Theoretical and Experimental Investigation of Alginate Microtube Extrusion for Cell Culture Applications

Abstract

A novel cell culture technology, consisting of hollow alginate tubes, OD ~550 μm, ID ~450μm containing a cell suspension, provides stress-free conditions. Cells reach confluency in approximately ten days with cell densities of 0.5 – 1 billion cells per mL. Tubes are manufactured in a tri-axial needle extruder with three concentric flows. The cell suspension flows in the inner needle (N1), the alginate solution flows in the annulus between N1 and the second needle (N2) and a CaCl₂ solution is the sheath fluid between the second and third needle (N3). Beyond the tip of N2, the sheath solution is in contact with the alginate and Ca²⁺ diffuses into the alginate solution and crosslinks it to form an alginate microtube around the core fluid. The cross-linked layer moves radially inwards like a front, starting at the sheath/annulus interface and ends at the annulus/core interface. A mathematical model is used to find the minimum length z_C of direct contact between the CaCl₂ solution and the alginate solution to complete the cross-linking. Experimental results support the theoretical findings that stable tubes can only be manufactured if the contact length exceeds z_C. Experiments also show that the extruder configuration N3>N2 is best for alginate tube manufacture.

Graphical abstract

1. Introduction

Cell therapy is becoming a powerful method to treat chronic diseases and cell damage from injuries¹. Stem cells, which can be expanded, differentiated, and modified, offer treatments for brain injuries, diabetes, macular degeneration and numerous other afflictions^2,3,4. Cell therapy can be broadly segmented in autologous and allogeneic applications. Autologous cell therapy is attractive because rejection factors are eliminated, but the cost associated with personalized cell therapy is high¹. Genetically engineered chimeric-antigen-receptor T cells, (CAR T cells) have achieved extraordinary success in treating B cell malignancies and have shown high potential for treating solid tumors^5,6,7. Other applications of human pluri-potent stem cells (hPSCs) include their use to test drugs and treatments^8,9, as well as the possibility of growing organs or tissues^10,11,12.

A limiting factor, both in feasibility and from a cost perspective, is the expansion of cells to produce clinically relevant doses, whilst still preserving phenotype and properties and not affecting cell viability, fate, and proliferation negatively. Current technologies are plagued by consistency of product and difficulties associated with scaling up (allogeneic applications) and scaling out (autologous applications). Although 2D culture systems are still used (e.g. mesenchymal stem cells) for certain applications, significant effort has gone into improving consistencies of 3-D culture methods. However, 3D reactors have their own problems. The most common 3D culture systems involve cells suspended in a growth medium that is constantly stirred or shaken. Growth medium, which is perfused through the reactor to provide nutrition and remove metabolites, is stirred to create more homogeneous conditions. Cells which form clusters are prone to cell death if the clusters become too large¹³ – one practice is to use stirring to control cluster size. In combination, these practices expose cells to varying hydrodynamic conditions. Shear and dilatational stresses affect cells in a variety of ways, but oftentimes these mechanical interactions with cells could cause phenotypical changes and/or apoptosis. This problem does not only present itself in scale-up applications, but scale-out applications face similar problems, due to the difficulties in replicating hydrodynamic conditions.

Methods to expand cells in an environment where they are shielded from fluctuations in hydrodynamic stresses have led to the development of hollow fiber reactors^14,15,16. Nutrients are pumped through a thin capillary with a permeable wall. Cells are initially seeded on the exterior wall where they continue to grow as a cluster. Alternatively, one can take the opposite approach and place the cells inside a capillary tube with permeable walls and the growth medium on the outside^17-23. This alternative approach has the advantages that cluster formation is limited to the inner diameter (ID) of the alginate tubes (AlgTubes) and the cells are shielded from hydrodynamic stresses. The tube diameter can be matched with nutrient transport to assure that cells at the center of the tube still meet their nutritional and oxygen needs. This approach was pioneered by Takai and co-workers¹⁷ with the objective to prepare materials for tissue engineering. Later^18,19 tubes of several meters long were manufactured, also with tissue engineering applications in mind.

The use of AlgTubes for cell manufacture, including expansion and modification, was developed by Lei²³. Some examples of cell manufacture in AlgTubes by his group are briefly discussed. Lin³⁶ suspended T cells in AlgTubes, the tubes were placed in a perfusion culture vessel and cells were expanded over a period of two weeks. The shielding of cells from shear stress and other hydrodynamic perturbations led to high cell viability, low DNA damage, high growth rate (≈320-fold expansion over 14 days), high purity (≈98% CD3+), and high yield (≈3.2 × 10⁸ cells mL⁻¹ hydrogel.) In another study²⁰, glioblastoma tumor-initiating cells (TICs) were expanded in AlgTubes for the purpose of developing target-specific drugs. The cost-effective production of large numbers of high-quality glioblastoma TICs for drug discovery with current cell culturing technologies remains very challenging. The culture of glioblastoma TICs in AlgTubes, allowed long-term culturing (~50 days) of glioblastoma TICs with high growth rate (~700-fold expansion), high cell viability and high volumetric yield (~3.0 × 10⁸ cells/mL) without losing the stem cell properties. A final example³⁷ was the manufacture of human pluripotent stem cells. hPSCs enjoy great interest due to their importance in cell therapies and tissue engineering. Cells were suspended in AlgTubes and placed in a culture vessel. The culture medium was exchanged daily. The cells were expanded 50 fold after 9 days and over 94% of the cells still expressed the pluripotency marker Oct4.

