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Research article

Risk assessment and integrated process modeling–an improved QbD approach for the development of the bioprocess control strategy

  • Received: 31 May 2020 Accepted: 03 August 2020 Published: 10 August 2020
  • A Process characterization is a regulatory imperative for process validation within the biopharmaceutical industry. Several individual steps must be conducted to achieve the final control strategy. For that purpose, tools from the Quality by Design (QbD) toolbox are often considered. These tools require process knowledge to conduct the associated data analysis. They include cause and effect analysis, multivariate data analysis, risk assessment and design space evaluation. However, this approach is limited to the evaluation of single unit operations. This is risky as the interactions of the operations may render the control strategy invalid. Hence, a holistic process evaluation is required. Here, we present a novel workflow that shows how simple data analysis tools can be used to investigate the process holistically. This results in a significant reduction of the experimental effort and in the development of an integrated process control strategy. This novel QbD workflow is based on a novel combination of risk assessment and integrated process modeling. We demonstrate this workflow in a case study and show that the herein presented approach can be applied to any biopharmaceutical process. We demonstrate a workflow that can reduce the number of factors and increase the amount of responses within a Design of Experiments (DoE). Consequently, this result demonstrates that experimental costs and time can be reduced by investing more time in thoughtful data analysis.

    Citation: Daniel Borchert, Diego A. Suarez-Zuluaga, Yvonne E. Thomassen, Christoph Herwig. Risk assessment and integrated process modeling–an improved QbD approach for the development of the bioprocess control strategy[J]. AIMS Bioengineering, 2020, 7(4): 254-271. doi: 10.3934/bioeng.2020022

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  • A Process characterization is a regulatory imperative for process validation within the biopharmaceutical industry. Several individual steps must be conducted to achieve the final control strategy. For that purpose, tools from the Quality by Design (QbD) toolbox are often considered. These tools require process knowledge to conduct the associated data analysis. They include cause and effect analysis, multivariate data analysis, risk assessment and design space evaluation. However, this approach is limited to the evaluation of single unit operations. This is risky as the interactions of the operations may render the control strategy invalid. Hence, a holistic process evaluation is required. Here, we present a novel workflow that shows how simple data analysis tools can be used to investigate the process holistically. This results in a significant reduction of the experimental effort and in the development of an integrated process control strategy. This novel QbD workflow is based on a novel combination of risk assessment and integrated process modeling. We demonstrate this workflow in a case study and show that the herein presented approach can be applied to any biopharmaceutical process. We demonstrate a workflow that can reduce the number of factors and increase the amount of responses within a Design of Experiments (DoE). Consequently, this result demonstrates that experimental costs and time can be reduced by investing more time in thoughtful data analysis.


    Fixed point theory is an extensively used mathematical tool in various fields of science and engeneering [1,2,3] Many researchers have generalized Banach contraction principle in various directions. Some have generalized the underlying space while some others have modified the contractive conditions [4,5,6,7].

    Zadeh [8] initiated the notion of fuzzy set which lead to the evolution of fuzzy mathematics. Kramosil and Michalek [9] generalized probabilistic metric space via concept of fuzzy metric. George and Veeramani [10] defined Hausdorff topology on fuzzy metric space after slight modification in the definition of fuzzy metric presented in [9]. Heilpern [11] defined fuzzy mapping and establish fixed point result for it. Subsequently many concepts and results from general topology were generalized to fuzzy topological space.

    Nˇadˇaban [12] generalized b-metric space by introducing fuzzy b-metric space in the setting of fuzzy metric space initiated by Michalek and Kramosil. Faisar Mehmood et al.[13] generalized fuzzy b-metric by introducing the concept of extended fuzzy b-metric. In this article we present the idea of μ-extended fuzzy b-metric space which extends the concepts of fuzzy b-metric and extended fuzzy b-metric spaces. We also establish a Banach-type fixed point result in the context of μ-extended fuzzy b-metric space.

    First we recollect basic definitions and results which will be used in the sequel.

    Definition 1.1. [14] A binary operation :[0,1]2[0,1] is said to be continuous t-norm if ([0,1],,) is an ordered abelian topological monoid with unit 1.

    Some frequently used examples of continuous t-norm are xLy=max{x+y1,0}, xPy=xy and xMy=min{x,y}. These are respectively called Lukasievicz t-norm, product t-norm and minimum t-norm

    Definition 1.2. [9] A fuzzy metric space is 3-tuple (S,ς,), where S is a nonempty set, is continuous t-norm and ς is a fuzzy set on S×S×[0,) which satisfies the following conditions, for all p,q,rS,

    (KM1)ς(p,q,0)=0;

    (KM2)ς(p,q,)=1, for all >0 if and only if p=q;

    (KM3)ς(p,q,)=ς(q,p,);

    (KM4)ς(p,r,+t)ς(p,q,)ς(q,r,t), for all ,t>0;

    (KM5)ς(p,q,.):[0,)[0,1] is non-decreasing continuous;

    (KM6)limς(r,y,)=1.

