*2.1.2 Ionic gelation*

This method is based on the electrostatic interaction of the two aqueous phases and the nanospheres are prepared from natural polymers. Polyelectrolytes form hydrogel beads by cross-linking with counterions. These hydrogel beads are also called gelispheres. Gelispheres spreads into the polymer structure and forms a cross-linked cage. In this way, biomolecules can be trapped in this lattice structure. Examples of synthetic monomers and polymers used in this method include hydroxyethyl methacrylate, N-vinyl-2-pyrrolidone, N-isopropylacrylamide, vinyl acetate, acrylic acid, polyethylene glycol acrylate/methacrylate, polyethylene glycol diacrylate/dimethacrylate. Chitosan, gelatin, alginate fibrin, collagen, hyaluronic acid, dextran can be used as a natural polymer. Multivalent cations are generally

magnesium, sodium, calcium, potassium, ferric, aluminum, barium and zinc ions. In the experimental stage, polymer and electrolyte concentrations, pH, temperature and biomolecule concentrations are important parameters to be considered.

by this method have more ability to convert poorly soluble and poorly absorbed

*The Components of Functional Nanosystems and Nanostructures*

to the literature. **Figure 4** schematizes the synthesized gold nanospheres.

barriers, cosmetics and many other areas [1].

**2.2 Quantum dots**

Enhanced lithium storage

Hydrogen production and photocatalytic activity

For sensing trace cysteine in

For application in water

For high-performance supercapacitors

*Some examples of nanospheres, according to the literature.*

Water treatment and improved lubricating performance

properties

HeLa cells

treatment

**Table 1.**

**117**

Nanospheres are used effectively in controlled drug delivery systems, tumortargeted treatment methods, tumor-targeted treatment methods, epithelial cell therapy, genetic engineering studies, treatment methods targeting blood–brain

Quantum mechanics is the starting point of nanotechnology. These nano-sized semiconductor crystals are called quantum dots. Quantum dots are giant atomic structures that contain thousands of atoms. When substances are nano-sized, they

**Purpose of usage Nanospheres References**

For lithium-ion batteries Pyrite/carbon nanospheres [12]

For PSA detection Polydopamine nanospheres loaded with L-cysteine-

coated cadmium sulfide quantum dots

Ultra-small ZnFe2O4 nanosphere [10]

High specific surface area TiO2 nanosphere [11]

Gold-silver nanospheres [13]

Alkyl-capped copper oxide nanospheres [14]

MoO3 nanospheres [16]

CuS nanospheres [17]

[15]

Particularly particle size and surface distribution properties are essential characteristic analyzes for the characterization of nanospheres. Because particle sizes and surface distributions also illuminate the properties of in vivo distribution, biological fate, toxicity and targeting ability. For the characterization of these properties, scanning electron microscopy (SEM), transmission electron microscopy (TEM), photon correlation spectroscopy methods are used. The Zeta potential analysis provides insightful information about load stability and particle collection. Zetasizer is used for this analysis. Fourier transform infrared spectroscopy (FTIR) analyses are performed to reveal the chemical bonding between the active substance and the polymers. The physical state of the active substance in the nanospheres is determined by differential scanning calorimetry (DSC) analysis after the lyophilization of the nanospheres. It can be applied many analysis methods such as in vitro drug release studies, drug release kinetics and stability study. Stability studies help to examine the effects of nanospheres on the physicochemical parameters of their formulations. In this way, suitable storage conditions are determined. Absorption and storage at room temperatures are performed for approximately 6 months and the observed changes in physicochemical parameters are recorded. Appropriate storage conditions are determined according to the results obtained. **Table 1** shows the intended use of the nanospheres synthesized according

drugs into better deliverables.

*DOI: http://dx.doi.org/10.5772/intechopen.92027*

### *2.1.3 Salting out*

This method is based on a salt interaction such as a polymer, organic solvent, and magnesium chloride hexahydrate or magnesium acetate tetrahydrate. An emulsification mechanism is performed. Salting out method is based on increasing hydrophobic effect as a result of electronic repulsion of dissolved anions with highdensity loads. This resulting hydrophobic effect increases the uniformity of the water-soluble intermediate phase structure, reduces entropy and causes agglomeration of the solvent. This is because the presence of high-charge salts in the system decreases entropy by increasing the regularity between similar surfaces, this result is also desirable. Anions reducing water solubility; OH ≈ SO4 2 ≈ CO3 2 > ClO3 ≈ Cl ≈ OAc ≈ IO3 > Br ≈ I > NO3 and the cations: Na+ > K+ > Li+ ≈ Ba2+ ≈ Rb+ ≈ Ca2+ ≈ Co2+ ≈ Mg2+ ≈ Fe2+ ≈ Zn2+ ≈ Cs+ ≈ Mn2+ ≈ Al3+ > NH4+ > H+ .

#### *2.1.4 Controlled gelification*

In this method, gelation is formed using calcium chloride and sodium alginate. A suitable mixture of these two compounds results in gelling. Poly-L-lysine is added to the resulting solution as a polymer and a polyelectrolyte mixture is formed by mixing. Subsequently, nanospheres are synthesized by centrifugation [8].

#### *2.1.5 Solvent evaporation*

This method is based on the principle of emulsifying the active substance in the polymer and an organic solvent and removing the solvent by reducing the temperature and pressure. In this method, polyvinylchloride or gelatin may be used as the emulsifying agent.

#### *2.1.6 Solvent displacement*

It is one of the most widely used methods for the synthesis of nanoparticles. The solvent displacement is based on the displacement of a semi-polar solvent with the polymer interface. In this method, the organic phase containing the active substance and the polymer structure in the aqueous phase is self-emulsified. The polymer and active ingredient are dissolved in an organic solvent such as watermiscible ethanol, methanol or acetone. The organic phase is injected into the aqueous phase containing the active ingredient. The nanospheres are synthesized by precipitating the polymer in which the organic phase is dispersed in the aqueous phase.

#### *2.1.7 Desolvation technique*

This method is particularly preferred for obtaining nanospheres from natural polymers. The active substance is added to the polymer and solvent, and crosslinking is performed. Crosslinking agents must be added to effect crosslinking. The suspension is lyophilized by centrifugation, and the nanospheres are synthesized [9].

The most common and advantageous method used in the synthesis of nanospheres is the solvent displacement method. This method will provide great advantages especially in controlled drug release systems. Nanospheres synthesized


**3. Radiographic evaluation**

The classification and subsequent treatment of acetabular fractures are based on imaging studies that have been derived from a thorough understanding of the

**Figure 6.** *The greater sciatic notch is divided by the piriformis.*

#### *Essentials in Hip and Ankle*

anatomy of the innominate bone [17–19]. Two limbs of an inverted *"Y"* of bone support the articular surface of acetabulum. These columns are connected to the sacroiliac articulation by a thick strut of bone lying above the greater sciatic notch known as the sciatic buttress.

The radiographic anatomy of the acetabulum can be determined using AP pelvis and two 45-degree oblique radiographs as proposed by Judet and Letournel. Therefore, three radiographic projections of the pelvis that are used to evaluate the fractures of acetabulum are as follows:

These plain films are interpreted based on the understanding of normal radiographic landmarks of the acetabulum, and disruption of these landmarks represents a fracture involving that portion of the bone. These landmarks are referred to as "lines". They are generated by the tangency of the applied x-ray beam to a region of cortical bone.


**8. General toxicity**

486 Non-Viral Gene Therapy

*et al.*, 1999; Fischer *et al.*, 2002; Brus *et al.*, 2004; Neu *et al.*, 2005; Gary *et al.*, 2007). PEI 25 kDa serves as gold standard for in vitro transfection experiments (Godbey *et al.*, 2000). The mechanism of cell entry and action for gene delivery is intensively analyzed. To enhance the endosomal release of endocytosed polyplexes PEI uses the so-called "proton-sponge" effect (Boussif *et al.*, 1995; Behr, 1997) Due to the high buffer capacity of PEI amino groups in PEI molecules will be protonated at lower pHs like in the endosomal-lysosomal environment, additional chloride influx into the vesicles increases the osmolarity and the vesicles begin to swell and under the increased osmotic pressure the vesicle will be disrupted and the nucleic acid protected from PEI will be released into the cytoplasm (Godbey *et al.*, 1999; Akinc *et al.*, 2005; Nel *et al.*, 2009). PEI has been used for many years to facilitate nucleic acid delivery (Demeneix and Behr, 2005). However, due to toxicity and variable performance a lot of research is undertaken to reduce the toxicity of PEI and maintain or improve the efficacy and specificity by modification PEI backbone and/or conjugation of hydrophilic molecules like polyethylene glycol (PEG) (Petersen *et al.*, 2002a; Petersen *et al.*, 2002b), disulfide linkages (Breunig *et al.*, 2008), or for specific targeting molecules like transferrin, galactose, TAT-peptide, RGD-motifs (Ogris *et al.*, 1999; Kunath *et al.*, 2003a; Kunath *et al.*, 2003b; Kleemann *et al.*, 2005). Other approaches are reduction of the molecular weight of PEI 25 kDa or purification of PEI 25 kDa via gel filtration (Boeckle *et al.*, 2004; Urban-Klein *et al.*, 2005; Werth, 2006; Fahrmeir *et al.*, 2007) or using instead of the branched PEI 25 kDa the linear form PEI22kDa (Breunig *et al.*, 2005). Thomas and colleagues showed that full deacylation of linear PEI dramatically improves the efficacy but on cost of increased cytotoxicity due to increased numbers of protonatable nitrogens in the PEI molecule

Modifications of PEI with the hydrophilic poly(ethylene glycol) (PEG) reduces dramatically the cytotoxicity of PEI 25 kDa but in part on cost of efficacy and increased immunomodulatory and proinflammatory effects (Kichler *et al.*, 2002; Petersen *et al.*, 2002b; Mao *et al.*, 2005; Glodde *et al.*, 2006; Beyerle *et al.*, 2010a; Beyerle *et al.*, 2010b). PEG provides polyplexes with improved solubility, lower surface charge, diminished aggregation, lower

Glodde et al. synthesized a series of PEG-PEI copolymers and found that the molecular weight of PEG was found to be the major determinant of polyplex size, via its influence on particle aggregation and polyplex stability (Glodde *et al.*, 2006). Transfection efficiency was correlated to polyplex stability and low molecular weight PEI 2 kDa grafted with PEG showed higher activity than their counterparts with high molecular weight PEI 25 kDa (Williams *et al.*, 2006). In contrast, Petersen and Mao showed good transfection efficiencies for PEI 25 kDa - PEG copolymers with high molecular weight PEG and low numbers of grafting on PEI backbone compare to low molecular weight PEG with high grafting numbers on PEI 25 kDa (Mao *et al.*, 2005; Merkel *et al.*, 2009; Beyerle *et*

Grayson and colleagues investigated the siRNA transfection efficacy of different PEI polymers (branched 800 Da, branched 25 kDa and linear 22 kDa) in HeLa derivative cell line (Grayson *et al.*, 2006). They showed that the siRNA delivery and activity was mainly dependent on the biophysical and structural characteristics of the polyplexes and only

cytotoxicity, and possibly improved "stealth effect" in the bloodstream.

