8. Surgery and IP/HIPEC chemotherapy for recurrent ovarian cancer

The Cochrane review on the use of intraperitoneal (ip) chemotherapy for primary OC [58] concluded that this treatment prolonged PFS and OS. While there is evidence of a survival benefit for IP chemotherapy/HIPEC after cytoreductive surgery in primary disease, there are fewer reports on its use and efficacy in recurrent disease [59]. No mention was made of this type of treatment in the Cochrane review on recurrent ovarian cancer [60] nor in the review by the Fifth Ovarian cancer Consensus Conference of the Gynecologic Cancer InterGroup [7].

Boisen et al. [61] reported on a retrospective study of 25 patients treated with iv/ip chemotherapy but without secondary cytoreductive surgery. The study period was over 6 years on a selected group of patients and 10 of 25 had an improved treatment-free interval. In a feasibility study of ip chemotherapy in 56 patients with platinum-sensitive recurrent disease all of whom had had prior secondary cytoreductive surgery (67.9% to <1 cm), 79% tolerated 6 cycles of ip platinum. No difference in outcome was noted related to the completeness of secondary surgery and the median overall survival was 51 months; no clinical factors associated with improved PFS or OS were identified [62]. The data from other studies report that the main indicators for response to ip chemotherapy are (1) volume of residual disease and (2) platinumsensitive disease [63, 64]. Fujiwara, in contrast reported responses in patients with suboptimal surgical resection [65].

Ansaloni et al. [66] provided one of the first reports on HIPEC following cytoreductive surgery in 30 patients with recurrent disease. In this small study, HIPEC was considered safe and there was a trend to improved survival with complete cytoreduction and HIPEC. A more recent study [67] reported a survival benefit in what they described as randomised trial on the use of HIPEC in recurrent ovarian cancer. However, there were a number of deficiencies in study design and questions were raised about the validity of the results and the efficacy of HIPEC as reported in that study [68]. In another retrospective review [69], Cripe et al. reported on 32 patients that CRS and HIPEC were feasible. However, they also noted 65.6% grade 3 or 4 toxicity (morbidity) and that troublesome pleural effusions were associated with diaphragmatic stripping and/or resection. As a number of chemotherapeutic agents were used with varying dwell times and temperatures, it is unclear what regimen to recommend. As with primary disease, a key component in the use of HIPEC is complete cytoreduction or minimal residual disease (<5 mm deposits). A recent report on a retrospective cases series from China on 46 patients with advanced (n = 16) or recurrent (n = 30) ovarian cancer reported a survival benefit with HIPEC but only when there was complete surgical cytoreduction [70]. However, the adjuvant treatment included iv/ip chemotherapy and it is not clear what contribution HIPEC and ip chemotherapy made to improved survival. In contrast, in a study on secondary cytoreductive surgery in EOC, 50 patients underwent surgery only and 29 also had HIPEC, although there were no deaths in the latter group and two in the former group, the addition on HIPEC did not confer an advantage on median disease-free survival [71]. Data were not presented, however, on overall survival or disease-specific survival. In a larger retrospective multi-centre Italian study on 226 patients with primary ovarian cancer and 285 with ROC treated over 16 years, HIPEC was of benefit in patients with ROC who had had complete surgical resection and platinum-sensitive disease [72]. In a large French study of HIPEC in primary and recurrent ovarian cancer, no difference was noted in overall survival between patients with platinum-sensitive and platinum-resistant disease and the main prognostic factor for survival and DFI was the extent of disease, or tumour burden, as measured on the peritoneal cancer index [73]. In the studies showing benefit of CSC and HIPEC, it is still unclear what, if any, additional benefit HIPEC can achieve over CSC. There is still ongoing debate about the role of HIPEC, with the view that HIPEC should be offered only in clinical trials [74]. In fact a number of trials of ip chemo and HIPEC in recurrent ovarian cancer are recruiting [75].

presence of metastatic disease at other sites. In assessing the role of specialised surgery for recurrent metastatic ovarian cancer, factors to be considered include—morbidity of surgery, likelihood of resecting disease, likelihood of palliating symptoms by surgical resection, and the patient's prognosis, with and without surgery. There is also some evidence that patients treated with IP chemotherapy and then subsequently with bevacizumab have a greater propensity to develop unusual sites of metastastic recurrence [80]. Patients with a BRCA mutation compared to those who do not have a BRCA mutation more often develop unusual sites of

Surgery for Recurrent Ovarian Cancer http://dx.doi.org/10.5772/intechopen.71587 285

Most patients with EOC present with advanced stage disease and most will develop recurrence. A common presentation of recurrent disease is relapsing and remitting bowel obstruction, the course of which is more often chronic than acute [81, 82]. Invariably the development of bowel obstruction indicates recurrent (or progressive) disease, even if the tumour markers are not elevated and there is no radiological evidence of disease. The management is conservative, at least initially with fasting, intravenous fluids and pharmacological manipulation [81, 82]. Involvement of the palliative care team is important. Surgical intervention is associated with significant morbidity and mortality and not all patients, perhaps only about two-thirds benefit from surgery in terms of resumption of adequate oral intake. Despite this common problem in recurrent ovarian cancer, QoL data on surgical and non-surgical intervention are

