# A Novel Framework for Parameter and State Estimation of Multicellular Systems Using Gaussian Mixture Approximations

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Modeling of the Multicellular Dynamics

#### 2.2. Observability of Multicellular Systems Dynamics

#### 2.3. Estimator Design

- (I)
- Prediction step, a priori state and error covariance estimates:$$\begin{array}{c}\hfill {\widehat{\mathbf{x}}}_{k}^{-}={F}_{\mathrm{d}is}{\widehat{\mathbf{x}}}_{k-1}^{+}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}{P}_{k}^{-}={F}_{\mathrm{d}is}{P}_{k-1}^{+}{F}_{\mathrm{d}is}^{T}+V\end{array}$$
- (II)
- Computation of the estimator gain:$$\begin{array}{c}\hfill {K}_{k}={P}_{k}^{-}{H}_{\mathrm{d}is}^{T}{\left(\right)}^{{H}_{\mathrm{d}is}}-1\end{array}$$
- (III)
- Correction using current measurement ${\mathbf{y}}_{k}$, posterior estimates:$$\begin{array}{c}\hfill {\widehat{\mathbf{x}}}_{k}^{+}={\widehat{\mathbf{x}}}_{k}^{-}+{K}_{k}\left(\right)open="("\; close=")">{\mathbf{y}}_{k}-{H}_{\mathrm{d}is}{\widehat{\mathbf{x}}}_{k}^{-}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.277778em}{0ex}}{P}_{k}^{+}=\left(\right)open="("\; close=")">I-{K}_{k}{H}_{\mathrm{d}is}& {P}_{k}^{-}\end{array}$$

- (I)
- Initialization: A set of ${N}_{P}$ initial estimates (particles) is drawn from the initial state PDF:$$\begin{array}{c}\hfill {\widehat{\mathbf{x}}}_{0,j}^{+}\sim p\left({\mathbf{x}}_{0}\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}j=1,\dots ,{N}_{P}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$Subsequently, the following steps are executed recursively for $k>0$:
- (II)
- Propagation: Each particle is propagated from ${t}_{k-1}$ to ${t}_{k}$ with Equation (12):$$\begin{array}{c}\hfill {\widehat{\mathbf{x}}}_{k,j}^{-}=\underset{{t}_{k-1}}{\overset{{t}_{k}}{\int}}\mathbf{f}>({\mathbf{x}}_{k-1,j}^{+}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{\mathbf{v}}_{k-1})\mathrm{d}t\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}{\widehat{\mathbf{y}}}_{k,j}^{-}=\mathbf{h}>({\mathbf{x}}_{k,j}^{-}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{\mathbf{w}}_{k-1})\end{array}$$
- (III)
- Weighting: The relative likelihood ${q}_{j}$ of each particle conditioned on the current measurement is computed by evaluating $p>({\mathbf{y}}_{k,j}\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}{\mathbf{x}}_{k,j}^{-})$ using Equation (13). Furthermore, all weights are normalized.$$\begin{array}{c}\hfill {q}_{j}={q}_{j}{\left(\right)}^{\sum _{j=1}^{{N}_{P}}}-1\phantom{\rule{0.166667em}{0ex}}.\end{array}$$
- (IV)
- Regularization and resampling: The posterior state PDF $p>({\mathbf{x}}_{k}\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}{\mathbf{y}}_{k})$ is approximated by a sum of weighted kernel functions, and ${N}_{P}$ a posteriori particles are generated from this PDF.$$\begin{array}{c}\hfill p>({\mathbf{x}}_{k}\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}{\mathbf{y}}_{k})=\sum _{j=1}^{{N}_{P}}{q}_{j}K>({\widehat{\mathbf{x}}}_{k,j}^{+})\phantom{\rule{0.166667em}{0ex}}.\end{array}$$From the posterior PDF, any desired statistical measures, e.g., mean and covariance, can be determined.

## 3. Results and Discussion

#### 3.1. Comments on the Computational Implementation

#### 3.2. Linear Dynamics: Protein Expression

#### 3.3. Nonlinear Dynamics: Intracellular Oscillator Modeled by Lotka–Volterra Dynamics

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

NDF | Number density distribution function |

ODE | Ordinary differential equation |

PBE | Population balance equation |

PBM | Population balance model |

PDE | Partial differential equation |

Probability density function | |

SDE | Stochastic differential equation |

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**Figure 1.**(

**Left**) Two-dimensional cell distribution; only individual size is measurable, but not growth rate; (

**right**) model-based online estimation.

**Figure 2.**Benchmark 1: protein expression, ${N}_{GMD}=2$; reconstruction of integral quantities of GMDs using Kalman filters and the Hungarian algorithm for data association.

**Figure 3.**Benchmark 2: Lotka–Volterra dynamics, ${N}_{GMD}=2$; reconstruction of integral quantities of GMDs using particle filter: modes of the PDF (red) represent estimates of the GMDs (only each second measurement (blue) is depicted to improve visibility).

Parameter | Value |
---|---|

k | 2 |

d | $0.01$ |

V | ${10}^{-2}\xb7\mathrm{d}iag\left([2.1,\phantom{\rule{0.166667em}{0ex}}2.73,\phantom{\rule{0.166667em}{0ex}}2.11,\phantom{\rule{0.166667em}{0ex}}4.2,\phantom{\rule{0.166667em}{0ex}}0.04,\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}0.04,\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}0.01]\right)$ |

W | ${10}^{-3}\xb7\mathrm{d}iag\left([1,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}0]\right)$ |

Parameter | Value | Parameter | Value |
---|---|---|---|

${k}_{1}$ | 0.8 | V | ${10}^{-4}\xb7\mathrm{d}iag\left([0.6,\phantom{\rule{0.166667em}{0ex}}0.05,\phantom{\rule{0.166667em}{0ex}}1.2,\phantom{\rule{0.166667em}{0ex}}{10}^{-3},\phantom{\rule{0.166667em}{0ex}}5\times {10}^{-4},\phantom{\rule{0.166667em}{0ex}}{10}^{-4}]\right)$ |

${k}_{2}$ | 1.2 | W | ${10}^{-4}\xb7\mathrm{d}iag\left([0.6,\phantom{\rule{0.166667em}{0ex}}1.2,\phantom{\rule{0.166667em}{0ex}}5\times {10}^{-4}]\right)$ |

${k}_{3}$ | 0.4 | ${N}_{P}$ | 200 |

${k}_{4}$ | 0.5 | $K\left(\mathbf{x}\right)$ | $\mathcal{N}(\mathbf{x},\phantom{\rule{0.166667em}{0ex}}0.01\xb7V)$ |

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**MDPI and ACS Style**

Dürr, R.; Waldherr, S.
A Novel Framework for Parameter and State Estimation of Multicellular Systems Using Gaussian Mixture Approximations. *Processes* **2018**, *6*, 187.
https://doi.org/10.3390/pr6100187

**AMA Style**

Dürr R, Waldherr S.
A Novel Framework for Parameter and State Estimation of Multicellular Systems Using Gaussian Mixture Approximations. *Processes*. 2018; 6(10):187.
https://doi.org/10.3390/pr6100187

**Chicago/Turabian Style**

Dürr, Robert, and Steffen Waldherr.
2018. "A Novel Framework for Parameter and State Estimation of Multicellular Systems Using Gaussian Mixture Approximations" *Processes* 6, no. 10: 187.
https://doi.org/10.3390/pr6100187