(Thomas *et al.*, 2005).

*al.*, 2011a).

**6. Modifications of PEI**

Hornung et al. described that any rupture or leakage of the endosomal or lysosomal membrane will release cathepsin B, which leads to an inflammasome activation associated with IL-1 production and apoptosis (Hornung *et al.*, 2008). Beyerle et al. found that application PEI/siRNA complexes caused release of proinflammatory cytokines like IL-6, G-CSF, TNF-a, IP-10 in murine lung cell lines (Beyerle *et al.*, 2010a; Beyerle *et al.*, 2010b; Beyerle *et al.*, 2011a and Beyerle *et al.*, 2011c). Cytokine release upon PEI/nucleic acid polyplex treatment has been also described by Gautam and Kawakami et al. (Gautam *et al.*, 2001; Kawakami *et al.*, 2006). Cubillos-Ruis and co-workers investigated linear PEI/siRNA complexes for antitumor immunity and identified linear PEI as TLR 5 agonist of mouse and human. They found that linear PEI/siRNA complexes induced a pattern of inflammatory cytokines which are triggered in vivo by flagellin in a TLR5 dependent manner (Cubillos-

Toxicity of Polymeric-Based Non-Viral Vector Systems for Pulmonary siRNA Application 489

Fig. 1. Structure-function-relationships of PEG-PEI copolymers

investigated molecules.

Overview of the structure-function relationships of PEG modified PEI copolymers (B-C) in comparisonto PEI 25kDa (A) with regard to cytotoxic (v,w), oxidative stress (x,y) and proinflammatory responses (z). Arrows represent the up- or downregulation of the

able to pass through biological membranes/barriers and transfer the desired information to target sites with minimal impact on the integrity of the target cell or tissue (Forrest and Pack, 2002; Omidi *et al.*, 2008). Viral vectors possess high efficacy accompanied by stimulation of the immune systems which is a limitation of these systems to deliver nucleic acids and human use. Therefore, non-viral vector systems should overcome these adverse side effects and represent safer and more efficient alternatives with improved bioavailability and reduced cellular toxicity in the clinics (Akhtar *et al.*, 2000; Somia and Verma, 2000; Panyam and Labhasetwar, 2003). It has been shown that cationic polymers and lipid-based transfection reagents could elicit cellular gene expression changes and complexation with siRNA increased these changes (Omidi *et al.*, 2003; Omidi *et al.*, 2005; Fedorov *et al.*, 2006; Hollins *et al.*, 2007; Tagami *et al.*, 2007; Tagami *et al.*, 2008). Beyerle et al. analyzed the expression changes of genes related to cytotoxicity, inflammation and oxidative stress in a pathway focused qRT-PCR array system upon treatment with different PEI-PEG copolymers in murine lung epithelial cells (LA-4 cell line) and could show that PEGylated PEI copolymers altered the gene expression profile on cost of upregulation of genes involved in inflammatory and oxidative stress processes while PEI 25 kDa mainly induced genes related to cytotoxicity and apoptosis (Beyerle *et al.*, 2010a). In addition, the potential of PEI and PEI-PEG copolymers to induce DNA damage and therefore their genotoxic potential was investigated in a lung epithelial cell line derived from the MutaMouse, but no indication for

Ruiz *et al.*, 2009). Thus, for in vivo use a lot of effort should be made to avoid the high proinflammatory effects caused by the rupture or leakage of the endosome caused by PEI. Godbey classified PEI-mediated toxicity in an immediate toxicity, associated with free PEI and a delayed form, connected with cellular processing of PEI/DNA polyplexes (Godbey *et al.*, 2001). To form stable and protective PEI nucleic acid polyplexes an excess of PEI polymer is needed, 60-80% PEI remains in a free form after nucleic acid escape and is mainly attributed to PEI toxicity. The high positively charged PEI molecule is able to disrupt cell membranes, disruption of the endosome is on one hand favourable with respect to the intended cytoplasmatic delivery, but on the other hand disruption of other cell membranes (e.g., lysosomal membranes, mitochondrial membrane, plasma membrane) is not favourable as it will cause stress responses or even apoptotic or necrotic cell death. In this context it has been shown that PEI causes apoptosis in an unspecific manner in all kinds of cells (Beyerle *et al.*, 2010a; Merkel *et al.*, 2011) which should be avoided with regard to human use. Therefore, a purification approach of the PEI polymer before and after complexation with nucleic acid is one possibility to reduce PEI-related toxicity (Boeckle *et al.*, 2004; Werth, 2006; Fahrmeir *et al.*, 2007).


**5. References**

and vitamin-K antagonists(u). The diagnosis of anomalies in the inferior vena cava influences the strategy for prevention the pulmonary embolism and long them maintenance treatment. The use of mechanical device as caval filter is clearly limited by the anomalous anatomy of the inferior vena cava and, generally, is-not indicated. On the other hand the use of oral anticoagulant (commonly warfarin) should be adjusted to maintain a target international normalized ratio of 2.5 (range 2-3) and extended indefinitely in absence of main contra-

At present the introduction of new drugs as the factor Xa antagonists (rivarixaban, apixaban, edoxaban, ect) and the direct thrombin inhibitors as dabigatran etexilate could improve the therapeutic options. The promising results of the recent clinical studies in terms of efficacy and safety, suggest that these new drugs may allow a reduction of the length of hospital stay after an acute DVT, and a better adherence to guidelines in the long term treatment. The principal advantages of these drugs are the absence of the need of a routine coagulation monitoring and a therapeutic activity not influenced by dietary regimen and by drugs as NSADIs and statins(u). Potential limitations are the lack of specific antidotes (however the hal-life of these drugs is relative short) and the absence of a simple assay for

In conclusion these interesting pharmacological characteristics could improve the benefit-

The complexity of the ontogeny of the IVC, with numerous anastomosis formed between the three primitive paired veins, can lead to a wide array of variations in the basic plan of venous return from the abdomen and lower extremity. Some of these anomalies have significant clinical implications. Although vascular structures can usually be readily identified on contrast-enhanced CT scans, identification of unusual venous arrangements may be difficult in those cases in which intravenous contrast material is contraindicated. In such patients, MR imaging may be used to distinguish aberrant vessels from masses by demonstrating flow voids or flow-related enhancement. The echo-scanning may suggest the presence of venous anomalies but usually it insufficient for a detailed diagnosis. A

risk balance of long-term anti-coagulant therapy and the overall clinical outcome.

knowledge of IVC and renal vein anomalies is essential to avoid diagnostic pitfalls.

venous thrombosis. *Thromb Haemost*. 1997;78:1-6.

en pacientes jóvenes. *Med Clin (Barc).* 2004;123:217-9.

[1] Rosendaal F. Thrombosis in the young: epidemiology and risk factors. A focus on

[2] García-Fuster MJ, Fernández C, Forner MJ, & Vayá A. Estudio prospectivo de los

[3] Ruggeri M, Tosseto A, Castaman G & Rodeghiero F. Congenital absence of the inferior

[4] M. Bianchi, D. Giannini, A. Balbarini, M.G. & Castiglioni. Congenital hypoplasia of

factores de riesgo y las características clíni- cas de la enfermedad tromboembólica

vena cava: a rare risk factor for idiopathic deep vein thrombosis. *Lancet*.

inferior vena cava and inherited thrombophilia: rare associated risk factor for

indications(d).

**4. Conclusions**

**5. References**

2001;357:441.

quantification of activity or plasma level.

idiopathic deep vein thrombosis. A case report. *J Cardiovasc Med* (Hagerstown). 2008 Jan;9(1):101-4.

**4**

**Endovascular Therapies in Acute DVT**

Deep venous thrombosis of the lower limb is a common disease with an incidence of 80 per 10000 (Patel et al 2011) and has potential fatal consequences in the form of pulmonary

It is usually seen in patients undergoing major surgery particularly orthopaedic surgery, trauma, prolonged immobilisation or hypercoagulable states (such as in the context of malignancy). There are associations with drugs such as the oral contraceptive pill and

Deep venous thrombosis of the lower extremity can occur anywhere from the ankle to the IVC, however it is those that occur between the IVC and femoral veins that most often lead to venous hypertension, resulting in the more severe symptoms. They are also more likely to

The clinical spectrum can range from being completely asymptomatic to post thrombotic

Traditionally, DVTs have been treated with the use of oral anticoagulation medication (such as warfarin) for a period of 6 months. However, this is associated with a high risk for recurrent thrombosis, and approximately one third will develop post thrombotic syndrome despite treatment (Prandoni et al 1996). The recurrence of DVT is thought to be related to damage to the venous valves during an episode of thrombosis and the low rate of recanalisation particularly in caval/iliac/femoral venous thrombosis, which leads to obstruction and venous hypertension. It has been suggested that anticoagulation therapy alone may be inadequate to prevent the damage to the venous valves in the setting of caval/iliac/femoral DVT. Moreover, it has been shown that early thrombus removal is associated with a lower incidence of symptoms related to post thrombotic syndrome (Sharafuddin 2003). These factors have prompted use of invasive techniques such as catheter directed thrombolysis and mechanical thrombectomy, particularly in the acute setting, for

There are currently two large randomised controlled trials (TORPEDO and ATTRACT) underway investigating the efficacy of these invasive techniques and early results suggest

hormone replacement therapy (tamoxifen) that predispose to hypercoagulability.

**1. Introduction**

**2. Pathology and clinical presentation**

syndrome: ulceration, pain, and intractable oedema.

recur (Vendatham 2006).

thrombus removal.

embolism.