Surgical intervention includes—placement of a gastrostomy tube [83], by pass procedures, but most often formation of a diverting stoma. As the disease is often more extensive in the pelvis with serosal and mesenteric disease, more often an ileostomy is raised rather than a colostomy, although often when a loop ileostomy is performed it is necessary to defunction the large bowel by raising a mucous fistula. If a recto-vaginal fistula develops from extensive pelvic disease, a colostomy may provide successful palliation but typically to a limited extent. That is, the patient will continue to have other problems related to the pelvic disease—including pelvic pain, discharge and vaginal or rectal bleeding. It is important to discuss with the patient the likely palliative benefit of surgery, as it is to discuss the outcomes from the surgical and non-

There are fewer reports on the role of surgery for second, third, etc. relapse of EOC. Intuitively the factors that are important in surgical decision making for first recurrence should also be important in surgical decision making in patients with second and subsequent recurrence. It is clear too that if surgery is contemplated for such relapses the patients are highly selected and more often than not surgical intervention will be for palliation (e.g. bowel obstruction) rather than for complete cytoreduction. More usually in clinical practice patients with second and

10. Recurrent ovarian cancer and bowel obstruction

recurrence.

notably absent from most reports.

surgical management of bowel obstruction.

11. Surgery for second recurrence and beyond

#### 9. Recurrent ovarian cancer outside the abdomen and pelvis

With the improvement in overall survival in ovarian cancer, and better understanding of cancer genetics, targeted therapies and improved surgery, it is now more common to see patients with unusual or atypical sites of recurrent disease [76]. Sites include breast, brain, bone (including vertebral spine), chest wall, skin (other than port site metastasis) and lymph nodes such as the axillary nodes [77–79]. Given the unusual location of metastasis it is important to exclude other sites of disease and commonly PET-CT is used. Biopsy is often necessary to exclude another cancer. In contrast, histologic confirmation of recurrent OC in the pelvis and/or abdomen is not usual clinical practice. Management of the recurrence will include general supportive measures such as pain relief, radiotherapy (e.g. with vertebral metastasis) and chemotherapy, trial drugs and specialised surgery, for example, neurosurgery. The surgery may be indicated for symptom relief and may be considered necessary, even life-saving, in the presence of metastatic disease at other sites. In assessing the role of specialised surgery for recurrent metastatic ovarian cancer, factors to be considered include—morbidity of surgery, likelihood of resecting disease, likelihood of palliating symptoms by surgical resection, and the patient's prognosis, with and without surgery. There is also some evidence that patients treated with IP chemotherapy and then subsequently with bevacizumab have a greater propensity to develop unusual sites of metastastic recurrence [80]. Patients with a BRCA mutation compared to those who do not have a BRCA mutation more often develop unusual sites of recurrence.


**Table 3.**

*Biomarkers of oxidative stress in KD.*

*Role of Oxidative Stress in the Cardiovascular Complications of Kawasaki Disease DOI: http://dx.doi.org/10.5772/intechopen.86781*

defined as activated pro-coagulant platelets [74] and (ii) annexin V negative platelets, characterized by an increased mitochondrial membrane potential, prone to become pro-coagulant when in contact with adenosine diphosphate (ADP) and thromboxane, mediators normally released from activated platelets [75]. It has been hypothesized that in KD patients, activated pro-coagulant platelets could contribute to the increased thrombotic risk detected in these patients. Implication of oxidative stress in KD is depicted in **Figure 2**. Biomarkers of oxidative stress in KD patients are summarized in **Table 3**.


**1. Introduction**

An essential step in the development of viable quantum technologies is to achieve precise control over quantum dynamics [1,2]. In many situations, optimal performance relies on the ability to create particular target states. However, in dynamically reaching such states, the quantum adiabatic theorem [3] poses a formidable challenge since finitetime driving inevitably causes parasitic excitations [4–7]. Acknowledging and addressing this issue, the field of "shortcuts to adiabaticity" (STA) [8–11] has developed a variety of techniques that permit to facilitate effectively adiabatic dynamics in finite time.

Recent years have seen an explosion of work on, for instance, counterdiabatic driving [12–19], the fast-forward method [20–23], time-rescaling [24,25], methods based on identifying the adiabatic invariant [26–29], and even generalizations to classical dynamics [30–32]. For comprehensive reviews of the various techniques, we refer to the recent literature [9–11].

Among these different paradigms, counterdiabatic driving (CD) stands out, as it is the only method that forces evolution through the adiabatic manifold at all times. However, experimentally realizing the CD method requires applying a complicated control field, which often involves non-local terms that are hard to implement in many-body systems [15,17]. This may be particularly challenging if the system is not readily accessible, due to, for instance, geometric restrictions of the experimental set-up.

In the present paper, we propose an alternative method to achieve transitionless quantum driving by leveraging the system's (realistically) inevitable interaction with the environment. Our novel paradigm is inspired by "envariance," which is short for entanglement-assisted invariance. Envariance is a symmetry of composite quantum systems, first described by Wojciech H. Zurek [33]. Consider a quantum state |*ψ*SE - that lives on a composite quantum universe comprising the system, S, and its environment, E. Then, |*ψ*SE - is called envariant under a unitary map *u*S ⊗ IE if and only if there exists another unitary IS ⊗ *u*E acting on E such that the composite state remains unaltered after applying both maps, i.e., (*u*S ⊗ IE )|*ψ*SE - = |*φ*SE - and (IS ⊗ *u*E )|*φ*SE - = |*ψ*SE -. In other words, the state is envariant if the action of a unitary on S can be inverted by applying a unitary on E.

