multi-stage fatigue model

multi-stage fatigue model 

 

Microstructure-based Multistage Fatigue Modeling of Aluminum Alloy 7075-T651

https://icme.hpc.msstate.edu/mediawiki/index.php/Microstructure-based_Multistage_Fatigue_Modeling_of_Aluminum_Alloy_7075-T651.html

Fatigues models uses the 3D print microstructures features to predict how many cycles of of specific strains that a materials can withstand. This is very similar to the replicative aging model of yeast cells. 

Braunewell Bornholdt, 2007, Superstability of the yeast cell-cycle dynamics

[PB07JTB] J Theor Biol. 2007 Apr 21;245(4):638-43. Epub 2006 Nov 21. Superstability of the yeast cell-cycle dynamics: ensuring causality in the presence of biochemical stochasticity

Braunewell S, Bornholdt S.

In their 2009 JTB paper, the author cited a measure of reliability in this 07JTB paper. I searched the entire paper for reliability, but did find one hit in the abstract. In the main text, the author mentioned  "stability of the systems under strong noise", termed "stability criterion" (basically robustness or reliability. Based on its explanation below, this is a rather context-specific criterion. 

It seems that PB07 and PB09 are based on the Li04PNAS paper, a boolean network model on yeast cell cycle. 

Braunewell and Bornholdt, 2009, reliability of network

PB09JTB

investigate the interplay of topological structure and dynamical robustness.

reliability of attractors

boolean network dynamics

The reliability criteriont was used to show the robustness of the yeast cell-

cycle dynamics against timing perturbations (Braunewell and Bornholdt, 2007) 

[PB07JTB] Braunewell, S., Bornholdt, S., 2007. Superstability of the yeast cell-cycle dynamics: ensuring causality in the presence of biochemical stochasticity. J. Theor. Biol. 245 (4), 638–643.

See also 

http://www.researchgate.net/profile/Stefan_Braunewell

Block, Li, Savits, 2003, initial and final behaviour of failure rate functions for mixtures and systems

The initial approximation of mixture failure rate is  straight forward. For each subpopulation pdf = failure rate *Viability.  Because viability approaches 1 initially, mixture of pdf leads to mixture of failure rates.

Reference:

Initial and Final Behaviour of Failure Rate Functions for Mixtures and Systems
Author(s): Henry W. Block, Yulin Li and Thomas H. Savits Source: Journal of Applied Probability, Vol. 40, No. 3 (Sep., 2003), pp. 721-740 Published by: Applied Probability Trust

Stable URL: http://www.jstor.org/stable/3215946 .

Notes, Lai & Xie, 2006, stochastic ageing and dependence for reliability

page 48, mixture failure rate r(t):
 r(t) = weighted sum of subpopulation pdf / weighted sum of subpopulation survival function
 This can be found by the definition of r(t).  Failure rate is mortality rate in biology.

page 57, initial failure mixture rate r(0+):
 r(0+) = weighted sum of subpopulation r_i(0+),
shown by Block, Li, Savit, 2003, initial and final behavior of failure rate function for mixtures and systems, Journal of Applied Probability, 40, 721-740
 I am surprised that this basic formula has a 2003 citation.  This formula was in GG91 book, and they cited the Barlow, Proschan, (Hunter) book. 

A general definition of failure intensity function

It starts with a discrete definition, and then extend to continuous time. $\bar{F(x)}= 1 - F(X)$ is the survival function.


http://132.187.98.10:8080/encyclopedia/en/failureIntensityfunction.pdf

mixture models, pdf, CDF, and intensity functions, GG01, GG91, !!!!!!!!!!

Mixture models frequently are applied to the probability density function, not the cumulative functions. But, based on wikipedia entry on "mixture density" , mixture can be defined in both pdf and CDF, and it seems to interchangeable.  In fact, this is quit natural, because mixture is a linear form. Derivative of linear combinations of CDFs lead to linear combinations of pdfs.

GG01 applied binomial formula to the mortality rate functions, not the viability functions. GG91 used the same approach, page 258-261, section 6.6.

What would happen if we apply binomial formula to the survival functions? Interestingly, GG91 page 265-266, section 6.7, applied binomial formula to the failure CDF function, $F(x)$ in GG01. So, proof changed from GG91 to GG01.

The failure CDF for binomial active change of $q$ of $n$ elements is:
 $ F(x) = (1- q exp(-kx))^n $,
which is a pleasantly clean form. 

In contrast, Witten85 applied weights to the reliability (viability) function, though his formulation of the Gompertz model did not use this approach.

The more that I compare GG91 and GG01, the more it looks like linear rules are the same between viability function and intensity functions (mortality rates). However, I have not been able to prove this myself. In fact, I found it otherwise, somehow. 

Reference:

http://en.wikipedia.org/wiki/Mixture_density

Whose notes are these?

