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As blood donor numbers decrease, while demand for platelets increases, hospitals worldwide are becoming increasingly vulnerable to critical platelet shortages. Alternative methods of supplying platelets are therefore required. One approach is to engineer platelets
To engineer functional cell products, bioreactors enable precise control of the operating conditions, with the aim of mimicking
Our focus here is on the fluid dynamics of a bioreactor system used to harvest platelets from megakaryocytes seeded onto a biomaterial scaffold, as an alternative to platelets obtained from donors. Platelet transfusions are often used during surgical procedures, as well as for patients with haematological diseases or undergoing chemotherapy or stem cell treatment for cancer (
In both the processes of proplatelet elongation and platelet production, fluid shear stress is important and the focus of ongoing experimental research. For example, it has been shown
To mimic the local mechanical environment of megakaryocytes
Bioreactors for platelets are relatively novel in that their product are delicate cells that are washed out of the system, rather than a tissue growing on a construct. The critical challenges in the operation of platelet bioreactors are to control flow to ensure the megakaryocytes experience a sufficient level of shear stress, while keeping the product concentration as high as possible, so as to minimise the amount of expensive postprocedure concentration of suspended platelets that has to occur. Additionally, excessive backflow in the direction of outlet to inlet should be avoided, to reduce the risk of transporting cells into the wrong parts of the bioreactor.
Our focus is an
The bioreactor has a collagen scaffold with graduated porosity, to retain the megakaryocyte cells [16.4 − 22.4
To address this question, we employ mathematical modelling. For examples, of mathematical and computational modelling of bioreactors, see
The bioreactor system is characterised by disparate lengthscales enabling the identification of small parameters, e.g., the ratio of tube diameter to length. While the presence of small parameters makes a purely computational approach to solving the full system to governing equations challenging, they can be exploited in a systematic asymptotic analysis to derive simpler reduced models that retain the key physical mechanisms while being tractable and amenable to numerical methods. The primary aim of this work is to use such a reduced mathematical modelling approach to determine valve dynamics that address the challenges mentioned above. The mathematical model presented in this paper may also be viewed as a framework for any bioreactor build from long thin tubes, porous scaffolds, and pinch valves; thus may readily be adapted to similar bioreactors, such as those used by
We use the NavierStokes equations and Darcy’s law, which have both been widely used in bioreactor modelling, see, for example, (
Darcy’s law is appropriate for a rigid scaffold in which the pores are connected, and therefore good for a simplestfirst modelling approach. To couple flow in the channels and scaffold, appropriate interface conditions must be specified at the interface. We impose continuity of normal stress, normal flux, and the BeaversJoseph slip condition. This last condition relates the shear stress of the channel flow to the discontinuity in slip velocities of the two regions, through a slip coefficient that depends on the geometric properties of the interface and the permeability of the scaffold. Since its initial empirical justification, this condition has been analytically justified for laminar flow in a channel, which is close to our physical setup (
The paper is structured as follows. We first detail the geometry and operation of the bioreactor in
A schematic of the perfusion bioreactor setup is shown in
Schematic of the valvecontrolled, gravity driven platelet bioreactor designed by
Throughout the bioreactor, materials with low platelet reactivity are used to reduce occurrence of platelets being preemptively activated: polycarbonate for the channel and scaffold walls, and medical grade polyvinyl chloride for the tubing.
To seed the bioreactor scaffold with megakaryocytes,
We model the cell culture media as a Newtonian viscous fluid, with viscosity
In this paper, we present a simple model that captures the flow in the numerous different regions of the bioreactor system. In particular, we assume that the aspect ratio and flow rate are everywhere small enough that we can adopt the lubrication approximation, and thus we neglect inertia in the system. We note that for junctions in which there are sharp changes in crosssectional area, the lubrication approximation breaks down. However, as discussed at the end of
The (equal) centreline arclengths of the upper and lower tube systems are denoted
The upper and lower tube systems have four regions differing in their cross section, shown in
Simplified modelling domain, showing crosssectional shapes of each of the valve tubing, valve, resistor tubing, channel, and scaffold regions.
We denote by
The elliptical crosssections of the valves have semimajor axes
The velocities of the walls in the bioreactor are
The scaffold has centreline arclength,
The crosssections of the upper tube system, lower tube system, and scaffold are labelled as Ω_{
u
}, Ω_{
ℓ
} and Ω_{
s
}, respectively, with boundaries
Typical values for
Model parameters, chosen to mimic the bioreactor of
Name  Symbol  Bioreactor value(s)  Units  Source 

