Cartilage Tissue Engineering Calculator

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For reference of full discription of the model, please see the paper Zhou, S., Z. Cui, et al. (2008). "Nutrient gradients in engineered cartilage: metabolic kinetics measurement and mass transfer modeling." Biotechnol Bioeng 101(2): 408-21.

Model development

Physical Model and  Simplifications

It is important to identify nutrient limitation and consider this in the design of tissue engineering and advise appropriate protocols of culture conditions. The concentrations of different nutrients, metabolites and growth factors are usually low and the interactions between different species can be safely neglected and hence the transport of each species can be analysed independently.  The transient distribution of nutrients and metabolites within a three dimensional engineered cartilage can then be mathematically described using perfusion-diffusion-reaction model, i.e.

                                  ...........(1)

where  i = 1,2, … N with N the number of concerned substances (nutrients, metabolites, growth factors ect),  and C is the concentration, V the perfusion velocity vector,  D  is the effective diffusivity within the gowing tissue depending on scaffold structure and degradation, cell density, deposition of extracullar matrix. g is the consumption rate or production rate of i per unit volume.  And

                                     g i = f [C1, C2, ……, CN,  ρ(x,y,z), P(t), T, etc].........(2)

with ρ(x,y,z)  is the cell density distribution, T is the temperature, and P(t) is the hydrostatic pressure time profile.  Eqation (2) will introduce non-linearity and inter-coupling of variables.  With determined boundary condition (e.g. convection of the medium, and perfusion rate if any) and initial conditions (scaffold features, cell seeding density distribution), the partial differential equation group (Equation 1) can be solved numerically, the results giving the transient distributions of individual concentrations.  Post-processing of these data can indicate whether anywhere in the construct the concentration of any key nutrients becomes critical.

    In practical applications, Equation (1) and Equation (2) can be simplified by analysing the operation conditions, for example,  (i) many constructs can be treated as one-dimensional either in Cartesian coordinate or polar coordinate, (ii)  cell proliferation and matrix deposition are slow process, and often the transient feature can be simulated with a series of  pseudo-steady state simulations, (iii) the perfusion term does not exist in common culture process where nutrients simply diffuse into the construct at the boundary.  Perfusion through the construct is sometimes used to overcome these gradients, but there is conflicting data on whether this is useful or not  (Obradovic et al. 2000).

    Here we have considered engineered cartilage as a porous construct with homogeneous and uniform properties.  Chondrocytes are uniformly distributed within the porous structure and the cells consume glucose, oxygen and produce lactic acid and extracellular proteins.  There is no convective fluid flow within the construct and all mass transport within the construct is due to diffusion.

    The gradients modeled arise from the interaction of rates of nutrient or metabolite transport and consumption/production rates.  The magnitude of the gradients depends on the properties of construct (size, porosity and hence effective solute diffusivity), cell density and metabolic kinetics of the cultured cells, and boundary conditions which depend on culture conditions.  In the model it is necessary to make some simplifications: (i) One-dimensional models were used for calculation of gradients across engineered constructs.  This is valid when the construct is flat and its length and width are much greater than its height.  Experimental measurements by Malda et al. show that the assumption of one dimensional transport holds even for a short cylindrical construct (4 mm in diameter and 5 mm height); they found oxygen gradients along axis did not differ markedly from those at different radial positions (centre, 1 mm to centre and 1.5 mm to centre)  (Malda et al. 2004a).  (ii) The effect of cells on the overall diffusion coefficient of the construct was neglected since the volume fraction of cells is very low (for cell density of 10 million cells/ml, volume fraction of cells is about 1%).  (iii) The mass transfer resistance outside the cartilage construct in perfused and suspended culture is neglected.   Although a stagnant layer always exists near the surface of the construct and the estimation of this convection mass transfer coefficient is possible using empirical correlations or computational fluid dynamics (CFD), this is deemed unnecessary as the mass diffusional resistance within the constructs outweighs the convective resistance.  Most previous modeling studies for cartilage tissue engineering have also adopted a similar approach   (Malda et al. 2004a; Obradovic et al. 2000; Pisu et al. 2004; Sengers et al. 2005).  (iv) The size of the scaffold and diffusivity will change due to the deposition of matrix.  We did not predict the change of the size and diffusivity with time.  However, we did consider how change of diffusivity can influence cell viability in selected instances.