I bought a used copy of The Biology of Life Span: A Quantitative Approach, by LA Gavrilov and NS Gavrilova.  This used book contains two pages of notes.  There are not many people working on the theoretic and quantitative aging modeling in the world. So, I wonder who do they belong to.

GG91, 6.2, the need for a critical attitude to mathematical models of lifespan

GG91 discussed several instances of 'blatant errors' in 6.2.

The first is EA Murphy 1978, "Genetics of longevity in man". Murphy78 proposed a $k$ subsystems in a serial configuration, and each subsystems break down after $n$ random damages. GG91 shows an error of differentiation in the published results.

The second example is Skurnick and Kemeny, 1978, Mechanisms of Ageing and Development. SK78 argued that 'an organism' as a 'chain' whose strength is determined by the 'weakest link'. GG91 connected SK78 with the 'Bingo model'. SK78 used the extreme value distributions. SK78 inferred the Weibull model of aging and Gompertz model in old ages.  GG91 stated that SK78's approximation of Gompertz model is incorrect because maximum approximation rule was used for minimum approximation. GG91 argues that this is a case that intuition and common sense would prevent such errors from happening.

The third example is Koltover 1983, Progress in Modern Biology. Koltover83 argued that death of an organism is the result of damage to at least one of the $Q$ blocks of genes in the genome. GG91 described this assumption as 'stupidity', because GG91 argued that every cell has its own genome.  This critique seems to be a misplaced because K83 model can be viewed as a 'genetic interaction model', which is commonly used in evolutionary genetics.

K83 used $m_c$ for a critical value of 'disfunction' and seems to be deterministic in nature and lead to simulanenous death of a homogeneous population. K83 then introduced heterogeneity using a truncated exponential distribution to describe different populations.

The fourth example is Witten 1985, Mechanisms of Ageing and Development. GG91 shows that the derived Gompertz coefficient in W85 cannot be positive if its assumption were also positive. In other words, W85 seemed to be working on a exponential growth model if the final Gompertz derivation holds.

Witten 1985 MAD, critical elements in a graph and Gompertz model

Witten 1985, Mech Ageing and Dev. 1985.

Witten85 used $R(t)$ as the survival function. $h_0$ as the initial mortality rate, $\gamma$ as the Gompertz coefficient.  Witten85 seems to treat probability = reliability (page 142), which is different from Leemis's approach.

Witten85 assumed $m$ critical components in a graph model of a cell. There existed a critical number $m_c$ that is the threshold for the system (cell) to fail. This is equivalent to the parallel block configuration used by GG01.

Witten85 used Markov transitional states to model the failure of each critical component $M*$.  I did not see how  these transitional states are used for his derivation of the Gompertz model.

Witten85 used 'deviation' concept from Witten83. Witten85 assumes exponential failure function for each critical component (page 148), similar to GG01. It seems that although 'deviation' argument was used, it was not incorporated into the building toward Gompertz model.

In Eq 21, Witten85 shows that system viability (reliability) $R_SYS$ is basically the viability function of a serial system. Witten85 assumed a 'critical time' $t*$ (page 148), and approximate the binomial form in Eq 21 to obtain the Gompertz form using the exactly approximation used by GG91. 

It is perplexing to me that Eq21 shows a serial configuration (based on Eq 14), but Witten85 argues for parallel configuration in Eq 13.  For serial system, the product form also means that any single failure of the critical components will lead to system failure.  If 'all of the critical elements must fail' for a system to fail, it should be a parallel configuration.

Witten85 introduced 'cost' in Eq 30 on page 152: Cost ~ m^beta, where $m$ is the number of critical elements.

Witten 1983, MAD,

Witten83MAD argues for a systems approach and argues that a 'local model' is need for mechanistic insights. 

Without loss of generality, let us consider a single cell as our system of interest. When a new cell comes into being, it must function in some "average normal" manner (we'll assume that we are examining normal cells). In order for a cell to function in this "average normal manner" it must have some idea of what functions it must perform. That is to say, if it is a pancreatic islet cell then it must do all those things a "normal" pancreatic islet cell must do. This knowledge must be abstractly embedded in the cell as programming an internal model/rules and laws - whatever you choose. In brief, the cell has a set of internal perceptions as to what its normal function in a normal environment must be. (page 71, Witten83MAD).

If, as clock time passes, the cell's external environment changes in some substantial
manner, then the deviation between the cell's programmed picture of the normal world environment and the real world environment becomes quite large. The consequence of this deviation could well be the development of reversible and non-reversible agerelated effects. In the following paragraphs, we investigate how these concepts may be formalized. This will lead us to a definition of senescence in terms of these deviations. (page 72).

Witten83 use $R_SYS$ as survival function, and $\lambda$ as mortality rate. Witten83 formulated the normal deviation $\alpha_N$ and aging devivation $\alpha_R(t)".

