# **The Price of Uncertainty in Present-Biased Planning**

Susanne Albers and Dennis Kraft(B)

Department of Computer Science, Technical University of Munich, Munich, Germany<EMAIL_ADDRESS>
**Abstract.** The tendency to overestimate immediate utility is a common cognitive bias. As a result people behave inconsistently over time and fail to reach long-term goals. Behavioral economics tries to help affected individuals by implementing external incentives. However, designing robust incentives is often difficult due to imperfect knowledge of the parameter β ∈ (0, 1] quantifying a person's present bias. Using the graphical model of Kleinberg and Oren [8], we approach this problem from an algorithmic perspective. Based on the assumption that the only information about β is its membership in some set B ⊂ (0, 1], we distinguish between two models of uncertainty: one in which β is fixed and one in which it varies over time. As our main result we show that the conceptual loss of efficiency incurred by incentives in the form of penalty fees is at most 2 in the former and 1 + max B/ min B in the latter model. We also give asymptotically matching lower bounds and approximation algorithms.

**Keywords:** Approximation algorithms · Behavioral economics Heterogeneous agents · Incentive design · Penalty fees Variable present bias

### **1 Introduction**

Many goals in life such as losing weight, passing an exam or paying off a loan require long-term planning. But while some people stick to their plans, others lack self-control; they eat unhealthy food, delay their studies and take out new loans. In behavioral economics the tendency to change a plan for no apparent reason is known as *time-inconsistent behavior*. The questions are, what causes these inconsistencies and why do they affect some more than others? A common explanation is that people make present biased decisions, i.e., they assign disproportionately greater value to the present than to the future. In this simplifying model a person's behavior is the mere result of her present bias and the setting in which she is placed. However, the interplay between these two factors is intricate and sometimes counter-intuitive as the following example demonstrates:

Consider two runners Alice and Bob who have two weeks to prepare for an important race. Each week they must choose between two types of workout.

Work supported by the European Research Council, Grant Agreement No. 691672. c The Author(s) 2017

N. R. Devanur and P. Lu (Eds.): WINE 2017, LNCS 10674, pp. 325–339, 2017. https://doi.org/10.1007/978-3-319-71924-5\_23

Type A always incurs an effort of 1, whereas type B incurs an effort of 3 in the first and 9 in the second week. Since A offers less preparation than B, Alice and Bob's effort in the final race is 13 if they consistently choose A and 1 if they consistently choose B. Furthermore, A and B are incompatible in the sense that switching between the two will result in an effort of 16 in the final race. Figure 1 models this setting as a directed acyclic graph G with terminal nodes s and t. The intermediate nodes vX and vXY represent a person's state after completing the workouts X, Y ∈ {A, B}. To move forward with the training, Alice and Bob must perform the tasks associated with the edges of G, i.e., complete workouts and run the race. Looking at G it becomes clear that two consecutive workouts of type B are the most efficient routine in the long run. However, this is not necessarily the routine a present biased person will choose.

For instance, assume that Alice and Bob discount future costs by a factor of a = 1/2−ε and b = 1/2+ε respectively. We call a and b their present bias. At the beginning of the first week Alice and Bob compare different workout routines. From Alice's perspective two workouts of type A are strictly more preferable to two workouts of type B as she anticipates an effort of 1 + a(1 + 13) = 8 − 14ε for the former and 3 + a(9 + 1) = 8 − 10ε for the latter. A similar calculation for Bob shows that he prefers two workouts of type B. Considering that neither Alice nor Bob finds a mix of A and B particularly interesting at this point, we conclude that Alice chooses A in the first week and Bob B. However, come next week, Bob expects an effort of 1 + b16 = 8 + 16ε for A and 9 + b = 19/2 + ε for B. Assuming ε is small enough, A suddenly becomes Bob's preferred option and he switches routines. Alice on the other hand has no reason to change her mind and sticks to A. As a result she pays much less than Bob during practice and in the final race. This is remarkable considering that her present bias is only marginally different from Bob's. Moreover, it seems surprising that only Bob behaves inconsistently, although he is less biased than Alice.

#### **1.1 Related Work**

Traditional economics and game theory are based on the assumption that people maximize their utility in a rational way. But despite their prevalence, these assumptions disregard psychological aspects of human decision making observed in empirical and experimental research [5]. For instance, time-inconsistent behavior such as procrastination seems paradox in the light of traditional economics. Nevertheless, it can be explained readily by a tendency to overestimate immediate utility in long-term planning, see e.g. [13]. By studying such cognitive biases, behavioral economics tries to obtain more realistic economic models.

A significant amount of research in this field has been devoted to *temporal discounting* in general and *quasi-hyperbolic discounting* in particular, see [6] for a survey. The quasi-hyperbolic discounting model proposed by Laibson [11] is characterized by two parameters: the *present bias* β ∈ (0, 1] and the *exponential discount rate* δ ∈ (0, 1]. People who plan according to this model have an accurate perception of the present, but scale down any costs and rewards realized t ≥ 1 time units in the future by a factor of βδt . To keep our work clearly delineated in scope, we adopt Akerlof's model of quasi-hyperbolic discounting [1] and make

**Fig. 1.** Task graph of the running scenario

the following two assumptions: First, we focus on the present bias β and set the exponential discount rate to δ = 1. Secondly, we assume people to be *naive* in the sense that they are unaware of their present bias and only optimize their current perceived utility when making a decision. Note that Alice and Bob from the previous example behave like agents in Akerlof's model for a present bias of β = 1/2 − ε and β = 1/2 + ε respectively.

Until recently the economic literature lacked a unifying and expressive framework for analyzing time-inconsistent behavior in complex social and economic settings. Kleinberg and Oren closed this gap by modeling the behavior of naively present biased individuals as a planning problem in task graphs like the one depicted in Fig. 1 [8]. We introduce this framework formally in Sect. 2. As a result of Kleinberg and Oren's work, an active line of research at the intersection of computer science and behavioral economics has emerged. For instance, the graphical model has been used to systematically analyze different types of quasi-hyperbolic discounting agents such as *sophisticated* agents who are fully or partially aware of their present bias [9] and agents whose present bias varies randomly over time [7]. Furthermore, the graphical model was used to shed light on the interplay between temporal biases and other types of cognitive biases [10].

The graphical model is of particular interest to us as it provides a natural framework for a design problem frequently encountered in behavioral economics. Given a certain social or economic setting, the problem is to improve a timeinconsistent person's performance via various sorts of *incentives*, such as monetary rewards, deadlines or penalty fees, see e.g. [12]. Using the graphical model, Kleinberg and Oren demonstrate how a strategic choice reduction can incentivize people to reach predefined goals [8]. To implement their incentives, they simply remove the corresponding edges from the task graph. However, there is a computational drawback to this approach. As we have shown in previous work, an optimal set of edges to remove from a task graph with n nodes is NP-hard to approximate within a factor less than √n/3 [2]. A more general form of incentives avoiding these harsh complexity theoretic limitations are penalty fees. In the graphical model penalty fees are at least as powerful as choice reduction and admit a polynomial time 2-approximation [3].

#### **1.2 Incentive Design for an Uncertain Present Bias**

Frederick, Loewenstein and O'Donoghue have surveyed several attempts to estimate people's temporal discount functions [6]. But as estimates differ widely across studies and individuals, the difficulty of predicting a person's temporal discount function becomes apparent. Clearly, this poses a serious challenge for the design of reliable incentives. After all, Alice and Bob's scenario demonstrates how arbitrarily small changes in the present bias can cause significant changes in a person's behavior. In this work we address the effects of incomplete information about a person's present bias in two different notions of uncertainty.

In Sect. 3 we consider naive individuals whose exponential discount rate is δ = 1, but whose present bias β is unknown. The only prior information we have about β is its membership in some larger set B. Our goal is to construct incentives that are robust with respect to the uncertainty induced by B. More precisely, we are interested in incentives that work well for any present bias contained in B. An alternative perspective is that we try to construct incentives which are not limited to a single person, but serve an entire population of individuals with different present bias values. A simple instance of this problem in which a single task must be partitioned and stretched over a longer period of time has been studied by Kleinberg and Oren [8]. But like most research on incentivizing *heterogeneous* populations, see e.g. [12], Kleinberg and Oren's results are restricted to a very specific setting. They themselves suggest the design of more general incentives as a major research direction for the graphical framework [8].

Using penalty fees as our incentive of choice and a fixed reward to keep people motivated, we present the first results in this area. Our contribution is twofold. On the one hand, we try to quantify the conceptual loss of efficiency caused by incomplete knowledge of β. For this purpose we introduce a novel concept called *price of uncertainty*, which denotes the smallest ratio between the reward required by an incentive that accommodates all β ∈ B and the reward required by an incentive designed for a specific β ∈ B. We present an elegant algorithmic argument to prove that the price of uncertainty is at most 2. Remarkably, this bound holds true independent of the underlying graph G and present bias set B. To complement our result, we construct a family of graphs G and present bias sets B for which the price of uncertainty converges to a value strictly greater than 1. On the other hand, we consider the computational problem of constructing penalty fees that work for all β ∈ B, but require as little reward as possible. Drawing on the same algorithmic ideas we used to bound the price of uncertainty yields a polynomial time 2-approximation. Furthermore, we present a non-trivial proof to show that the decision version of the problem is contained in NP. Since all hardness results of [3] also apply under uncertainty, we know that there is no 1.08192-approximation unless P = NP.

### **1.3 Incentive Design for a Variable Present Bias**

In Sect. 4 we generalize our notion of uncertainty to individuals whose present bias β may change arbitrarily over time within the set B. This model is inspired by work of Gravin et al. [7], except that we do not rely on the assumption that β is drawn independently from a fixed probability distribution. Instead, our goal is to design penalty fees that work well for all possible sequences of β over time. We believe this to be an interesting extension of the fixed parameter case as the variability of β may capture changes in a person's temporal discount function caused by unforeseen cognitive biases different from her present bias. As a result we obtain more robust penalty fees.

Again, our contribution is twofold. On the one hand, we introduce the *price of variability* to quantify the conceptual loss of efficiency caused by unpredictable changes in β. Similar to the price of uncertainty, we define this quantity to be the smallest ratio between the reward required by an incentive that accommodates all possible changes of β ∈ B over time and the reward required by an incentive designed for a specific and fixed β ∈ B. However, unlike the price of uncertainty, the price of variability has no constant upper bound. Instead, the ratio seems closely related to the *range* τ = max B/ min B of the set B. By generalizing our algorithm from Sect. 3 we obtain an upper bound of 1+τ for the price of variability. To complement this result, we construct a family of graphs G for which the price of variability converges to τ /2. On the other hand, we consider the computational aspects of constructing penalty fees for a variable β. As a result of the unbounded price of variability, we are not able to come up with a constant polynomial time approximation. Instead, we obtain a (1 + τ )-approximation. However, by using a sophisticated reduction from VECTOR SCHEDULING, we prove that no efficient constant approximation is possible unless NP = ZPP. We conclude our work by studying a curious special case of variability in which individuals may temporarily lose their present bias. For this scenario, which is characterized by the assumption that 1 ∈ B, optimal penalty fees can be computed in polynomial time.

