Modeling analysis of ant system with multiple tasks and its application to spatially adjacent cell state estimate

Abstract

The problem of multi-cell tracking plays an important role in studying dynamic cell cycle behaviors. In this paper, a novel ant system with multiple tasks is modeled for jointly estimating the number of cells and individual states in cell image sequences. In our ant system, in addition to pure cooperative mechanism used in traditional ant colony optimization algorithm, we model and investigate another two types of ant working modes, namely, dual competitive mode and interactive mode with cooperation and competition to evaluate the tracking performance on spatially adjacent cells. For adjacent ant colonies, dual competitive mode encourages ant colonies with different tasks to work independently, whereas the interactive mode introduces a trade-off between cooperation and competition. In simulations of real cell image sequences, the multi-tasking ant system integrated with interactive mode yielded better tracking results than systems adopting pure cooperation or dual competition alone, both of which cause tracking failures by under-estimating and over-estimating the number of cells, respectively. Furthermore, the results suggest that our algorithm can automatically and accurately track numerous cells in various scenarios, and is competitive with state-of-the-art multi-cell tracking methods.

Appendix

Proof

The function q(t) in Eq. (9) can be rewritten as

$$ q(t) = P\bigl[R + q(t - 1)\bigr] $$

(14)

$$ \tau (t + 1) = \rho \tau (t) + q(t) + R $$

(15)

Take a constant λ, let λ−Pλ=PR, then we have

$$ \lambda = \frac{P}{1 - P}R $$

(16)

Then, the Eq. (14) is reduced to

$$ q(t) - \lambda = P\bigl[q(t - 1) - \lambda \bigr] $$

(17)

Due to q(0)=q ₀, then

$$ q(t) = \lambda + (q_{0} - \lambda )P^{t} $$

(18)

Substituting Eq. (18) into Eq. (15), we have

$$ \tau (t + 1) = \rho \tau (t) + AP^{t} + B $$

(19)

where A=q ₀−λ,B=R+λ are constants, let ν is constant

$$ \nu = \frac{B}{1 - \rho} $$

(20)

The Eq. (18) can be rewritten as

$$ \frac{\tau (t + 1) - \nu}{P^{t}} = \frac{\rho}{P} \cdot \frac{\tau (t) - \nu}{P^{t - 1}} + A $$

(21)

Case 1: If ρ≠P, let κ is constant

$$ \kappa = \frac{A}{1 - \frac{\rho}{P}} $$

(22)

Then Eq. (21) can be rewritten as

$$ \frac{\tau (t + 1) - \nu}{P^{t}} - \kappa = \frac{\rho}{P}\biggl( \frac{\tau (t) - \nu}{P^{t - 1}} - \kappa \biggr) $$

(23)

The above form can be rewritten as

$$ \frac{\tau (t) - \nu}{P^{t - 1}} - \kappa = C\biggl(\frac{\rho}{P} \biggr)^{t} $$

(24)

where C is constant, due to τ ₀=τ(0), then C=P(τ ₀−ν)−κ, so we have

$$ \tau (t) = \nu + \kappa P^{t - 1} + \frac{C}{P} \rho^{t} $$

(25)

Due to ρ,P∈(0,1), when t>0, we have

$$ \bigl\vert \tau (t) \bigr\vert \le \vert \nu \vert + \biggl\vert \frac{\kappa}{P} \biggr\vert + \biggl\vert \frac{C}{P} \biggr\vert $$

(26)

So, we say,τ(t) is bounded

Case 2: If ρ=E, Then Eq. (21) can be rewritten as

$$ \frac{\tau (t + 1) - \nu}{P^{t}} = \frac{\tau (t) - \nu}{P^{t - 1}} + A $$

(27)

The above form can be rewritten as

$$ \frac{\tau (t) - \nu}{P^{t - 1}} = At + C $$

(28)

where C is constant, then

$$ \tau (t) = \nu + (At + C)P^{t - 1} $$

(29)

Due to P∈(0,1), we say τ(t) is bounded.