The cell density achieved in the AlgTubes vary from one cell type to the other, but typically, cell densities of 300 million to one billion cells/mL can be reached in the tubes^22,23. A direct consequence of the high cell density is that bioreactors can be made much smaller than conventional stirred tank reactors and still produce the same number of cells. An important advantage of the AlgTube technology is that it lends itself to scale-up and scale-out applications. When the size of a bioreactor is increased, one does not need to be concerned with replicating the hydrodynamic conditions in the growth medium, since conditions outside the AlgTubes do not affect the cells. Turning the focus to autologous applications, the extrusion of AlgTubes into a small vessel and expanding the cells to reach confluency, can lead to the development of personalized bioreactors (PBRs), which will advance this segment of cell therapy. The logical next step is to combine the AlgTube technology with a small vessel that can supply all nutrition and oxygen needs for the cells. If the PBR with an integrated extruder can be operated as a closed system, then it holds a big advantage from a sterility viewpoint. Cell harvesting is also straightforward²²; by adding t-EDTA to the PBR at the end of the culture period, the alginate is depolymerized. The PBR is placed in a centrifuge, and cells collect at the bottom from where they are readily retrieved. A closed system requires that the AlgTubes are extruded into the PBR. In this study some key aspects of extruding AlgTubes directly into the PBR are analyzed.

Li²² used a biaxial extruder to manufacture AlgTubes. The core solution and the alginate solution (annulus) flow into a container with a 100 mM CaCl₂ solution. The cross-linking occurs in the container (i.e. the pool). The contents of the beaker were gently stirred with a magnetic stirrer to sweep the concentric fluids away from the extruder tip, since the biaxial extruder had the propensity to form clogs at the tip. When the extrusion was complete, the AlgTubes were transferred to a culture vessel. Details on the manufacture of the biaxial extruder and AlgTube manufacture can be found in Li²². The use of a biaxial extruder in a PBR to achieve a closed system is not practical. The pool must be swirled, which is impractical for extrusion into a closed PBR, and if clogs form, they must be manually removed before extrusion can resume, and this is also not possible in the closed PBR. Due to the clogging problems (cf. Figures 1a), the bi-axial extruder was not considered in this analysis. The triaxial extruder has proven to operate without forming clogs, therefore the analysis is focused on the two options for the triaxial extruder.

Figure 1:

(a) Bi-axial needle extruder, the tip is immersed in CaCl₂ solution. (b) Triaxial Needle extruder, flow as in Option 2, Fig. 1(c).

The stability of primary laminar flow, consisting of axisymmetric fluids that flow concentrically in a straight tube, has been the topic of many investigations^27,28,29,35. One of the first analyses was presented by Hickox²⁷. Both asymmetric and axisymmetric perturbations were considered, and solutions were sought in the form of a perturbation series, using the wavenumber as the expansion parameter; this approach limited the analysis to long wavelength perturbations. There always existed an unstable solution to either the asymmetric or the axisymmetric perturbations, regardless of the Reynolds number. The most important parameter is the difference in viscosity between the two fluids. Joseph^28,35 presented a stability analysis of the same concentric flow problem. The primary concentric laminar flow is more stable if the viscosity of the core fluid is larger than the annulus fluid, but only to a point; if the volume fraction of the annulus fluid increases, the primary flow becomes unstable³⁵. One aspect of the linear stability analyses that is specifically important to AlgTube manufacture is to assure that the viscosity of the core fluid is larger than that of the alginate solution, therefore the core solution is constitutively modified by adding methyl cellulose, to slightly exceed the viscosity of the alginate solution.

The key question we want to address in this analysis is to determine the contact length z_C between the sheath flow and alginate flow to assure that cross-linking is completed. The analysis focuses on the diffusion of Ca²⁺ ions from the sheath solution into the alginate solution and the formation of the cross-linked membrane. The problem is closely related to the entry length Graetz problem^24,25,26. The Ca²⁺ ions rapidly cross-link the sodium alginate. Initially we used a bi-axial extruder (core and annulus flows) and positioned the tip of the extruder in a CaCl₂ solution. This approach caused problems with clogging, seen in Figure 1(a). (The core flow was colored blue; the annulus flow was colored yellow and the resulting alginate tube appears green.) The triaxial extruder circumvents the clogging problem, because the CaCl₂ solution is also flowing and the AlgTubes are swept away from the extruder tip, Figure 1(b).

2. Materials and Methods

2.1. Mathematical Model

In Figure 2(a), a schematic of the triaxial extruder is shown. The triaxial extruder comprises of three concentric tubes. The inner tube separates the core fluid from the annulus fluid. Where this tube ends, the annulus fluid is layered directly around the core fluid in a stable concentric flow configuration. The middle tube separates the annulus fluid from the sheath fluid. Downstream from the tip of the middle tube, the cross-linking of the alginate occurs. The outer tube envelopes the sheath fluid.

Figure 2:

(a) Cross-sectional schematic of custom Ramé-Hart triaxial needle extruder, (b) Needle termination schematic of Option 1, (c) Needle termination schematic of Option 2. z_C is the minimum distance of contact between the CaCl₂ solution and the alginate solution to assure complete cross-linking. In the case of option 1, it is also the minimum distance between the needle tip and the pool surface, in option 2 it is the difference in length of N3 and N2.

