We present a search for a new particle T′ decaying to top quark via T′→t+X, where X is an invisible particle. In a data sample with 4.8  fb-1 of integrated luminosity collected by the CDF II detector at Fermilab in pp̅ collisions with √s=1.96  TeV, we search for pair production of T′ in the lepton+jets channel, pp̅ →tt̅ +X+X→ℓνbqq′b+X+X. We interpret our results primarily in terms of a model where T′ are exotic fourth generation quarks and X are dark matter particles. Current direct and indirect bounds on such exotic quarks restrict their masses to be between 300 and 600  GeV/c2, the dark matter particle mass being anywhere below mT′. The data are consistent with standard model expectations, and we set 95% confidence level limits on the generic production of T′T̅ ′→tt̅ +X+X. For the dark matter model we exclude T′ at 95% confidence level up to mT′=360  GeV/c2 for mX≤100  GeV/c2.