upper reservoir height 

0−3 (0.5)  m  – 
lower reservoir height 

0−0.5 (0.25)  m  – 
scaffold height 

0.004  m 

tube system length 

1.4  m  – 
scaffold length 

0.01  m 

valve tubing length 

0.5  m  – 
valve length 

0.005  m  – 
resistor tubing length 

0.15  m  – 
channel length 

0.09  m  – 
scaffold halfwidth 

7.5 × 10^{–4}  m 

valve tubing radius 

0.001  m  – 
resistor tubing radius 

2 × 10^{–4}  m  – 
channel halfheight 

0.0025  m  – 
channel halfwidth 

7.5 × 10^{–4}  m  – 
valve timescale 

10^{–5} − 10  s  – 
interface slip coefficient 

1.3–7 (4)  − 

cell culture viscosity 

0.001  Pa s 

scaffold permeability 

10^{–15} − 10^{–9} (10^{–11})  m^{2} 

scaffold tortuosity 

1.49−1.99 (1.74)  − 

scaffold porosity 

0.88–0.91 (0.9)  − 

average pore diameter 

4 × 10^{–5} − 8 × 10^{–5} (6 × 10^{–5})  m 

lab atmospheric pressure 

1.025 × 10^{5}  Pa  – 
cell culture density 

1,000  kg/m^{3} 

gravitational acceleration 

10  m/s^{2}  – 
We consider flow of a viscous Newtonian fluid of viscosity
Here,
At the solid walls of the tube systems we impose the usual noslip and noflux conditions given by
We prescribe normal stress and no tangential velocity at the inlet and outlet of the upper and lower tube systems,
At the interfaces Γ_{
u
} and Γ_{
ℓ
} between scaffold and channels, we prescribe continuity of normal flux, continuity of normal stress, and the BeaversJosephSaffman condition:
We exploit the slender geometry of the bioreactor to derive systematically a reduced model, where we consider the lubrication regime in which the aspect ratio and reduced Reynolds number are small in each tubing section. For the purposes of nondimensionalisation, we introduce the aspect ratio
The reduced pressures are nondimensionalised with respect to the pressure head across the upper tube system. We choose the velocity scaling
The governing
The scale for the reduced pressure is the same as in the tube systems, motivated by the continuity of stress condition (
In addition to
Dimensionless parameters corresponding to the bioreactor geometry of
Name  Symbol  Formula  Value(s) 

upper reservoir height 


1 
lower reservoir height 


0.5 
scaffold height 


20 
tube system length 


1 
scaffold length 


0.0071 
valve tubing length 


0.36 
valve length 


0.0036 
resistor tubing length 


0.11 
channel length 


0.064 
scaffold halfdepth 


7.5 
valve tubing radius 


5 
resistor tubing radius 


1 
channel halfbreadth 


12.5 
channel halfdepth 


3.25 
reduced Reynolds number 


2.6 × 10^{–4} 
dimensionless slip coefficient 


1.8 × 10^{–7} 
dimensionless permeability 


20−2.0 × 10^{7} 
Strouhal number 


1–80 
aspect ratio 


1.4 × 10^{–4} 
fraction of valves left open for numerical purposes 

−  0.0005 
Dimensionless coordinates, separating valve tubing, valve, resistor tubing, channel, and scaffold regions, computed from dimensional values are
Coordinate 