    Note that ς(p,q,) indicates the degree of closeness between p and q with respect to 0.

    Remark 1.1. For pqand>0, it is always true that 0<ς(p,q,)<1.

    Lemma 1.1. [15] Let S be a nonempty set. Then ς(p,q,.) is nondecreasing for all p,qS.

    Example 1.1. [16] Let S be a nonempty set and ς:S×S×(0,)[0,1] be fuzzy set defined on a metric space (S,d) such that

    ς(x,y,)=pqpq+rd(x,y),x,ySand>0,

    where p,q and r are positive real numbers, and is product t-norm. This is a fuzzy metric induced by the metric d. The above fuzzy metric is also defined if minimum t-norm is used instead of product t-norm.

    If we take p=q=r=1, then the above fuzzy metric becomes standard fuzzy metric.

    Definition 1.3. [12] Let S be a non-empty set and b1 be a given real number. A fuzzy set ς:S×S[0,) is said to be fuzzy b-metric if for all p,q,rS, the following conditions hold:

    (FbM1)ς(p,q,0)=0;

    (FbM2)ς(p,q,)=1, for all >0 if and only if q=p;

    (FbM3)ς(p,q,)=ς(q,p,);

    (FbM4)ς(p,r,b(+t)ς(p,q,)ς(q,r,t), for all ,t>0;

    (FbM5)ς(p,q,.):(0,)[0,1] is continuous and limς(p,q,)=1.

    Faisar Mehmood et al. [13] defined extended fuzzy b-metric as.

    Definition 1.4. [13] The ordered triple (S,ς,) is called extended fuzzy b-metric space by function α:S×S[1,), where S is non-empty set, is continuous t-norm and ς:S×S[0,) is fuzzy set such that for all x,y,zS the following conditions hold:

    (FbM1)ςα(p,q,0)=0;

    (FbM2)ςα(p,q,)=1, for all >0 if and only if q=p;

    (FbM3)ςα(p,q,)=ςα(q,p,);

    (FbM4)ςα(p,r,α(p,r)(+t)ςα(p,q,)ςα(q,r,t), for all ,t>0;

    (FbM5)ςα(p,q,.):(0,)[0,1] is continuous and limςα(p,q,)=1.

    The authors in [13] established the following Banach type fixed point result in the setting of extended fuzzy b-metric space.

    Theorem 1.1. Let (S,ςα,) be an extended fuzzy-b metric space by mapping α:X×S[1,) which is G-complete such that ςα satisfies

    limtςα(p,q,t)=1, p,qSandt>0. (1.1)

    Let f:SS be function such that

    ςα(fp,fq,kt)ςα(p,q,t), p,qSandt>0, (1.2)

    where k(0,1). Moreover, if for b0S and n,pN with α(bn,bn+p)<1k, where bn=fnbo. Then f will have a unique fixed point.

    Motivated by the concept presented in [13], we present μ-extended fuzzy b-metric space and generalize Banach contraction principle to it using the approach of Grabiec [17].

    Definition 2.1. Let α,μ:X×X[1,) defined on a non-empty set X. A fuzzy set ςμ:X×X×[0,)[0,1] is said to be μ-extended fuzzy b-metric if for all p,q,rX, the following conditions hold:

    (μE1)ςμ(p,q,0)=0;

    (μE2)ςμ(p,q,)=1, for all >0 if and only if q=p;

    (μE3)ςμ(p,q,)=ςμ(q,p,);

    (μE4)ςμ(p,r,α(p,r)+μ(p,r)ȷ)ςμ(p,q,)ςμ(q,r,ȷ), for all ,t>0;

    (μE5)ςμ(p,q,.):(0,)[0,1] is continuous and limςμ(p,q,)=1.

    And (X,ςμ,,α,μ) is called μ-extended fuzzy b-metric space.

    Remark 2.1. It is worth mentioning that fuzzy b-metric and extended fuzzy b-metric are special types of μ-extended fuzzy b-metric when α(x,y)=μ(x,y)=b, for some b1 and α(x,y)=μ(x,y), respectively.

    In the following we exemplify Definition 2.1.