**Citation:** Touil, A.; Deffner, S. Environment-Assisted Shortcuts to Adiabaticity. *Entropy* **2021**, *23*, 1479. https://doi.org/10.3390/e23111479

Academic Editor: Ronnie Kosloff

Received: 16 October 2021 Accepted: 5 November 2021 Published: 9 November 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Envariance was essential to derive Born's rule [33,34], and in formulating a novel approach to the foundations of statistical mechanics [35]. Moreover, experiments [36,37] showed that this inherent symmetry of composite quantum states is indeed a physical reality, or rather a "quantum fact of life" with no classical analog [34]. Drawing inspiration from envariance, we develop a novel method for transitionless quantum driving. In the following, we will see that instead of inverting the action of a unitary on S, we can suppress undesirable transitions in the energy eigenbasis of S by applying a control field on the environment E. In particular, we consider the unitary evolution of an ensemble of composite states {| *ψ*SE -} on a Hilbert space HS ⊗ HE of arbitrary dimension, and we determine the general analytic form of the time-dependent driving on HE , which suppresses undesirable transitions in the system of interest S. This general driving on the environment E guarantees that the system S evolves through the adiabatic manifold at all times. We dub this technique environment-assisted shortcuts to adiabaticity, or "EASTA" for short. In addition, we prove that the cost associated with the EASTA technique is exactly equal to that of counterdiabatic driving. We illustrate our results in a simple two-qubit model, where the system and the environment are each described by a single qubit. Finally, we conclude with discussing a few implications of our results in the general context of decoherence theory and quantum Darwinism.

#### **2. Counterdiabatic Driving**

We start by briefly reviewing counterdiabatic driving to establish notions and notations. Consider a quantum system S, in a Hilbert space HS of dimension *d*S, driven by the Hamiltonian *H*0(*t*) with instantaneous eigenvalues {*En*(*t*)}*n*∈-0, *d*S−1 and eigenstates {|*n*(*t*)-}*n*∈-0, *d*S−1. For slowly varying *H*0(*t*), according to the quantum adiabatic theorem [3], the driving of S is transitionless. In other words, if the system starts in the eigenstate |*n*(0)-, at *t* = 0, it evolves into the eigenstate |*n*(*t*)- at time *t* (with a phase factor) as follows:

$$|\psi\_n(t)\rangle \equiv \mathcal{U}(t)|n(0)\rangle = e^{-\frac{i}{\hbar}\int\_0^t \mathbb{E}\_n(s)ds - \int\_0^t \langle n|\partial\_s n\rangle ds}|n(t)\rangle \equiv e^{-\frac{i}{\hbar}f\_n(t)}|n(t)\rangle. \tag{1}$$

For arbitrary driving *H*0(*t*), namely for driving rates larger than the typical energy gaps, the system undergoes transitions. However, it was shown [12–14] that the addition of a counterdiabatic field *H*CD(*t*) forces the system to evolve through the adiabatic manifold. Using the following total Hamiltonian,

$$H = H\_0(t) + H\_{\infty}(t) = H\_0(t) + i\hbar \sum\_{n} (|\partial\_l n\rangle\langle n| - \langle n \mid \partial\_l n\rangle\langle n| \langle n|) ),\tag{2}$$

the system evolves with the corresponding unitary *U*CD(*t*) = ∑*n* |*ψn*(*t*)*n*(0)| such that the following holds:

$$
\langle \mathcal{U}\_{\rm cp}(t) | n(0) \rangle = e^{-\frac{i}{\hbar} f\_n(t)} | n(t) \rangle. \tag{3}
$$

This evolution is exact no matter how fast the system is driven by the total Hamiltonian. However, the counterdiabatic driving (CD) method requires adding a complicated counterdiabatic field *H*CD(*t*) involving highly non-local terms that are hard to implement in a many-body set-up [15,17]. Constructing this counterdiabatic field requires determining the instantaneous eigenstates {|*n*(*t*)-}*n*∈-0, *d*S−1 of the time-dependent Hamiltonian *H*0(*t*). Moreover, changing the dynamics of the system of interest (i.e., adding the counterdiabatic field) requires direct access and control on S.

In the following, we will see how (at least) the second issue can be circumvented by relying on the environment E that inevitably couples to the system of interest. In particular, we make use of the entanglement between system and environment to avoid any transitions in the system. To this end, we construct the unique driving of the environment E that counteracts the transitions in S.