### **2 The Model**

In the following we introduce Kleinberg and Oren's graphical framework [8]. Let G = (V,E) be a directed acyclic graph with n nodes that models some long-term project. The start and end states are denoted by the terminal nodes s and t. Furthermore, each edge e of G corresponds to a specific task whose inured effort is captured by a non-negative cost c(e). To finish the project, a present biased agent must sequentially complete all tasks along a path from s to t. However, instead of following a fixed path, the agent constructs her path dynamically according to the following simple procedure:

When located at any node v different from t, the agent tries to evaluate the minimum cost she needs to pay in order to reach t. For this purpose she considers all outgoing edges (v, w) of her current position v. Because the tasks associated with these edges must be performed immediately, the agent assesses their cost correctly. In contrast, all future tasks, i.e., tasks on a path from v to t not incident to v, are discounted by her present bias of β ∈ (0, 1]. As a result, we define her *perceived cost* for taking (v, w) to be dβ(v, w) = c(v, w) + βd(w), where d(w) denotes the cost of a cheapest path from w to t. Furthermore, we define dβ(v) = min{c(v, w) + βd(w) | (v, w) ∈ E} to be the agent's *minimum perceived cost* at v. Since the agent is oblivious to her own present bias, she only traverses edges (v, w) for which dβ(v, w) = dβ(v). Ties are broken arbitrarily. Once the agent reaches the next node, she reiterates this process.

To motivate the agent, a non-negative reward r is placed at t. Because the agent must reach t before she can collect r, her *perceived reward* for reaching t is βr at each node different from t. When located at v = t, the agent is only motivated to proceed if dβ(v) ≤ βr. Otherwise, if dβ(v) > βr, she quits. We say that G is *motivating*, if she does not quit while constructing her path from s to t. Note that sometimes the agent can construct more than one path from s to t due to ties in the perceived cost of incident edges. In this case, G is considered motivating if she does not quit on any such path.

For the sake of a clear presentation, we will assume throughout this work that each node of G is located on a path from s to t. This assumption is sensible because the agent can only visit nodes reachable from s. Furthermore, she is not willing to enter nodes that do not lead to the reward at t. Consequently, only nodes that are on a path from s to t are relevant to her behavior. All nodes not satisfying this property can be removed from G in a simple preprocessing step.

### **2.1 Alice and Bob's Scenario**

To illustrate the model, we revisit Alice and Bob's scenario. The task graph G is depicted in Fig. 1. Remember that a = 1/2−ε and b = 1/2 +ε denote Alice and Bob's respective present bias. For convenience let 0 < ε ≤ 1/54. Furthermore, assume that a reward of r = 27 is awarded upon reaching t.

We proceed to analyze Alice and Bob's walk through G. At their initial position s they must decide whether they move to vA or vB. For this purpose they try to find a path that minimizes the perceived cost. As the more present biased person, Alice's favorite path is s, vA, vAA, t with a perceived cost of da(s) = da(s, vA)=8−14ε. By choice of ε this cost is covered by her perceived reward ar = 27/2 − 27ε. Consequently, she is motivated to traverse the first edge and moves to vA. A similar argument shows that Bob moves to vB. Once they reach their new nodes, Alice and Bob reevaluate plans. From Alice's perspective vA, vAA, t is still the cheapest path to t. Bob, however, suddenly prefers vB, vAB, t to his original plan. Nevertheless, both of their perceived cost remains covered by their perceived reward and they move to vAA and vAB respectively. At this point the only option is to take the direct edge to t. For Alice the perceived cost at vAA is sufficiently small to let her reach t. In contrast, Bob's perceived cost of db(vAB) = 16 exceeds his perceived reward of br = 27/2 + 27ε and he quits.

### **2.2 Cost Configurations**

Bob's behavior in the previous example demonstrates how present biased decisions can deter people from reaching predefined goals. To ensure an agent's success it is therefore sometimes necessary to implement external incentives such as penalty fees. In the graphical model, penalty fees allow us to arbitrarily raise the cost of edges in G. More formally, let ˜c be a so called *cost configuration*, which assigns a non-negative extra cost ˜c(e) to all edges e of G. The result is a new task graph Gc˜, whose edges e have a cost of c(e)+˜c(e). A present biased agent navigates through Gc˜ according to the same rules applying in G. We say that ˜c is motivating if and only if Gc˜ is. To avoid ambiguity we annotate our notation whenever we consider a specific ˜c, e.g., we write dc˜ and dβ,c˜ instead of d and dβ.

We conclude this section with a brief demonstration of the positive effects penalty fees can have in Alice and Bob's scenario. Let ˜c be a cost configuration that assigns an extra cost of ˜c(vB, vAB)=1/2 to (vB, vAB) and ˜c(e) = 0 to all other edges e = (vB, vAB). Note that G and Gc˜ are identical task graphs except for the cost of (vB, vAB). Because Alice does not plan to take (vB, vAB) on her way through G and has even less reason to do so in Gc˜, we know that ˜c does not affect her behavior. For similar reasons, ˜c does not affect Bob's choice to move to vB. However, once Bob has reached vB his perceived cost of the path vB, vAB, t is db,c˜(vB, vAB) = 19/2 + 16ε, whereas his perceived cost of vB, vBB, t is only db,c˜(vB, vBB) = 19/2 + ε. Since the latter option appears to be cheaper and is covered by his perceived reward, Bob proceeds to vBB and then onward to t. As a result ˜c yields a task graph that is motivating for Alice and Bob alike. This is a considerable improvement to the original task graph.

## **3 Uncertain Present Bias**

In this section we consider agents whose present bias β is uncertain in the sense that our only information about β is its membership in some set B ⊂ (0, 1]. We call B the *present bias set*. For technical reasons we assume that B can be expressed as the union of constantly many closed subintervals from the set (0, 1]. This way the intersection of B with a closed interval is either empty or contains an efficiently computable minimal and maximal element. To measure the degree of uncertainty induced by B, we define the range of B as τ = max B/ min B.

### **3.1 A Decision Problem**

Our goal is to construct a cost configuration ˜c that is motivating for all β ∈ B, but requires as little reward as possible. To assess the complexity of this task, let UNCERTAIN PRESENT BIAS (UPB) be the following decision problem:

**Definition 1 (UPB).** *Given a task graph* G*, present bias set* B *and reward* r > 0*, decide whether a cost configuration* c˜ *motivating for all* β ∈ B *exists.*

If τ = 1, i.e., B only contains a single present bias parameter, UPB is identical to the decision problem MOTIVATING COST CONFIGURATION (MCC) studied in [3]. Since MCC is NP-complete, UPB must be NP-hard. But unlike MCC it is not immediately clear if UPB is also contained in NP. The reason is that proving MCC ∈ NP only requires to verify whether a given cost configuration is motivating for a single value of β; a property that can be checked in polynomial time [2]. However, proving UPB ∈ NP requires to verify whether a given cost configuration is motivating for all β ∈ B. Taking into account that B may very well be an infinite set, it becomes clear that we cannot check all values of β individually. Interestingly, we do not have to; checking a finite subset B ⊆ B of size O(n2) turns out to be sufficient.

**Proposition 1.** *For any task graph* G*, reward* r *and present bias set* B *a finite subset* B ⊆ B *of size* O(n2) *exists such that* G *is motivating for all* β ∈ B *if it is motivating for all* β ∈ B *.*

The above proposition is related to a theorem by Kleinberg and Oren, which bounds the number of paths an agent takes as β varies over (0, 1] by O(n2) [8]. Kleinberg and Oren's argument does not only establish existence of B , but also yields a polynomial time algorithm to construct B , which in turn implies that UPB ∈ NP. Due to space constraints, we refer to the full version of this paper for a corresponding proof of Proposition 1 as well as all other omitted proofs.

**Corollary 1.** *UPB is NP-complete.*

### **3.2 The Price of Uncertainty**

Since UPB is NP-complete, it makes sense to consider the corresponding optimization problem UPB-OPT. For this purpose, let r(G, B) be the infimum over all rewards admitting a cost configuration motivating for all β ∈ B and define:

**Definition 2 (UPB-OPT).** *Given a task graph* G *and present bias set* B*, determine* r(G, B)*.*

Clearly, UPB-OPT must be at least as hard as the optimization version of MCC. Consequently, we know that UPB has no PTAS and is NP-hard to approximate within a ratio less than 1.08192 [3]. But does the transition from a certain to an uncertain β reduce approximability?

Setting complexity theoretic considerations aside for a moment, an even more general question arises: How does the transition from a certain to an uncertain β affect the efficiency of cost configurations assuming unlimited computational resources? To quantify this conceptual difference in efficiency, we look at the smallest ratio between optimal cost configurations motivating for all β ∈ B and optimal cost configurations motivating for a specific β ∈ B. We call this ratio the *price of uncertainty*.

**Definition 3 (Price of Uncertainty).** *Given a task graph* G *and a present bias set* B*, the price of uncertainty is defined as* r(G, B)/ sup{r(G, {β}) | β ∈ B}*.*

Let us illustrate the price of uncertainty by going back to Alice and Bob's scenario and assume that B = {a, b} with a = 1/2−ε and 1/2+ε. In other words, the agent either behaves like Alice or she behaves like Bob, but we do not know which. It is easy to see that in either case the agent minimizes her maximum perceived cost on the way from s to t by taking the path P = s, vB, vBB, t. This minmax cost, which is either da(vB, vBB) = 19/2−ε or db(vB, vBB) = 19/2+ε, provides two lower bounds for the necessary reward when divided by the respective present bias. More formally, it holds true that r(G, {a}) ≥ (19/2 − ε)/(1/2 − ε) and r(G, {b}) ≥ (19/2 + ε)/(1/2 + ε). However, as we have seen in Sect. 2, neither Alice nor Bob are willing to follow P without external incentives. To discourage the agent from leaving P, we assign an extra cost of ˜c(s, vA)=5ε to **Algorithm 1.** UncertainPresentBiasApprox

 b ← min B; P ← minmax path from s to t w.r.t d*b*(e); α ← max{d*b*(e) | e ∈ P}; **foreach** v ∈ V \ {t} **do** ς(v) ← successor of v on a cheapest path from v to t; T = {(v, ς(v)) | v ∈ V \ {t}}; **foreach** e ∈ E **do** c˜(e) ← 0; **foreach** e ∈ E \ (P ∪ T) **do** c˜(e) ← 2α/b + 1; **foreach** (v, w) ∈ T **such that** v ∈ P **and** w /∈ P **do 7** P- ← v, ς(v), ς(ς(v)),...,t; u ← first node of P different from v that is also a node of P; c˜(v, w) ← cost of most expensive edge of P between v and u; **return** c˜;

(s, vA), ˜c(vB, vAB)=1/2 + 16ε to (vB, vAB) and ˜c(e) = 0 otherwise. This extra cost does not affect the agent's maximum perceived cost along P, which she still experiences at (vB, vBB). As a result, our bounds for r(G, {a}) and r(G, {b}) are tight and we get sup{r(G, {β}) | β ∈ B} = r(G, {a}). Moreover, because we have used the same cost configuration ˜c to derive r(G, {a}) and r(G, {b}), it must hold true that r(G, B) = sup{r(G, {β}) | β ∈ B}, implying that the price of uncertainty in Alice and Bob's scenario is 1.