 Its aging model in linear form is:

Its threshold-like aging model is:

Witten83 then use mortality rate and survival function to find a probability density function, called a "general form of mortality" (page 75). Witten83 parametrized $\alpha_R(t)$ for detailed studies.

Bibtex, reliability and network (in progress)

Title:	Network Reliability
Authors:	Ball, Michael O. Colbourn, Charles J. Provan, J.S.
Department/Program:	ISR
Type:	Technical Report
Keywords:	algorithms, combinatorics, computational complexity, graph theory, reliability, network reliability, networks, Systems Integration
Issue Date:	1992
Series/Report no.:	ISR; TR 1992-74
Abstract:	This paper provides a detailed review of the state of the art in the field of network reliability analysis. The primary model treated is a stochastic network in which arcs fail randomly and independently with known failure probabilities. The inputs to the basic network reliability analysis problem consist of the network and a failure probability for each are in the network. The output is some measure of the reliability of the network. The reliability measures treated most extensively in this paper are: the two terminal measure, the probability that there exists a path between two specified nodes; the all-terminal measure the probability that the network is connected and the k-terminal measure, the probability that a specified node subset, K, is connected. In all cases the results concerning each problem's computational complexity, exact algorithms, analytic bounds and Monte Carlo methods are covered. The paper also treats more complex reliability measures including performability measures and stochastic shortest path, max flow and PERT problems. A discussion is provided on applications and using the techniques covered in practice.
URI:	http://hdl.handle.net/1903/5255
Appears in Collections:	Institute for Systems Research Technical Reports

@ARTICLE{4335422,
author={Ball, M.O.},
journal={Reliability, IEEE Transactions on},
title={Computational Complexity of Network Reliability Analysis: An Overview},
year={1986},
volume={35},
number={3},
pages={230-239},
abstract={This paper presents an overview of results related to the computational complexity of network reliability analysis problems. Network reliability analysis problems deal with the determination of reliability measures for stochastic networks. We show how these problems are related to the more familiar computational network problems of recognizing certain subnetworks, finding optimal subnetworks, and counting certain subnetworks. We use these relationships to show that the k-terminal, the 2-terminal, and the all-terminal network reliability analysis problems are at least as hard as the renowned set of computationally difficult problems, NP-Complete. Finally, we discuss the impact of these results on how one should approach problem solving in this area.},
keywords={Algorithm design and analysis;Computational complexity;Computer network reliability;Computer networks;Educational institutions;Graph theory;Problem-solving;Reliability theory;Seismic measurements;Stochastic processes},
doi={10.1109/TR.1986.4335422},
ISSN={0018-9529},}



@INPROCEEDINGS{12999,
author={Ray, G. A. and Dunsmore, J. J.},
booktitle={INFOCOM '88. Networks: Evolution or Revolution, Proceedings. Seventh Annual Joint Conference of the IEEE Computer and Communcations Societies, IEEE},
title={Reliability of network topologies},
year={1988},
pages={842-850},
abstract={The authors present some analytical results which can be used to compute network reliability. Some simple techniques to make asymptotic approximations based on parameters of the network topology are derived. Explicit reliability formulae are computed for four network topologies, including the star and counterrotating ring. Actual manufacturer's data and failure models for laser diodes and LEDs are used to compare the effects of transmitter reliability on the whole network.<>},
keywords={computer networks;graph theory;network topology;reliability;LAN;LEDs;asymptotic approximations;counterrotating ring;failure models;laser diodes;network reliability;network topologies;network topology;reliability formulae;star topology;transmitter reliability;Communication networks;Computer aided manufacturing;Computer network reliability;Computer networks;Light emitting diodes;Network topology;Optical transmitters;Reliability theory;Telecommunication network reliability;Virtual manufacturing},
doi={10.1109/INFCOM.1988.12999},}

@ARTICLE{5220476,
author={Abraham, J.A.},
journal={Reliability, IEEE Transactions on},
title={An Improved Algorithm for Network Reliability},
year={1979},
volume={R-28},
number={1},
pages={58-61},
abstract={Boolean algebra has been used to find the probability of communication between a pair of nodes in a network by starting with a Boolean product corresponding to simple paths between the pair of nodes and making them disjoint (mutually exclusive). A theorem is given, the use of which enables the disjoint products to be found much faster than by existing methods. An algorithm and results of its implementation on a computer are given. Comparisons with existing methods show the usefulness of the algorithm for large networks.},
keywords={Algorithm design and analysis;Boolean algebra;Computer network reliability;Computer networks;Ducts;Fault trees;Telecommunication network reliability;Boolean expressions;Disjoint products;Reliability algorithm;Terminalpair reliability},
doi={10.1109/TR.1979.5220476},
ISSN={0018-9529},}

Reliability network equivalent approach to distribution systems reliability evaluation, R. Billongton, P. Wong

Reliability network equivalent approach to distribution systems reliability evaluation,
R. Billongton, P. Wong
IEE, Proc. Gener. Transm. Distrib. Vol 145, No2, March 1998

This paper grouped subnetwork into large blocks, which is similar to the way that I am modeling the interacting loci for QTL study.