A custom triaxial needle (Rame-Hart Instrument Co.) serves as the extruder. One or more of these needles are placed in the PBR lid and multiple tubes can be extruded in parallel. Referring to Figure 2(a), the core flow (red) is the cell suspension. The annulus flow (blue) is a 1% wt. sodium alginate solution, and the sheath solution (green) is a 100 mM solution of CaCl₂. The sodium alginate is crosslinked by the divalent Ca²⁺ions, and a hydrogel casing is formed around the cell suspension. The flow rates of the core and annulus solutions are controlled with Fluigent MicroFluidics pumps; their ratio determines the ratio of tube wall thickness to tube diameter. The sheath solution is delivered with a peristaltic pump. The extrusion process takes less than 30 minutes, afterwards the PBR content is replaced with growth medium. Growth medium is perfused through the PBR and a micro sparger (Mott Corp.) provides oxygen as required; the sparger also has the secondary function of keeping the AlgTubes (which have close to neutral buoyancy) suspended in the growth medium. At the end of the culture period, a solution of t-EDTA is introduced to the PBR and the AlgTubes are dissolved. The PBR has a conical bottom and when the contents are centrifuged, the cells settle in the bottom of the PBR from where they can be retrieved.

There are two options to configure the extruder tip. Option 1, seen in Figure 2(b), is to keep the outer tube the same length as the middle tube, the concentric flow is stress-free beyond this point. All three fluids have the same constant velocity. The stand-off distance, z_C, between the tip and the liquid surface in the PBR must be long enough to ensure the tube is completely cross-linked upon entering the pool. Option 2 (see Figure 2(c)), has an outer tube that is longer than the middle tube (length N₃ > N₂). Here z_C is the difference in length of N3 and N2 that will lead to complete cross-linking before the AlgTube enters the pool. The concentric flow is contained in the outer tube, like Hagen-Poiseuille flow, but with the difference that cross-linking is occurring which changes the parabolic velocity profile as shown in Figure 5. The triaxial extruder, shown in Figure 1(b) is an example of option 2 in operation.

Figure 5:

Solutions of eqns.(9) and (23). The alginate region lies between s₁ = 0.4 and s₂ = 0.6.

In the case of option 1, the problem reduces to an eigenvalue problem with constant coefficients and the solution can be sought in the form of Bessel functions. The solution converges slowly at the extruder tip; therefore, we also present a Leveque approximation which gives a reasonable solution at and close to the tip. The second option presents a more challenging problem. The eigenvalue problem has variable coefficients, because of the laminar flow profile. We present two approaches to solve this problem and show that both provide eigenvalues which are close to the numerical results.

The mathematical model analyses the cross-linking reaction in the concentric flows consisting of the core solution/alginate solution/ CaCl₂ solution; the alginate solution, which forms an annulus around the core solution and the CaCl₂ solution, which forms an annulus around the alginate solution. This configuration is enabled by the triaxial extruder. The fluid properties of the three coaxial flows are assumed to be similar. The sheath fluid (R₂ ≤ r ≤ R₃) is the CaCl₂ solution, the annulus fluid (R₁ ≤ r ≤ R₂) is the sodium alginate solution and the core fluid is the cell suspension (0 ≤ r ≤ R₁). Use the outer radius R₃ to scale radial and axial positions. The Ca²⁺ concentration is scaled with the initial concentration C₀ of the CaCl₂ solution. The diffusion coefficient D is assumed to be the same for all the fluids. This is a reasonable assumption, even when the alginate has become cross-linked.

Set the origin of the axial position (z = 0) at the tip of the middle tube. This marks the point where Ca²⁺ starts to diffuse into the alginate layer. In the case of option 1, all three fluids flow at the same axial velocity, V_z = V₁ when z ≥ 0. For option 2, the axial velocity has a profile as shown in Figure 4, let V₂ denote the maximum velocity in this case. The Peclet number for this model is

$$P{e}_m=\frac{V_{1,2}{R}_3}{D}.$$ (1)

Figure 4:

The axial velocity for option 2, with s₂ = 0.5 and r_F = 0.2. The alginate is crosslinked in the region r_F ≤ r ≤ s₂.

For option 1, the governing equations for the Ca²⁺ concentration in the sheath and the core are the same, namely

$$\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial C}{\partial r}\right)=\frac{V_1{R}_3}{D}\frac{\partial C}{\partial z}=P{e}_m\frac{\partial C}{\partial z}$$ (2a)

A cross-linking reaction occurs in the alginate and its governing equation should account for the reaction term.