#### **3.3 Bounding the Price of Uncertainty**

As Alice and Bob's scenario demonstrates, cost configurations designed for an uncertain β are not necessarily less efficient than those designed for a specific β. Therefore one might wonder whether scenarios exist in which a real loss of efficiency is bound to occur, i.e., can the price of uncertainty be greater than 1? The following proposition shows that such scenarios indeed exist.

**Proposition 2.** *There exists a family of task graphs and present bias sets for which the price of uncertainty converges to* 1.1*.*

As the price of uncertainty can be strictly greater than 1, the question for an upper bound arises. Ideally, we would like to design a cost configuration ˜c motivating for all β ∈ B assuming the reward is set to r(G, {b}) for some constant factor > 1 and b = min B. Clearly, the existence of such a ˜c would imply a constant bound of for the price of uncertainty independent of G and B. Using a generalized version of the approximation algorithm we proposed in [3], it is indeed possible to construct a ˜c with the desired property for = 2.

The main idea of UncertainPresentBiasApprox is simple: First, the algorithm computes a value α such that α/b is a lower bound on the reward necessary for agents with present bias b, i.e., r(G, {b}) ≥ α/b. In particular, this bound implies sup{r(G, {β}) | β ∈ B} ≥ α/b. Next the algorithm constructs a ˜c such that a reward of 2α/b is sufficiently motivating for all β ∈ B, i.e., r(G, B) ≤ 2α/b. As a result the price of uncertainty can be at most 2. In the following we try to convey the intuition behind the algorithm in more detail.

We begin with the computation of α. For this purpose let P be a path minimizing the maximum cost an agent with present bias b perceives on her way from s to t. We call P a *minmax path* and define α = max{db(e) | e ∈ P} to be the maximum perceived edge cost of P. Since cost configurations cannot decrease edge cost, it should be clear that α is a valid lower bound on the reward required for the present bias b, i.e., r(G, {b}) ≥ α/b.

We proceed with ˜c. The goal is to assign extra cost in such a way that any agent with a present bias β ∈ B traverses only two kinds of edges. The first kind of edges are those on P. It is instructive to note that each such edge (v, w) ∈ P is motivating for a reward of α/b if β ≥ b. The reason is that

$$d\_{\beta}(v, w) = \beta \left(\frac{c(v, w)}{\beta} + d(v, w)\right) \le \beta \left(\frac{c(v, w)}{b} + d(v, w)\right) = \beta \frac{d\_b(v, w)}{b} = \beta \frac{\alpha}{b}.$$

In particular, P is motivating for each present bias β ∈ B. The second kind of edges are on cheapest paths to t. To identify these edges, the algorithm assigns a distinct successor ς(v) to each node v ∈ V \ {t} such that (v, ς(v)) is the initial edge of a cheapest path from v to t. Since we assume t to be reachable from all other nodes of G at least one suitable successor must exist. By definition of ς, we know that P = v, ς(v), ς(ς(v)),...,t is a cheapest path from v to t. We call P the ς*-path* of v and T = {(v, ς(v)) | v ∈ V \ {t}} a *cheapest path tree*.

Remember that we try to keep agents on the edges of P and T. For this purpose, we assign an extra cost of ˜c(e)=2α/b + 1 to all other edges. This raises their perceived cost to at least 2α/b + 1; a price no agent is willing to pay for a perceived reward of β2α/b. However, since we have not assigned any extra cost to T so far, the perceived cost of edges in P and T is unaffected by the current ˜c. In particular, all edges of P are still motivating for a reward of α/b and any present bias β ∈ B. To keep agents from entering costly ς-paths P = v, ς(v), ς(ς(v)),...,t, we assign an extra cost to the out-edges (v, ς(v)) of P, i.e., v ∈ P but ς(v) ∈/ P. The extra cost ˜c(v, ς(v)) is chosen to match the cost of a most expensive edge on P between v and the next intersection of P and P. It is easy to see that the resulting ˜c can no more than double the perceived cost of any edge in P, see the proof of Theorem 1 for a precise argument. Furthermore, the perceived cost of any out-edge (v, ς(v)) of P is either high enough to keep agents on P or they do not encounter edges exceeding the perceived cost of (v, ς(v)) until they reenter P. We conclude that a reward of 2α/b is sufficiently motivating, leading us to one of the central results of our work.

### **Theorem 1.** *The price of uncertainty is at most* 2*.*

It is interesting to note that UncertainPresentBiasApprox can be executed in polynomial time. Furthermore, in the proof of Theorem 1 we argue that α/b ≤ r(G, B) ≤ 2α/b. As a result we have also found an efficient constant factor approximation of UPB-OPT.

**Corollary 2.** *UPB-OPT admits a polynomial time* 2*-approximation.*

### **4 Variable Present Bias**

So far we have considered agents with an unknown but fixed present bias. We now generalize this model to agents whose β may vary arbitrarily within B as they progress through G. It is convenient to think of β as a *present bias configuration*, i.e., an assignment of present bias values β(v) ∈ B to the nodes v of G. Whenever the agent reaches a node v, she acts according to the current present bias value β(v). We say that G is motivating with respect to a present bias configuration β if and only if the agent does not quit on a walk from s to t.

To illustrate the consequences of a variable present bias we revisit Alice and Bob's scenario once more. Recall that the agent in this scenario is either like Alice with a present bias of a = 1/2 − ε or like Bob with a present bias of b = 1/2 + ε, i.e., B = {a, b}. But while she had to commit to one present bias before, she is now free to change between a and b. For instance, her present bias could be b at s and vB, but a otherwise, i.e., β(v) = b for v ∈ {s, vB} and β(v) = a for v ∈ V \ {s, vB}. In this case she walks along the same path Bob would take, i.e., s, vB, vAB, t. However, there is a subtle difference. At vAB the agent behaves like Alice and needs strictly more reward than Bob to remain motivated while traversing (vAB, t). Under closer examination, which we will not go into detail here, it is in fact easy to see that the variability of β makes our agent more expensive to motivate than any agent with a fixed present bias from B.

#### **4.1 Computational Consideration**

Let G be an arbitrary task graph and B a suitable present bias set. We want to construct a cost configuration ˜c that is motivating for all present bias configuration β ∈ BV , but requires as little reward as possible. Using arguments similar to those of Sect. 3, the computational challenges of this task are readily apparent. In particular, the corresponding decision problem VARIABLE PRESENT BIAS (VPB) is equivalent to MCC whenever B only contains a single element.

**Definition 4 (VPB).** *Given a task graph* G*, present bias set* B *and reward* r > 0*, decide whether a cost configuration* c˜ *motivating for all* β ∈ BV *exists.*

Because MCC is NP-complete [3], it immediately follows that VPB is NP-hard. A proof that VPB ∈ NP can be found in the full version of this paper.

#### **Corollary 3.** *VPB is NP-complete.*

As it is NP-hard to find optimal cost configurations for general B, we turn to the optimization version of the problem. For this purpose let r(G, BV ) be the infimum over all rewards admitting a cost configuration ˜c motivating for all β ∈ BV and define VPB-OPT as:

**Definition 5 (VPB-OPT).** *Given a task graph* G *and present bias set* B*, determine* r(G, BV )*.*

Interestingly, approximating VPB-OPT seems to be much harder than UPB-OPT. The reason why the 2-approximation for UPB-OPT, i.e., UncertainPresentBiasApprox, does not work anymore is simple. Recall that the cost configuration ˜c returned by the algorithm lets the agent take shortcuts along cheapest paths to t. To ensure that these shortcuts do not become too expensive, ˜c assigns extra cost to their initial edge. This way the perceived cost within a shortcut should not be greater than that for entering. As long as the present bias is fixed, this works fine. However, if the present bias can change, the agent may become more biased within a shortcut and require higher rewards to stay motivated. One way to fix this problem is to let the assigned extra cost depend on τ , i.e., the range of B. More precisely, we multiply the cost assigned in line 9 of Algorithm 1 by τ and change line 5 to assign a cost of ˜c(e) = (1 + τ )α/b + 1. As a result we obtain a new algorithm VariablePresentBiasApprox with an approximation ration of 1 + τ .

### **Theorem 2.** *VPB-OPT admits a polynomial time* (1 + τ )*-approximation.*

Although VariablePresentBiasApprox yields a good approximation for a moderately variable present bias, it does not provide constant approximation bounds like UncertainPresentBiasApprox. Surprisingly, a sophisticated reduction from VECTOR SCHEDULING (VS) [4], shows that VPB-OPT cannot have an efficient constant factor approximation unless ZPP = NP.

**Theorem 3.** *No polynomial time algorithm can approximate VPB-OPT within a constant factor* > 1*, unless* NP = ZPP*.*

### **4.2 Occasionally Unbiased Agents**

Although VPB is hard to solve in general, a curious special case consisting of all present bias sets B for which 1 ∈ B is not. Note that agents whose present bias varies within such a B becomes temporarily unbiased whenever 1 is drawn. For this reason we call these agents *occasionally unbiased*. A behavioral pattern unique to occasionally unbiased agents is that they may start to walk along a cheapest path at any point in time whenever their present bias becomes 1. As a result we can reduce VPB to a decision problem we call CRITICAL NODE SET (CNS) for occasionally unbiased agents.

**Definition 6 (CNS).** *Given a task graph* G*, present bias set* B *and reward* r*, decide the existence of a critical node set* W*.*

We consider a node set W *critical* if the following properties hold: (a) s ∈ W. (b) Each node v ∈ W has a path P to t that only uses nodes of W. (c) All edges e of P satisfy db(e) ≤ br with b = min B. As it turns out, such a W contains exactly those nodes an occasionally unbiased agent may visit with respect to a motivating cost configuration. This allows us to reduce VPB to CNS.

**Proposition 3.** *If* 1 ∈ B*, then VPB has a solution if and only if CNS has one.*

**Algorithm 2.** DecideCriticalNodeSet δ(t) ← 0; **foreach** v ∈ V \ {t} **in reverse topological order do** U ← {w | (v, w) ∈ E and c(v, w) + βδ(w) ≤ b}; **if** U = ∅ **then** δ(v) ← ∞; **else** δ(v) ← min{c(v, w) + δ(w) | w ∈ U}; **if** δ(s) < ∞ **then return** "yes" **else return** "no";

All that remains to show is that CNS is decidable in polynomial time. A straight forward approach to this simple algorithmic problem is DecideCriticalNodeSet. We therefore conclude that VPB is efficiently solvable for occasionally unbiased agents.