#### **3. Open System Dynamics and STA for Mixed States**

We start by stating three crucial assumptions: (i) the joint state of the system S and the environment E is described by an initial wave function |*ψ*SE (0)- evolving unitarily, according to the Schrödinger equation; (ii) the environment's degrees of freedom do not interact with each other; (iii) the S-E joint state belongs to the ensemble of singly branching states [38]. These branching states have the following general form:

$$|\psi\_{\mathcal{SE}}\rangle = \sum\_{n=0}^{N-1} \sqrt{p\_n} |n\rangle \bigotimes\_{l=1}^{N\_{\mathcal{E}}} |\mathcal{E}\_n^l\rangle\_{\prime} \tag{4}$$

where *pn* ∈ [0, 1] is the probability associated with the *n*th branch of the wave function, with orthonormal states |*n*-∈HS and *N*E *l*=1 |E*ln*-∈HE .

Without loss of generality, we can further assume √*pn* = 1/√*N* for all *n* ∈ -0, *N* − 1 since if √*pn* = 1/√*N* we can always find an extended Hilbert space [33,34] such that the state |*ψ*SE - becomes even. Thus, we can consider branching states |*ψ*SE - of the simpler form as follows:

$$|\psi\_{\mathcal{SE}}\rangle = \frac{1}{\sqrt{N}} \sum\_{n=0}^{N-1} |n\rangle \bigotimes\_{l=1}^{N\_{\mathcal{E}}} |\mathcal{E}\_n^l\rangle. \tag{5}$$

In the following, we will see that EASTA can actually only be facilitated for even states (5). In Appendix B, we show that EASTA cannot be implemented for arbitrary probabilities (i.e., (∃ *n*); √*pn* = 1/√*N*).

#### *3.1. Two-Level Environment* E

We start with the instructive case of a two-level environment, cf. Figure 1. To this end, consider the following branching state:

$$|\psi\_{\mathcal{SE}}(0)\rangle = \frac{1}{\sqrt{2}}|\mathbf{g}(0)\rangle \otimes |\mathcal{E}\_{\mathbb{X}}(0)\rangle + \frac{1}{\sqrt{2}}|\varepsilon(0)\rangle \otimes |\mathcal{E}\_{\varepsilon}(0)\rangle,\tag{6}$$

where the states |E*g*(0)- and |E*e*(0)- form a basis on the environment E, and the states |*g*(0)- and |*e*(0)- represent the ground and excited states of S at *t* = 0, respectively.

It is then easy to see that there exists a unique unitary *U* such that the system evolves through the adiabatic manifold in each branch of the wave function as follows:

$$(\exists!\,\mathcal{U}');\ (\mathcal{U}\otimes\mathcal{U}')|\psi\_{\mathcal{S}\mathcal{E}}(0)\rangle = (\mathcal{U}\_{\text{cp}}\otimes\mathbb{I}\_{\mathcal{E}})|\psi\_{\mathcal{S}\mathcal{E}}(0)\rangle.\tag{7}$$

Starting from the above equality, we obtain the following:

$$\begin{split} \langle \mathcal{U}|\mathcal{G}(0)\rangle \otimes \mathcal{U}'|\mathcal{E}\_{\mathcal{S}}(0)\rangle + \mathcal{U}|\varepsilon(0)\rangle \otimes \mathcal{U}'|\mathcal{E}\_{\varepsilon}(0)\rangle &= e^{-\frac{i}{\hbar}f\_{\mathcal{S}}(t)}|\mathcal{G}(t)\rangle \otimes |\mathcal{E}\_{\mathcal{S}}(0)\rangle \\ &+ e^{-\frac{i}{\hbar}f\_{\varepsilon}(t)}|\varepsilon(t)\rangle \otimes |\mathcal{E}\_{\varepsilon}(0)\rangle. \end{split} \tag{8}$$

Projecting the environment E into the state "|E*g*(0)-", we have

$$
\langle \mathcal{U} | \mathcal{G}(0) \rangle \langle \mathcal{E}\_{\mathcal{S}}(0) | \mathcal{U}' | \mathcal{E}\_{\mathcal{S}}(0) \rangle + \mathcal{U} | \varepsilon(0) \rangle \langle \mathcal{E}\_{\mathcal{S}}(0) | \mathcal{U}' | \mathcal{E}\_{\mathcal{E}}(0) \rangle = e^{-\frac{1}{\hbar} f\_{\mathcal{S}}(t)} | \mathcal{g}(t) \rangle,\tag{9}
$$

equivalently written as

$$
\langle (\mathcal{U}'\_{\mathcal{S}\circ \mathcal{S}}) \mathcal{U} | \mathcal{g}(0) \rangle + (\mathcal{U}'\_{\mathcal{S}^c}) \mathcal{U} | \varepsilon(0) \rangle = e^{-\frac{i}{\hbar} f\_{\mathcal{S}}(t)} | \mathcal{g}(t) \rangle,\tag{10}
$$

which implies the following:

$$(\mathcal{U}'\_{\mathcal{S},\emptyset})|\mathcal{g}(0)\rangle + (\mathcal{U}'\_{\mathcal{S},\mathcal{e}})|\mathcal{e}(0)\rangle = e^{-\frac{i}{\hbar}f\_{\mathcal{S}}(t)}\mathcal{U}^{\dagger}|\mathcal{g}(t)\rangle. \tag{11}$$