Jeff Tam and Jim Koukounaras

*The Alfred Hospital*

*Australia*

[21] Mavrakanas T, Bounameaux H. The potential role of new oral anticoagulants in the prevention and treatment of thromboembolism. *Pharmacology & Therapeutics* 130 (2011) 46–58


**4.1 UI definition**

The concept of UI can be based on:

The ICS definitions and terminologies of UI according to the above descriptions have been revised several times (Abrams et al., 1988, Abrams et al., 2002, Haylen et al., 2010). The current definition of UI symptoms is "Complaint of involuntary loss of urine" (Haylen et al., 2010). In the 2002 definition, UI symptoms were not enough to set the UI diagnose; UI signs were needed. Today the majority of studies on UI define UI according to UI symptoms. Studies on UI have used the definitions at the time. As definitions change, prevalence estimates will also change.


*Article* **The n-Pythagorean Fuzzy Sets**

## **Anna Bryniarska**

Department of Computer Science, Opole University of Technology, Proszkowska 76, 45-758 Opole, Poland<EMAIL_ADDRESS>
Received: 16 September 2020; Accepted: 23 October 2020; Published: 26 October 2020

**Abstract:** The following paper presents deductive theories of n-Pythagorean fuzzy sets (n-PFS). N-PFS objects are a generalization of the intuitionistic fuzzy sets (IFSs) and the Yager Pythagorean fuzzy sets (PFSs). Until now, the values of membership and non-membership functions have been described on a one-to-one scale and a quadratic function scale. There is a symmetry between the values of this membership and non-membership functions. The scales of any power functions are used here in order to increase the scope of the decision-making problems. The theory of n-PFS introduces a conceptual apparatus analogous to the classic theory of Zadeh fuzzy sets, consistently striving to correctly define the n-PFS algebra.

**Keywords:** fuzzy set; n-Pythagorean; n-PFS algebra; triangular norms

## **1. Introduction**

Zadeh [1] introduced the fuzzy set idea, which generalizes the theory of classical sets. In fuzzy sets, there is a membership function *µ*, which assigns a number from the set [0, 1] to each element of the universe. This determines how much this element belongs to this universe, where 0 means no belonging and 1 means full belonging to the set that is under consideration. Other values between 0 and 1 mean the degree of belonging to this set. This membership function is defined to describe the degree of belonging of an element to some class. The membership function in the fuzzy sets replace the characteristic function that is used in crisp sets. Since the work of Zadeh, the fuzzy set theory has been used in different disciplines such as management sciences, engineering, mathematics, social sciences, statistics, signal processing, artificial intelligence, automata theory, and medical and life sciences.

Atanassov studied the intuitionistic fuzzy sets (IFSs) [2,3]. IFSs have values for two functions: the membership function *µ* and the non-membership function *v*. Additionally, there is a constraint 0 ≤ *µ* + *v* ≤ 1. This is a symmetric relationship between the values and the membership function. In order to create model for imprecise information, the model of Pythagorean fuzzy sets (PFSs) was proposed by Yager [4,5]. This model is different than the IFSs model because it uses the condition 0 ≤ *µ* 2 + *v* 2 ≤ 1. Moreover, there is also the Pythagorean fuzzy number (PFN) idea established by Zhang and Xu [6]. In decision-making problems, there are also applications of PFSs proposed by Garg [7,8].

The decision-making problems in the model of Pythagorean fuzzy sets will significantly increase the application range of solving these problems than in the model of intuitionistic fuzzy sets. It is because more pairs (*µ*, *v*) satisfy the condition 0 ≤ *µ* 2 + *v* 2 ≤ 1 than the condition 0 ≤ *µ* + *v* ≤ 1. Is there any other data scale in decision-making problems that will help to extend its applicability even further? Yes, with the condition 0 ≤ *µ n* + *v n* ≤ 1, for any natural number *n* > 2.

In articles using local deduction regarding IFSs [3,9–12] and concerning PFSs [4,5,13–17], it was noted that there is a series of mathematical and logical inaccuracies. That is why the conceptual apparatus should be generalized and refined by formulating the deductive theory of n-Pythagorean fuzzy sets with the condition 0 ≤ *µ n* + *v n* ≤ 1, for any natural number *n*i0. The theory is presented below.

#### **2. Triangular Norms**

**Definition 1.** *The operation* •*t* : [0, 1] × [0, 1] → [0, 1] *is called a t-norm in the set* [0, 1]*, or a triangular norm, when it meets the following conditions (for any numbers x*, *y*, *z* ∈ [0, 1]*):*

*1. boundary conditions*

$$0 \bullet\_t y = 0,\\ y \bullet\_t 1 = y,\tag{1}$$

*2. monotonicity*

$$\text{If } \mathbf{x} \bullet\_t \mathbf{y} \le z \bullet\_t \mathbf{y}, \text{ when } \mathbf{x} \le z,\tag{2}$$

*3. commutativity*

$$\mathbf{x} \bullet\_{t} y = y \bullet\_{t} \mathbf{x},\tag{3}$$

*4. associativity*

$$\propto \bullet\_t \left( y \bullet\_t z \right) = \left( \mathbf{x} \bullet\_t y \right) \bullet\_t z,\tag{4}$$

Since there is sup{*t* ∈ [0, 1] : *x* •*t t* ≤ *y*}, there can be specified the operation →*t* : [0, 1] × [0, 1] → [0, 1], such that for any numbers *x*, *y* ∈ [0, 1]:

$$\mathbf{x} \to\_{l} y = \sup \{ t \in [0, 1] : \mathbf{x} \bullet\_{l} t \le y \}. \tag{5}$$

This operation is called *t-residuum* in the set [0, 1]. The operation •*s* : [0, 1] × [0, 1] → [0, 1], described by formula for any numbers *x*, *y* ∈ [0, 1]:

$$\propto \bullet\_s y = 1 - (1 - x) \bullet\_t (1 - y),\tag{6}$$

is called a *s-norm* or *triangular conorm*.

Using the definitions of t-norm and s-norm, after simple calculations we get (where the names of conditions are given analogously to the definition of t-norm):

**Theorem 1.** *For any numbers x*, *y*, *z* ∈ [0, 1]*:*

• *boundary conditions*

$$0 \bullet\_s y = y,\\ y \bullet\_s 1 = 1,\tag{7}$$

• *monotonicity*

$$\propto \bullet\_s y \le z \bullet\_s y,\text{ when } \ge z,\tag{8}$$

• *commutativity*

$$\mathfrak{x}\bullet\_{\mathfrak{s}}\mathfrak{y} = \mathfrak{y}\bullet\_{\mathfrak{s}}\mathfrak{x} \tag{9}$$

• *associativity*

$$\propto \bullet\_{\rm s} \left( y \bullet\_{\rm s} z \right) = \left( \propto \bullet\_{\rm s} y \right) \bullet\_{\rm s} z \tag{10}$$

Further, only continuous t-norms and s-norms are considered. The general discussion on the construction of triangular norms, using the results of functional equations, leads to the theorem from paper [18]:

#### **Theorem 2.**

*1. There is a continuous and strictly decreasing function for each continuous t-norm ft* : [0, 1] → [0, +∞) *such that ft*(1) = 0, *ft*(0) = 1 *and for any x*, *y* ∈ [0, 1]*:*

$$\mathbf{x} \bullet\_{\mathbf{t}} \mathbf{y} = \begin{cases} f\_{\mathbf{t}}^{-1} [f\_{\mathbf{t}}(\mathbf{x}) + f\_{\mathbf{t}}(\mathbf{y})]\_{\prime} & f\_{\mathbf{t}}(\mathbf{x}) + f\_{\mathbf{t}}(\mathbf{y}) \in [\mathbf{0}, \mathbf{1}] \\ \mathbf{0}, & \text{otherwise.} \end{cases} \tag{11}$$

*2. There is a continuous and strictly increasing function for each continuous s-norm fs* : [0, 1] → [0, +∞) *such that fs*(0) = 0, *fs*(1) = 1 *and for any x*, *y* ∈ [0, 1]*:*

$$\mathbf{x} \bullet\_{s} y = \begin{cases} f\_{s}^{-1} [f\_{s}(\mathbf{x}) + f\_{s}(y)], & f\_{s}(\mathbf{x}) + f\_{s}(y) \in [0, 1] \\ 1, & \text{otherwise.} \end{cases} \tag{12}$$

*3. For any x* ∈ [0, 1]

$$f\_{\mathbf{s}}(\mathbf{x}) = f\_{\mathbf{t}}(\mathbf{1} - \mathbf{x}). \tag{13}$$

Functions *ft* , *fs* are called *generators* of t-norm and s-norm, respectively.

**Example 1.** *For the t-norm x* •*t y* = min{*x*, *y*} *and any x*, *y* ∈ [0, 1]*, the generator is ft*(*x*) = 1 − *x. For the s-norm x* •*s y* = max{*x*, *y*} *and any x*, *y* ∈ [0, 1]*, the generator is fs*(*x*) = *x.*

**Example 2.** *For the t-norm x* •*t y* = 1 − min{1,((1 − *x*) *p* + (1 − *y*) *p* ) 1/*p*}, *p* ≥ 1 *and any x*, *y* ∈ [0, 1]*, the generator is ft*(*x*) = 1 − *x p .*

*For the s-norm x* •*s y* = min{1,(*x p* + *y p* ) 1/*p*}, *p* ≥ 1 *and any x*, *y* ∈ [0, 1]*, the generator is fs*(*x*) = *x p .*

**Theorem 3.** *Let ft* , *fs be generators of the triangular norms* •*t* , •*s . Then there exist operations* •*p* : [0, 1] × [0, 1] → [0, 1], •*l* : [0, 1] × [0, 1] → [0, 1] *defined by formulas:*

$$
\lambda \bullet\_p \ge \begin{cases}
f\_t^{-1}[\lambda f\_t(\mathbf{x})], & \text{for } \lambda f\_t(\mathbf{x}) \in [0, 1] \\
0, & \text{otherwise}.
\end{cases}
\tag{14}
$$

$$\lambda \bullet\_l \mathbf{x} = \begin{cases} f\_s^{-1}[\lambda f\_s(\mathbf{x})], & \text{for } \lambda f\_s(\mathbf{x}) \in [0, 1] \\ 1, & \text{otherwise.} \end{cases} \tag{15}$$

**Proof of Theorem 3.** Because *ft* is a strictly decreasing function, there is only one value *f* −1 *t* [*λ ft*(*x*)] ∈ [0, 1], when *λ ft*(*x*) ∈ [0, *fs*(0)].