**Corollary 4.** *If* 1 ∈ B*, then VPB can be solved in polynomial time.*

#### **4.3 The Price of Variability**

To conclude our work, we take a step back from computational considerations and look at the implications of variability from a more general perspective. Our goal is to quantify the conceptual loss of efficiency incurred by going from a fixed and known present bias to an unpredictable and variable one. Similar to the price of uncertainty we define the *price of variability* as the following ratio.

**Definition 7 (Price of Variability).** *Given a task graph* G *and a present bias set* B*, the price of variability is defined as* r(G, BV )/ sup{r(G, {β}) | β ∈ B}*.*

It seems obvious that the price of variability depends closely on the structure of G and B. Nevertheless, we would like to find general bounds for the price of variability much like we did in Sect. 3 for the price of uncertainty. As a first step, it is instructive to note that the price of uncertainty is a natural lower bound for the price of variability. The reason for this is that each cost configuration that motivates an agent whose present bias varies arbitrarily in B must also motivate an agent whose present bias is a fixed value from B. Therefore it holds true that r(G, BV ) ≥ r(G, B), which immediately implies the stated bound. Sometimes this bound is tight. Consider for instance Alice and Bob's scenario. As we have shown in Sect. 3, it is possible to construct a cost configuration ˜c verifying a price of uncertainty of 1. Using similar arguments, it is easy to see that ˜c remains motivating if we allow the present bias to vary, implying an identical price of variability. However, for general instances of G and B this tight relation between the price of uncertainty and the price of variability is lost. In fact, we can show that unlike the price of uncertainty, which has a constant upper bound of 2, the price of variability may become arbitrarily large as the range of B increases.

**Proposition 4.** *There exists a family of task graphs and present bias sets for which the price of variability converges to* τ /2*.*

Although Proposition 4 implies that the price of variability can become substantially larger than the price of uncertainty, it should be noted that the task graph constructed in the proof of this proposition is close to a worst case scenario. In particular, we can show that the price of variability cannot exceed τ +1, which is roughly twice the value obtained by Proposition 4. To verify this upper bound, it is helpful to recall the proof of Theorem 2. In the process of establishing the approximation ratio of VariablePresentBiasApprox we have argued that the cost configuration ˜c returned by the algorithm motivates any agent with a present bias configuration β ∈ BV for a reward of at most (τ +1)r(G, {min B}). Consequently, it holds true that r(G, BV ) ≤ (τ + 1)r(G, {min B}), implying that the price of variability cannot exceed τ + 1.

**Corollary 5.** *The price of variability is at most* τ + 1*.*

# **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.


**Author details**

Chang-Jin Ma1\* and Mariko Yamamoto1,2

\*Address all correspondence to<EMAIL_ADDRESS>
1 Department of Environmental Science, Fukuoka Women's University, Fukuoka, Japan

2 Fukuoka Prefectural Munakata High school, Munakata, Japan


*Perspective Chapter: Delivering LVAD Care to the Local Community DOI: http://dx.doi.org/10.5772/intechopen.111381*

occur in the local community, leading to improved access for patients with financial difficulties. This also improved patient compliance and diligence in attending all appointments, including seeking care as soon as issues arise. Other centers in the U.S. have demonstrated similar favorable outcomes with the shared-care model. A study published in JACC evaluated several key measures for LVAD patients implanted at the University of Utah Hospital. The authors included 336 patients implanted between 2007 and 2018, and categorized them based on level of care and resources utilized for their post implantation care. Overall, the rates of infection, bleeding, death, and pump thromboses were similar between care provided at the implanting institution (traditional care model) and outpatient care with LVAD specific training (the sharedcare model). Rates were higher when outpatient services that were utilized did not have LVAD specific training, highlighting the importance of ensuring proper LVADspecific training and resources are available at shared-care facilities [41].

There are no established criteria for shared care programs. However, a group of authors suggested several criteria that were published in the American Heart Association Journal Circulation, which focused on three broad categories: personnel, education, and equipment (**Table 1**). It is generally agreed that a shared care center should have an appointed local LVAD specialist—typically a heart failure cardiologist, though formal training in advanced heart failure is not a Joint Commission requirement for LVAD implantation facilities. Moreover, a nurse coordinator is another essential team member, who often serves as the primary contact for patients and other providers in the shared care team. Regarding education, LVAD specific training for all team members is essential. Preceptorships and LVAD vendors can provide the on-site education needed to learn the principles of LVAD management. Basic equipment should be garnered by the shared care facility and appropriate staff trained on its use. Device consoles, spare parts such as controllers, power cables, controller batteries, battery charges, and driveline equipment should ideally be available for timely replacement and troubleshooting [42].

Every follow-up visit starts with a review of systems concerning potential LVAD complications (**Table 2**). Next, is a blood pressure check which in continuous flow

#### **Table 1.** *Potential criteria for shared-care partners.*

#### **Table 2.**

*Symptoms in an LVAD patient.*

devices is typically taken as a mean arterial pressure obtained via doppler. This is done by inflating the sphygmomanometer to 20 mmHg above flow occlusion and the opening pressure is considered the systolic blood pressure if the patient has pulsatility. In absence of pulsatility, as in most LVAD patients, this is considered the MAP or mean arterial pressure. Recommended MAP is 60–80 mmHg [43] and MAPs outside of this range warrant intervention. One of the common alarms is "low flow" due to decrease in pump flow in both hyper and hypotension. Guideline directed medical therapy or neurohormonal antagonism must be continued to appropriately control blood pressure.

An ECG is recommended in every LVAD patient to evaluate for ventricular arrhythmias. Prompt attention to any signs and symptoms of right heart failure should warrant further investigation. Alarm interrogation and documentation is also necessary. Critical alarms include "VAD stopped" and "critical battery" of 5 minutes which display as a red triangle. A yellow flashing triangle necessitates evaluation by a LVAD specialist as it suggests the pump exceeds its power threshold. Non-flashing alerts require non-emergent evaluation and can be evaluated at the shared care center.

Additionally, laboratory tests are followed closely to ensure proper function of the LVAD. Regular INR checks, lactate dehydrogenase (LDH), and plasma free hemoglobin or hematocrit are measured regularly to monitor for adequate anticoagulation, as well as to monitor for hemolysis [44]. In addition, echocardiography is used to monitor ventricular function. Aortic valve insufficiency symbolizes inadequate function and leads to adjustment to LVAD parameters [45].

Driveline exit sites must be thoroughly inspected and suspicion of infection should be followed up with site culture, blood cultures, and appropriate imaging studies. This can be performed at the local shared care site as it does not require hardware manipulation. As infection rates are higher in this population and linked to a higher 1-year mortality, prompt treatment of infections with appropriate antibiotics is warranted [46]. GI bleeding is another complication that needs prompt evaluation and can be completed at any shared care site with GI services. As arteriovenous malformations (AVMs) are frequently the source of GI bleeding, urgent endoscopy can serve as diagnostic and therapeutic modality [47]. Reversal of INR is not ideal and requires careful monitoring as subtherapeutic states lead to pump thrombosis and failure. Neurologic complications, both hemorrhagic and cardioembolic strokes account for 19% mortality following an LVAD implant [12]. Due to its increasing prevalence, guidelines for LVAD patients were added to the cardiopulmonary resuscitation (CPR) algorithm in 2017. All staff at shared care centers must be able to recognize situations in which this protocol must be initiated. If the device hum is present, then controller function and adequate power must be confirmed. In the absence of the device hum, with MAP <50 mmHg, CPR should be initiated. However, with increasing time of pump discontinuation, the likelihood of pump thrombosis increases.

#### *Perspective Chapter: Delivering LVAD Care to the Local Community DOI: http://dx.doi.org/10.5772/intechopen.111381*

Extracorporeal membrane oxygenation (ECMO) can be considered in these scenarios if CPR is successful and further care should be transferred to a tertiary center [48].

Follow up care in an LVAD patient is key to a successful outcome (**Figure 2**). Houston Methodist, another implanting center who successfully employs the shared care model, developed an extensive protocol to be used at shared care sites and specified the steps taken for equipment checks from patient education to clinician verification [49]. Educating the new shared care site on management of LVAD visits and emergencies can be a challenge. There are no current guidelines on the training required by ancillary staff to conduct such visits. Allowing for this care to occur near a patient's home can prove to be a psychosocial advantage, encouraging family members involvement and support. If a regional center is trained in the basics of LVAD management, it would decrease inpatient transfers for non-LVAD related hospitalizations, thereby improving patient experience while increasing community hospital revenue [50].

The success of the shared care model can also allow former non-implantation sites to naturally evolve into implantation sites themselves. The lessons learned by its participation as a shared care site enabled DHLC to develop its own LVAD transplant program further expanding patient access to needed durable mechanical support. New implant centers will then develop their own selection committee for candidacy in accordance with the International Society of Heart and Lung Transplant Committee standards and guidelines. The patient is scored on the

**Figure 2.** *Follow up evaluation of LVAD patient.*

Interagency Registry for Mechanically Assisted Circulatory Support (INTERMACS) scale with 1 noting critical cardiogenic shock and 3 as stable on continuous intravenous inotropic support. INTERMACS 4 through 7 are not typically considered for LVAD as they have not yet reached the severity level required and are managed with medications. Once the case is reviewed by the team, patients with an INTERMACS score greater than two can choose the location for further work up for implantation with a plethora of tests such as echocardiograms, CT scans, and colonoscopies which are used to maximize the patient's status for an optimal outcome. Once a LVAD is implanted, follow up care is transferred back to the referring heart failure specialist who can continue to manage the patient in conjunction with the LVAD team. While heart transplantation remains the goal for some patients, others have been rejected due to age or other comorbidities. With the increasing demand for LVADs as destination therapy, more shared care networks will likely make the transition to implantation centers as well.


Author details

Selvaraj Rajesh1 \* and Gladima Nisia T.2

1 Department of Information Technology, Mepco Schlenk Engineering, Sivakasi, India

2 Department of Computer Science Engineering, AAA College of Engineering and Technology, Sivakasi, India

\*Address all correspondence to<EMAIL_ADDRESS>
© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


**3. Materials and methods**

**Figure 1.** Apparatus for forming highly dissolved gas in water.

210 *Escherichia coli* Escherichia coli - Recent Advances on Physiology, Pathogenesis and Biotechnological Applications - Recent Advances on Physiology, Pathogenesis and Biotechnological Applications

**Figure 2.** Representative pictures of liquid film formation with various nozzle diameters at a normal pressure in the pipeline.

## **3.1. Microorganism preparation and enumeration**

Stock cultures of *E. coli* (ATCC 11303) were propagated in Luria-Bertani (LB) broth (Wako Chemical Co., Ltd., Osaka, Japan) containing 30 g L−1 sodium chloride and incubated for 24 h at 37°C by using a reciprocal shaker set to rotate at 150 rpm. The initial enumeration was approximately 109 –1010 CFU mL−1. The permanent stock was maintained in 20% glycerol at −80°C.