 

Governing equations

Common tissue culture methods were divided into static culture which is shown in Figure 1a, and medium-mixed culture, and the latter was further divided into perfusion (medium is supplied continuously) and suspended culture (medium is changed batch-wise) as shown in Figure 1b and 1c.

Figure 1.  Sketch of different culture conditions.  (a) Static culture, the medium is static and changed batch-wise;  (b) Perfusion culture, the medium is continuously perfused along the surface of the construct;  (c) Suspended culture.  The construct is suspended in the medium.  The medium is well stirred and changed batch-wise.

 

(a) Static culture  The typical case for the static culture condition shown in Figure 1a is a layer of hydrogel containing cells  on the bottom of the culture plate covered by a certain amount of medium, or a  cell seeded flat polymer scaffold covered by medium.  In this condition, the medium was assumed to be stationary.  Gradients of nutrients in the medium could not then be neglected.  Also, diffusion of nutrients and wastes through the construct and medium is not at steady-state but varies with time.  Change of concentration of nutrients and wastes in the medium can be described as:

                  ..............................(3)

where C is concentration (mM), t is time (hr), Daqu is diffusion coefficient in aqueous solution (mm2/hr), X is diffusion distance.  Subscript i represents glucose, lactate or oxygen.

    In the construct, since cells consume nutrients and produce wastes, Equation (3) becomes

                   .........................(4)

where g  is consumption/production rate of nutrients and wastes, r is cell density.

    Explicit central finite difference scheme was used to solve Equation (3) and (4) numerically with a ΔX = 0.1 mm and Δt=1.6 s.   These step sizes were chosen considering convergence, calculation time and possible accumulation of calculation errors.

                    ............................(5)

                ..........................(6)

    Usually engineered constructs are cultured under 21% oxygen.  Oxygen is thus less likely to be the key factor limiting cell growth and activity in comparison to native cartilage.  For the boundary conditions, here oxygen tension at the surface of the medium was assumed to be constant at 21%.  Gradients of glucose and lactate at the medium-air interface were taken as  as no glucose or lactate leaves the medium.  Gradients of glucose, lactate and oxygen at the bottom of the construct i.e. where it is in contact with the impermeable surface were also taken as .  The initial conditions were simply defined as at t = 0, the concentrations of nutrients in the constructs and in the medium are those in the fresh medium, and lactic acid concentration is set to be zero. 

(b) Perfusion culture   In this system medium is perfused along the surface of the construct parallel to it.  If the construct is perfused from both sides, then the diffusion distance is a half of the thickness.  Here perfusion rate was assumed large enough so that concentrations of glucose, lactate and oxygen in the medium would not change significantly.  Thus the concentrations of glucose, lactate and oxygen at the surface of the construct were set to be the same as in the medium (e.g., 5.56 mM, 0 mM and 21%).  Gradients of glucose, lactate and oxygen at the bottom of the construct (or centre, if perfused for both sides) are zero.  This condition could thus be modeled as a pseudo-steady state system and described by:

                                 ............................................(7)

and                          ............................ (8)

    Diffusion coefficients were also based on 1.2% alginate gel and was 84% of those in aqueous solution.

 

(c) Suspended culture   In this system, the construct is suspended in a defined volume of medium in a bioreactor.  The medium is mixed by direct stirring or by rotating the bioreactor.  In this model, medium was assumed to be well mixed and homogenous.  However, it is different from the steady state as described in (b) since the medium is changed batch-wise and the concentrations of glucose and lactate will change continuously between medium changes.  The finite difference approach was also used here.  Change of concentrations of glucose, lactate and oxygen was described by Equation (4) and (6).  Since the construct is supplied from both sides, the diffusion distance was taken as half of its thickness.  Boundary conditions used were the same as in perfusion culture.  That is, gradients in the centre are zero and concentrations at the surface of the construct were equal to those in the medium .   However, unlike situation in the perfusion culture,  changed with time as described by

               .........................(9)

where  is concentration of glucose or lactate in the medium.   is time difference.  AA is the area of construct surface.  Di is diffusion coefficient of glucose or lactate in the construct.   is gradient of glucose or lactate at the surface of the construct, V is the volume of the medium.  The initial conditions are similar to those in static culture.