For the second option, the governing equations in the sheath and unreacted alginate solution have variable coefficients, due to the parabolic laminar flow profile

$$\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial C}{\partial r}\right)=\frac{V_2(1-r^2)R_3}{D}\frac{\partial C}{\partial z}=P{e}_m(1-r^2)\frac{\partial C}{\partial z}$$ (2b)

The dimensionless radius of the annulus is s₂ and the core radius is s₁. At z = 0, the concentration is:

$$C(r,0)=\{\begin{array}{cc}1,\hfill & s_2\le r\le 1\hfill \\ 0,\hfill & 0\le r\le s_2\hfill \end{array}\phantom{\}}$$ (3)

The boundary conditions are

$$\frac{\partial C}{\partial r}=0,r=0,1.$$ (4a,b)

The cross-linking reaction that occurs in the alginate solution is known to be fast. For that reason, we model it in a similar way to a solid fuel combustion reaction^30,31,32, where the combustion rate depends on the thermal conduction at the front. Thence, the cross-linked state progresses as a sharp front from r = s₂, inwards to r = s₁. The rate at which this front travels, depends on the rate at which Ca²⁺ diffuses into the front. The finite element model of the cross-linking problem confirms that the reaction progresses as a steep front³³ (see their Figure 4). Let S denote the (initial) dimensionless concentration of cross-linking sites. The mobility of these sites, associated with the bulky polymers, is much smaller than that of the Ca²⁺. Therefore, the sites will be stationary on the time scale that the front moves. Again, note the similarity to solid fuel combustion; the concentration of crosslinking sites is S ahead of the front and zero behind the front. The dimensionless rate at which the front moves is

$$\frac{d{r}_F}{dt}=\frac{-1}{S}{\frac{\partial C}{\partial r}}_{r=r_F}$$ (5)

The governing equation for the alginate is defined by eqns. (2a, b) for s₂ > r > r_F. The concentration (Ca²⁺) is zero ahead of the reaction front, i.e. r < r_F.

Once the front reaches s₁, the reaction ceases, and the concentration dynamics only depends on diffusion. First consider the stress-free flow of option 1.

Stress-free flow (option 1)

Solving eq. (2a) leads to a Sturm-Liouville problem, and the eigenfunctions of the SL problem are

$$u_n(r)=J_0({\beta }_{n}r)$$

To satisfy eq. (4b), β_n are the roots of J₁(β_n) = 0. The analytical solution for the Ca²⁺ concentration is

$$C(r,z)=C_s+\sum _{n=1}{A}_n{J}_0({\beta }_{n}r)e^{-{\beta }_{n∕P{e}_m}^{2}z}.$$ (6)

The (far-field, z ≫ 0) steady state is C_s,

$$C_s=1-(s_2^2-s_1^2)-S$$ (7)

Note that C_s does not depend on r. If S > 1 – ($S>1-(s_2^2-s_1^2)$), then there is insufficient Ca²⁺ to cross-link all the sites and C_s = 0.

The coefficients of solution (6) are obtained by substituting the solution (6) into (3), applying the orthogonality property of SL eigenfunctions.

$$A_n=\frac{-2s_2}{{\beta }_n}{J}_1({\beta }_n{s}_2)∕[J_0({\beta }_n{)}^2]$$

One can use this solution to calculate the flux of Ca²⁺ that enters the inner shell s₁ ≤ r ≤ s₂. We assume that the cross-linking kinetics is fast, thus the polymerized alginate wall moves inwards as a sharp front. Rewrite eq. (5) in terms of axial displacement, instead of time. The kinematic equation that describes the radial position of the front as a function of the axial distance is

$$\frac{d{r}_F}{dz}=\frac{1}{P{e}_{m}S}\sum _{n=1}{A}_n{\beta }_n{J}_1({\beta }_n{r}_F)e^{-{\beta }_n^2∕P{e}_{m}z}$$ (8)

Integrating the kinematic equation from z = 0, r_F = s₂ to r_F = s₁ at z = z_C, we can determine the length of the stress-free flow. If we consider only the first term of the series, then we will obtain a simpler expression that will also provide conservative estimates of z_C (i.e. z_C (one term) > z_C (multiple terms)). Note that the first term β₁ = 3.8317 decays slowest with z.

$$\frac{d{r}_F}{dz}=\frac{A_1{\beta }_1}{P{e}_{m}S}{J}_1({\beta }_1{r}_F)e^{-\frac{{\beta }_1^2}{P{e}_m}z}$$ (9)

Complete cross-linking will be reached at a critical (minimum) length

$$z_C=\frac{P{e}_m{R}_3}{{\beta }_1^2}\text{ln}\left[1+\frac{{\beta }_1^{2}S}{A_1}{\int }_{s_1}^{s_2}\frac{d{r}_F}{J_1({\beta }_1{r}_F)}\right]$$ (10)

The following observations can be made; the critical length increases linearly with axial velocity, and logarithmically with a higher density of cross-linking sites.

In the following section we consider option 2.

Tube-constrained flow (option 2)

Problem (2b) with inlet condition (3) and boundary conditions (4a, b) is just a variation on the classic Graetz problem. The solution has the form,

$$C(r,z)=\sum _{n=1}{B}_n{W}_n(r)e^{-{\beta }_{n∕P{e}_m}^{2}z}$$ (11)

The eigenvalue problem that results upon substituting this form into eq. (2b) is,

$$\frac{1}{r}\frac{d}{dr}\left(r\frac{d{W}_n}{dr}\right)=-{\beta }_n^2(1-r^2)W_n$$ (12)

with boundary conditions

$$\frac{d{W}_n}{dr}=0,r=0,1$$ (13a,b)

The variable coefficient adds to the complexity, but eq. (12) can be transformed into Kummer’s equation which can be solved analytically. Define

$$v=\beta r^2$$

and

$$\theta (v)e^{-v∕2}=W(r).$$

Eq. (12) is rewritten as

$$v\frac{d^2\theta }{d{v}^2}+(1-v)\frac{d\theta }{dv}-\left(\frac{1}{2}-\frac{\beta }{4}\right)\theta =0.$$