It is noted that *λ* •*p x* =*d f f* −1 *t* [*λ ft*(*x*)], for *λ ft*(*x*) ∈ [0, *ft*(0)], and *λ* •*p x* =*d f* 0 otherwise. Similarly, we define the operation •*l* .

**Definition 2.** *Operations* •*p*, •*l specified in Theorem 3 are called p-norm (with properties similar to the power functions) and the l-norm (with properties similar to the linear function), respectively, and the system* **SYager** = h[0, 1], •*t* , •*s* , •*p*, •*l* , 0, 1i *is called the Yager system of the triangular norms.*

The following notation agreement is accepted:

$$
\lambda \bullet\_p \mathfrak{x} =\_{df} \mathfrak{x}^{\lambda},
\tag{16}
$$

$$
\lambda \bullet\_l \mathfrak{x} =\_{df} \lambda \mathfrak{x}.\tag{17}
$$

**Fact 1.** *If xλ* = *f* −1 *t* [*λ ft*(*x*)]*, for λ ft*(*x*) ∈ [0, *ft*(0)]*, then ft*(*x λ* ) = *λ ft*(*x*)*. If λx* = *f* −1 *s* [*λ fs*(*x*)]*, for λ fs*(*x*) ∈ [0, *fs*(1)]*, then fs*(*λx*) = *λ fs*(*x*)*.*

**Theorem 4.** *In the system* **SYager** = h[0, 1], •*t* , •*s* , •*p*, •*l* , 0, 1i *operations* •*p*, •*l satisfy the following conditions (for any x*, *y*, *λ*, *λ*1, *λ*2 ∈ [0, 1]*):*

$$
\lambda(\mathfrak{x}\bullet\_{\mathfrak{s}}\mathfrak{y}) = \lambda\mathfrak{x}\bullet\_{\mathfrak{s}}\lambda\mathfrak{y},\tag{18}
$$

$$(\mathfrak{x}\bullet\_{t}\mathfrak{y})^{\lambda}=\mathfrak{x}^{\lambda}\bullet\_{t}\mathfrak{y}^{\lambda},\tag{19}$$

$$(\lambda\_1 + \lambda\_2)\mathfrak{x} = \lambda\_1 \mathfrak{x} \bullet\_{\mathfrak{s}} \lambda\_2 \mathfrak{x},\tag{20}$$

$$
\mathfrak{x}^{\lambda\_1 + \lambda\_2} = \mathfrak{x}^{\lambda\_1} \bullet\_t \mathfrak{x}^{\lambda\_2}.\tag{21}
$$

#### **Proof of Theorem 4.**

#### **3. n-Pythagorean Fuzzy Set and Yager Aggregation Operators**

**Definition 3.** *Let F be a set of all fuzzy sets for the nonempty space X. Any function p* : *X* → [0, 1] × [0, 1] *defined for any µp*, *vp* ∈ *F is:*

$$p = \{ \langle \mathbf{x}, \langle \mu\_p(\mathbf{x}), v\_p(\mathbf{x}) \rangle \rangle : \mathbf{x} \in X \}. \tag{22}$$

*It is called the n-Pythagorean fuzzy set (n-PFS) if the following condition is satisfied (for any natural number n* > 0 *):*

$$0 \le (\mu\_p(\mathbf{x}))^n + (v\_p(\mathbf{x}))^n \le 1,\text{ for any } \mathbf{x} \in \mathbf{X}.\tag{23}$$

Let **n-PFS** mean set of all n-PFS.

The fuzzy sets *µp*, *vp* indicate the membership and non-membership functions. Zhang and Xu [6] considered *p*(*x*) = h*µp*(*x*), *vp*(*x*)i as n-Pythagorean fuzzy number (n-PFN) represented by *p* = h*µp*, *vp*i. The notation is used:

$$\mathbf{n}\text{-FFN} =\_{df} \{ \langle \mu, v \rangle \in [0, 1] \times [0, 1] : 0 \le \mu^n + v^n \le 1 \}.\tag{24}$$

**Fact 2.**

$$\mathfrak{n}\text{-PFS} = \{p(\mathfrak{x}) : \mathfrak{x} \in \mathcal{X}, p \in \mathfrak{n}\text{-PFS}\}.\tag{25}$$

When *n* = 1, then the 1-Pythagorean fuzzy sets are the intuitionistic fuzzy sets (IFS), which were studied by Atanassow [2]. Moreover, when *n* = 2, then the 2-Pythagorean fuzzy sets are the PFS of Yager [4].

Simple arithmetic properties of inequalities 0 ≤ *µ n*+1 + *v n*+1 ≤ *µ n* + *v n* ≤ 1 result from:

**Theorem 5.** *For any natural number n* > 1

$$n\text{-PFN} \subseteq (n\text{-1})\text{-PFN} \subseteq [0,1] \times [0,1]. \tag{26}$$

Thus, entering a power scale for a value of the membership and non-membership functions allows to replace h*µ*, *v*i ∈ [0, 1] × [0, 1] such that *µ* + *v* > 1, by h*µ n* , *v n* i ∈ **1-PFN**, for some *n*. As a result, the aggregation operations on the *IFS* can be extended to the aggregation operations on the *n-PFS*.

**Theorem 6.** *In any system* **SYager** = h[0, 1], •*t* , •*s* , •*p*, •*l* , 0, 1i*, for any* h*µ*1, *v*1i,h*µ*2, *v*2i ∈ **1***-PFN and a number λ* ∈ [0, 1]*, the following conditions are satisfied:*

$$
\langle \mu\_1 \bullet\_s \mu\_2, v\_1 \bullet\_t v\_2 \rangle \in \mathbf{1} \text{-PFN},\tag{27}
$$

$$<\langle \mu\_1 \bullet\_t \mu\_2, v\_1 \bullet\_s v\_2 \rangle \in \mathbf{1} \text{-PFN},\tag{28}$$

*Furthermore, in some systems* **SYager** *(not in all - see Proof of Theorem 3,4 and Remark 1) there are additional conditions:*

$$
\langle \lambda \bullet\_{\mathsf{I}} \mu\_{1}, \lambda \bullet\_{\mathsf{p}} v\_{1} \rangle \in \mathbf{1} \text{-PFN}\_{\mathsf{I}} \tag{29}
$$

$$
\langle \lambda \bullet\_p \mu\_1 \,\mu\_1 \,\lambda \bullet\_l v\_1 \rangle \in \mathbf{1} \text{-PFN}.\tag{30}
$$

#### **Proof of Theorem 6.**

1. For Equation (27): h*µ*1, *v*1i,h*µ*2, *v*2i ∈ **1-PFN** iff *µ*1 + *v*1 ≤ 1, *µ*2 + *v*2 ≤ 1 if and only if *µ*1 ≤ 1 − *v*1, *µ*2 ≤ 1 − *v*2.

Hence and from the monotonicity of the s-norm and its determination by t-norm:

*µ*1 •*s µ*2 ≤ (1 − *v*1) •*s* (1 − *v*2) = 1 − (1 − (1 − *v*1) •*s* (1 − *v*2) = 1 − *v*1 •*t v*2 iff *µ*1 •*s µ*2 + *v*1 •*t v*2 ≤ 1 iff h*µ*1 •*s µ*2, *v*1 •*t v*2i ∈ **1-PFN**.

$$\begin{aligned} \langle \mu\_1 v\_1 \rangle &\in \mathbf{1} \mathbf{P} \mathbf{N} \text{ iff } \mu\_1 + v\_1 \le 1; \\ \lambda \bullet\_l \mu\_1 &= f\_s^{-1}[\lambda f\_s(\mu\_1)] = \lambda \mu\_1 \text{ and } \lambda \bullet\_p v\_1 = f\_s^{-1}[\lambda f\_s(v\_1)] = 1 - \lambda(1 - v\_1); \\ \lambda \bullet\_l \mu\_1 + \lambda \bullet\_p v\_1 &= \lambda \mu\_1 + 1 - \lambda(1 - v\_1) = 1 - \lambda + \lambda(\mu\_1 + v\_1); \\ \text{Since } \lambda(\mu\_1 + v\_1) &\le \lambda, \text{ so } 0 \le 1 - \lambda + \lambda(\mu\_1 + v\_1) \le 1. \end{aligned}$$

4. For Equation (30): let *µ* •*s v* = max{*µ*, *v*}, *µ* •*t v* = min{*µ*, *v*}, then *fs*(*x*) = *x*, *ft*(*x*) = 1˘*x*; h*µ*1, *v*1i ∈ **1-PFN** iff *µ*1 + *v*1 ≤ 1 iff h*v*1, *µ*1i ∈ **1-PFN**. Hence, and from point 3, there is h*λ* •*l v*1, *λ* •*p µ*1i ∈ **1-PFN**, which is equivalent to h*λ* •*p µ*1, *λ* •*l v*1i ∈ **1-PFN**.

$$\square$$

**Remark 1.** *Assuming generators of triangular norms from the Example 2 λ* •*l µ*1 = *λ* 1/*pµ*1, *λ* •*p v*1 = 1 − *λ* 1/*p* (1 − *v*1)*, for µ*1 = 1/2, *v*1 = 7/8, *λ* = 1/4 *and p* = 2*, it is obtained that* h*λ* •*l µ*1, *λ* •*p v*1i = h1/8, 15/16i ∈/ **1***-PFN*,(1 < 1/8 + 15/16)*.*

**Theorem 7.** *Let the system* **SYager** = h[0, 1], •*t* , •*s* , •*p*, •*l* , 0, 1i *conditions from Equations (27)–(30) of the Theorem 6 apply. Then, for any natural number n* > 1*, for any* h*µ*1, *v*1i,h*µ*2, *v*2i ∈ *n-PFN, and number λ* ∈ [0, 1] *the following conditions are satisfied:*

$$<\langle (\mu\_1^n \bullet\_s \mu\_2^n)^{1/n}, (v\_1^n \bullet\_t v\_2^n)^{1/n} \rangle \in \mathfrak{n} \text{-PFN},\tag{31}$$

$$<\langle (\mu\_1^n \bullet\_t \mu\_2^n)^{1/n}, (v\_1^n \bullet\_s v\_2^n)^{1/n} \rangle \in \mathfrak{n}\text{-PFN},\tag{32}$$

$$<\langle (\lambda \bullet\_l \mu\_1^n)^{1/n}, (\lambda \bullet\_p v\_1^n)^{1/n} \rangle \in \mathfrak{n} \text{-PFN},\tag{33}$$

$$<\langle (\lambda \bullet\_p \mu\_1^n)^{1/n}, (\lambda \bullet\_l v\_1^n)^{1/n} \rangle \in \mathfrak{n}\text{-PFN}.\tag{34}$$

#### **Proof of Theorem 7.**

$$
\langle \mu\_1, v\_1 \rangle, \langle \mu\_2, v\_2 \rangle \in \mathbf{n} \cdot \mathbf{FFN} \text{ iff } \langle \mu\_1^{\text{n}}, v\_1^{\text{n}} \rangle, \langle \mu\_2^{\text{n}}, v\_2^{\text{n}} \rangle \in \mathbf{1} \cdot \mathbf{FFN}.\tag{35}
$$

Then, the conditions of Theorem 6 are satisfied, which are equivalent to the above conditions (31)–(34).