    The oxygen tension in the medium was regarded as constant and set to be 21%.  Any deformation of the construct possibly caused by movement of the medium was also neglected.

    A hydrogel is not likely to be used as scaffold in a rotating or stirred bioreactor due to its unsuitable mechanical properties.  Scaffolds used are usually made from polymers such as poly lactic acid (PLA) and poly glycolic acid (PGA).    In the modeling work by Obradovic et al. (2000), diffusion coefficients in PGA comparable to those in native cartilage (50% of the value in water) were used, so  in this work.

 

Simplication for website

The programs for the models were programmed with Matlab.  They can not be run in the website.  The calculators in this website were based on Javascript.  It can not show the live gradients.  Thus the lowest concentrations of glucose or oxygen were picked, since what concerns us is how low the nutrients can fall in the deep zone of the construct.  The relationship between the lowest concentration, cell density, construct dimension and diffusivity was fitted into approximated functions.  Once glucose concentration drops to critical level, we will consider the cell viability may be compromised.  Based on this, we can get a relationship between maximum viable cell density, scaffold dimension and diffusivity.  This relationship was also fitted into approximated functions.

Static culture  

Typically, when 5 ml medium is added to a well of a 6-well plate (3.5 mm in diameter), the height of the medium is 5.2 mm.  For this condition, profiles of glucose, lactate, oxygen and pH across the medium and construct throughout a 48-hour static culture  are shown in Figure 2.  The time difference between lines in the figure is 2 hours (This applies to all other figures).  The construct thickness was 2 mm and cell density was 4 million cells/ml.  It can be seen that in general the gradients are very flat.  For most of time, glucose concentration remained well above 1 mM.  However after 36 hours, concentrations in the construct fell to below 1mM and to about 0.1 mM at the bottom of the construct at the end of the culture period.  Oxygen tension reached a steady level by 4 hours culture but began to fall towards the end of the culture period.  This is due to the increase of oxygen consumption induced by low glucose.  The oxygen level remained above 15% throughout the culture.  Lactate concentration increased to 11 mM and pH dropped to 7.0 at the bottom of the construct after 48 hours.

Figure 2.  Profiles of (a)glucose, (b)oxygen, (c)lactate and (d)pH across the medium and the construct during a 48-hour static culture.  It was assumed that 5 ml medium was put in 3.5 mm culture plate, resulting a medium depth of 5.2 mm.  The construct was assumed to be 1.2 % alginate gel of 2 mm thick.  Cell density was 4 million/ml.  The red line represents the interface between the medium and the construct.  


 

    However, if the cell density increased to 20 million, the situation was very different.  The glucose concentration in the deep zone of the construct dropped to zero in about 10 hours.  Oxygen tension fell to about 2% over the same period, as shown in Figure 3.  Lactate concentration rose to about 11 mM and pH dropped to around 7.0 (Not shown).   With 5.56 mM glucose in the medium, lactate concentration could only reach a maximum of 11 mM and pH thus could only fall to about pH 7.0 even when glucose in the medium was completely consumed. 

   

Figure 3.  Profiles of (a) glucose and (b) oxygen across the medium and the construct during a 48-hour static culture.  It was assumed that 5 ml medium was put in 3.5 mm culture plate, resulting a medium depth of 5.2 mm.  The construct was assumed to be 1.2 % alginate gel of 2 mm thick.  Cell density was 20 million/ml.  The red line represents the interface between the medium and the construct.

 

    One may argue that the fall in glucose could arise because there is insufficient medium.  However, the fall in oxygen tension reveals that supply of nutrients under these conditions is diffusion-limited since oxygen supply was assumed to be abundant at the surface of medium.   We can further analyze this by assuming that the construct is covered by an infinite amount of medium.  This condition was simulated by keeping concentrations of the medium constant at some position far from the construct surface.  Here it was assumed that in the medium which is 50 mm from the surface of the construct, glucose concentration was maintained at 5.56 mM, lactate concentration was zero and oxygen tension was 21% (this will overestimate glucose and oxygen concentration and underestimate lactate concentration in the construct slightly).  The results are shown in Figure 4.  Glucose and oxygen dropped to nearly zero in about 12 hours.  This indicates that for a certain thickness of construct, the cell density that can be supported by static culture is limited, no matter how much medium is used.

