Predictability and identifiability assessment of models for prostate cancer
under androgen suppression therapy

Zhimin Wu; Tin Phan; Javier Baez; Yang Kuang; Eric J. Kostelich; Zhimin Wu; Tin Phan; Javier Baez; Yang Kuang; Eric J. Kostelich

doi:10.3934/mbe.2019176

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 3512-3536. doi: 10.3934/mbe.2019176

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Research article Special Issues

Predictability and identifiability assessment of models for prostate cancer under androgen suppression therapy

School of Mathematical and Statistical Sciences, Arizona State University, 901 S. Palm Walk, Tempe, AZ 85287-1804, USA

Received: 27 December 2018 Accepted: 21 February 2019 Published: 19 April 2019

The past two decades have seen the development of numerous mathematical models to study various aspects of prostate cancer in clinical settings. These models often contain large sets of parameters and rely on limited data sets for validation. The quantitative analysis of the dynamics of prostate cancer under treatment may be hindered by the lack of identifiability of the parameters from the available data, which limits the predictive ability of the model. Using three ordinary differential equation models as case studies, we carry out a numerical investigation of the identifiability and uncertainty quantification of the model parameters. In most cases, the parameters are not identifiable from time series of prostate-specific antigen, which is used as a clinical proxy for tumor progression. It may not be possible to define a finite confidence bound on an unidentifiable parameter, and the relative uncertainties in even identifiable parameters may be large in some cases. The Fisher information matrix may be used to determine identifiable parameter subsets for a given model. The use of biological constraints and additional types of measurements, should they become available, may reduce these uncertainties. Ensemble Kalman filtering may provide clinically useful, short-term predictions of patient outcomes from imperfect models, though care must be taken when estimating ``patient-specific'' parameters. Our results demonstrate the importance of parameter identifiability in the validation and predictive ability of mathematical models of prostate tumor treatment. Observing-system simulation experiments, widely used in meteorology, may prove useful in the development of biomathematical models intended for future clinical application.

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Citation: Zhimin Wu, Tin Phan, Javier Baez, Yang Kuang, Eric J. Kostelich. Predictability and identifiability assessment of models for prostate cancerunder androgen suppression therapy[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3512-3536. doi: 10.3934/mbe.2019176

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Abstract

A correction on

Numerical investigation and improvement of the aerodynamic performance of a modified elliptical-bladed Savonius-style wind turbine. By Sri Kurniati, Sudirman Syam and Arifin Sanusi. AIMS Energy, 2023, Volume 11, Issue 6: 1211–1230. Doi: 10.3934/energy.2023055

The authors would like to make the following corrections to the published paper [1].

On page 1213, we updated the contents of "one symbol statement: ρ" in section 2. The updated contents are as follows:

- ρ is the the density of air,

On page 1215, we updated the contents of "Eq 16" in section 2. The updated contents are as follows:

ϕ ϕ

$\frac{{{\bf{∂}} {\bf{ \pmb{\mathsf{ ϕ}} }}}_{{\bf{ℓ}}}}{{\bf{∂}} \boldsymbol{t}}+{\bf{∇}} \left({{\varnothing }}_{{\bf{ℓ}}}\overrightarrow{\boldsymbol{V}}\right)-{\bf{∇}} \left({\bf{\Gamma }}_{{\bf{ℓ}}}\overrightarrow{\boldsymbol{V}}{{\varnothing }}_{{\bf{ℓ}}}\right) = {\boldsymbol{R}}_{{\bf{ℓ}}}$

(16)

On page 1216, we updated the contents of "Table 2" in section 2. The updated contents are as follows:

Table 2. The terms in the general transfer equation Eq 16.

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \frac{{\partial {\rm{ \mathsf{ ϕ} }}}_{\ell }}{\partial t}+\nabla \left({\varnothing }_{\ell }\overrightarrow{V}\right)-\nabla \left({\mathrm{\Gamma }}_{\ell }\overrightarrow{V}{\varnothing }_{\ell }\right)={R}_{\ell } \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(16)$
$l$	${\varnothing }_{\ell }$	${\mathrm{\Gamma }}_{\ell }$
1	U	$vt$	$-\frac{1}{\rho }\frac{\partial \rho }{\partial x}+\frac{\partial }{\partial x}\left(vt\frac{\partial u}{\partial x}\right)+\frac{\partial }{\partial y}\left(vt\frac{\partial v}{\partial x}\right)+\frac{\partial }{\partial z}\left(vt\frac{\partial w}{\partial x}\right)+gx$
2	V	$vt$	$-\frac{1}{\rho }\frac{\partial \rho }{\partial y}+\frac{\partial }{\partial x}\left(vt\frac{\partial u}{\partial y}\right)+\frac{\partial }{\partial y}\left(vt\frac{\partial v}{\partial y}\right)+\frac{\partial }{\partial z}\left(vt\frac{\partial w}{\partial y}\right)+gy$
3	W	$vt$	$-\frac{1}{\rho }\frac{\partial \rho }{\partial z}+\frac{\partial }{\partial x}\left(vt\frac{\partial u}{\partial z}\right)+\frac{\partial }{\partial y}\left(vt\frac{\partial v}{\partial z}\right)+\frac{\partial }{\partial z}\left(vt\frac{\partial w}{\partial z}\right)+gz$
4	1	0	0
5	K	$vt/$	$G-\varepsilon$
6	$\varepsilon$	$vt/$	${C}_{1}\frac{\varepsilon }{k}G-{c}_{2}\frac{\varepsilon }{k}\varepsilon$

| Show Table

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Conflict of interest

All authors declare no conflicts of interest in this paper.

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