Value  0  0.357  0.361  0.468  0.496  0.504  0.532  0.639  0.643  1 
The reduced Reynolds number,
We work in the physically relevant regime in which the reduced Reynolds number
We reduce the system to a network of pressure—crosssectional area—flux relations at leading order as follows. Integrating the continuity
Here, the crosssectional area, axial flux and net flux out of the adjoining scaffold are given by
The pressureflux relations in the upper and lower tube systems are determined as follows. The transverse components of the momentum
To determine
We solve (
Substitution of Darcy’s law into the continuity equation in (
We solve (
We solve Poisson’s
Network diagram of the reduced bioreactor system. At the edges we impose the reduced system (
To determine the scaffold pressure, and hence the net flux out of the scaffold into the adjoining channel
The dimensionless crosssectional areas (
The crosssectional area along the centreline, when the valves are closed, is plotted in
To summarise, the pressure—crosssectional area—flux relations are given by
Using the parameters given in
We analytically solve the system of second order differential equations for the velocities and pressures in each of the 18 regions, with each pair of velocities and pressures having two degrees of freedom. By imposing the conditions at the inlets, outlets, and junctions between regions, as described above, we obtain a system of 36 linear algebraic equations relating the 36 degrees of freedom. These are readily solved numerically in
Within the bioreactor, the aim is to ensure the MKs are subject to sufficient levels of shear stress to ensure effective platelet production, while the shear stress stays below the level at which platelet activation and aggregation occurs. Although the precise values of these bounds are unknown,
We estimate the shear stress distributions from predictions of the Darcy velocity through the porous scaffold following a model that has previously been used in the literature to relate flux to shear stress in porous scaffolds that are modelled using Darcy’s law
We idealise the pores as cylinders of diameter
We consider various norms of the stress as part of our analysis, and define the following shear stress metrics:
In addition to shear stress metrics, we also report the following fluxes:
The fluid flow generated within the bioreactor fulfils three functions: to provide the crossflow through the scaffold that exerts shear stress on megakaryocytes; to wash platelets out of the system to be collected; and to supply nutrients to the scaffold and remove waste products. These three functions can be promoted by choosing which valves are open or shut. To understand how the valves influence each of these aspects, we first consider valves in different combinations of static configurations: fully open, partially open, or shut. For static valves, we first use the parameters of
We compute the average scaffold shear stress,
Trends in fluxes and shear stress for static valves.
In
We compute the average shear stress
For scaffold permeabilities between 10^{–15} m^{2} and 10^{–12} m^{2}, opening the lower inlet valve, in addition to the upper inlet and lower outlet valves, significantly decreases scaffold shear stress, as shown by the blue dashed line in
The average scaffold shear stress is also affected by varying the tube system length
After platelets break off from their parent cell, they should be washed out of the system quickly, so as to minimise risk of damage or activation. This requires sufficient flux down the scaffold and out of the lower outlet. In
To allow diffusive nutrient transport, where there is a relatively high flux along the upper channel, while avoiding excessive collection volume at the lower outlet (i.e., dilute product), the configuration with the upper valves open and lower valves closed can be used. As seen in
To move between static configurations, valves must be opened and closed. This should be done by optimising some combination of shear stress, nutrient supply, and platelet collection, and without inducing backflow in the scaffold, upper channel, or lower channel.
In
Dimensionless fluxes
Although we impose the valve area to vary sinusoidally in time, as shown in
Broadly,
When the valves move on a slow timescale with
The above principles seen from moving individual valves can be applied to determine what valve combinations should be used in transitioning between static configurations. There is freedom to choose the ordering of valve movements, and the extent to which different valves move in sync with each other, by choosing when each valve starts opening or closing. We first discuss valve ordering for fast valves, and then examine slow valves.
To illustrate valve ordering for fast valves, suppose we wish to maximise shear stress in the scaffold when transitioning from the configuration with only upper valves open (promoting diffusive nutrient transport) to the configuration with only lower valves open (promoting platelet collection). Then, to take advantage of the enhanced scaffold fluxes shown in
