,in = 0.4 mM For the two different channels, the growth of the EMT6/Ro tumour spheroid is calculated under these conditions. The initial radius of the tumour is r0 = μm. Figure 10 depicts the increases in radius of the tumour with three different voltages (−2 V, −5 V and −10 V). It is clear that the MTS grow faster in the large channel than the small channel provided other operational conditions are the same. For example in the large channel it takes 23.4 h for EMT6/Ro tumour spheroid to grow from r0 = 50 μm to r0 = 70 μm while this time is 24.5 h in the small channel. Because here we consider the low nutrient concentration in the bulk liquid; if the applied voltage or the flow velocity is low, a smaller channel cross-section implies limited nutrients available around the spheroid and hence affects the spheroid growth. However, the growth rate is less sensitive to the channel size for higher voltages or higher flow velocity. In addition, as the MTS diameter increases, the growth rate becomes more sensitive to the channel size.
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9460-z/MediaObjects/10544_2010_9460_Fig10_HTML.gif
Fig. 10

Effect of the voltage (flow velocity) on tumour growth. (a) Ve = −2 V, (b) Ve = −5 V (c) Ve = −10 V

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 Co,in = 0.175 mM and Cg,in = 0.4 mM. The MTS tumour with the initial radius r0 = 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.

Figure 11 depicts the glucose concentration distributions at the spheroid surface. The concentration data are sampled on the central plane (the side view of plane s-s, see Fig. 1) from the upstream to the downstream. Similar figures can be plotted for oxygen and other nutrient components. The front-surface of the tumour has a higher average concentration than its back-surface. Figure 12 illustrates the front- and back surface growth (red line) and the average growth (blue line) of the tumour (the side view of the profile at the plane s-s, as indicated on Fig. 1). From these figures, it is obvious that there is a deviation between the front- and back-surface and the average growth of the tumour. Especially for low voltages (low flow velocities), the front part of the tumour grows faster than the back part. From these figures, it can conclude that in the cases of low nutrient concentrations one cannot assume that tumour growth is uniform and remains spherical.
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9460-z/MediaObjects/10544_2010_9460_Fig11_HTML.gif
Fig. 11

Nutrients concentration distributions around the spheroid surface. The concentration data are sampled on the central plane (Plane s-s, see Fig. 1) from the upstream to the downstream

https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9460-z/MediaObjects/10544_2010_9460_Fig12_HTML.gif
Fig. 12

Comparison of the tumour’s front- and back-face growth (blue line) with average growth (red line) at different velocities (voltages) from side view

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.

Copyright information

© Springer Science+Business Media, LLC 2010