The *E. coli* inoculum for each disinfection experiment was prepared by inoculating 100 μL of bacterial glycerol stock into 100 mL of LB broth containing 30 g L−1 sodium chloride. The culture was then incubated for 20 h at 37°C with continuous shaking at 150 rpm. Cells were harvested and washed three times with a 0.9% (w/v) saline solution followed by centrifugation (10 min at 8000 g at room temperature) in a CF15D2 centrifuge (Hitachi, Japan). The pellet was re-suspended in 100 mL saline solution.

*E. coli* were enumerated by using the plate count technique. Briefly, the samples were diluted into a series of 10-fold dilutions by using autoclaved artificial seawater at 3.4% salinity, and 100 μL of either a diluted or an undiluted sample was plated on LB agar (Wako). For samples with a low number of viable cells, 1 mL of the undiluted sample was poured into agar maintained at 45°C. Colonies growing on each plate were counted after incubating the plates overnight at 37°C. Each sample was analyzed in triplicate.

## **3.2. Seawater sample preparation**

The artificial seawater was prepared by adding artificial sea salt (GEX Inc., Osaka, Japan) to distilled water to obtain a final salinity of 3.4%, as measured with a salinity meter (YK-31SA, Lutron Electronic Enterprice Co., Ltd., Taiwan). As for the preparation of filtered natural seawater, natural seawater (pH = 8.3, salinity 3.3%) was first filtered through a glass fiber filter (GA-100, Advantec, Toyo); then, the seawater was filtered through a membrane filter with a pore size of 0.45 μm (Millipore, Ireland). For all experiments, prepared *E. coli* cultures were added into the artificial/filtered seawater to obtain a bacterial concentration of 5–6 log10 CFU mL−1. The solution was stirred for 30 min to acclimatize the bacteria before starting the experiments. For each batch mode operation, 12 L of samples were prepared, of which 4–5 L were used to restart the system. The pH and temperature of samples were measured with a pH meter (Horiba D-51, Japan).

### **3.3. Experimental setup**

Disinfection experiments were conducted in batch mode (**Figure 4**). Sample water, as the influent, was pumped in one shot into the device. Following the first influx of water, pressurized CO2 was also injected into the main chamber. System pressure was adjusted by a gas pressure regulator and gas exhaust valve. The fluid was then circulated by pumping inside the system for 25 min. A pump was used to apply a higher pressure than that inside the main chamber to accelerate gas solubilization in water. During the treatment period, the outer wall of the device was kept in contact with cool water by using a water jacket to maintain the initial temperature of the sample at ±1.0°C. The treated water was then collected from a bottom valve of the device.

*3.4.2. Experimental procedure for investigating the effect of pressure cycling*

pression [8, 10, 13, 15]. However, such high pressure and CO2

tions (8–550 MPa) and with CO2

**Figure 4.** Setup of the water treatment apparatus.

volume/flow rate.

sures (<1 MPa) and no discharge of CO2

In previous works, the pressure cycling procedure was conducted with high-pressure opera-

*Escherichia coli* Inactivation Using Pressurized Carbon Dioxide as an Innovative Method for Water Disinfection

an economic standpoint. In order to overcome the above disadvantages, in the present study, we employed a process involving pressure cycling for *E. coli* inactivation but used lower pres-

To investigate the effect of pressure cycling, two pumps (0.20 kW, Iwaya-WPT-202, Japan; 0.75 kW, 32 mm × 32 mm SUP-324 M, Toshiba, Japan) and nozzles with various sizes (15 mm height × 4–8 mm diameter) were used to change the flow rate and pressure power of the input (a treatment without a nozzle was also used, whereby the diameter of the pipeline inlet was 15 mm). Pumping pressure and system pressure were measured by pressure gages. The pressure difference ΔP = pumping pressure (MPa) − pressure inside the main chamber (MPa). The water flow rate was measured by a flow meter (GPI, Nippon Flow Cell Co., Ltd., Japan). The recycle number was calculated in relation to the treatment time and HRT, wherein HRT = sample

discharges between each cycle of decompression and com-

between each cycle of raised and lowered pressure.

release are undesirable from

http://dx.doi.org/10.5772/intechopen.68310

213

### **3.4. Procedure for disinfection experiments**

### *3.4.1. Experimental procedure for investigating the effects of pressure and temperature*

To investigate the effects of pressure and temperature, 7 L of sample were pumped into the main chamber by using a 0.2 kW pump (Iwaya-WPT-202), and the fluid was circulated inside the system at a flow rate of 14 L min−1 (hydraulic retention time, HRT = 0.5 min). The pump was used to apply 0.12 MPa higher pressure than that inside the main chamber. The sensitivity of bacteria to pressurized CO2 treatments under different conditions was determined by varying the CO2 pressure (0.2–0.9 MPa) and seawater temperature (11–28°C) for a 25-min treatment period [23]. Each experiment was conducted in triplicate.

*Escherichia coli* Inactivation Using Pressurized Carbon Dioxide as an Innovative Method for Water Disinfection http://dx.doi.org/10.5772/intechopen.68310 213

**Figure 4.** Setup of the water treatment apparatus.

*E. coli* were enumerated by using the plate count technique. Briefly, the samples were diluted into a series of 10-fold dilutions by using autoclaved artificial seawater at 3.4% salinity, and 100 μL of either a diluted or an undiluted sample was plated on LB agar (Wako). For samples with a low number of viable cells, 1 mL of the undiluted sample was poured into agar maintained at 45°C. Colonies growing on each plate were counted after incubating the plates over-

212 *Escherichia coli* Escherichia coli - Recent Advances on Physiology, Pathogenesis and Biotechnological Applications - Recent Advances on Physiology, Pathogenesis and Biotechnological Applications

The artificial seawater was prepared by adding artificial sea salt (GEX Inc., Osaka, Japan) to distilled water to obtain a final salinity of 3.4%, as measured with a salinity meter (YK-31SA, Lutron Electronic Enterprice Co., Ltd., Taiwan). As for the preparation of filtered natural seawater, natural seawater (pH = 8.3, salinity 3.3%) was first filtered through a glass fiber filter (GA-100, Advantec, Toyo); then, the seawater was filtered through a membrane filter with a pore size of 0.45 μm (Millipore, Ireland). For all experiments, prepared *E. coli* cultures were added into the artificial/filtered seawater to obtain a bacterial concentration of 5–6 log10 CFU mL−1. The solution was stirred for 30 min to acclimatize the bacteria before starting the experiments. For each batch mode operation, 12 L of samples were prepared, of which 4–5 L were used to restart the system. The pH and temperature of samples were measured with a pH

Disinfection experiments were conducted in batch mode (**Figure 4**). Sample water, as the influent, was pumped in one shot into the device. Following the first influx of water, pres-

pressure regulator and gas exhaust valve. The fluid was then circulated by pumping inside the system for 25 min. A pump was used to apply a higher pressure than that inside the main chamber to accelerate gas solubilization in water. During the treatment period, the outer wall of the device was kept in contact with cool water by using a water jacket to maintain the initial temperature of the sample at ±1.0°C. The treated water was then collected from a bottom valve

To investigate the effects of pressure and temperature, 7 L of sample were pumped into the main chamber by using a 0.2 kW pump (Iwaya-WPT-202), and the fluid was circulated inside the system at a flow rate of 14 L min−1 (hydraulic retention time, HRT = 0.5 min). The pump was used to apply 0.12 MPa higher pressure than that inside the main chamber. The sensitiv-

treatments under different conditions was determined by

pressure (0.2–0.9 MPa) and seawater temperature (11–28°C) for a 25-min

*3.4.1. Experimental procedure for investigating the effects of pressure and temperature*

treatment period [23]. Each experiment was conducted in triplicate.

was also injected into the main chamber. System pressure was adjusted by a gas

night at 37°C. Each sample was analyzed in triplicate.

**3.2. Seawater sample preparation**

meter (Horiba D-51, Japan).

**3.3. Experimental setup**

surized CO2

of the device.

varying the CO2

**3.4. Procedure for disinfection experiments**

ity of bacteria to pressurized CO2

### *3.4.2. Experimental procedure for investigating the effect of pressure cycling*

In previous works, the pressure cycling procedure was conducted with high-pressure operations (8–550 MPa) and with CO2 discharges between each cycle of decompression and compression [8, 10, 13, 15]. However, such high pressure and CO2 release are undesirable from an economic standpoint. In order to overcome the above disadvantages, in the present study, we employed a process involving pressure cycling for *E. coli* inactivation but used lower pressures (<1 MPa) and no discharge of CO2 between each cycle of raised and lowered pressure.

To investigate the effect of pressure cycling, two pumps (0.20 kW, Iwaya-WPT-202, Japan; 0.75 kW, 32 mm × 32 mm SUP-324 M, Toshiba, Japan) and nozzles with various sizes (15 mm height × 4–8 mm diameter) were used to change the flow rate and pressure power of the input (a treatment without a nozzle was also used, whereby the diameter of the pipeline inlet was 15 mm). Pumping pressure and system pressure were measured by pressure gages. The pressure difference ΔP = pumping pressure (MPa) − pressure inside the main chamber (MPa). The water flow rate was measured by a flow meter (GPI, Nippon Flow Cell Co., Ltd., Japan). The recycle number was calculated in relation to the treatment time and HRT, wherein HRT = sample volume/flow rate.

### *3.4.3. Experimental procedure for investigating the effect of the working volume ratio*

The WVR is defined as the ratio between the sample volume and apparatus volume. To examine the effect of WVR, different sample volumes (5, 6, 7, and 8 L) were used to vary the sample volume ratios (50, 60, 70, and 80%). The experiment was conducted with the following two flow rate levels: 14 and 25 L min−1. The water level was measured by using a gauge to evaluate the effect of WVR on the bubble-generating shield inside the main chamber. The HRT and recycle number were calculated as described in Section 3.4.2.

### **3.5. Scanning electron microscopy**

Changes in cell morphology after pressurized CO2 treatment were assessed by using SEM. The pellets of *E. coli* were immobilized with 2.5% glutaraldehyde in phosphate buffered saline (PBS) for 3 h at 4°C and then rinsed with PBS three times. Next, the samples were soaked in 1.0% osmium tetroxide in cacodylate buffer for 90 min and then washed three times with cacodylate buffer for removal of the fixative. After fixation, the cells were dehydrated by consecutive soaking in increasing concentrations of ethanol solutions (50, 70, 80, 90, 95, and 100%), and this was followed by an ethanol/t-butyl alcohol (v/v = 1:1) treatment for 30 min. The prepared cells were then soaked in t-butyl alcohol two times for 1 h, freeze-dried for 2 h, and sputter coated with gold-palladium. Finally, the cells were examined by using a scanning electron microscope (QuantaTM 3D, FEI Co., USA) at 20 kV [23].