If the aim is relatively simple, then the principles explained above may be sufficient to pinpoint an appropriate set of valve configurations. More generally, when moving between any two static configurations, we may wish to optimise some function of
Suppose we move from the static configuration promoting shear stress, advective nutrient transport, and platelet collection, to the static configuration promotive diffusive nutrient transport, as shown in the upper right panel of
In moving from the lower outlet valve closing first (
In
We have so far studied advantages and disadvantages from varying the synchronisation of valves when the valves are moving quickly
Finally, we examine how valve speed affects shear stress, illustrated in
We have constructed a lubrication model of the gravitydriven, valvecontrolled platelet bioreactor of
We exploit the small aspect ratio throughout the bioreactor, assuming that crosssectional lengthscales are small compared to axial lengthscales. We consider the regime in which the following two parameters are order one: the dimensionless permeability, measuring the relative velocities in the scaffold and tubing systems, and the Strouhal number, measuring the velocity of fluid driven by the pressure head, relative to the valve wall velocity. We further consider the following two parameters to be small: the reduced Reynolds number, measuring the ratio of inertial to viscous effects, and the dimensionless interfaceslip coefficient, which depends on the permeability and geometry of the scaffold material. This means that we can employ the lubrication approximation for the NavierStokes equations, which we solve subject to no slip and continuity of normal velocity and pressure at the scaffoldchannel interfaces, and no slip on the rigid walls. We obtain equations relating the flux through each crosssection of bioreactor tubing to axiallydependent pressure gradients,
The aim of our modelling is to understand how different bioreactor geometries and valve configurations affect fluxes in the bioreactor, and shear stress in the scaffold, and to control valve dynamics so as to optimise these quantities in a computationallyefficient manner.
We have tested different static valve configurations, and measured their effects on fluxes and scaffold shear stresses. Having computed the flux along the upper channel, lower channel, and the scaffold, we have demonstrated that there are some static valve configurations that should be avoided, as they induce significant backflow in the scaffold. Additionally, we have ranked static configurations in order of highest to lowest scaffold shear stress, and shown that the greatest scaffold shear stress arises from an open upper inlet and lower outlet, and closed upper outlet and lower inlet.
Geometric parameters affect shear stress as follows. First, shear stress increases linearly with the upper reservoir height, which provides the driving pressure for the flow; this is reflected in the model
We have investigated the effect of valve movement on the bioreactor flow. When transitioning between two static valve configurations with valves moving slowly (opening and closing over 40 s), fluxes vary monotonically. Therefore, to determine what valve combinations should be used in transitioning between static configurations, it is sufficient to pay attention to any intermediate configurations that are passed through en route (for example, to avoid passing through configurations that induce high backflow).
In contrast, when opening and closing valves more quickly (say over 0.5 s), timedependent effects are more significant. When a valve is closing, flow away from it is enhanced, and when it is opening, flow towards it is enhanced. To avoid huge spikes in backflow, and take advantage of large spikes in scaffold shear stress, nutrient transport, or platelet collection, our model can be used to sweep through dynamic valve configurations that transition between static valve configurations. We have illustrated two examples of such transitions, and the model can in future be used to quickly obtain the exact configuration of valve movement that optimises some given function of tubing and scaffold fluxes, and scaffold shear stress.
The work thus far suggests further extensions. A restriction that may quite readily be lifted is the scaffold being long and thin. In the bioreactor of
We have exploited a reduced mathematical model, retaining the key physics, to characterise the flow conditions in a valvecontrolled bioreactor for platelet production. We have demonstrated how design parameters and operating conditions affect the bioreactor fluxes and shear stresses experienced by cells. This model can be used to aid experimentalists to mimic the
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.
SW, JO, DH, CG, SB, and RC initiated the project. HS, SW, and JO constructed the model, in discussion with DH, CG, SB, and RC. HS, SW, and JO wrote the manuscript. All authors reviewed the manuscript.
This publication is based on work supported by the EPSRC project grant 2100109.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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