**Definition 4.** *In the system* **SYager** = h[0, 1], •*t* , •*s* , •*p*, •*l* , 0, 1i*, the following aggregation operators are defined: the Yager operators on n-PFN: for any* h*µ*1, *v*1i,h*µ*2, *v*2i ∈ *n-PFN and number λ* ∈ [0, 1]*.*

*When conditions (27)–(30) of the Theorem 6 are satisfied:*

$$<\langle \mu\_1, v\_1 \rangle \oplus \langle \mu\_2, v\_2 \rangle = \langle (\mu\_1^n \bullet\_s \mu\_2^n)^{1/n}, (v\_1^n \bullet\_t v\_2^n)^{1/n} \rangle,\tag{36}$$

$$<\langle \mu\_1, \upsilon\_1 \rangle \oplus \langle \mu\_2, \upsilon\_2 \rangle = \langle (\mu\_1^n \bullet\_1 \mu\_2^n)^{1/n}, (\upsilon\_1^n \bullet\_s \upsilon\_2^n)^{1/n} \rangle,\tag{37}$$

$$
\langle \lambda \langle \mu\_1, v\_1 \rangle = \langle (\lambda \bullet\_{\mathfrak{l}} \mu\_1^n)^{1/n}, (\lambda \bullet\_{\mathfrak{p}} v\_1^n)^{1/n} \rangle,\tag{38}
$$

$$
\langle \mu\_1, v\_1 \rangle \lambda = \langle (\lambda \bullet\_p \mu\_1^n)^{1/n}, (\lambda \bullet\_l v\_1^n)^{1/n} \rangle. \tag{39}
$$

*when* h(*λ* •*l µ n* 1 ) 1/*n* ,(*λ* •*p v n* 1 ) 1/*n* i ∈/ *n-PFN, then:*

$$
\lambda \langle \mu\_1, v\_1 \rangle = \langle 1, 0 \rangle,\tag{40}
$$

*or when* h(*λ* •*p µ n* 1 ) 1/*n* ,(*λ* •*l v n* 1 ) 1/*n* i ∈/ *n-PFN, then:*

$$
\langle \mu\_1, v\_1 \rangle^{\lambda} = \langle 0, 1 \rangle. \tag{41}
$$

Hence, in any system **SYager** = h[0, 1], •*t* , •*s* , •*p*, •*l* , 0, 1i, using the Theorem 4, there is:

**Theorem 8.** *For any* h*µ*1, *v*1i,h*µ*2, *v*2i,h*µ*3, *v*3i ∈ *n-PFN and number λ* ∈ [0, 1]*:*

$$
\langle \mu\_1, \upsilon\_1 \rangle \oplus \langle 0, 1 \rangle = \langle \mu\_1, \upsilon\_1 \rangle,\\
\langle \mu\_1, \upsilon\_1 \rangle \oplus \langle 1, 0 \rangle = \langle 1, 0 \rangle,\tag{42}
$$

$$
\langle \mu\_1, v\_1 \rangle \oplus \langle \mu\_2, v\_2 \rangle = \langle \mu\_2, v\_2 \rangle \oplus \langle \mu\_1, v\_1 \rangle,\tag{43}
$$

$$(\langle \mu\_1, \upsilon\_1 \rangle \oplus \langle \mu\_2, \upsilon\_2 \rangle) \oplus \langle \mu\_3, \upsilon\_3 \rangle = \langle \mu\_1, \upsilon\_1 \rangle \oplus (\langle \mu\_2, \upsilon\_2 \rangle \oplus \langle \mu\_3, \upsilon\_3 \rangle),\tag{44}$$

$$
\langle \mu\_1, \upsilon\_1 \rangle \oplus \langle 1, 0 \rangle = \langle \mu\_1, \upsilon\_1 \rangle,\\
\langle \mu\_1, \upsilon\_1 \rangle \otimes \langle 0, 1 \rangle = \langle 0, 1 \rangle,\tag{45}
$$

$$<\langle \mu\_1, v\_1 \rangle \oplus \langle \mu\_2, v\_2 \rangle = \langle \mu\_2, v\_2 \rangle \oplus \langle \mu\_1, v\_1 \rangle,\tag{46}$$

$$(\langle \mu\_1, \upsilon\_1 \rangle \oplus \langle \mu\_2, \upsilon\_2 \rangle) \oplus \langle \mu\_3, \upsilon\_3 \rangle = \langle \mu\_1, \upsilon\_1 \rangle \oplus (\langle \mu\_2, \upsilon\_2 \rangle \oplus \langle \mu\_3, \upsilon\_3 \rangle),\tag{47}$$

$$
\lambda \langle \langle \mu\_1, \upsilon\_1 \rangle \oplus \langle \mu\_2, \upsilon\_2 \rangle \rangle = \lambda \langle \mu\_1, \upsilon\_1 \rangle \oplus \lambda \langle \mu\_2, \upsilon\_2 \rangle,\tag{48}
$$

$$(\langle \mu\_1, v\_1 \rangle \oplus \langle \mu\_2, v\_2 \rangle)^\lambda = \langle \mu\_1, v\_1 \rangle^\lambda \oplus \langle \mu\_2, v\_2 \rangle^\lambda,\tag{49}$$

$$
\langle \lambda\_1 + \lambda\_2 \rangle \langle \mu\_1, \upsilon\_1 \rangle = \lambda\_1 \langle \mu\_1, \upsilon\_1 \rangle \oplus \lambda\_2 \langle \mu\_1, \upsilon\_1 \rangle,\tag{50}
$$

$$
\langle \mu\_1, v\_1 \rangle^{\lambda\_1 + \lambda\_2} = \langle \mu\_1, v\_1 \rangle^{\lambda\_1} \oplus \langle \mu\_1, v\_1 \rangle^{\lambda\_2}.\tag{51}
$$

#### **4. Triangular Norms in the n-PFN and the n-PFS Algebra**

**Definition 5.** *For any* h*µ*1, *v*1i,h*µ*2, *v*2i ∈ *n-PFN:*

$$
\langle \mu\_1, v\_1 \rangle \le\_n \langle \mu\_2, v\_2 \rangle \text{ iff } \mu\_1 \le \mu\_2, v\_1 \ge v\_2. \tag{52}
$$

*The results of operations maximum and minimum for any A* ⊆ *n-PFN are described for the relation* ≤*n and are denoted by:* max*n A*, min*n A.*

**Fact 3.** *For any x*, *y* ∈ [0, 1]*:*

**Fact 4.** *There is:*

$$\mathfrak{n}\text{-PFN} = \{ \langle \mathfrak{x}, y \rangle \in [0, 1] \times [0, 1] : \langle 0, 1 \rangle \le\_{\mathfrak{n}} \langle \mathfrak{x}, y \rangle \le\_{\mathfrak{n}} \langle 1, 0 \rangle \}. \tag{53}$$

**Definition 6.** *There are:*

	- *(a) boundary conditions*

$$
\langle \emptyset, 1 \rangle \circ\_t \langle \mu\_1, \upsilon\_1 \rangle = \langle 0, 1 \rangle,\\
\langle \mu\_1, \upsilon\_1 \rangle \circ\_t \langle 1, 0 \rangle = \langle \mu\_1, \upsilon\_1 \rangle,\tag{54}
$$

*(b) monotonicity*

$$
\langle \mu\_1, \upsilon\_1 \rangle \circ\_t \langle \mu\_2, \upsilon\_2 \rangle \leq\_n \langle \mu\_3, \upsilon\_3 \rangle \circ\_t \langle \mu\_2, \upsilon\_2 \rangle, \text{ when } \langle \mu\_1, \upsilon\_1 \rangle \leq\_n \langle \mu\_3, \upsilon\_3 \rangle,\tag{55}
$$

*(c) commutativity*

$$
\langle \mu\_1, v\_1 \rangle \circ\_t \langle \mu\_2, v\_2 \rangle = \langle \mu\_2, v\_2 \rangle \circ\_t \langle \mu\_1, v\_1 \rangle,\tag{56}
$$

*(d) associativity*

$$
\langle \mu\_1, \upsilon\_1 \rangle \circ\_t (\langle \mu\_2, \upsilon\_2 \rangle \circ\_t \langle \mu\_3, \upsilon\_3 \rangle) = (\langle \mu\_1, \upsilon\_1 \rangle \circ\_t \langle \mu\_2, \upsilon\_2 \rangle) \circ\_t \langle \mu\_3, \upsilon\_3 \rangle. \tag{57}
$$

	- *(a) boundary conditions*

$$
\langle \mathbf{1}, \mathbf{0} \rangle \circ\_{\sf s} \langle \mu\_1, \upsilon\_1 \rangle = \langle \mathbf{1}, \mathbf{0} \rangle,\\
\langle \mu\_1, \upsilon\_1 \rangle \circ\_{\sf s} \langle \mathbf{0}, \mathbf{1} \rangle = \langle \mu\_1, \upsilon\_1 \rangle,\tag{58}
$$

*(b) monotonicity*

$$
\langle \mu\_1, \upsilon\_1 \rangle \circ\_s \langle \mu\_2, \upsilon\_2 \rangle \leq\_n \langle \mu\_3, \upsilon\_3 \rangle \circ\_s \langle \mu\_2, \upsilon\_2 \rangle, \text{ when } \langle \mu\_1, \upsilon\_1 \rangle \leq\_n \langle \mu\_3, \upsilon\_3 \rangle,\tag{59}
$$

*(c) commutativity*

$$
\langle \mu\_1, v\_1 \rangle \circ\_s \langle \mu\_2, v\_2 \rangle = \langle \mu\_2, v\_2 \rangle \circ\_s \langle \mu\_1, v\_1 \rangle,\tag{60}
$$

*(d) associativity*

$$
\langle \mu\_1, \upsilon\_1 \rangle \circ\_{\sf s} (\langle \mu\_2, \upsilon\_2 \rangle \circ\_{\sf s} \langle \mu\_3, \upsilon\_3 \rangle) = (\langle \mu\_1, \upsilon\_1 \rangle \circ\_{\sf s} \langle \mu\_2, \upsilon\_2 \rangle) \circ\_{\sf s} \langle \mu\_3, \upsilon\_3 \rangle. \tag{61}
$$

**Theorem 9.** *The Yager operator* ⊗ *on the n-PFN is a t-norm in the set n-PFN and the operator* ⊕ *is a s-norm in the set n-PFN.*

**Proof of Theorem 9.** Conditions (45)–(47) of the Theorem 8 proof that the operator ⊗ satisfies conditions (a),(c), and (d) of the Definition 6 (1) of the t-norm in the set **n-PFN**. It is enough to prove that this operation is monotonous.