### **3.6. Statistical analysis**

The statistical analysis was done by using the statistical computer program R (version 3.2.2, available at http://cran.R-project.org). Multicollinearity regression was performed to evaluate statistically significant variables of the system with a significance level of 0.05. Predicted values of inactivation efficacy were based on the following first-order regression model:

$$\mathbf{y}\_{i} = \boldsymbol{\beta}\_{o} + \sum \boldsymbol{\beta}\_{i} \mathbf{x}\_{i} \tag{1}$$

Approximately 5.4–5.7 log reductions of the *E. coli* load were achieved within 10–25 min by

Operating conditions: 0.3–0.9 MPa, 20 ± 1°C, and a working volume ratio (WVR) of 70%. Asterisks (\*) and (\*\*) indicate

*Escherichia coli* Inactivation Using Pressurized Carbon Dioxide as an Innovative Method for Water Disinfection

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215

only 0.4–0.9 log reductions were achieved after 25 min by the pressurized air treatment; these

after the first few minutes of exposure time, whereas the pH of pressurized air-treated seawater remained around 8.3 during the treatment period (**Figure 5b**). It has been hypothesized

bacterial inactivation process [12, 20, 21, 25]. However, Dang et al. [24] demonstrated that the low pH alone is not the main cause of the bactericidal activity. Perhaps with the concomitant

treatment (this involved complete inactivation of bacterial cells), whereas

and pressurized air on (a) *E. coli* inactivation and (b) the pH of seawater (SW).

reduced the pH of both filtered seawater and artificial seawater to around 5.0

is probably a major factor driving the

, the low pH prompted the *E. coli* cells to become more

penetration into the cells [24].

the pressurized CO2

**Figure 5.** Effect of pressurized CO2

Pressurized CO2

tests involved pressures of 0.3–0.9 MPa (**Figure 5a**).

that the *E. coli* load was completely inactivated after 25 and 10 min, respectively.

that the decrease in pH caused by pressurized CO2

permeable, thereby stimulating the process of CO2

presence of pressure and dissolved CO2

where *y*i represents the predicted responses, *x*i is a parameter, β0 is the model intercept, and βi is the linear coefficient.


aged care 302; providers in Australia 308; services 303; transitions 310–311 Age-friendly Cities and Communities (AFCC) project: Cocagne, Canada 426–428; definition 420; Mormal community, France 424–426; participatory approaches 430–432; Quebec, Canada 421–424; Wallonia, Belgium 428–430 age-friendly village 353 "Ageing in Europe" international conference 371 ageism 83, 126, 264, 273; countering 244; definition of 233–234; propensity for 301; reducing 82–83 Agency for Quality of Life (AVIQ) 428 agency, productive forms of 81 Aktion Demenz Moosburg (Action Dementia Moosburg) 68 Alfred Kinsey's report 214 all-party parliamentary groups (APPGs) 398 allusions, cultural use of 98 Alzheimer Society of Canada 316 Alzheimer Society Research Program (ASRP) in Canada 317 analysis skills 47, 50, 52 An apple is a complicated fruit (case study) 128–130 Anganwadi (pre-school) 114 Anglia Ruskin University in Cambridge 50 anxiety 166, 222, 274, 314, 324 APFA *see* Aula Permanente de FormaciónAbierta (APFA) APPGs *see* all-party parliamentary groups (APPGs) applied ageing research: citizen science 70; communication strategies and exchange formats 70–73; concepts, goals, and research design 61–62; description of 59–60; equitable inclusion and ongoing engagement 64–65; interdisciplinary research strategy, development of 62–64; PAR 68–70, 73–74; participation, stages of 60, 66–68; project goals 61, *61*; reaching hard-to-reach 65; stakeholder analysis 60–61 appreciative inquiry 48–49 Archipelago Lab 217, 220 "Archipelago of Generations" educational course 216–217 artifacts 113, *114,* 115, 210–211; design of 205 artificial intelligence 359–361 arts-based activities 115 arts-based approaches 127–128, 248 arts-based methods 113, 126, 142, 248

ATAT project *see* Adapt Tech, Accessibly Technology (ATAT) project

audio and community radio production 266

audio approaches 249

50; improvement 316; personnel 207; processes of 199, 201; research on 4, 88; safety and quality of 308; services 89–90; in sustainable communities 88

CareComLabs project 406; co-creation potentials and pitfalls 354–355; collective pride, sense of 356; COVID-19 as obstacle 357; objective 352; personal reflexions 355–356; phases 353; pillars 352

caring community 352, 354, 356; academyled development 407; assembling local allies 407–408; awareness, creation of 408; Bachdorf case study 407–409; qualitative interview study 408–409; technology-based innovations 384–385

Caring Community Living Labs (CareComLabs) 406

programme 234–236; co-production (*see* co-production); digital forms of 262; discourses of 263; examples of 244; form of 264; further 242–243; gerontological research on 263; in later life 269; methods contributing to critical pedagogy 239–241; participatory action research and 262–263; political types of 263; traditional forms of 262, 264

climate change, casualties of 107

161, 340, 346, 349; proposals 164; in public policy discourse 159; qualitative methodology of 341; questions 101–102; research 5, 341, 348–350; *see also* CareComLabs project; Socially Excluded Older Adults: Voices and Experiences (SEVEN)

co-creative knowledge exchange 337

co-productive relationships 336

researcher-older people collaboration 88; singularities 89

desired areas of participation **37** desires, elicit needs and *see* elicit needs and desires developing/advancing research questions 22 development project 70 diary/diaries: completion 203; interview method 202; narratives in 203 DigiGen project: communication, age groups 362; communication, challenges in 360–361; design and implementation 361–362; digitalization and artificial intelligence 359; dropouts 362; lessons learned 363–364; participatory approach 358–360; positive and negative experiences 360–363; recruitment and persistence 362; time as asset 360 digital citizenship 262–264, 270; concept of 263–264; focusing on 267 digital communication 273, 285 digital competence 175, 177 digital divide/exclusion 391; ATAT project 391–399; description 390; older adults' experiences 392–393 digital environments, creation of 270 digital interface 112 digitalization 358–359 Digital Literacy Project (DLP) 127–128 digital participation 267 digital platforms 200, 264 digital poverty 392–393 digital skills 262–264 digital spaces 262, 264, 270 digital technologies 263–264, 267, 269–270 discourse analysis, qualitative analytic tools for 98 discrimination 264, 343–344, 346; against older people 82; prevention of 291 discussion groups 86–87, 235, 422–423 dissemination 248, 251, 307; forms of 255, 258; participatory approaches to 249; participatory forms of 251; of research 316 divergences of views 163 diversity 284, 305; sensitive participation 66 DIY media making 251 dressmaking skills 241–242 dynamic reflexivity model 21, 26–28, *27* ecological principle 152 Economic and Social Research Council (ESRC) 275; Festival of Social Science 279; funded research project 249 economic deprivation 92, 236 education: in civic inclusion 233; opportunity 85; role in exclusion 232–233; social interaction and 177 edutainment 175; concept of 176

effective engagement 315, 323 effective project communication 62 effective public involvement 332 EIT Food RIS Consumer Engagement Labs project 158, 166 elicit needs and desires: description of 199; design research 200; home care 200–201, 205–207; medications and packaging 201; methodological challenges **211,** 212; product ideas 207–212; solicited diaries and photography 201–205; user participation, levels and modes of 200 emergent nature consumers 161 emotional labour 24 empirical analyses, shortage of 217 employing citizen science for older adults: 'Adapt Tech, Accessibly Technology' (ATAT) project 391; benefits of approaches 396; co-production 393–395; description of 390–391; ethical approval and informed consent 391; experiences of digital exclusion 392–393; lessons learned 396–398 empowering partnership 322 empowerment 71; of older adults 2–3; processes of 113; questions of 258–259 Empowerment Project 332–334, 337–338; experiences of 333; partners 335 energy: technologies 108, 119; transformation process 108; transition 107 Engineering and Physical Sciences Research Council (EPSRC) 108 EPSRC *see* Engineering and Physical Sciences Research Council (EPSRC) equal inclusion 66 equitable inclusion 60, 64–65, 68 equitable partnership 5, 19, 216, 218 equity 65–66, 71, 105, 270 ergonomic equipment 205 ESRC *see* Economic and Social Research Council (ESRC) ethical sensitivity and responsibility 322 "ethic of care" framework 46 ethnic groups 234 European Commission 289 evidence-based knowledge 291 evidence-based practice 308 evidence-informed practice, continuous improvement of 300 evidence to practice 308–309 exclusion: consequences of 96; in older age 231; role of education in 232–233; from social relations 244; socio-cultural and civic forms of 231; socio-cultural aspects of 233–234; socio-cultural dimensions of 243

experiential expertise 232 experiential knowledge 3, 404, 406 expert analyses, verifying and enriching 45–46 Expert Panel (EP) group members 112

	- making 250–252; description of 248–249; empowerment and community voice, zine-making and questions of 258–259; research context 249–250; zine-making process, value of 255–258; zines as products 252–255

#### *Index*

inclusive research 1, 17, 19–20, 27, 93 in-depth interview (IDI) 220 indigenous elders, co-creating social measures with: co-created questions 101–102; culturally specific scale, developing 98–99; description of 92–93; development of further co-created scales 102–103; initial study co-creation meetings 99–101; participatory inquiry into social indicators 95–96; participatory process 96–97; socially excluded cultural groups 93–94; social measures and scales 94 individualism/individuality/individuals 93, 302–303, 340–341; loss of 274; memory 138; participants 153; senior researchers 219; skills 82; volunteer 280 inequalities, risk of reproducing 165–166 informal education 72, 85, 215, 226 information and communications technologies (ICT) functional skills 175 Information and Consent process 344–345 informed advocacy 309–310 informed consent 148, 166, 188, 223, 314, 391, 409 informed participatory research 141 initial content analysis 46 Innovation in Dementia and Aging (IDEA) lab 317–318 Institute for Applied Research on Ageing (IARA) 59–60, 68, 71, 73–74 Institute of Applied Social Sciences 216–217, 222 institutional agendas 82 institutions 284 interdisciplinary managers 64 interdisciplinary participatory ageing research, principles for 73–74 interdisciplinary research strategy, development of 62–64 interest in science 35 intergenerational caregiving 88 intergenerational co-creation 139 intergenerational collaboration 88, 127 intergenerational experiences 154, 156 intergenerational groups, participants in 89 intergenerational learning project 359–360; *see also* DigiGen project intergenerational perception 87 intergenerational project "DigiGen" in Germany: description of 358; lessons learned from 364–365; positive and negative experiences 360–363; project's participatory approach 358–360 intergenerational research 153, 267, 359 intergenerational working groups 88–89 intergroup contact hypothesis 274

international research project 248 interpersonal intimacy 185 interpersonal reflexivity 24 intervention, forms of 234 intra/intergenerational interactions 147 involvement of older people 2, 4–5, 188, 291, 293–295, 315; critical reflection on 20; in findings' dissemination 6 Israel's Bar-Ilan University Brookdale Programme 50 itinerary 129, 138, 334–335

