  Angiogenesis is a key phenomenon for tumour progression, diagnosis and
treatment in brain tumours and more generally in oncology. Presently, its
precise, direct quantitative assessment can only be done on whole tissue
sections immunostained to reveal vascular endothelial cells. But this is a
tremendous task for the pathologist and a challenge for the computer since
digitised whole tissue sections, whole slide images (WSI), contain typically
around ten gigapixels.
  We define and implement an algorithm that determines automatically, on a WSI
at objective magnification $40\times$, the regions of tissue, the regions
without blur and the regions of large puddles of red blood cells, and
constructs the mask of blur-free, significant tissue on the WSI. Then it
calibrates automatically the optical density ratios of the immunostaining of
the vessel walls and of the counterstaining, performs a colour deconvolution
inside the regions of blur-free tissue, and finds the vessel walls inside these
regions by selecting, on the image resulting from the colour deconvolution,
zones which satisfy a double-threshold criterion. A mask of vessel wall regions
on the WSI is produced. The density of microvessels is finally computed as the
fraction of the area of significant tissue which is occupied by vessel walls.
  We apply this algorithm to a set of 186 WSI of paediatric brain tumours from
World Health Organisation grades I to IV. The segmentations are of very good
quality although the set of slides is very heterogeneous. The computation time
is of the order of a fraction of an hour for each WSI on a modest computer. The
computed microvascular density is found to be robust and strongly correlates
with the tumour grade.
  This method requires no training and can easily be applied to other tumour
types and other stainings.