For any h*µ*1, *v*1i,h*µ*2, *v*2i,h*µ*3, *v*3i ∈ **n-PFN** h*µ*1, *v*1i ⊗ h*µ*2, *v*2i = h(*µ n* 1 •*t µ n* 2 ) 1/*n* ,(*v n* 1 •*s v n* 2 ) 1/*n* i, h*µ*3, *v*3i ⊗ h*µ*2, *v*2i = h(*µ n* 3 •*t µ n* 2 ) 1/*n* ,(*v n* 3 •*s v n* 2 ) 1/*n* i. Let h*µ*1, *v*1i ≤*n* h*µ*3, *v*3i. Then *µ*1 ≤ *µ*3, *v*1 ≥ *v*3. Hence, and from the monotonicity of the t-norm and s-norm, it is obtained that: *n n n n n n n n*

*µ* 1 •*t µ* 2 ≤ *µ* 3 •*t µ* 2 and *v* 3 •*s v* 2 ≤ *v* 1 •*s v* 2 iff (*µ n* 1 •*t µ n* 2 ) 1/*n* ≤ (*µ n* 3 •*t µ n* 2 ) 1/*n* and (*v n* 1 •*s v n* 2 ) 1/*n* ≥ (*v n* 3 •*s v n* 2 ) 1/*n* iff h(*µ n* 1 •*t µ n* 2 ) 1/*n* ,(*v n* 1 •*s v n* 2 ) 1/*n* i ≤*n* h(*µ n* 3 •*t µ n* 2 ) 1/*n* ,(*v n* 3 •*s v n* 2 ) 1/*n* i iff h*µ*1, *v*1i ⊗ h*µ*2, *v*2i ≤*n* h*µ*3, *v*3i ⊗ h*µ*2, *v*2i.

Proof that the operator ⊕ satisfies conditions of the Definition 6 (2) about the s-norm in the set **n-PFN** is analogical.

Summarizing, for the knowledge of operations and relationships introduced in the **n-PFN**, the following operations and conclusion relationships can be determined for the **n-PFS**:

**Definition 7.** *For any p*1, *p*2 ∈ *n-PFS, and number λ* ∈ [0, 1]*:*

$$p\_1 \oplus p\_2 = \{ \langle \mathbf{x}, \langle \mu\_{p\_1}(\mathbf{x}), v\_{p\_1}(\mathbf{x}) \rangle \oplus \langle \mu\_{p\_2}(\mathbf{x}), v\_{p\_2}(\mathbf{x}) \rangle \rangle : \mathbf{x} \in \mathbf{X} \},\tag{62}$$

$$p\_1 \otimes p\_2 = \{ \langle \mathbf{x}, \langle \mu\_{p\_1}(\mathbf{x}), \upsilon\_{p\_1}(\mathbf{x}) \rangle \otimes \langle \mu\_{p\_2}(\mathbf{x}), \upsilon\_{p\_2}(\mathbf{x}) \rangle \rangle : \mathbf{x} \in \mathcal{X} \},\tag{63}$$

$$
\lambda p\_1 = \{ \langle \mathbf{x}, \lambda \langle \mu\_{p\_1}(\mathbf{x}), v\_{p\_1}(\mathbf{x}) \rangle \rangle : \mathbf{x} \in \mathbf{X} \}, \tag{64}
$$

$$p\_1^{\lambda} = \{ \langle \mathbf{x}, \langle \mu\_{p\_1}(\mathbf{x}), v\_{p\_1}(\mathbf{x}) \rangle^{\lambda} \rangle : \mathbf{x} \in X \},\tag{65}$$

$$\text{supp}(p\_1) = \{ \mathbf{x} \in X : \mu\_{p\_1}(\mathbf{x}) > 0, v\_{p\_1}(\mathbf{x}) > 0 \},\tag{66}$$

$$p\_1 \subseteq\_n p\_2 \text{ iff for any } \mathbf{x} \in X, \langle \mu\_{p\_1}(\mathbf{x}), \upsilon\_{p\_1}(\mathbf{x}) \rangle \le\_n \langle \mu\_{p\_2}(\mathbf{x}), \upsilon\_{p\_2}(\mathbf{x}) \rangle. \tag{67}$$

The system n-PFS, with defined in the Definition 7 operations (t-norm, s-norm, p-norm, l-norm, and support) and inclusion relations, is called the **n-PFS algebra**.

Let **1** =*d f* h0, 1i, **0** =*d f* h1, 0i. Then from the Definition 7 and the Theorem 8 there are:

**Theorem 10.** *In the algebra n-PFS, for any p*1, *p*2, *p*3 ∈ *n-PFN and the number λ* ∈ [0, 1]*:*

$$p\_1 \oplus \mathbf{0} = p\_1, p\_1 \oplus \mathbf{1} = \mathbf{1},\tag{68}$$

$$p\_1 \oplus p\_2 = p\_2 \oplus p\_{1\prime} \tag{69}$$

$$(p\_1 \oplus p\_2) \oplus p\_3 = p\_1 \oplus (p\_2 \oplus p\_3),\tag{70}$$

$$p\_1 \otimes \mathbf{1} = p\_1, p\_1 \otimes \mathbf{0} = \mathbf{0},\tag{71}$$

$$p\_1 \otimes p\_2 = p\_2 \otimes p\_1. \tag{72}$$

$$(p\_1 \otimes p\_2) \otimes p\_3 = p\_1 \otimes (p\_2 \otimes p\_3),\tag{73}$$

$$
\lambda(p\_1 \oplus p\_2) = \lambda p\_1 \oplus \lambda p\_2.\tag{74}
$$

$$(p\_1 \otimes p\_2)^{\lambda} = p\_1^{\lambda} \otimes p\_2^{\lambda},\tag{75}$$

$$(\lambda\_1 + \lambda\_2)p\_1 = \lambda\_1 p\_1 \oplus \lambda\_2 p\_1. \tag{76}$$

$$p\_1^{\lambda\_1 + \lambda\_2} = p\_1^{\lambda\_1} \otimes p\_1^{\lambda\_2}.\tag{77}$$

#### **5. Conclusions**

This paper presents the elements of deductive theory of the n-PFS, where the membership degree *µ* and the non-membership degree *v* determine not only in the square scale, but in any power scale, i.e., 0 ≤ *µ n* + *v n* ≤ 1. As a result, any local deductions in the n-PFS range can be formulated. It may be interesting to use the described model to create similar models but based on other functional scales, for example for the function of scale *fn*(*x*) = *x*/*n* : 0 ≤ *µ*/*n* + *v*/*n* ≤ 1 or *fn*(*x*) = 1 + log*n* (*x* + 1), for *x* ∈ [0, 1], where *f*(*x*) < *x* : 0 ≤ (*l* + log*n* (*µ* + 1)) + (1 + log*n* (*v* + 1)) ≤ 1. In the research, the n-PFS theory can be used to describe n-PFS as a system of information granules [19].

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**

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**Botulinum Toxin Injections and Electrical Stimulation for Spastic Paresis Improve Active Hand Function Following Stroke**

**Jong-Min Lee 1, Jean-Michel Gracies 2, Si-Bog Park 3, Kyu Hoon Lee 3, Ji Yeong Lee 1 and Joon-Ho Shin 1,\***

**\*** Correspondence<EMAIL_ADDRESS>Tel.: +82-2-901-1884; Fax: +82-901-1590

Received: 22 September 2018; Accepted: 21 October 2018; Published: 25 October 2018

**Abstract:** Botulinum toxin type A (BTX-A) injections improve muscle tone and range of motion (ROM) among stroke patients with upper limb spasticity. However, the efficacy of BTX-A injections for improving active function is unclear. We aimed to determine whether BTX-A injections with electrical stimulation (ES) of hand muscles could improve active hand function (AHF) among chronic stroke patients. Our open-label, pilot study included 15 chronic stroke patients. Two weeks after BTX-A injections into the finger and/or wrist flexors, ES of finger extensors was performed while wearing a wrist brace for 4 weeks (5 days per week; 30-min sessions). Various outcomes were assessed at baseline, immediately before BTX-A injections, and 2 and 6 weeks after BTX-A injections. After the intervention, we noted significant improvements in Box and Block test results, Action Research Arm Test results, the number of repeated finger flexions/extensions, which reflect AHF, and flexor spasticity. Moreover, significant improvements in active ROM of wrist extension values were accompanied by marginally significant changes in Medical Research Council wrist extensor and active ROM of wrist flexion values. In conclusion, BTX-A injections into the finger and/or wrist flexors followed by ES of finger extensors improve AHF among chronic stroke patients.

**Keywords:** spasticity; spastic paresis; botulinum toxin; electrical stimulation; stroke management; rehabilitation; hand

**Key Contribution:** Botulinum toxin type A injections into the finger and/or wrist flexor muscles followed by electrical stimulation of finger extensors targeting spastic paresis improve active hand function and impairment among patients with chronic stroke.

#### **1. Introduction**

Upper limb spasticity (ULS) is a common impairment after stroke, which can cause abnormal posture, pain, reduced movement, and functional impairment [1–4]. Various interventions, including occupational therapy, physical therapy, oral medications, orthoses, and focal neurolysis, have been used in the management of ULS.