Just Therapy 341

```
know-do gap 306
```
knowledge 182; authority of 92; creation 248; creators 60; disseminating and translating 249; effective methods of transferring 176; exchange 249; and experiences 215; of experts-byexperience 17–18; of experts-by-training 17; generation 20; and perspectives 348–349; production of 3, 17, 23, 61; of sexual practices 223; sharing of 216; and skills 53, 182; of smartphones/iPhones 175; of technologies 164; transfer 113, 308; translation 251; utilization 337

La Fameuse 129, *130*

language, use of 337

library services, usefulness of 50

The Lipstick Caryatids (case study) 137, *138*

literacy 130, 144

#### *Index*

National Health System (NHS) 2

negative stereotyping 291

neighborhood websites 296

non-academics 23; perspective and experiences

of 22; in research 17; scope of

224–226; views and needs of 32 older individuals, sexual problems of 225–226 older learners 84, 241

older people, research and 81; agenda-setting process 82; reducing ageism 82–83; socio-cultural context of 244

PA *see* productive ageing (PA)

paper-and-pencil-based survey 33

PAR *see* Participatory Action Research (PAR)

2; voices and experiences of older coresearchers 11 participatory design (PD) 1, 393; divergent interests of 201 participatory research 32–34, 65, 70–71, 302; advantages of 69–70; approaches 68–69; challenges of 216; community members in 196; culture of 183; definition of 35; negative and positive aspects of **40**; obstacles **38**; practice, inclusivity of 27; projects 41; style 2–3; topics for **38** Patient and Family Engagement Committee 317 patient and public involvement (PPI) 1 Patient-Centered Outcomes Research Institute (PCORI) 2 patient engagement activities 315 Patient Engagement In Research Scale (PEIRS) 319 patient involvement, power of 325 patient-oriented research (POR) 316, 324; literature on 326; possibilities for 316 patient partners 315, 318–321, 325; individualized assistance for 319–320; during lockdowns 322; perspectives of 316; in research 319 Pays de Mormal Communauté de Communes (CCPM) 424 PD *see* participatory design (PD) peer researcher training (PRT) 22–24 peer research methodology 293–297 peer-reviewed journals 307 pension 59 people living with dementia 46, 66, 248–249, 316, 331–337; everyday lives of 11; experiences of neighbourhoods for 10; graphic magazines with (*see* graphic magazines with people living with dementia) people with disabilities 317 personal contacts in public health 110 personal diaries 201–202 personal enrichment 79 personal reflexivity 23 person-centred aged care 302 person-centred interviews 148 phone befriending: calls, impact of 190, **190**; content and methodology 188–189; conversations 188; methodology 189, 195–196; regular activities 186; service design of 196; sustainable and effective 185–186 photography 201–202; approaches 48; method 202–204; process of writing solicited diary 204–205 photovoice method 5, 48

physical artifacts, refinement of 159

physical mobility 162

	- Involvement and Engagement (PIPERS)

planned learning 110

Principles and Strategy for Public Involvement and Engagement (PIPERS) 113, 115–116

	- 323; partners 62

Promenade Parlante project *135*; An apple is a complicated fruit (case study) 128–130; background 126; The Cabot Square project (case study) 131–133; conceptual approach 126–127; The Crazy Walk! (case study) 130; description of 125–126; Lipstick Caryatids (case study) 137; living memory and artsbased approaches 127–128; Manoir du Souvenir (case study) 134–137, *136*; public performance of 131; The Seville: a phantasmagoria (case study) 134–136; shifting mentorship 133

safe care 205 Science Year 2022 375 SDAP *see* social development action plan (SDAP) SDG *see* sustainable development goals (SDG) secondary analysis 46; context of 53; of interview/focus group data 46 secular institutions 93 self-analysis 48 self-care routines 202, 210 self-discovery 225 self-expression 200 semi-structured interviews 142, 144, 147, 154 senior researchers 219, 222, 225; conviction of 221; female 226; social competencies of 219 Senior University Zurich (UZH3) 353 sense living libraries 284–285 sense of agency 224, 307 sense of belonging 185–186 sense of ownership 294–295 sense of security 185–186 senses framework 303 SEOA *see* socially excluded older adults (SEOA) service: improvement 302; provider 302–303, 308; provision 341; users 45, 337 SEVEN project *see* Socially Excluded Older Adults: Voices and Experiences (SEVEN) The Seville: a phantasmagoria (case study) 134–136, *136* sex/sexual/sexuality 217, 223, 225; activity 214–215, 217, 222, 226; and ageing 225; behaviours 214, 220; discussions 221–222; experiences 220; health 214; during interview 223; life 215, 223–224; old age 214; orientation research 350; of people 214; practices 220–221, 224–225; subject of 216–217 shared authority 139 shared learning 108 "shoebox" workshop 128 singularities: CUIDÁNDONOS 89; Frank J. manning certificate in gerontology 84; OFECUM 86–87 skills of establishing relations 221 Smarter Health Care 406 smartphones 177; challenges and barriers, usage 394; handling 371; icons used 397–398 Smart VitAALity project 62; interconnections **63,** 64; knowledge exchange **63,** 64; outline **63**; research implementation strategy **63,** 64 SNSF *see* Swiss National Science Foundation (SNSF)

society 199; active participation in 251

topics of 38, **38**; limitations 41; national citizen science survey 33; negative and positive aspects of 39, **40**; obstacles perceived by people **39,** 39–40; previous participation in research projects 35; research interest 32–33


**Author details**

Marise Neijman, R.T. Karsten, L. van der Molen, O. Lapid2 and M.W.M. van den Brekel1,3\*

1 Netherlands Cancer Institute, Amsterdam, The Netherlands

2 Amsterdam University Medical Centre, University of Amsterdam, The Netherlands

3 University of Amsterdam, The Netherlands

\*Address all correspondence to<EMAIL_ADDRESS>
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Lipofilling in Post-Treatment Oral Dysfunction in Head and Neck Cancer Patients DOI: http://dx.doi.org/10.5772/intechopen.101824*


**3. Imposing Condition and Geometry**

Initially, (when *t* = 0) the FBF is at rest in an infinite circular pipe with radius *R*( > 0) as shown in Figure 1. After time *t* = 0+, the pipe suddenly starts to rotate about its Z-axis having the angular velocity Ω*tp*. According to mathematical situation, appropriate initial and boundary conditions are

$$F(\mathbf{r},0) = \frac{\partial F(\mathbf{r},0)}{\partial t} = 0, \ \mathbf{r}(\mathbf{r},0) = 0; \ \mathbf{r} \in [0,R]. \tag{8}$$

$$F(\mathbb{R}, t) = R\Omega t^p; \qquad t > 0, \quad p \in \mathbb{N}, \ p > 0,\tag{9}$$

where Ω denotes the angular constant and *N* is the set of natural numbers.

**Figure 1.** Geometry of the problem.

#### *3.1. Velocity Field Due to Rotating Circular Pipe*

By applying the Laplace transform to Equations (5) and (9),

$$\frac{\partial^2 \overline{F}(\mathbf{r}, s)}{\partial \mathbf{r}^2} + \frac{1}{\mathbf{r}} \frac{\partial \overline{F}(\mathbf{r}, s)}{\partial \mathbf{r}} - \left[\frac{1}{\mathbf{r}^2} + a(s)\right] \overline{F}(\mathbf{r}, s) = 0,\tag{10}$$

and

$$\mathbb{F}(\mathbb{R}, s) = \frac{R \Omega p \, ! \,}{\text{s}^{p+1}} \, , \tag{11}$$

where

$$a(\mathbf{s}) = \frac{s(1 + \lambda\_1^{\alpha}\mathbf{s}^{\alpha} + \lambda\_2^{2\alpha}\mathbf{s}^{2\alpha})}{\nu(1 + \lambda\_3^{\beta}\mathbf{s}^{\beta})}.$$

Also *F*(r,*s*) and *F*(*R*,*s*) are the Laplace transforms of the functions *F*(r, *t*) and *F*(*R*, *t*) respectively. By applying the variable transformation *z* = r*a*(*s*) in Equation (10)

$$z^2 \frac{\partial^2 \overline{F}}{\partial \mathbf{r}^2} + z \frac{\partial \overline{F}}{\partial \mathbf{r}} - (\mathbf{1}^2 + z^2)\overline{F} = 0. \tag{12}$$

The above Equation (12) is the modified Bessel equation of order 1. So, the general solution of this equation is given by the successive equation

$$\overline{F}(z,s) = \mathbb{C}\_1 I\_1(z) + \mathbb{C}\_2 K\_1(z),\tag{13}$$

where *C*1 and *C*2 are constants and *I*1, *K*1 are the modified Bessel functions of first and second kind of order 1. For a finite solution at r = 0(*z* = 0), the constant *C*2 should be zero i.e., *C*2 = 0. So, Equation (13) implies

$$\mathsf{F}(z,s) = \mathsf{C}\_1 I\_1(z). \tag{14}$$

The value of *C*1 is calculated from Equation (14) by taking Equation (11) into account

$$\mathcal{C}\_1 = \frac{\Omega R p!}{s^{p+1} I\_1(\mathcal{R}\sqrt{a(s)})}.\tag{15}$$

Therefore the Equation (14) implies

$$\overline{F}(\mathbf{r}, \mathbf{s}) = \frac{\Omega R \, p!}{\mathbf{s}^{p+1}} \times \frac{I\_1(\mathbf{r}\sqrt{a})}{I\_1(R\sqrt{a})}.\tag{16}$$

The solution in Equation (16) is in the complex form of the first and second kind of Bessel functions. Normally, it is difficult to find out the solution of such complex expressions with an ordinary inverse Laplace transformation. However, an alternately inverse Laplace of Equation (16) was numerically calculated by using the "Stehfest's algorithm" [27]

$$f\_n(\mathbf{x}) = \ln(2)\mathbf{x}^{-1} \sum\_{i=1}^{2n} a\_i(n) F(i\ln(2)\mathbf{x}^{-1}), \qquad n \ge 1, \mathbf{x} > 0,\tag{17}$$

where *ai*(*n*) are coefficients defined as follows

$$a\_i(n) = \frac{(-1)^{n+i}}{n!} \sum\_{s=\lfloor \frac{(i+1)}{2} \rfloor}^{\min(i,n)} s^{n+1} \, ^\text{\textquotedblleft C}\_s \, ^\text{\textquotedblleft C}\_{i-s} \, ^\text{\textquotedblright C}\_{i-s} \, \qquad n \ge 1, 1 \le i \le 2n,\tag{18}$$

and "MATHCAD" software.