Therefore,

$$\mathcal{U}'\_{\mathcal{G},\mathcal{E}} = e^{-\frac{i}{\hbar}f\_{\mathcal{E}}(t)} \langle \mathcal{g}(0) | \mathcal{U}^{\dagger} | \mathcal{g}(t) \rangle,\text{ and }\mathcal{U}'\_{\mathcal{G},\mathcal{E}} = e^{-\frac{i}{\hbar}f\_{\mathcal{E}}(t)} \langle e(0) | \mathcal{U}^{\dagger} | \mathcal{g}(t) \rangle. \tag{12}$$

Additionally, by projecting E into the state "|E*e*(0)-" we obtain the following:

$$\mathcal{U}\_{\varepsilon,\xi}^{\prime} = e^{-\frac{i}{\hbar}f\_{\varepsilon}(t)} \langle g(0) | \mathcal{U}^{\dagger} | e(t) \rangle, \text{ and } \mathcal{U}\_{\varepsilon,\varepsilon}^{\prime} = e^{-\frac{i}{\hbar}f\_{\varepsilon}(t)} \langle e(0) | \mathcal{U}^{\dagger} | e(t) \rangle. \tag{13}$$

It is straightforward to check that the operator *U*, which reads as follows:

$$\mathcal{U}' = \begin{pmatrix} \mathcal{U}'\_{\mathcal{K}:\mathcal{S}} & \mathcal{U}'\_{\mathcal{K}:\mathcal{E}} \\ \mathcal{U}'\_{\mathcal{C}:\mathcal{S}} & \mathcal{U}'\_{\mathcal{C}:\mathcal{E}} \end{pmatrix} \tag{14}$$

is indeed a unitary on E.

In conclusion, we have constructed a unique unitary map that acts only on E, but counteracts transitions in S. Note that coupling the system and environment implies that the state of the system is no longer described by a wave function. Hence the usual counterdiabatic scheme evolves the density matrix *ρ*S (0) to another density *ρ*S (*t*) such that both matrices have the same populations and coherence in the instantaneous eigenbasis of *H*0(*t*) (which is what EASTA accomplishes, as well).

(**b**) Environment-assisted shortcut scheme.

**Figure 1.** Sketch of the two different schemes of applying STA in a branching state of the form presented in Equation (6). In both panels, the state preparation involves a Hadamard gate (H) applied on S, and coupling with the environment through a c-not operation. In panel (**a**), we describe the "usual" counterdiabatic scheme. As shown in Section 3, local driving on E suppresses any transitions of S in the instantaneous eigenbasis of *H*0(*t*). The latter scheme is illustrated in panel (**b**).

#### *3.2. N-Level Environment* E

We can easily generalize the two-level analysis to an *N*-level environment. Similar to the above description, coupling the system to the environment leads to a branching state of the following form:

$$|\psi\_{S\mathcal{E}}(0)\rangle = \frac{1}{\sqrt{N}}\sum\_{n=0}^{N-1}|n(0)\rangle \otimes |\mathcal{E}\_n(0)\rangle,\tag{15}$$

where the states {|E*n*(0)-}*n* form a basis on the environment E. We can then construct a unique unitary *U* such that the system evolves through the adiabatic manifold in each branch of the wave function as follows:

$$<\langle \exists! \, \mathcal{U} \rangle ; \ (\mathcal{U} \otimes \mathcal{U}') | \psi\_{\mathcal{S} \mathcal{E}}(0) \rangle = (\mathcal{U}\_{\text{cp}} \otimes \mathbb{I}\_{\mathcal{E}}) | \psi\_{\mathcal{S} \mathcal{E}}(0) \rangle. \tag{16}$$

The proof follows the exact same strategy as the two-level case, and we find the following:

$$(\forall \ (m, n) \in [0, N - 1]^2); \ l\!\!/^{\prime}\_{m, n} = e^{-\frac{i}{\hbar}f\_m(t)} \langle n(0) | \!\!/ \!L^{\dagger} | m(t) \rangle. \tag{17}$$

The above expression of the elements of the unitary *U* is our main result, which holds for any driving *H*0(*t*) and any *N*-dimensional system.

#### *3.3. Process Cost*

Having established the general analytic form of the unitary applied on the environment, the next logical step is to compute and compare the cost of both schemes: (a) the usual counterdiabatic scheme and (b) the environment-assisted shortcut scheme presented above (cf. Figure 1). More specifically, we now compare the time integral of the instantaneous cost [39] for both driving schemes [39–43], (a) *C*CD(*t*)=(1/*τ*) *t* 0 \$*H*CD(*s*)\$*ds* and (b) *C*env(*t*)=(1/*τ*) *t* 0 \$*H*env(*s*)\$*ds* (\$.\$ is the operator norm), where the driving Hamiltonian on the environment can be determined from the expression of *U* (*t*), *H*env(*t*) = *ih*¯ *dU* (*t*) *dtU*†(*t*).

In fact, from Equation (17) it is not too hard to see that the field applied on the environment *H*env(*t*) has the same eigenvalues as the counterdiabatic field *H*CD(*t*), since there exists a similarity transformation between *H*env(*t*) and *H*∗CD(*t*). Therefore, the cost of both processes is exactly the same, *C*CD = *C*env, for any arbitrary driving *H*0(*t*). Details of the derivation can be found in Appendix A. Note that for *t* = *τ*, the above definition of the cost becomes the total cost for the duration "*τ*" of the process.


