  Carbon ion beam radiotherapy enables a very localised dose deposition.
However, already small changes in the patient geometry or positioning errors
can significantly distort the dose distribution. A live monitoring system of
the beam delivery within the patient is therefore highly desirable and could
improve patient treatment. We present a novel three-dimensional imaging method
of the beam in the irradiated object, exploiting the measured tracks of single
secondary ions emerging under irradiation. The secondary particle tracks are
detected with a TimePix stack, a set of parallel pixelated semiconductor
detectors. We developed a three-dimensional reconstruction algorithm based on
maximum likelihood expectation maximisation. We demonstrate the applicability
of the new method in an irradiation of a cylindrical PMMA phantom of human head
size with a carbon ion pencil beam of 226MeV/u. The beam image in the phantom
is reconstructed from a set of 9 discrete detector positions between -80 and 50
degrees from the beam axis. Furthermore, we demonstrate the potential to
visualise inhomogeneities by irradiating a PMMA phantom with an air gap as well
as bone and adipose tissue surrogate inserts. We successfully reconstructed a
3D image of the treatment beam in the phantom from single secondary ion tracks.
The beam image corresponds well to the distribution expected from the beam
direction and energy. In addition, cylindrical inhomogeneities with a diameter
of 2.85cm and density differences down to 0.3g/cm$^3$ to the surrounding
material are clearly visualised. This novel 3D method to image a therapeutic
carbon ion beam in the irradiated object does not interfere with the treatment
and requires knowledge only of single secondary ion tracks. Even with detectors
with only a small angular coverage, the 3D reconstruction of the fragmentation
points presented in this work was found to be feasible.