    Example 2.1. Let S={1,2,3} and α,μ:S×S[1,) be two functions defined by α(m,n)=1+m+n and μ(m,n)=m+n1. If ςμ:S×S×[0,)[0,1] is a fuzzy set defined by

    ςμ(m,n,)=min{m,n}+max{m,n}+,

    where contiuous t-norm is defined as t1t2=t1×t2, for all t1,t2[0,1] We show that (S,ςμ,,α,μ) is μ-extended fuzzy b-metric space. Clearly α(1,1)=3,  α(2,2)=5,  α(3,3)=7,α(1,2)=α(2,1)=4,  α(2,3)=α(3,2)=6,  α(1,3)=α(3,1)=5, and μ(1,1)=1,  μ(2,2)=3,  μ(3,3)=5, μ(1,2)=μ(2,1)=2,  μ(2,3)=μ(3,2)=4,  μ(1,3)=μ(3,1)=3. One can easily verify that the conditions (μE1),(μE2),(μE3) and (μE5) hold. In order to show that (S,ςμ,×,α,μ) is μ-extended fuzzy b-metric space, it only remains to prove that (μE4) is satisfied for all m,n,pS. For for all ,ȷ>0, it is clear that

    ςμ(1,2,α(1,2)+μ(1,2)ȷ)=1+4+2ȷ2+4+2ȷ2+ȷ+2+ȷ9+3ȷ+3+ȷ=ςμ(1,3,)ςμ(3,2,ȷ),
    ςμ(1,3,α(1,3)+μ(1,3)ȷ)=1+5+3ȷ3+5+3ȷ2+ȷ+2+ȷ6+ȷ+2+ȷ=ςμ(1,2,)ςμ(2,3,ȷ),

    and

    ςμ(2,3,α(2,3)+μ(2,3)ȷ)=2+6+4ȷ3+6+4ȷ1+ȷ++ȷ6+2ȷ+3+ȷ=ςμ(2,1,)ςμ(1,3,ȷ).

    Hence ςμ is μ-extended fuzzy b-metric.

    Example 2.2. Let S={1,2,3} and α,μ:S×S[1,) be two functions defined by α(m,n)=max{m,n} and μ(m,n)=min{m,n}. If ςμ:S×S×[0,)[0,1] is a fuzzy set defined by

    ςμ(m,n,)={1,  m=n,0,  =0,2,  ,0<<2,max{m,n}+1,  2<<3,max{m,n},  3<,1+1,  S,

    where continuous t-norm is defined to be the minimum, that is t1t2=min(t1,t2). Obviously conditions (μE1),(μE2),(μE3) and (μE5) trivially hold. For p,q,rS notice the following:

    Case 1: When 0<+ȷ2<1. Then

    ςμ(1,2,α(1,2)+μ(1,2)ȷ)=+ȷ2min{2,ȷ2}=ςμ(1,3,)ςμ(3,2,ȷ).

    Case 2: When 1<2+ȷ<32. Then

    ςμ(1,2,α(1,2)+μ(1,2)ȷ)=22+ȷ+1min{2,ȷ2}=ςμ(1,3,)ςμ(3,2,ȷ).

    Case 3: When 2+ȷ>3 such that =0 and ȷ>3. Then

    ςμ(1,2,α(1,2)+μ(1,2)ȷ)=2ȷ>0=min{0,3ȷ}=ςμ(1,3,)ςμ(3,2,ȷ).

    Case 4: When 2+ȷ>3 such that >3 and ȷ=0. Then

    ςμ(1,2,α(1,2)+μ(1,2)ȷ)=1>0=min{3,0}=ςμ(1,3,)ςμ(3,2,ȷ).

    Similarly it can be easily verified that condition (μE4) is satisfied for all the remaining cases. Hence (S,ςμ,,α,μ) is μ-extended fuzzy b-metric space.

    Before establishing an analog of Banach contraction principle in setting of μ-extended fuzzy b-metric space, we present the following concepts in the setting of μ-extended fuzzy b-metric space.

    Definition 2.2. Let (S,ςμ,,α,μ) be a μ-extended fuzzy b-metric space and {an} be a sequence in S.

    (1) {an} is a G-convergent sequence if there exists a0S such that

    limnςμ(an,a0,)=1, >0.

    (2) {an} in X is called G-Cauchy if

    limnςμ(an,an+p,)=1,foreachpNand>0.

    (3) S is G-complete, if every Cauchy sequence in S converges.

    Next, we prove Banach fixed point Theorem in μ-extended fuzzy b-metric space.

    Theorem 2.1. Let (S,ςμ,,α,μ) be a G-complete μ-extended fuzzy b-metric space with mappings α,μ:S×S[1,) such that

    limtςμ(u,v,t)=1, u,vSandt>0. (2.1)

    Let f:SS be a mapping satisfying that there exists k(0,1) such that

    ςμ(fu,fv,kt)ςμ(u,v,t), u,vSandt>0. (2.2)

    If for any b0S and n,pN,

    max{supp1limiα(bi,bi+p),supp1limiμ(bi,bi+p)}<1k,

    where bn=fnbo, then f has a unique fixed point.