*4.1. Batch Mode EM Algorithm*

The batch mode parameter restoration using EM algorithm is quite simple and has been utilized in many researches. A dominated way to do this is using the well-known Baum-Welch algorithm. This is an algorithm that makes the expectation step and maximization step recursively. Here we simply describe how to extend the expectation and maximization steps to SemiTMC-GMM model, within one iteration of the EM algorithm. A simple recall of the forward and backward calculations [7] are displayed below:

$$\begin{split} a\_{1}(k) &= p\left( (\mathfrak{v}\_{1}, d\_{1}) = k, \mathfrak{y}\_{1} \right), \\ a\_{n}(k) &= \sum\_{(\mathfrak{v}\_{n-1}, d\_{n-1}) \in \Lambda \times \Gamma \times L} a\_{n-1}(k) \cdot p\left( (\mathfrak{v}\_{n}, d\_{n}) | (\mathfrak{v}\_{n-1}, d\_{n-1}) \right) \cdot p\left( \mathfrak{y}\_{n} | (\mathfrak{v}\_{n}, d\_{n}) \right), \\ \beta\_{N}(k) &= 1, \\ \beta\_{n}(k) &= \sum\_{(\mathfrak{v}\_{n+1}, d\_{n+1}) \in \Lambda \times \Gamma \times L} p\left( (\mathfrak{v}\_{n+1}, d\_{n+1}) | (\mathfrak{v}\_{n}, d\_{n}) \right) \cdot p\left( \mathfrak{y}\_{n+1} | (\mathfrak{v}\_{n+1}, d\_{n+1}) \right) \cdot \beta\_{n+1}(k). \end{split} \tag{12}$$

In the above equations, the *αn*(*k*) and *βn*(*k*) are the forward and backward calculations, *p* ((*vn*, *dn*)|(*vn*−1, *dn*−1)) is the state transition probability that has been described in Equations (8)–(10), *p* (*yn*|(*vn*, *dn*)) = *p* (*hn*|*vn*) *p* (*yn*|*vn*, *hn*) is the conditioned observation density.

Then, the algorithm requires the following probabilities:

$$\gamma\_n(k) = p((\sigma\_n, d\_n) = k | \mathcal{y}\_1^N) = \frac{a\_n(k)\beta\_n(k)}{\sum\_{k' \in \Lambda \times \Gamma \times L} a\_n(k')\beta\_n(k')},\tag{13}$$

$$\tilde{\gamma}\_{\mathbb{H}}(i) = \sum\_{d\_{\mathbb{H}}} \gamma\_{\mathbb{H}}((i, d\_{\mathbb{H}})) = \sum\_{d\_{\mathbb{H}}} p(\mathfrak{v}\_{\mathbb{H}} = i, d\_{\mathbb{H}} | y\_1^N), \tag{14}$$

$$\tilde{\gamma}\_n(i,j) = \tilde{\gamma}(i) \cdot \frac{c\_{ij}p(y\_n|\mathbf{v}\_n = i, h\_n = j)}{\sum\_{j' \in K} c\_{ij'} p(y\_n|\mathbf{v}\_n = i, h\_n = j')},\tag{15}$$

$$\mathfrak{F}\_{\mathfrak{H}}(l,k) = \frac{a\_{\mathfrak{n}}(l) \cdot p\left(y\_{\mathfrak{n}+1}h\_{\mathfrak{n}+1}(\mathfrak{v}\_{\mathfrak{n}+1},d\_{\mathfrak{n}+1}) - k \vert y\_{\mathfrak{n}}h\_{\mathfrak{n}}(\mathfrak{v}\_{\mathfrak{n}},d\_{\mathfrak{n}}) - l\right) \cdot \mathfrak{h}\_{\mathfrak{n}+1}(k)}{\sum\_{\mathfrak{l}',\mathfrak{l}' \in \Lambda \times \Gamma \times \perp} \left\{ a\_{\mathfrak{n}}(l') \cdot p\left(y\_{\mathfrak{n}+1}h\_{\mathfrak{n}+1}(\mathfrak{v}\_{\mathfrak{n}+1},d\_{\mathfrak{n}+1}) - k' \vert y\_{\mathfrak{n}}h\_{\mathfrak{n}}(\mathfrak{v}\_{\mathfrak{n}},d\_{\mathfrak{n}}) - l'\right) \cdot \mathfrak{h}\_{\mathfrak{n}+1}(k')\right\}}. \tag{16}$$