Let $\frac{1}{2}-\frac{\beta }{4}=\alpha$. The general solution is

$$\theta (v)=1+\alpha v+\frac{\alpha (\alpha +1)}{2{!}^2}{v}^2+\frac{\alpha (\alpha +1)(\alpha +2)}{3{!}^2}{v}^3+\dots$$

Eq. (12b) is satisfied when

$$\theta (\beta ∕2)=\frac{1}{2}{\theta }^{\prime }(\beta ∕2)$$

$({\theta }^{\prime }=\frac{d\theta }{dv})$. The first root is

$${\beta }_1=5.06$$

This result is in close agreement with the numerical result of Siegel²⁶. Figure 3 shows plots of the eigenfunctions W₁(β₁r) and J₀(β₁r), associated with the first eigenvalues 5.06 and 3.83 of options 2 and 1 respectively.

Figure 3:

The eigenfunctions for the cases of stress-free flow (eq. (2a)) and confined flow (eq.(2b)).

The computation of the double factorial and large exponents of Kummer’s solution is a source of numerical errors. We propose an alternative method to solve eq. (12). The rationale is to start with the solution of the freeflow eigenvalue problem,

$$\frac{1}{r}\frac{d}{dr}\left(r\frac{d{W}_n}{dr}\right)=-{\beta }_n^2{W}_n$$ (14)

which is W = J₀(βr), and multiply it with a power series in r. Naturally we must add a power series with J₁(βr) in the expansion since it is generated by the Laplace operator.

The alternative solution takes the form

$$W=J_0(\beta r)(1+a_2{r}^2+a_4{r}^4+...)+J_1(\beta r)(b_{1}r+b_3{r}^3+...)$$ (15)

Substitute (15) into (12), the coefficients can be determined by comparing powers. Using four terms in each power series of (15), the solution is as follows.

Let

$$G=\frac{1}{(72∕(49{\beta }^2)+20∕21)}$$

then the first coefficient is

$$a_2=\frac{{\beta }^2}{(16G+6{\beta }^2)}$$

and all others can be expressed in terms of a₂;

$$a_4=a_{2}G;a_6=\left(\frac{3G}{14}-\frac{{\beta }^2}{12}\right)a_2;b_1=-\frac{2}{\beta }{a}_2;b_3=\beta a_2;b_5=\left(\frac{2}{5}-\frac{27G}{35\beta }\right)a_2;b_7=\frac{\beta G}{14}{a}_2.$$

The first eigenvalue that satisfies the Neuman boundary condition at r = 1, is β₁ = 5.052. This result is close to the numerical value of Siegel²⁶, β₁ = 5.067. A plot of the first eigenfunction W₁(r) is identical to the result shown in Figure 3 (constrained flow). This approach offers an alternative to the confluent hypergeometric series. The second solution that is independent to (15) is

$$V=Y_0(\beta r)(1+a_2{r}^2+a_4{r}^4+...)+Y_1(\beta r)(b_{1}r+b_3{r}^3+...)$$ (16)

The solution of the front propagation (5) is more complicated than for the stress-free case. The alginate behind the reaction front is cross-linked, and the velocity of the cross-linked alginate is the same as the velocity of the sheath fluid at r = s₂. A schematic of the velocity profile is shown in Figure 4.

The Ca²⁺ transport in the sheath is described by eq. (2b), but in the cross-linked region r_F ≤ r ≤ s₂ the concentration is described by eq. (2a). The solutions are limited to the first terms. In the sheath region it is

$$C_{sheath}=\left(AW(r)+BV(r)\right)e^{-{\beta }_{1∕P{e}_m}^{2}z}$$ (17)

And in the cross-linked region it is

$$C_{xlink}=H{J}_0({\beta }_{1}r)e^{-{\beta }_{1∕P{e}_m}^{2}z}$$ (18)

At the interface r = s₂, continuity of the concentrations is imposed.

$$AW(s_2)+BV(s_2)=H{J}_0({\beta }_1{s}_2)$$ (19)

The first derivatives W′ and V′ are calculated using eqns. (15,16). Continuity of flux gives

$$A{W}^{\prime }(s_2)+B{V}^{\prime }(s_2)=-H{\beta }_1{J}_1({\beta }_1{s}_2)$$ (20)

The Neumann condition at r = 1 is

$$A{W}^{\prime }(1)+B{V}^{\prime }(1)=0$$ (21)

The existence condition that guarantees a non-trivial solution of the system (19-21) also constitutes the transcendental equation, and its solutions are the eigenvalues. The interface position s₂ is a parameter of the transcendental equation. To demonstrate, consider three different values s₂ = (0,2; 0.5; 0.8). The first eigenvalues for s₂ = (0,2; 0.5; 0.8) are

$${\beta }_1=(5.07;4.97;4.55)$$

One notes that the eigenvalues decrease as the sheath gets thinner. The first eigenfunction of the sheath region is

$$W_1({\beta }_{1}r)=\left(W({\beta }_{1}r)-\frac{W^{\prime }(1)}{V^{\prime }(1)}V({\beta }_{1}r)\right)$$

By projecting W₁(r) onto the inlet condition (3), the first coefficient B₁ the series approximation of the solution of (2b) can be calculated.

