Mathematical Models of Adipose Tissue Dynamics - The Mechanobiology of Obesity and Related Diseases - page 13

adipose cell-size distributions at discrete time points can be used to theoretically
examine the dynamic processes (e.g., cell recruitment, growth/shrinkage, and
death) in adipose tissues, because cell-size distributions reflect what adipose cells
experience during their life span. We introduce a Bayesian method to deduce
longitudinal information from changes of the cross-sectional information of
adipose cell-size distributions measured at different times. Bayesian inference has
been applied to understand not only the adipose tissue dynamics [ 16 , 17 ], but also
pancreatic islet development [ 18 ].
Given the mean and uncertainty of measured frequencies, m i and dm i at the ith
bin,

the maximum entropy principle [ 19 ] gives the likelihood of predicted

frequencies n i ð x Þ of a model M associated with a set of parameters x as

exp m i n i ð x Þ 2
2dm i

¼ exp½ E ð x Þ;

P ð D j x ; M Þ/ Y
i

ð 1 Þ

where the mismatch between the measurement and prediction is quantified as a
cost,

m i n i ð x Þ 2
2dm i

E ð x Þ ¼ X
i

: ð 2 Þ

In this chapter, the model M will be a dynamic model that predicts the evolution
of adipose cell-size distributions m i : The set of parameters x represents physical
processes such as recruitment, growth, death rates, and their size dependences.
Bayes' rule (or product rule in probability theory) gives the posterior proba-
bility distribution of the parameter set x ; given data D and model M [ 20 ]:

P ð x j D ; M Þ ¼P ð x j M Þ P ð D j x ; M Þ

P ð x j M Þ exp½ E ð x Þ
R dx P ð x j M Þ exp½ E ð x Þ ;

P ð D j M Þ ¼

ð 3 Þ

where P ð x j M Þ is the prior distribution of x, usually set as constant with the
assumption of complete ignorance. Using the probability P ð x j D ; M Þ , tis
straightforward to compute the mean and uncertainty of parameter x:
x ¼ Z dx xP ð x j D ; M Þ; ð 4 Þ

dx 2 ¼ Z

dx x 2 P ð x j D ; M Þ x 2 :

ð 5 Þ

Monte Carlo (MC) methods are usually used to compute these, because the
update in MC is determined by the probability P ð x j D ; M Þ .
Generally we can propose several hypothetical models to explain the given
data. Bayesian model comparison is particularly appropriate in such a context
because Bayes' rule balances model complexity and goodness of fit. An overly