Remote sensing (RS) of solar-induced chlorophyll fluorescence (SIF) has a great potential for monitoring plant photosynthetic activity. Radiative transfer models (RTM) are essential to better interpret and extract information from SIF signals. DART is one of the most comprehensive and accurate 3D RTMs. Its standard mode DART-FT simulates SIF using a discrete ordinates method but is not adapted to large landscapes due to computational constraints. DART-Lux, the new mode based on a bi-directional path tracing algorithm, greatly improves DART computational efficiency for simulating images. This paper presents the theory of a novel SIF modelling algorithm in DART-Lux. We verified its accuracy with DART-FT and the SCOPE model for three types of canopies: turbid medium, maize field and forest. DART-Lux closely matches DART-FT (relative difference < 2%) with much better computational efficiency depending on the scene complexity, number of spectral bands and needed accuracy. For example, simulation time is reduced by a factor of ≈48, and memory usage by ≈50 for a maize field at 1 cm resolution. It allowed to simulate SIF images of large scenes as the 3 x 3 km$^2$ Ripperdan agricultural site that DART-FT could not simulate. The new SIF modelling algorithm opens new horizons for RS studies of large and complex landscapes. It is available as part of released DART versions (v1152 onwards) (https://dart.omp.eu/).