   

Figure 4.  Profiles of (a) glucose and (b) oxygen across the medium and the construct during 48 hours of static culture, assuming glucose, oxygen and lactate concentrations were constant at 50 mm from the surface of the construct.  Construct thickness was 2 mm and cell density was 20 million/ml.  The red line represents the interface between the medium and the construct.

 

    We can not run the program on the webpage, so it needs some simplification.   For each construct thickness, the higher the cell density, the lower the glucose and oxygen concentration. From the calculations shown in Figs 2-4, glucose appears the limiting nutrient. Figure 5a shows the relationship between the lowest glucose concentration and cell density in a construct of 2 mm thick.  The lowest glucose concentration was reached after 48 hours culture even if the construct is covered by vast amount of medium.  Chondrocytes begin to die when glucose concentrations falls to 0.2-0.3 mM  (Zhou 2005).  Here we use 0.3 mM as the critical concentration of glucose.  From Figure 5a, it can be estimated that the maximum cell density can be supported by static culture for 2 mm thick construct fed from one side is about 8.3 million cells/ml.  At higher cell densities, glucose concentrations will fall to below critical levels, initially in the construct depth.

A similar analysis for different thicknesses of construct provides the relationship of construct thickness and the maximum cell density that can be supported and results are shown in Figure 5b.  As the thickness of the construct increases, the maximum cell density that can be supported decreases.  This relationship was found to fit the exponential decay equation well:  

              (R2=0.993)……..(10)

Where rmax is the maximum cell density that can be supported and h is the thickness of the construct.  This equation provides the basic criteria for design of static culture.

Figure 5.  Limitation of glucose supply and cell density in static culture.  (a) Relationship between lowest glucose concentration in the centre of a construct and its cell density.  The construct was assumed to be 2 mm thick and covered by vast amount of medium.  Glucose concentration in the medium was 5.56 mM and pH was 7.4.  (b) Relationship between the thickness of a construct and the maximum cell density that can be supported in static culture, assuming the construct is immersed in vast amount of medium with glucose concentration of 5.56 mM and pH 7.4.

Using medium with higher glucose concentration will certainly relieve the condition of low glucose in the construct.  For the condition in Figure 4, if the glucose concentration in the medium was increased to 25 mM, results shows that after 48 hours, the lowest glucose concentration in the 2-mm construct was well above critical levels at 10 mM.  However, in this case, the pH in the construct fell to pH6.4, well below the physiological range and will adversely affect metabolism or cause cell death .  This indicates that for static culture, cell viability cannot be maintained purely by adding more glucose.  Factors such as oxygen, pH, thickness and cell density are all inter-related and must all be considered.

The gradients were calculated based on the initial diffusivities, which are 84% of those in free solutions.  However, when more and more extracellular matrix is deposited, the diffusivities will decrease (diffusivities in cartilage are about 40% of those in free solutions  (Maroudas et al. 1968)).  This study did not predict the change of diffusivities.  However, we evaluated how change of diffusivities can affect nutrient and metabolite concentrations in the construct.  To give an example, a construct of 2 mm thick and a cell density of 8 million/ml was assumed to be immersed in vast amount of medium containing 5.56 mM gluose. The lowest glucose concentrations (in the deep zone) were calculated according to different diffusivities.  As shown in Figure 7, the glucose concentration was above critical level of 0.3 mM when initial diffusivities were used.  But when diffusivities decreased, the lowest glucose concentration could drop to below the critical level and thus threatening cell viability.

 

Figure 6.  Effect of diffusivity on the lowest glucose concentration in a construct with 8 million cell/ml.  The construct was assumed to be 2 mm thick and covered by vast amount of medium.  Glucose concentration in the medium was 5.56 mM and pH was 7.4.

 

 

Perfusion culture  

Diffusion and consumption of nutrients were regarded as steady-state.  Concentrations of glucose and lactate and oxygen tension at the surface were regarded as constant at 5.56 mM, 0 and 21%, respectively.  Compared with static culture, this culturing system can support much higher cell density. 

(1) Lowest glucose concentration

The relationship between the lowest glucose concentration and cell density can be fitted into a polynomial function well.  For instance, for a construct of 2 mm thick, and the diffusivities are 0.84 of those in aqueous solution, the relationship between the lowest glucose concentration and cell density could be described as

………………….(11)

as shown in Figure 7.