$${\beta }_1=\frac{{\int }_{s_2}^1{W}_1({\beta }_{1}r)r(1-r^2)dr}{{\int }_0^1{W}_1({\beta }_{1}r{)}^{2}r(1-r^2)dr}$$

The concentration in the cross-linked region is

$$C_{xlink}=\frac{B_1\left(W(s_2)-\frac{W^{\prime }(1)}{V^{\prime }(1)}V(s_2)\right)}{J_0({\beta }_1{s}_2)}{J}_0({\beta }_{1}r)e^{-{\beta }_{1∕P{e}_m}^{2}z}$$ (22)

The front propagation is described by

$$\frac{d{r}_F}{dz}=\frac{1}{P{e}_{m}S}\frac{d{C}_{xlink}({\beta }_1{r}_F)}{dr}{e}^{-\frac{{\beta }_1^2}{P{e}_m}z}.$$ (23)

To summarize this section, we have presented two options to operate the extruder. The eigenvalue problem is straightforward for option 1, but it resembles the Graetz problem for option 2. The classic solution of the latter eigenvalue problem involves Kummer’s equation. But we also present an alternative solution which is not computationally demanding. Expressions for the front propagation have also been derived for both cases. The solutions of eqns. (9,23) provide important information on the minimum distance between the extruder tip and the buffer pool into which the alginate tubes are extruded (option 1) and the length of the outer needle (N3) that extends beyond N2 (option 2). Note that in the case of option 2, the outer needle extends into the buffer pool, as shown in Figure 2(c).

2.2. Materials

Reagents and their suppliers: Methylcellulose stock solution (cat # HSC001, R&D system), Sodium Alginate (cat # 194-13321, 80~120cp, Wako Chemicals), Calcium Chloride Dihydrate (cat # 10035-04-8, Thermo Scientific Fairlawn Chemicals).

Masterflex L/S Digital Drive with Multichannel Pump Head for L/S Precision Tubing (cat. #: EW-77921-85, Masterflex), Pulse Dampener (cat. #: HV-07596-20, Masterflex), Fluigent LineUP Flow EZ (cat. #: LU-FEZ-2000, Fluigent), Fluigent Flow Unit (cat. #: FLU-M-D, Fluigent), Fluigent P-Cap Reservoir (cat. #: P-CAP15-HP, Fluigent), Fluigent Low Pressure Generator (cat. #: FLPG005, Fluigent), Fluigent Link (cat. #: LU-LNK-0002, Fluigent), custom Ramé-Hart triaxial needle.

2.3. Experimental Methods

A methylcellulose solution, alginate solution, and 100 mM calcium chloride buffer were flown through the Ramé-Hart triaxial needle into water as the core, shell, and sheath flow, respectively. The methyl cellulose flowrate and alginate flowrate were controlled by PID control through the Fluigent pressure control system. The calcium chloride sheath flowrate was set by the Masterflex peristaltic pump with a flow stabilizer to negate pulsating flow. The tubes were extruded into an aqueous solution, after which they were collected and examined under a microscope. Tubes were extruded with the tip of the triaxial extruder suspended above the surface of the water (option 1) and with the tip of the needle submerged beneath the surface (option 2).

3. Results and Discussion

3.1. Theoretical Results

In this section we will compare the results for the cases of stress-free and constrained flow. Choosing the parameter values Pe_m = 1,000 and S = 0.1, we will vary the radii s₁ and s₂ of the core and annulus fluids. For the first comparison, set the inner and outer radii of the alginate region at s₁ = 0.4 and s₂ = 0.6. The position of the cross-linking front moves inwards from s₂ = 0.6 as the axial distance from the extruder increases. The results are plotted in Figure 5 – keep in mind that the ordinate in these plots, labeled ‘Reaction front’, is the distance from z = 0.

z_C denotes the axial length when cross-linking is complete. The length z_C is longer in the case of free flow (i.e. option 1). To interpret the results in absolute terms, z_C should be multiplied with the length scale R₃. In the following two figures the alginate region lies between s₁ = 0.2 and s₂ = 0.4 (Figure 6) and s₁ = 0.6 and s₂ = 0.8 (Figure 7).

Figure 6:

Axial position of reaction front as a function of cross-linking front position. The alginate region lies between s₁ = 0.2 and s₂ = 0.4.

Figure 7:

Axial position of reaction front as a function of cross-linking front position. The alginate region lies between s₁ = 0.6 and s₂ = 0.8.

The constrained flow exhibits greater variability with position of the alginate region. When the alginate region lies closer to the center (Figure 6), z_C~7 for constrained flow, and it increases to z_C~21 when the region lies closer to the outer boundary. Practically speaking, the thinner the sheath of the CaCl₂ solution, the longer it takes to achieve complete cross-linking.

A shortcoming of the analysis is to limit the series solutions to the first term. This approximation lessens the accuracy near the extruder tip. Traditionally, researchers have turned to the Leveque solution to study behavior near the extruder tip (or channel entrance), but the Leveque solution is based on the assumption that the curvature can be neglected because the boundary layer is still thin²⁵. Consider the following function

$$f(r,t)=\frac{f_0}{\sqrt{t}}{e}^{-\frac{(R-r{)}^2}{4t}}$$

The function satisfies the diffusion equation, provided the curvature is neglected.

