  It is known that a Bruen chain of the three-dimensional projective space
$\mathrm{PG}(3,q)$ exists for every odd prime power $q$ at most $37$, except
for $q=29$. It was shown by Cardinali et. al (2005) that Bruen chains do not
exist for $41\le q\leq 49$. We develop a model, based on finite fields, which
allows us to extend this result to $41\leqslant q \leqslant 97$, thereby adding
more evidence to the conjecture that Bruen chains do not exist for $q>37$.
Furthermore, we show that Bruen chains can be realised precisely as the
$(q+1)/2$-cliques of a two related, yet distinct, undirected simple graphs.