*γn*(*k*) is the probability of (*vn*, *dn*) conditioned on all observed data *yN* 1 . *γ*˜*n*(*k*) is the marginal probability of *γn*(*k*) over *dn*, this probability is the one that we are looking for to estimate the concerning hidden state *vn*. *γ*˜*n*(*i*, *j*) is the probability of each Gaussian component *w.r.t. γ*˜*n*(*k*); this probability helps to compute the parameters related to Gaussian mixture, i.e., *ckj*, *μkj*, Σ*kj*. *ξn*(*l*, *k*) is the joint probability of (*vn*, *dn*) = *l* and (*vn*+1, *dn*+1) = *k* conditioned on *yN* 1 . Here we give the formula of parameter update by using Equations (13)–(16):

$$
\zeta\_k = \gamma\_1(k),
\tag{17}
$$

$$a\_{lk} = \frac{\sum\_{n=1}^{N-1} \xi\_n(l,k)}{\sum\_{n=1}^{N-1} \gamma\_n(l)},\tag{18}$$

$$c\_{i\bar{j}} = \frac{\sum\_{n=1}^{N} \tilde{\gamma}\_n(i, j)}{\sum\_{n=1}^{N} \tilde{\gamma}\_n(i)},\tag{19}$$

$$\boldsymbol{\mu}\_{ij} = \frac{\sum\_{n=1}^{N} \tilde{\gamma}\_n(\boldsymbol{i}, \boldsymbol{j}) \boldsymbol{y}\_n}{\sum\_{n=1}^{N} \tilde{\gamma}\_n(\boldsymbol{i}, \boldsymbol{j})},\tag{20}$$

$$\Sigma\_{ij} = \frac{\sum\_{n=1}^{N} \tilde{\gamma}\_n (i\_\prime j) (y\_n - \mu\_{ij})^\top (y\_n - \mu\_{ij})}{\sum\_{n=1}^{N} \tilde{\gamma}\_n (i\_\prime j)}. \tag{21}$$

In fact, Equations (13)–(16) are the expectation step in one iteration of EM algorithm, while Equations (17)–(21) are the maximization step. Then, the parameter can be learned by recursively performing the two steps until the iteration number exceeds a pre-defined value, maximally 100 iterations, for example.


*Article* **Solvent Effect on the Structure and Properties of RGD Peptide (1FUV) at Body Temperature (310 K) Using Ab Initio Molecular Dynamics**

**Khagendra Baral 1 , Puja Adhikari 1 , Bahaa Jawad 1 , Rudolf Podgornik 2,3,4,5 and Wai-Yim Ching 1,\***

**Abstract:** The structure and properties of the arginine-glycine-aspartate (RGD) sequence of the 1FUV peptide at 0 K and body temperature (310 K) are systematically investigated in a dry and aqueous environment using more accurate ab initio molecular dynamics and density functional theory calculations. The fundamental properties, such as electronic structure, interatomic bonding, partial charge distribution, and dielectric response function at 0 and 310 K are analyzed, comparing them in dry and solvated models. These accurate microscopic parameters determined from highly reliable quantum mechanical calculations are useful to define the range and strength of complex molecular interactions occurring between the RGD peptide and the integrin receptor. The in-depth bonding picture analyzed using a novel quantum mechanical metric, the total bond order (TBO), quantifies the role played by hydrogen bonds in the internal cohesion of the simulated structures. The TBO at 310 K decreases in the dry model but increases in the solvated model. These differences are small but extremely important in the context of conditions prevalent in the human body and relevant for health issues. Our results provide a new level of understanding of the structure and properties of the 1FUV peptide and help in advancing the study of RGD containing other peptides.

**Keywords:** RGD peptide (1FUV); ab initio molecular dynamics; total bond order; partial charge; dielectric function


**3. Controller synthesis method**

54 Petri Nets – Manufacturing and Computer Science

arc of the node *M* in *G*

 *M*

oriented cycles of *G.*

where is an oriented cycle of *G*,

from happening at *M* by a place *p* iff

**2.2. Theory of regions and synthesis problem13**

The theory of regions is proposed for the synthesis of pure nets given a finite TS15, which can be adopted to synthesize the liveness-enforcing net supervisor (LENS) for a plant model12-13.

 Assume that there exists a node *v* in *G* such that there exists a path from it to any node. The objective of the theory of regions is to find a pure PN (*N*, *M*0), having *T* as its set of transitions and characterized by its incidence matrix [*N*](*p*, *t*) and its initial marking *M*0, such that its reachability graph is *G* and the marking of node *v* is *M*0. In

Consider any marking *M* in net (*N*, *M*0). Because (*N*, *M*0) is pure, *M* can be fully

vector of path *M* . For any transition *t* that is enabled at *M*, i.e., *t* is the label of an outgoing

Consider now any oriented cycle of a reachability graph. Applying the state equation to a

*N pt t C*

( )*t* is a firing vector corresponding to

[ ]( , ) ( ) 0,

result, *M* can be arbitrarily chosen. The reachability of any marking *M* in *G* implies that

0 *Mp M p N p M G* ( ) ( ) [ ]( , ) 0, *M*

The above equation is called the reachability condition. Notably, (3) is necessary but not

It is clear that the cycle equations and reachability conditions hold for any place *p*. For each pair (*M*, *t*) such that *M* is a reachable marking of *G* and *t* is a transition not enabled at *M*, *t* should be prevented from happening by some place *p*. Since the net is pure, *t* is prevented

> 0 *M p N p N pt* ( ) [ ]( , ) [ ]( , ) 1 *M*

, (*M*, *M*

(2)

where *M*

) *G M* [ *t* > *M'* (1)

. There are several paths from *M*0 to *M*.

is the same for all these paths. As a

(3)

*M* from *M*0 to *M*. Applying (1)

(4)

is the firing

*,* and *C* is the set of

For convenience, our method follows the interpretation of the theory of regions in13.

the following, *M* denotes both a reachable marking and its corresponding node in *G*.

characterized by its corresponding incidence vector [*N*](*p*, ) *M*

(*p*) = *M*(*p*) + [*N*](*p*, ) *MM*

node in and summing them up give the following cycle equation:

*t T*

According to the definition of *G*, there exists an oriented path

sufficient. Hence, spurious markings are beyond this paper.

along the path leads to *M*(*p*) = *M*0(*p*) + [*N*](*p*,) *M*

Under the cycle equations, the product [*N*](*p*,) *M*

In this section, an efficient controller synthesis method is developed based on the theory of regions. Please note that all transitions of the PN models are regarded as controllable ones.