When r = R, and t = 0, f(1,0) → ∞. We approximate the initial state by defining $f_0=\sqrt{P{e}_m{z}_0}dR$, and requiring that Pe_mz₀ ≪ 1. The normalization factor is

$$C_0={\int }_{s_2}^{R_{max}}\frac{f_0{e}^{-\frac{(R-r{)}^2}{4P{e}_m{z}_0}}}{\sqrt{P{e}_m{z}_0}}dR$$

Note that the integral extends to R_max > 1. Additional sources are placed outside the domain to satisfy the Neumann boundary condition at r = 1. The calcium concentration is

$$C(r,z)={\int }_{s_2}^{R_{max}}\frac{f_0{e}^{-\frac{(R-r{)}^2}{4P{e}_{m}z}}}{\sqrt{P{e}_{m}z}}dR∕C_0$$

In Figure 8 plots of C(r, z) are shown at three distances from the extruder tip; z = 0.1, z = 1 and z = 10 . If R₃ = 375 μm, then the actual distances from the extruder tip are 37.5 μm, 375 μm and 3.75mm.

Figure 8:

Plots of calcium concentration at three distances from the extruder tip for the case s₂ = 0.5. The plots do not account for the consumption of calcium at r_F, w-here the concentration drops to zero, while cross-linking sites are still converted.

The flux is

$$J=-{\int }_{s_2}^{R_{max}}\frac{(R-r)}{2P{e}_{m}z}\frac{f_0{e}^{-\frac{(R-r{)}^2}{4P{e}_{m}z}}}{\sqrt{P{e}_{m}z}}dR∕C_0$$

The fluxes at axial positions z = 1 and z = 10 are shown in Figure 9, without accounting for the reaction. The actual fluxes would be the same as the plots for r ≥ r_F, but zero for r < r_F when the cross-linking reaction occurs. Suppose r_F = 0.3, when z = 10, then the green arrow in Figure 9 marks the point where the flux will drop to zero.

Figure 9:

Calcium fluxes at two distances from the extruder tip. The sheath flow starts at s₂ = 0.5. The plots do not account for the consumption of calcium at r_F. Ahead of the front the fluxes are zero.

The kinematic equation which uses the Leveque solution, can be used to describe the front dynamics close to the extruder tip;

$$\frac{d{r}_F}{dz}=\frac{J(r_F)}{P{e}_{m}S}$$ (24)

This expression should give more accurate results for small z values, i.e. close to the extruder tip than eq. (9). In Figure 10 the solution to eq. (24) is plotted. It compares quite well with the plot for the freeflow case shown in Figure 5.

Figure 10:

Solutions of eq. (24). The alginate region lies between s₁ = 0.4 and s₂ = 0.6.

As is evident from the mathematical model (Figures 5-7), the distance from initial contact between the core and annulus solution to the point where the annulus solution is fully crosslinked varies based on multiple parameters. The variability based on numerous variables poses an issue for the development of an optimal needle set up. The mathematical model can provide crucial insight for towards the design process. In the following experimental section, the practical implications of these variables are explored.

In the following section we present some experimental results for the two extruder options.

3.2. Experimental Results and Discussion

3.2.1. Option 1

Option 1 corresponds to a setup as shown in Figure 2(b). In Figure 11 a micrograph of the manufactured tube is shown. The inner radius of N3 is R₃ = 375 μm. One infers from Figure 11 that s₂ = 295/375 ≈ 0.8 and s₁ = 230/375 ≈ 0.6. The theoretical results for s₂ = 0.8 and s₁ = 0.6 are shown in Figure 7, the curve labeled as Free Flow. From Figure 7, the theoretical axial distance that is needed for complete cross-linking is z_C = 17, or in real terms 6.4 mm. Two qualitatively different outcomes were observed for Option 1. When the needle was positioned close to the surface of the water, z < z_C, the surface tension of the water formed a bridge with the tip of the needle. Cross-linking starts in this “bridge” zone, but the reaction is only completed in the pool, (see also Supplemental Materials, Option1). We will refer to the needle position z < z_C with the bridge formation as option 1a. To maintain option 1a, the needle needs to be continuously positioned within two to three millimeters from the fluid surface. This requires very precise control, and as the fluid level of the calcium chloride buffer rises, there is the risk that the tip of the needle could become submerged and then the concentric flows could become disrupted and irregular tubes form. We therefore conclude that option 1a is not a viable option.

Figure 11:

Example of a tube produced by Option 1a. The flow ratio for Core:Shell:Sheath flow was 1:1.5:25, which produced a tube with an OD of about 580 μm and a shell thickness of 60 μm.

When the needle was raised further above the liquid surface, which will now be called option 1b, surface tension was unable to form the “bridge”, and droplets began forming. In other words, we were not able to maintain stable concentric free flow of the three axisymmetric fluids at a tip distance z > z_C from the pool, instead the flow broke up into droplets. The lack of a continuous stream caused the tubes to break up, as shown in Figure 12 (see Supplemental Materials Option1). A continuous stream could be maintained by drastically increasing the flow of the CaCl₂ solution, however this would result in pinched tubes with thin alginate walls that aren’t satisfactory for cell culture conditions. Taken together, the experiments showed that option 1 would not produce AlgTubes of the desired qualities consistently.

Figure 12:

Example of a tube produced with option 1b where stringed together droplets were formed. The flow ratio of the core: annulus: sheath is 1:1.5:25. This produced about 1000 μm OD droplets with approximately 50 μm of shell thickness.

