Mesenchymal Stem Cell Heterogeneity and Ageing In Vitro: A Model Approach - Computational Modeling in Tissue Engineering - page 190

6 The Case of Stable, Inherited Heterogeneity: MSC Ageing

In order to define age dependent properties of the MSC populations we extended
our former approach and assumed that stem cell states become de-stabilised with
'cellular age' resulting in an increased tendency for spontaneous differentiation.
We model this scenario assuming that the noise amplitude of a cell r(a) depends
on its generation number m. Accordingly, each individual cell is characterised by
its noise amplitude, which depends on the cell's differentiation state a and age m:

r ð a ; m Þ¼ r 0 ½ 1 a f ð E Þ þ mr D ½ 1 2a 0 : ð 5 Þ

The first term on the right hand side of Eq. 5 defines the extrinsic, environmentally
determined noise amplitude (see Eq. 2 ). Here, r 0 denotes the noise amplitude for
initial stem cells, i.e. r(a = 0, m = 0). The second term on the right hand side
defines the effect of ageing, which is most obvious considering the noise ampli-
tudes in stem cell states r(a = 0):

r ð a ¼ 0 Þ¼ r 0 þ mr D : ð 6a Þ

This stem cell noise amplitude increases with each generation by the rate r D .
This assumption allows us to quantify the 'age' of a cell by its stem cell noise
amplitude r(a = 0). The noise amplitude of differentiated cells is given by:

r ð a ¼ 1 Þ¼ r 0 ½ 1 f ð E Þ mr D [ 0 ð 6b Þ

This amplitude decreases with m. For m RE = r 0 [1 - f(E)]/r D it is equal to zero.
In order to ensure that r(a = 1) [ 0, we assume that r becomes independent of m
for m [ m RE .
According to these assumptions the heterogeneity of a population can be
described by the probability distribution to find a cell of age m and in state a.
During expansion the average number of cell divisions grows, thus noise ampli-
tudes in differentiated states decrease and, consequently, cells accumulate in these
states. This effect is independent of the environment. It is determined by the ageing
rate r D . This rate determines also the age m RE of replicative senescence. Assuming
r D *2.5 9 10 -3 one obtains an upper limit of m RE of about 60. This can be seen
as an upper bound to the experimental findings so far (\30 at 20% pO 2 ,[ 95 ] and
\40 at 3% pO 2 ,[ 30 ]).
In simulations of the model we assumed for the environmental term of the noise
amplitude:

f ð E Þ¼ 2 ð 1 r E = r 0 Þ; ð 7 Þ

where r E is the mean noise amplitude defined by the environment.
Figure 8 shows results on the simulated age-structure and population hetero-
geneity applying the age model and assuming r e = r 0 = 0.075. After 20 days of