*3.5. Sustainability*

Sustainability was examined in relation to organizational conditions of leadership, staff participation, resources, and communication.

**Leadership** was vital for the sustainability of participatory programs. Administrative and supervisor support for wellness towards PIP teams and employees were indicated in many forms within the qualitative data. It included financial support, enabling staff to take time off for meetings and activities, providing meeting space, and verbal encouragement.

Both management and supervisor support were cited in the qualitative data by many employees and managers to be important for program sustainability. The effects of administrator support (and turnover) were mentioned as important in encouraging or discouraging team meetings and activities. Support from management (administrators and nursing directors) and from supervisors (department heads and unit/office managers) were examined separately.

Overall, the NPHP centers demonstrated higher management support than the PIP centers. Administrators in PIP centers discussed HP as part of their everyday work and considered their wellness champions and teams as part of their organizational structure (except in I-2).

Supervisor support was perceived to be higher in two PIP centers, I-1 and I-3, and low at I-2. At the NPHP center C-2, many employees described their supervisors as being supportive to HP by allowing flexibility in their staff schedules and even covering for them while they participated in activities. Other supervisors within this center were described to be supportive by participating in activities themselves and motivating their staff to participate.

PIP teams were able to provide more activities for the staff members with the presence of leadership support. When leadership support was absent, management in these centers talked about being faced with other pressures, needing to make decisions in favor of other more urgent projects and programs.

Management changes in two PIP centers led to combining the existing PIP teams with other, previously inactive committees ("staff appreciation committee" in site I-1, and "fun committee" in I-3). In both cases, the non-supervisory PIP teams were converted to administrator-directed committees with different agendas and priorities.

**Sta**ff **participation** in programs and activities is obviously a key measure of program impact as well as likely sustainability. Perceived lack of staff participation in PIP teams and activities was evaluated from the PIP team member interviews and focus groups.

All PIP team members stated in the focus groups that clinical staff had difficulty in getting time to attend team meetings or activities. Staff participation in team-sponsored activities was poor within center I-2. Researcher experience and logs showed that participation in PIP teams had been quite high at the start of the project, yet involvement of clinical staff diminished due to frustrations with their decision-making process and clinical responsibilities.

Employee participation in all three PIP teams dropped to a low number of non-supervisory clinical staff at the end of year 3 of the intervention. Low participation of clinical staff in activities were attributed to staffing shortages, time constraints, and clinical care responsibilities.

**Resources** mentioned in the interviews and focus groups included financial support from the corporation, in-house/in-kind personal effort, and outside support. Although wellness was a stated goal for both the PIP and NPHP centers, no funds were allocated in any of the facilities' budgets for employee wellness except in the first year (2008), when the corporation provided \$700 per year for each facility. After this line item was dropped, most facilities engaged in regular fund-raising for their programs and many of the key informants expressed frustration at the lack of funds for their programs.

In the PIP centers, several projects and especially the higher cost projects simply could not be implemented without adequate resources. At one PIP site (I-2), much of their past activity had focused on fund-raising, which detracted from effort and time that could otherwise have been spent on health and wellness activities.

Using existing in-house resources was an opportunity for the PIP teams to benefit from the knowledge and skills of individual staff members. At Center I-1, one of the nurses offered yoga classes and another offered massages; at I-3, the dietitian offered many in-house programs and services including a weight loss program, potlucks, and healthy recipes.

Team members in the PIP centers identified researcher involvement and guidance as a key outside resource and a vital link to sustainability of the teams. At center I-2, most members believed that the PIP team would not continue without researcher support. The research team provided material support to all three PIP centers, including ergonomic training sessions to staff members, tools and seedlings for the garden, and an exercise class instructor.

Among NPHP centers, the primary forms of outside support cited were discounts for gym membership, health information, and free health screenings by group health insurance companies. In Center C-2, respondents also cited outside support from community programs along with insurance company services.

**Communication** was mentioned in the qualitative data as factors that are essential for sustaining a participatory health program.

Communication between the PIP teams or wellness champions with management as well as with the employees was uneven in both the PIP and NPHP centers. Employees in the PIP centers reported having good communication except at the I-2 center, where both the employees and managers expressed lack of communication as one of the largest stressors and most critical barriers to successful program implementation. The administrator at center I-2 concurred about the lack of communication between the PIP team and management. There appeared to be a trend towards better communication in centers where there was focused attention on work organization issues. In centers where closed-door management meetings were opened to staff or where the wellness champion utilized several methods of communication (e.g., memos, flyers, announcements during meetings, etc.), it was viewed as a smaller problem or not a problem at all.