Interacting loci and QTL in network model of aging

Complex system configuration can be addressed by state vectors.

See PPT slides of J. Akipelu.

Synthetical lethal pairs is equivalent to parallel components

The network survivor function S(t) is based on network structure function (see Leemis09, example 3.9 figure 3.9 on page72). Mortality rate (hazard rate) can be found by definition:  h(t) = -S' / dS .

$S_a$ and $S_b$ follow the Weibull model in earlier aging based on GG01.

System mortality rate for serial components

This explains that product of survivor function leads to summation of mortality rate (hazard function).

Basic concepts and references on aging and reliability, Barlow & Proschan 1996

IFR: increasing failure rate
DFR: decreasing failure rate

The failure rate function $r(t)$ is defined as: $r(t) = f(t) / S(t) = f(t) / (1- F(t))$ in Barlow & Proschan96 on page 10. Function $f(t)$ should be the pdf of death,  and $F(t)$ is the failure distribution (page 2), and $S(t)$ should be viability.

Barlow&Proschan96 quote $r(t)$ as "force of mortality" by Steffernson 1930. The reference is missing in the book's reference list), "hazard rate", "intensity function".  I found a JSTOR examination of actuarial science in 1910 with a question of comparing 'force of mortality' and 'rate of mortality'.  So, this concept should goes back much earlier.


Parzen, E, 1962, stochastic process, holden-Day, San Francisco


Gumbel 1958, Statistics of extremes. Columbia Univ Press.
Discussed exponential distribution and limiting distribution.

Lawrence Leemis, Reliability-Probabilistic models and statistical methods, second edition, 2009.
Its explanation on series and parallel structure functions are really clear.


Gavrilov 1991

Notes, Leemis, Reliability, structure function, reliability function

Reference: Lawrence Leemis, Reliability-Probabilistic models and statistical methods, second edition, 2009.

The state of the component is defined by a vector x_i=0 or 1 for a failed and functioning component.
Structure function of a system is $\phi(x)$ = 0 when system failed, and 1 if the system is functioning.

A series system functions if and only if all of its components function. Its structure function is:
  \phi(x) = min{x_1, x_2, ..., x_n}  = product_of x_i

A parallel system functions if and only if one or more component functions. Its structure function is:
 \phi(x) = max{x_1, x_2, ..., x_n} = 1 - product_of (1 - x_i)

Although the minimal and maximal values are very intuitive,  the mathematical formula for the parallel system took me a few minutes to grasp. To solve the parallel \phi(x), we first look for the opposite state vector, i.e., (1 - x_i).  Product of (1 - x_i) indicate a failed parallel system. So, the function system is 1 - product_of (1 - x_i).

Combination of series and parallel sub-units can be used to study complex systems.

Is there a formal proof that combination of these two basic subunit will form any configuration? This could be a very difficult combinatorial problem, and I need to consult a combinatorial mathematician on this topic. In practice, any complex systems can at least be 'reduced' to combinations of these two basic sub-units under some assumptions. This would be a convenient argument to study complex gene networks using these two basic configurations. 

System reliability function $r$ seems to be the viability function in aging (section 2.3, page 31). No, this does not seem to be case. Leemis defines survivor function on page 54:   $S(t) = P[T>=t]$.
Nevertheless, on page 72, example 3.9 show that reliability function can be used to calculate the survivor function.

What component should be we focus on to improve a systems's reliability? For parallel system, the limiting factor is the most reliable one (Page 39).

Mixture of survivor function is discussed in Secion 3.5, page 77. This is directly related to my power-law network model.

Leemis09 discussed Gompertz distribution on page 112. Leemis09 stated that Gompertz hazard function is derived from the assumption that Mill's ratio, 1/ harzard function = resistance to death, and h(t) decreases over time at a rate proportional to itself:
   $d(1/h(t) / dt = k (1/h(t))$
Solution to this assumption leads to a exponential function.

One relationship map for continuous univariate lifetime distributions is provided in figure 4.9 on page 114.

To find the reliability of a system at any time t, the component survivor functions should be used as argument in the reliability function (page 70): 
  S_system = r( S1(t), S2(t), S3(t), ..., Sn(t)). 
Q:How about mortality rate (hazard function)? It can be found by definition:
   h(t) = - S' / S

Page 71 gave an example for a two component in parallel configuration, which is equivalent to  synthetic lethalilty.
Structure function $r = 1 - (1-p1)(1-p2)$
Survivor function $S = 1 - (1-S1)(1-S2)$