Integrating multi-source information has recently shown promising performance in predicting Alzheimer's disease (AD). Multiple kernel learning (MKL) plays an important role in this regard by learning the combination weights of a set of base kernels via the principle of margin maximisation. The latest research on MKL further incorporates the radius of minimum enclosing ball (MEB) of training data to improve the kernel learning performance. However, we observe that directly applying these radius-incorporated MKL algorithms to AD prediction tasks does not necessarily improve, and sometimes even deteriorate, the prediction accuracy. In this paper, we propose an improved radius-incorporated MKL algorithm for AD prediction. First, we redesign the objective function by approximating the radius of MEB with its upper bound, a linear function of the kernel weights. This approximation makes the resulting optimisation problem convex and globally solvable. Second, instead of using cross-validation, we model the regularisation parameter C of the SVM classifier as an extra kernel weight and automatically tune it in MKL. Third, we theoretically show that our algorithm can be reformulated into a similar form of the SimpleMKL algorithm and conveniently solved by the off-the-shelf packages. We discuss the factors that contribute to the improved performance and apply our algorithm to discriminate different clinic groups from the benchmark ADNI data set. As experimentally demonstrated, our algorithm can better utilise the radius information and achieve higher prediction accuracy than the comparable MKL methods in the literature. In addition, our algorithm demonstrates the highest computational efficiency among all the comparable methods.