    Proof. Without loss of generality, assume that bn+1bn n0. From (2.2), it follows that, for any n,qN,

    ςμ(bn,bn+1,kt)=ςμ(fbn1,fbn,kt)ςμ(bn1,bn,t)ςμ(bn2,bn1,tk)ςμ(bn3,bn2,tk2)ςμ(b0,b1,tkn1).

    That is

    ςμ(bn,bn+1,kt)ςμ(b0,b1,tkn1). (2.3)

    For any pN, applying (μE4) yields that

    ςμ(bn,bn+p,t)=ςμ(bn,bn+p,ptp)=ςμ(bn,bn+p,tp+pttp)ςμ(bn,bn+1,tpα(bn,bn+p))ςμ(bn+1,bn+p,pttpμ(bn,bn+p))ςμ(bn,bn+1,tpα(bn,bn+p))ςμ(bn+1,bn+2,tpμ(bn,bn+p)α(bn+1,bn+p))ςμ(bn+2,bn+p,pt2tpμ(bn,bn+p)μ(bn+1,bn+p)).

    From (2.3) and (μE4), it follows that

    ςμ(bn,bn+p,t)ςμ(b0,b1,tpα(bn,bn+p)kn)ςμ(b0,b1,tpμ(bn,bn+p)α(bn+1,bn+p)kn+1)ςμ(b0,b1,tpμ(bn,bn+p)μ(bn+1,bn+p)α(bn+2,bn+p)kn+2)ςμ(b0,b1,tpμ(bn,bn+p)μ(bn+1,bn+p)μ(bn+(p3),bn+p)α(bn+(p2),bn+p)kn+(p1)).

    Noting that for k(0,1), α(bn,bn+p)k<1 and μ(bn,bn+p)k<1 hold for all n,pN and letting n, applying Eq 3, it follows that

    limnςμ(bn,bn+p,t)=111=1,

    that is {bn} is Cauchy sequence. Due to the completeness of (S,ςμ,,α,μ) there exists some bS such that bnbasn. We claim that b is unique fixed point of f. Applying Eq (1.1) and condition (μE4), we have

    ςμ(fb,b,t)ςμ(fb,bn+1,t2α(fb,b))ςμ(bn+1,b,t2μ(fb,b))ςμ(b,bn,t2α(fb,b)k)ςμ(bn+1,b,t2μ(fb,b)).

    Thus ςμ(fb,b,t)=1 and hence b is a fixed point of f. To show the uniqueness, let c be another fixed point of f. Applying inequality (2.3) yields that

    ςμ(b,c,t)=ςμ(fb,fc,t)ςμ(b,c,tk)=ςμ(fb,fc,tk)ςμ(b,c,tk2)ςμ(b,c,tkn),

    which implies that ςμ(b,c,t)1, as n, and hence b=c.

    Remark 2.2. If α(u,v)=μ(u,v) for all u,vS, then Theorem 2.1 reduces to Theorem 1.1.

    The following example illustrates Theorem 2.1.

    Example 2.3. Let S=[0,1] and ςμ(u,v,t)=e|uv|t,  u,vS. It can be easily verified that (S,ςμ,,α,μ) is a G-complete μ-extended fuzzy b-metric space with mappings α,μ:S×S[1,) defined by α(u,v)=1+uv and μ(u,v)=1+u+v, respectively and continuous t-norm as usual product.

    Let f:SS be such that f(x,y)=113x. For all t>0 we have

    ςμ(fu,fv,12t)=e23|uv|t>e|uv|t=ςμ(u,v,t).

    That is all the conditions of Theorem 2.1 are satisfied. Therefore, f has unique fixed point 34[0,1]=S.

    We introduce the concept of μ-extended fuzzy b-metric space and established fixed point result which generalizes Banach contraction principle to this newly introduced space. The concept we presented may lead to further investigation and applications. As the class of of μ-extended fuzzy b-metric spaces is wider than those of the fuzzy b-metric spaces and extended fuzzy b-metric spaces, therefore results established in this framework will generalize many results in the existing literature.

    The authors are grateful to the editorial board and anonymous reviewers for their comments and remarks which helped to improve this manuscript.

    The third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017- 01-17.

    The authors declare that they have no competing interest.


    Acknowledgments



    This project has received funding from the Ministry of Economic Affairs under PPP-Allowance under the TKI-Programme Life Sciences and Health.

    Conflict of interest



    The authors declare no conflict of interest.

    Author contributions



    DBO and CHE had the idea for the improved QbD approach. DBO wrote the manuscript and did the case study. DSU and YTH assisted in writing a reviewed the manuscript. CHE assisted in writing the manuscript.

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