Figure 7. Relationship between the lowest glucose concentration and cell density in a perfused construct.  Thickness 2 mm.  Relative diffusivity 0.84.

 

For all dimensions, the relationship between the lowest glucose concentration and cell density can always be described by a polynomial function in the form of

………………….(12)

The constants a and b are dependent on dimension of the construct (thickness h).  By calculating a and b according to different h, the relationships between a-h and b-h can be determined.  a and b can be described as

…………….(13)

……………..(14)

with R2=1, as shown in Figure 8. 

Figure 8. Relationship between a, b value and thickness of the construct.  Diffusivities are 0.84 of those in aqueous solution.  

 

The constants aa and bb are dependent on the diffusivities.  Here relative diffusivity Dr is used, that is, the diffusivity as compared to that in aqueous solution.  For example, the relative diffusivity is 0.84 in alginate gel, and is about 0.4 in native cartilage.  The relationship between aa-Dr and bb-Dr can be described as

…………………(15)

…………………(16)

as shown in Figure 9

Figure 9. Relationship between aa, bb value and relative diffusivity  

 

Thus, the relationship between the lowest glucose concentration in a scaffold glumin and cell density r, thickness h, and relative diffusivity Dr can be described as

……………(17)

This is an approximation of the model.  In order to examine how closely it describes the model, 100 random sets of (r, h, Dr) were tested in the model and Equation (17).  The average difference between them was 0.024 mM, or 4%.

 

(2) Maximum viable cell density

The critical glucose concentration is 0.3 mM.  From Figure 7, we can see that in order to maintain glucose above the critical level, the maximum cell density rmax that can be supported is about 80 million/ml (this is for thickness 2 mm).  Doing the same calculation for thickness 1, 1.5, 2, 2.5, 3, 3.5 and 4 mm, we can get a relationship between rmax and thickness h.  This can be expressed as

Figure 10.  Relationship between the maximum viable cell density and thickness of the construct.  Relative diffusivity is 0.84.

 

 The constant n is dependent on diffusivities.  Doing the same calculation for relative diffusivities 0.4, 0.5, 0.6, 0.7 and 0.84, we can get a relationship between n and relative diffusivity Dr

as shown in Figure 11.

Figure 11.  Relationship between n and Dr.

 

Thus we can get an approximation of the model that can be expressed as

 

 

Suspended culture  

Similar to static and perfusion culture, a relationship between rmax and h can be obtained for suspended culture by analyzing the lowest glucose or pH occurring in the construct.  In this culture system, the relationship is affected by the ratio of medium volume to construct volume (j)

    If medium of low glucose concentration (5.56 mM) is used, pH is not likely to be the limiting factor in regard to cell viability since even all the glucose is converted into lactate, pH will not drop to a very low level.  If the culture medium is saturated with air, neither will oxygen be the limiting factor.  In this case glucose becomes the limiting factor.  Here we took the ratio of j=(volume of medium)/(volume of construct) as an important parameter and estimated the lowest concentration of glucose reached for different thickness of construct and cell density.  Medium was assumed to be replaced every 48 hours (100% v/v).  Critical glucose concentration was assumed to be 0.3 mM.  Note that the thickness here is from surface of construct to the centre, that is, half of the whole thickness of the construct.  Figure 12a shows the minimum j required to maintain glucose above the critical level for different thickness and cell density.  The glucose concentration in the construct is very sensitive to the diffusion distance.  When the thickness is 1 mm, the limitation on cell density and j is relatively loose.  However, when the thickness increases, it becomes more and more difficult to maintain glucose above ‘safe’ level.  For example, when the thickness is 2.5 mm and the cell density is 35, a j  of 10000 is required, which is unrealistic.

    If medium with high glucose (e.g. 25 mM) is used, then glucose will less likely to be a problem.  Rather, pH is of more concern.  The thickness of construct, cell density and j should be designed to maintain pH in the construct above a critical level.  As found previously, cell viability began to be compromised when pH dropped below pH6.8 (Zhou 2005).   This is used as the critical pH level here.  The relationships between construct thickness, cell density and minimum j is shown in Figure 12b.   The medium was assumed to contain 25 mM HEPES and 44 mM NaHCO3.