3.2.2. Option 2

Option 2 (see Figure 2(c)) was investigated with 2 different lengths of the sheath needles (N3). The difference in length between N3 and N2 defines the length of confined flow where cross-linking occurs. For the short N3, the cross-linking length is SN = 3 mm and for the longer sheath needle, LN = 30 mm (refer to these cross-linking lengths as SN and LN). Both SN and LN produced tubes with smooth walls, an example is shown in Figure 13. Although the outer needle was submerged, the exiting calcium stream did not immediately disperse in the surrounding fluid; instead, it maintained its integrity for a distance (see Supplemental Materials, Option 2) From eq. (23), the critical length was calculated as z_C ≈ 12, (for R₃ = 375 μm, this length is 4.5 mm.) For LN = 30 mm, the cross-linking reaction was completed in the extruder. For SN, the alginate tube had only partially formed in the extruder. Upon entering the pool, the CaCl₂ solution continued to flow concentrically around the alginate tube and cross-linking was completed. But it is desirable to complete cross-linking under the controlled conditions in the extruder. When flow rates increase and less cross-linking occurs in the extruder (case of SN = 3 mm), the concentric flows become more prone to instabilities in the pool. This point is underscored by the results shown in Figure 14.

Figure 13:

Tubes extruded using option 2 using the short needle. The flow ratios for Core: Shell: Sheath were 1:1.5:25. This produced a tube with an outer diameter of about 510 μm and a wall thickness of about 50 μm. The longer needle also produced tubes of the same quality.

Figure 14:

Tubes produced under the same conditions. The flow ratio for Core: Shell: Sheath is 1:1.5:25 with a total flowrate of 1.1 mL/min (a) Used the shorter needle. The oscillations produced an outer diameter that varied between about 540 μm to 620 μm with a wall thickness between 40 μm and 100 μm. (b) The longer needle produced a tube with an outer diameter of about 580 μm and a wall thickness of approximately 60 μm. Smooth walls indicate no instabilities present when tube was extruded.

If the flow rates are increased by 50%, then the residence time in the extruder is shorter and the critical length z_C increases. The AlgTube shown in Figure 14b has an outer diameter of 586 μm, an inner diameter of 452 μm which means s₂ ≈ 0.8 and s₂ ≈ 0.6. Solving eq. (23) for the increased flow rate, the critical length for constrained flow is z_C ≈ 31, which is equivalent to z_CR₃ = 11.63 mm. Since z_CR₃ < 30 mm, the alginate was completely cross-linked in the case of LN and the resulting AlgTube, shown in Figure 14b, exhibits regular, smooth walls. When the short N3 is used, the length SN = 3mm is much less than the required 11.63 mm. Only a small fraction of the alginate was cross-linked in the extruder – even smaller than in the previous case (Figure 13, with slower flow rate). Thus, most of the cross-linking occurred after the flow had exited from the extruder. As shown in Figure 14a, there are observable instabilities in the tubes. The tubes do not have a constant diameter, but they oscillate. The flow is reminiscent of the bamboo flow analyzed by Kouris and Tsamopoulos³⁴. This result proves that smooth tubes can be produced in LN’s over a range of flow rates.

4. Conclusions

The triaxial extruder is ideal to manufacture thin hollow alginate tubes with the center fluid seeded with suspended cells. The alternative manufacturing method is to use a bi-axial extruder where the tip is submerged in a CaCl₂ solution. The latter method suffers from three drawbacks, (i) the extruder tip is prone to clogging, (ii) the extruded cells are exposed to the CaCl₂solution, which places a time limit on the extrusion process and (iii) the alginate tube’s outer diameter cannot be changed unless another extruder is used. Several of these drawbacks are eliminated by the triaxial extruder. The extrusion can be done into a cell-friendly buffer, which dilutes the CaCl₂that enters via the sheath flow and most importantly, extends the total extrusion time. By adjusting the sheath flow, the alginate tubes outer diameter can be changed. Finally, the clogging problem can be avoided by using flow rates which would ensure that the cross-linking is completed before the alginate tubes enter the pool.

The cross-linking reaction is controlled by the radial diffusion of Ca²⁺into the alginate. We modeled it as a propagating front problem and the main results of the analysis are equations (9) and (23). The experiments have shown that Option 1, (for which eq. (9) is relevant) does not provide stable tube manufacturing conditions. Thence, eq. (23) is the useful design equation; its solution produces the critical length z_Cfor a choice of flow rates. z_C can be compared to the length that the CaCl₂solution will be in contact with the alginate inside the extruder; if z_C is less than this length, then stable manufacturing conditions exist.

Two conclusions can be drawn from this study. First, of all three options (1a, 1b, 2), option 2 with LN performs the best. The longer needle (LN) provides the most stable manufacturing condition; the tube is fully cross-linked when it exits the tube, thus its morphology is not affected by the pool and stable tubes can be made for a greater range of flow conditions, compared to SN. The theoretical values support the experimental findings, and it is concluded that eq. (23) is useful for extruder designs.

Highlights.

• Synthesis of alginate microtubes for novel cell culture method

• Critical parameter for complete crosslinking of alginate polymer

• New method to solve variable coefficient eigenvalue problem

• Experimental results to validate conclusions of mathematical model

• Enables manufacture of closed system Personalized Bioreactors (PBRs)