Intramuscular injection of botulinum toxin type A (BTX-A) is the recommended first-line treatment for focal ULS [5–9]. Botulinum toxin type A, which is produced by anaerobic, gram-positive, spore-forming microorganisms from the genus *Clostridium*, is the most potent toxin among various

serotypes, and its safety is well established [10,11]. Its neurospecific activity, reversibility, and limited diffusion on local injection make it a safe and successful agent for symptoms characterized by hyperfunction of peripheral nerve terminals [12,13]. Although BTX-A injections have been shown to improve muscle tone, range of motion, and physical global assessment findings, evidence regarding their efficacy for improving active upper limb function (e.g., moving/manipulating objects) is lacking [14,15]. Indeed, systematic reviews have demonstrated that BTX-A has greater effects on passive function than on active function, and some studies have failed to show any improvements in active function with BTX-A treatment [4,16]. These results might hinder active treatment with BTX-A, because the ultimate goal of rehabilitation is functional improvement.

Several explanations for the discrepancy between impairment and function have been proposed. First, most previous treatments have targeted spasticity per se, while factors other than hypertonicity are likely to be associated with active function, including stretch-sensitive paresis, overactivity of agonistic muscles, and learned non-use [2,3,15]. Second, optimal adjunct therapies following BTX-A injections have not been used, despite the need for adjuvant therapy to improve efficacy [17,18]. The combination of modified constraint-induced movement therapy or active task-specific training with BTX-A injections has been shown to improve upper limb function among patients with chronic spastic hemiparesis [19,20]. However, these approaches are not available for individuals with moderate-to-severe muscle weakness, prompting the need for commonly or easily applicable optimal adjunctive therapies to augment hand function. Third, although hand function is critical for upper limb function as an end-effector, few studies have targeted active function of the hand itself [21]. As hand spasticity is particularly disruptive, improvements in shoulder or elbow function may not be adequate to ensure overall functional improvement [22]. In addition, appropriate outcome measures related to both impairment and function, especially active hand function, have not been used [23–25]. Although previous studies have used various outcome measures for upper limb function (e.g., disability assessment scale, motor activity log, and global self-assessment), none have focused on active hand function [20,24,26]. Fourth, standardized techniques for administering BTX-A are yet to be established, and thus there are variations in muscle selection, injection methods, and medication doses.

Many patients having ULS exhibit difficulty with finger extension, which is critical for gripping, grasping, and releasing movements. Thus, we hypothesized that relief of finger flexor spasticity following BTX-A treatment would help patients regain finger extensor strength, which would in turn improve hand function. Based on the findings of previous studies, we further hypothesized that electrical stimulation (ES) is a simple yet effective form of adjuvant therapy, as it facilitates muscle reeducation, including antagonist strengthening and agonist lengthening [27,28]. Therefore, in the present preliminary study, we aimed to determine whether BTX-A injections targeting finger flexor spasticity with concomitant ES of finger extensors can improve active hand function and impairment among patients with chronic stroke.

#### **2. Results**

Among the 15 included patients, 7 experienced cerebral infarction and 8 experienced intracerebral hemorrhage. Thirteen patients experienced lesions in the right hemisphere. The mean time from stroke was 12.8 ± 3.7 months. All 15 patients completed the study, and none experienced adverse events.

#### *2.1. Primary Outcomes*

After the intervention, there were significant improvements in Box and Block (BB) (*χ*2 = 6.50, *p* = 0.039), Action Research Arm Test (ARAT)-total (*χ*2 = 6.29, *p* = 0.043), and ARAT-gross movement scores (*χ*2 = 8.64, *p* = 0.013), which reflect active hand function (Table 1). Post-hoc analysis confirmed that there were significant improvements in these outcomes in the T1–T3 phase. In addition, significant improvements in ARAT-total and ARAT-gross movement scores were observed in the T2–T3 and T1–T2 phases, respectively.

**Table 1.** Changes in primary outcome measures over time.

Data are presented as mean ± standard deviation. Outcomes were assessed at immediately before (T1), 2 weeks (T2), and 6 weeks of BTX-A injections (T3). \* Friedman test; † Post-hoc Wilcoxon signed-rank test between T1 and T2, between T2 and T3, and between T1 and T3. ARAT: Action Research Arm Test, BB: Box and Block.

#### *2.2. Secondary Outcomes*

Repeat-finger extensor/extension (FE) values showed significant increases (*χ*2 = 10.00, *p* = 0.007) in the T1–T3 and T2–T3 phases. Active-FE values showed marginally significant increases during the T2–T3 phase (*χ*2 = 5.615, *p* = 0.060).

Flexor spasticity significantly decreased (Modified Ashworth Scale for wrist flexor/flexion (MAS-WF): *χ*2 = 8.40, *p* = 0.015; MAS for finger flexor/flexion (MAS-FF): *χ*2 = 20.18, *p* < 0.001), and post-hoc tests revealed significant differences in the T1–T2 and T1–T3 phases, but not in the T2–T3 phase (Table 2). Significant increases in active range of motion for wrist extensor/extension (AROM-WE) values (*χ*2 = 9.08, *p* = 0.011) were accompanied by marginally significant increases in Medical Research Council WE (MRC-WE) (*χ*2 = 6.00, *p* = 0.05) and AROM-WF (*χ*2 = 6.00, *p* = 0.05) values. Post-hoc analyses demonstrated that AROM-WE values were higher at T3 than at T1 or T2 and that MRC-WE values significantly increased from T1 to T3. No significant changes were noted for other secondary outcomes, such as the Quick Disabilities of Arm, Shoulder, and Hand (QDASH) score (*χ*2 = 3.636, *p* = 0.162).

**Table 2.** Changes in secondary outcome measures over time.

Data are presented as mean ± standard deviation. Outcomes were assessed at immediately before (T1), 2 weeks (T2), and 6 weeks of BTX-A injections (T3). \* Friedman test; † Post-hoc Wilcoxon signed-rank test between T1 and T2, between T2 and T3, and between T1 and T3. FE: finger extensor/extension, FP: fingertip to the palm, MAS: Modified Ashworth Scale, WF: wrist flexor/flexion, WE: wrist extensor/extension, FF: finger flexor/flexion, MRC: Medical Research Council, AROM: active range of motion, RD: radial deviation, UD: ulnar deviation, QDASH: Quick Disabilities of Arm, Shoulder, and Hand, ◦: degree.

#### **3. Discussion**

In the present study, we attempted to examine the efficacy of BTX-A treatment targeting finger flexor spasticity and concomitant ES for improving active hand function in the chronic phase of stroke. We found that BTX-A injections into the flexor muscles of the fingers and wrist followed by ES improved active hand function, as indicated by significant improvements in BB, ARAT (ARAT-total, ARAT-gross movement), and Repeat-FE scores. These findings are in contrast to those of previous studies, which failed to demonstrate improvements in active function after BTX-A treatment among patients with chronic stroke [29–31].

These discrepancies may be explained by our concept of intervention, as we considered treatment according to spastic paresis rather than flexor muscle spasticity only [2,3]. Overactivity of flexor muscles due to spasticity impedes finger and wrist extension, aggravating stretch-sensitive paresis and inhibiting voluntary recruitment of extensors. Similarly, a previous study demonstrated co-activation of spastic flexors and weak extensors [32]. Rehabilitation strategies often focus on finger/wrist flexors, as most hand functions are related to flexor movement (e.g., grasping, pinching, and manipulating objects). However, many patients with ULS following stroke exhibit difficulty with finger extension, which is needed to initiate hand grip and grasp. Thus, we attempted to restore the balance between the flexors and extensors of the wrist/fingers by targeting asymmetrical spastic paresis in two main steps (BTX-A injections and ES with a wrist brace).

Botulinum toxin type A injections to finger/wrist flexors reduced involuntary activation of the flexor muscles and further led to "therapeutic weakness" [33]. However, such weakness is only observed when the doses of BTX-A administered at the flexor digitorum superficialis and profundus are much larger than the doses recommended by the current consensus report [34,35]. Therapeutic weakening of the agonistic finger/wrist flexors allowed for restoration of antagonistic finger/wrist extensor function by reducing reciprocal inhibition from the flexors, stretch-induced paresis, agonistic flexor overactivity, and reflex tone [2,3,36].

Improved balance between agonist and antagonist muscles was accomplished with proprioceptive discharge reduction via group-IA afferents [37,38]. Botulinum toxin type A decreased afferent input of the muscle spindles and their stretch sensitivity. Presynaptic acetylcholine release block with BTX-A was more profound and earlier in gamma-motoneurons than in alpha-motoneurons. From the viewpoint of cortical activity, this BTX-A effect on proprioception is important because abnormal spindle activity generating irregular proprioceptive input may result in abnormal intracortical inhibition and cortical excitability [39]. Similarly, BTX-A-induced spasticity relief was associated with central sensorimotor activation change, which was likely mediated by an altered afferent input from spastic muscles [40]. Moreover, BTX-A reduced the activation of the contralateral putamen and thalamus via modulation of basal ganglia activation [41].

On the other hand, fascicle selection was tailored to each patient, and a flexible dosing regimen was utilized to ensure the appropriate level of weakness without inadvertent over-weakness. This may explain why we observed significant changes in ARAT-gross movement and MAS-WF/finger flexor/flexion (FF) values, but not MRC-WF/FF values or functional deterioration, between T1 and T2.

We subsequently utilized ES to facilitate target muscle (finger/wrist extensor) contraction, which has been difficult to achieve owing to increased flexor tone. ES enabled reeducation, stretching, and strengthening of the extensor muscles [28]. This selective facilitation of agonistic extensor muscles resulted in significant improvements in AROM-WE between T2 and T3, rather than improvements in AROM-WF. Concordantly, active function-related improvements were observed in ARAT-total, ARAT-gross movement, Repeat-FE, and Active-FE values during this phase, but not between T1 and T2.

Such differences between T1–T2 and T2–T3 may explain the discrepancies between active and passive functions following BTX-A treatment [42]. Our findings suggest that spasticity control is insufficient for restoring active function and that concomitant interventions selectively facilitating agonist muscles are necessary to improve outcomes. Notably, cyclic ES of both finger flexors and extensors following BTX-A injections has not been shown to be associated with functional improvements [25], suggesting that such interventions should target agonist function rather than antagonist spasticity per se.