#### *3.2. Shear Stress Due to Rotating Circular Cylinder*

By taking the Laplace transform of Equation (6)

$$\overline{\mathbb{Q}}(\mathbf{r},s) = \sqrt{b(s)} \left[ \frac{\partial \overline{\mathbb{F}}(\mathbf{r},s)}{\partial \mathbf{r}} - \frac{1}{\mathbf{r}} \mathbb{F}(\mathbf{r},s) \right],\tag{19}$$

where

$$b(\mathbf{s}) = \frac{\mu(1 + \lambda\_3^{\beta}\mathbf{s}^{\beta})}{(1 + \lambda\_1^{\alpha}\mathbf{s}^{\alpha} + \lambda\_2^{2\alpha}\mathbf{s}^{2\alpha})}.$$

Putting Equation (16) into (17) and after taking the few steps of simplification,

$$\overline{Q}(\mathbf{r},s) = \frac{\sqrt{b}}{s^{p+1}} \times \frac{p!\Omega R}{I\_1(R\sqrt{a})} \left[ \sqrt{a}I\_0(\mathbf{r}\sqrt{a}) - \frac{2}{r}I\_1(\mathbf{r}\sqrt{a}) \right]. \tag{20}$$

Again it is in a complicated form. So "Gaver Stehfest's algorithm" and "MATHCAD" software were used for the inverse Laplace transformation of the result given in Equation (20) .

#### **4. Results and Discussions**

In this article, the main goal is to establish the numerical technique to develop the solutions of the Burgers' fluid for the rotational flow of Burgers' fluid within cylindrical domain. The expressions of velocity and shear stress are found for an incompressible non-integer order model of the Burgers' fluid in a circular pipe. The Laplace transformation is used to establish the solutions. As there are complicated expressions in Equations (16) and (20) with respect to the Laplace transformation, it was not easy to find out final results by using direct inverse Laplace on Equations (16) and (20). Although, "MATHCAD" is used along with the "Stehfest's algorithm" instead of the inverse Laplace transformation to find out the numerical results. The numerical solutions for Oldroyd-B, Maxwell, second grade and Newtonian fluids are also derived with generalization of main results i.e., Equations (16) and (20). To determine the impact of physical parameters, the graphical illustrations are made. The behaviour of time on the velocity field and stress is shown in Figures 2 and 3. These figures show that the influence of velocity and stress depending upon the parameters r, *λ*1, *λ*2, *λ*3, *α*, *β*, *ν* and *μ*. Figures 2 and 3 show that the velocity and shear stress increase with respect to the increase in the dependent variable *t* with fixed values of other dependent parameters. Similarly, a clear increase in velocity and stress can also be seen in Figures 4 and 5 for r with respect to the time variable *t* while fixing the other dependent parameters. It is also observed from Figure 4 that velocity has linear relation for r with respect to *t*.

**Figure 2.** The aspect of velocity *F*(r, *t*) for various values of *t* and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.7, *ν* = 0.053, *n* = 12, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 3.** The aspect of stress *Q*(r, *t*) for various values of *t* and fixed values for *μ* = 0.010, Ω = 2.5, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.7, *ν* = 0.053, *n* = 12, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 4.** The aspect of velocity *F*(r, *t*) for various values of r and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.7, *ν* = 0.053, *n* = 12, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 5.** The aspect of stress *Q*(r, *t*) for various values of r and fixed values for *μ* = 0.09, Ω = 1.5, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.7, *ν* = 0.053, *n* = 12, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

Also, to uncover the aspects of other physical and fraction parameters on velocity and stress the graphs drawn and presented in Figures 6–17. All the graphs for velocity and stress in Figures 6–17 are plotted for concerned parameter against the change factor "time". One can say that the fractional parameter *α*, relaxation parameters *λ*1 and *λ*2 have decreasing behaviour for velocity and stress with respect to *t*. This fact can be observed in Figures 6–9, 12 and 13 respectively. Whereas, the velocity and stress are also increasing functions for the retardation parameter *λ*3, fractional parameter *β*, viscosity *μ* and *ν* with respect to time *t* as shown in Figures 10, 11, 14–17. Figures 2–17 make it possible to check the point to point variations, increment or decrement in two parameters (among which the graphs are made) for velocity and stress profile.

Finally, the comparison among the different generalized cases of Burgers' fluid are made in Figure 18 for velocity function against time *t*. The results of these generalized cases are derived by the implementation of the "Stehfest's algorithm" and "MATHCAD" to Equation (16). It is observed that the second grade fluids have more velocity as compared to other Newtonian and non-Newtonian fluid

cases of this model while taking the same values of different dependent parameters and variation in time *t*. The Oldroyd-B fluid behaves as like the second grade fluid but have less velocity as compared to the second grade fluid's velocity. Similarly, Figure 18 also shows the behaviour of velocity for Newtonian and Maxwell fluids. The Maxwell fluids have minimum velocity for this flow model with prescribed conditions among all the Newtonian and non-Newtonian cases. As a description it is clearing that in all the Figures 2–18, the units of the material constants are SI units. The comparison of solutions that obtained from two different methods analytically [29,30] and numerically (derived from Equations (16) and (20)) are given in Tables 1 and 2 for two different cases Maxwell and Newtonian fluids. These tables showing clearly that two different methods for the same problem have the same results. Which shows the consistency of our numerical technique and results for fractional model of Burgers' fluid with already published literature [29,30].

**Figure 6.** The aspect of velocity *F*(r, *t*) for various values of *λ*1 and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.8, *ν* = 0.013, *n* = 12, r = 0.8, *λ*2 = 20 and *λ*3 = 55.

**Figure 7.** The aspect of stress *Q*(r, *t*) for various values of *λ*1 and fixed values for *μ* = 0.09, Ω = 2.5, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.8, *ν* = 0.013, *n* = 12, r = 0.8, *λ*2 = 20 and *λ*3 = 55.

**Figure 8.** The aspect of velocity *F*(r, *t*) for various values of *λ*2 and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.2, *β* = 0.7, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5 and *λ*3 = 55.

**Figure 9.** The aspect of stress *Q*(r, *t*) for various values of *λ*2 and fixed values for *μ* = 0.09, Ω = 1.2, *R* = 1, *p* = 1,*s* = 1, *α* = 0.2, *β* = 0.7, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5 and *λ*3 = 55.

**Figure 10.** The aspect of velocity *F*(r, *t*) for various values of *λ*3 and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.2, *β* = 0.8, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5 and *λ*2 = 20.

**Figure 11.** The aspect of stress *Q*(r, *t*) for various values of *λ*3 and fixed values for *μ* = 0.09, Ω = 1.2, *R* = 1, *p* = 1,*s* = 1, *α* = 0.2, *β* = 0.8, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5 and *λ*2 = 20.

**Figure 12.** The aspect of velocity *F*(r, *t*) for various values of *α* and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *β* = 0.7, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 13.** The aspect of stress *Q*(r, *t*) for various values of *α* and fixed values for *μ* = 0.09, Ω = 1.2, *R* = 1, *p* = 1,*s* = 1, *β* = 0.7, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 14.** The aspect of velocity *F*(r, *t*) for various values of *β* and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 15.** The aspect of stress *Q*(r, *t*) for various values of *β* and fixed values for *μ* = 0.09, Ω = 1.2, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 16.** The aspect of velocity *F*(r, *t*) for various values of *ν* and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.8, *n* = 12, r = 0.8, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 17.** The aspect of stress *Q*(r, *t*) for various values of *μ* and fixed values for Ω = 2.5, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.8, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 18.** Comparisons of fluid velocity for different fluid models for same value of parameters.

**Table 1.** Comparison of exact solutions [30] and numerical solutions (obtained from "Stehfest's algorithm") for the fractional order derivative model of the Maxwell fluid.

**Table 2.** Comparison of exact solutions [29] and numerical solutions (obtained from "Stehfest's algorithm") for the fractional order derivative model of the Newtonian fluid.


**15. Conclusions**

India is the world's largest democracy and an economic powerhouse. However, it is also home to stubborn poverty and some of the world's worst religious persecution, particularly with the rise of a nationalism that adversely affects devoutly religious individuals from both the majority and minority religious traditions. Might India's rich religious traditions and beliefs unleash a dynamism that could promote human progress, economic enterprise, and overall human flourishing? Any investigation into the positive potential of religion on human flourishing in India will require a steady and long-term investment in building capabilities and capacity in the country.

**Funding:** A significant portion of the research in this paper has been funded by the Templeton Religion Trust.


*Article* **Hydrothermal Pretreatment of Wheat Straw: Effects of Temperature and Acidity on Byproduct Formation and Inhibition of Enzymatic Hydrolysis and Ethanolic Fermentation**

**Dimitrios Ilanidis, Stefan Stagge , Leif J. Jönsson and Carlos Martín \***

Department of Chemistry, Umeå University, SE-901 87 Umeå, Sweden<EMAIL_ADDRESS>(D.I<EMAIL_ADDRESS>(S.S<EMAIL_ADDRESS>(L.J.J.)

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**Abstract:** Biochemical conversion of wheat straw was investigated using hydrothermal pretreatment, enzymatic saccharification, and microbial fermentation. Pretreatment conditions that were compared included autocatalyzed hydrothermal pretreatment at 160, 175, 190, and 205 ◦C and sulfuric-acid-catalyzed hydrothermal pretreatment at 160 and 190 ◦C. The effects of using different pretreatment conditions were investigated with regard to (i) chemical composition and enzymatic digestibility of pretreated solids, (ii) carbohydrate composition of pretreatment liquids, (iii) inhibitory byproducts in pretreatment liquids, (iv) furfural in condensates, and (v) fermentability using yeast. The methods used included two-step analytical acid hydrolysis combined with high-performance anion-exchange chromatography (HPAEC), HPLC, ultra-high performance liquid chromatographyelectrospray ionization-triple quadrupole-mass spectrometry (UHPLC-ESI-QqQ-MS), and pyrolysisgas chromatography/mass spectrometry (Py-GC/MS). Lignin recoveries in the range of 108–119% for autocatalyzed hydrothermal pretreatment at 205 ◦C and sulfuric-acid-catalyzed hydrothermal pretreatment were attributed to pseudolignin formation. Xylose concentration in the pretreatment liquid increased with temperature up to 190 ◦C and then decreased. Enzymatic digestibility was correlated with the removal of hemicelluloses, which was almost quantitative for the autocatalyzed hydrothermal pretreatment at 205 ◦C. Except for the pretreatment liquid from the autocatalyzed hydrothermal pretreatment at 205 ◦C, the inhibitory effects on *Saccharomyces cerevisiae* yeast were low. The highest combined yield of glucose and xylose was achieved for autocatalyzed hydrothermal pretreatment at 190 ◦C and the subsequent enzymatic saccharification that resulted in approximately 480 kg/ton (dry weight) raw wheat straw.

**Keywords:** wheat straw; hydrothermal pretreatment; bioconversion inhibitors; enzymatic hydrolysis; ethanolic fermentation