In the past few years, proton therapy has taken the centre stage in treating 
various tumour types. The primary contribution of this study is to investigate 
the tumour control probability (TCP), relapse time and the corresponding 
secondary cancer risks induced by proton beam radiation therapy. We incorporate 
tumour relapse kinetics into the TCP framework and calculate the associated 
second cancer risks. To calculate proton therapy-induced secondary cancer 
induction, we used the well-known biologically motivated mathematical model, 
initiation-inactivation-proliferation formalism. We used the available in vitro 
data for the linear energy transfer (LET) dependence of cell killing and 
mutation induction parameters. We evaluated the TCP and radiation-induced second 
cancer risks for protons in the clinical range of LETs, i.e. approximately 8 
$\mathrm{keV/\mu m}$ for the tumour volume and 1-3 $\mathrm{keV/\mu m}$ for the 
organs at risk. This study may serve as a framework for further work in this 
field and elucidates proton-induced TCP and the associated secondary cancer 
risks, not previously reported in the literature. Although studies with a 
greater number of cell lines would reduce uncertainties within the model 
parameters, we argue that the theoretical framework presented within is a 
sufficient rationale to assess proton radiation TCP, relapse and carcinogenic 
effects in various treatment plans. We show that compared with photon therapy, 
proton therapy markedly reduces the risk of secondary malignancies and for 
equivalent dosing regimens achieves better tumour control as well as a reduced 
primary recurrence outcome, especially within a hypo-fractionated regimen.