In this paper we develop a directional search-and-capture model of cytoneme-based
morphogenesis. We consider a single cytoneme nucleating from a source cell and searching for a set
of N target cells \Omega k \subset \BbbR d, k = 1, . . . , N, with d \geq 2. We assume that each time the cytoneme
nucleates, it grows in a random direction so that the probability of being oriented toward the kth
target is pk with \sum N
k=1 pk < 1. Hence, there is a nonzero probability of failure to find a target
unless there is some mechanism for returning to the nucleation site and subsequently nucleating in
a new direction. We model the latter as a 1D search process with stochastic resetting, finite return
times, and refractory periods. We use a renewal method to calculate the splitting probabilities and
conditional mean first passage times for the cytoneme to be captured by a given target cell. We then
determine the steady-state accumulation of morphogen over the set of target cells following multiple
rounds of search-and-capture events and morphogen degradation. This then yields the corresponding
morphogen gradient across the set of target cells whose steepness depends on the resetting rate. We
illustrate the theory by considering a single layer of target cells and discuss the extension to multiple
cytonemes.