In addition, we utilized a wrist brace to maximize finger extension during ES while reducing unwanted wrist or metacarpophalangeal joint hyperextension. Full finger extension is difficult during concomitant wrist extension owing to passive tension that results from the muscle insertion pattern of two-joint finger flexor muscles. The wrist brace maintained the wrist/metacarpophalangeal joints in a neutral position to allow sufficient extension of the distal or proximal interphalangeal joints, and it may have decreased stretch-sensitive paresis of the finger extensors.

In the present study, we attempted to utilize comprehensive hand-relevant outcome measures for both impairment (MRC, MAS, and AROM) and active function (ARAT and BB). We also utilized measures specific to hand function, particularly finger extension (Active FE, Distance-FP, Repeat-FE, and grip strength). These various measures assessed at different time points helped classify the effects of our intervention in terms of spastic paresis. Recently, goal attainment scales have been used to determine individualized, focal changes following BTX-A injections [43]. However, such scales do not measure objective outcomes but rather measure subjective achievement according to the opinions of the physicians and patients [44]. Therefore, we believe that the objective scales utilized in the present study are more appropriate than goal attainment scales for determining focal functional changes in clinical trials.

According to the QDASH results, no improvement in subjective upper limb function was observed, and this is likely due to the small yet significant objective change over the course of the intervention. Further studies should examine whether both subjective and objective improvements in upper limb function can be observed in select groups of patients, such as those in the subacute phase of stroke and those with some level of strength in the wrist/finger extensors.

The present study had several limitations. First, as this was not a randomized controlled trial, it is difficult to conclude whether our intervention would outperform other interventions or whether the combination of BTX-A injections and ES is more effective than either approach alone. Second, we did not measure outcomes at 12 weeks, when the effects of BTX-A can no longer be observed [45]. Finally, we did not take into account thumb function. Considering that thumb function accounts for 40% of hand performance, future studies should investigate the efficacy of our intervention with regard to thumb function [46].

#### **4. Conclusions**

Our findings indicate that BTX-A injections into the finger and/or wrist flexor muscles followed by ES of finger extensors improve active hand function and impairment among patients with chronic stroke. The present results support the notion that interventions targeting spastic paresis, rather than spasticity per se, may improve active hand function.

#### **5. Materials and Methods**

The present open-label, pilot study included 15 patients (14 men; 14 right-handed individuals; mean age, 44.7 ± 15.0 years; age range, 22–74 years; Table 3) from the stroke unit of a rehabilitation hospital between March 2016 and February 2017. Patients engaged in inpatient rehabilitation programs at the time of enrollment were allowed to continue therapy throughout the intervention, and anti-spasticity medications were continued at current or reduced doses or were discontinued. The study was approved by the ethics review board of our rehabilitation hospital (NRC-2016-02-011), and all participants provided written informed consent prior to enrollment. The study was registered with ClinicalTrials.gov (number NCT 03549975, 29 March 2016).

**Table 3.** Clinical characteristics of the study population.

AP: adductor pollicis, BB: biceps brachii, FCU: flexor carpi ulnaris, PT: pronator teres, FDP: flexor digitorum profundus, FDS: flexor digitorum superficialis, FPB: flexor pollicis brevis, FPL: flexor pollicis longus, Rt.: right, Lt.: left.

#### *5.1. Inclusion and Exclusion Criteria*

The inclusion criteria were as follows: (1) age older than 18 years; (2) hemiplegic ULS secondary to unilateral ischemic or hemorrhagic stroke; (3) finger and wrist flexor spasticity graded at least 1+ on the MAS; and (4) stroke occurrence at least 6 months prior to enrollment. The exclusion criteria were as follows: (1) fixed contracture of the wrist or hand; (2) previous ULS treatment with neurolytic therapy, surgery, or BTX-A injection; (3) presence of any active implanted device; (4) presence of any neurological disorder causing motor deficit or spasticity, other than stroke; (5) inability to cooperate for all outcome measure-related tasks secondary to cognitive impairment or aphasia; (6) pregnancy, planned pregnancy, or lactation; and (7) contraindication for BTX-A treatment.

#### *5.2. Intervention*

#### 5.2.1. BTX-A Administration

Patients received BTX-A (onabotulinumtoxin A, Botox; Allergan Inc., Irvine, KY, USA) injections at week 0. The dilution of BTX-A was standardized, such that each vial of BTX-A (100 U) was diluted with 2 mL of normal saline (5 U/0.1 mL). A physician with experience in BTX-A treatment performed clinical evaluations of ULS and freely selected the target muscle and dose according to the pattern and severity of ULS in each patient. BTX-A injections were then administered under ultrasonographic guidance (ACCUVIX XG; Medison, Seoul, Korea). Injections at the flexor digitorum superficialis and profundus were administered by tailoring fascicle selection to each patient and by adopting a flexible dosing regimen. In each patient, the total BTX-A dose did not exceed 360 U. The dosages and injection sites are listed in Table 3.

#### 5.2.2. ES with a Wrist Brace

After 2 weeks of BTX-A injections, patients received ES of finger extensors while wearing a wrist brace for 4 weeks. ES was administered 5 days per week during 30-min sessions by trained occupational therapists, using a Novastim CU-FS1 unit (CU, Medical, Seoul, Korea). During ES, each patient was seated in a chair with the affected arm resting on a table and with a small pillow supporting the pronated forearm. Cyclic ES was administered to induce muscle contractions, using biphasic pulses (frequency, 50 Hz; pulse duration, 200 μs) delivered for 6 s (ramp up, 1 s; ramp down, 2 s), with a 12-s burst-off. Two surface electrodes (5 × 5 cm) were placed over the extensor digitorum communis. The amplitude of stimulation was adjusted to elicit adequate finger extension for a sufficient range of motion against gravity, without pain.

The wrist brace (National Rehabilitation Center, Seoul, Korea) used during ES was made of soft fabric, and it covered the area from the wrist to the metacarpophalangeal joint. Two metal bars were attached to the volar and dorsal regions of the brace to immobilize the wrist in a neutral position. This immobilization ensured that ES could induce selective distal or proximal interphalangeal joint extension while overcoming unwanted simultaneous wrist or metacarpophalangeal joint extension. A Velcro strap was used to restrain the dorsal and volar sides of the brace.

#### *5.3. Outcome Measures*

We recorded baseline characteristics, including sex, age, stroke type, brain lesion side, time since stroke, and handedness. Primary and secondary outcomes were assessed at baseline (week −1, T0), immediately before BTX-A injections (week 0, T1), and 2 weeks (week 2, T2) and 6 weeks (week 6, T3) after BTX-A injections, by an experienced research occupational therapist. For further screening, we evaluated changes in outcomes between T0 and T1. Patients exhibiting changes between these two time points were excluded from the study. We examined the effects of BTX-A at 2 weeks, as previous studies have indicated that BTX-A shows maximum efficacy between 1 and 2 weeks after treatment [47]. Outcomes at T3 represented the combined effects of BTX-A and ES with a wrist brace (Figure 1).

**Figure 1.** Experimental design. Primary and secondary outcomes are assessed at baseline (week 1, T0), immediately before BTX-A injections (week 0, T1), and 2 weeks (week 2, T2) and 6 weeks (week 6, T3) after BTX-A injections, by an experienced research occupational therapist. Patients exhibiting changes in outcomes between T0 and T1 were excluded. BTX-A: botulinum toxin type A, ES: electrical stimulation.

#### 5.3.1. Primary Outcomes

The primary outcomes were changes in patient results in the BB test and ARAT between T1 and T3, which represent changes in the active function of the upper limb following the present intervention. The BB test is a reliable and validated measure of unilateral gross manual dexterity, in which participants are required to pick up a 1-inch block, lift it over a partition, and then release it within a target area as many times as possible within 60 s [48]. The ARAT is a reliable and validated measure of arm function using observational methods with the following four subsections: grasp, grip, pinch, and gross movement. It comprises 19 items rated using a 4-point ordinal scale (0–3). Thus, the total scores on the ARAT range from 0 (failure to perform all movements; patients cannot perform any part of the test) to 57 (all movements are performed normally) [49].

#### 5.3.2. Secondary Outcomes

Hand-specific active function was assessed according to active finger extension (Active-FE), the maximum distance from the fingertip to the palm (Distance-FP), the number of repeated finger flexion/extension movements within 20 s (Repeat-FE), the rating of active thumb opposition (Thumb-opposition), and grip strength as measured using the Jamar dynamometer (Sammons Preston, Bolingbrook, IL, USA). Active-FE was rated using a 5-point ordinal scale (0 = no movement, 4 = functional extension of all digits) by reviewing a video clip [50]. Distance-FP (cm) values were obtained by measuring the distance from the middle fingertip to the mid-palmar crease [51]. Thumb-opposition was rated using a 7-point scale (1 = to the distal phalanx of the index finger, 7 = to the medial phalanx of the small finger) [51].

Upper limb impairment was assessed according to MAS scores for the wrist and finger flexors/extensors (MAS-WF/WE/FF/FE), MRC grades for wrist and finger flexor/extensor strength (MRC-WF/WE/FF/FE), and active range of motion for wrist flexion/extension (AROM-WF/WE) and radial/ulnar deviation (AROM-RD/UD) as determined through goniometric measurement. Modified Ashworth Scale grades of spasticity were converted from 1+, 2, 3, and 4 to 2, 3, 4, and 5, respectively [52].

Subjective general upper limb function was evaluated using the self-reported QDASH scale. The QDASH scale consists of 11 items regarding physical function and symptoms during the previous week, and the total scores range from 0 (no disability) to 100 (most severe disability) [53].

#### *5.4. Statistical Analysis*

For all outcome measures, the Friedman test was used to determine the overall change from pre-intervention (T1) to post-intervention (T2)/follow-up (T3). When the Friedman test yielded significant results, various pairwise comparisons were performed using post-hoc Wilcoxon signed-rank tests in order to determine the effect of each intervention (BTX-A/ES). All statistical analyses were performed using SPSS 20.0 (IBM SPSS Statistics, IBM Corporation, Armonk, NY, USA). The level of statistical significance was set at *p* < 0.05.

**Author Contributions:** Conceptualization, J.-M.L., J.-M.G. and J.-H.S.; methodology, J.-H.S.; software, J.Y.L.; validation, S.-B.P. and K.H.L.; formal analysis, J.-M.L.; investigation, J.Y.L.; resources, S.-B.P.; data curation, K.H.L.; writing—original draft preparation, J.-M.L.; writing—review and editing, J.-H.S.; visualization, J.-M.L.; supervision, J.-H.S.; project administration, J.-H.S.; funding acquisition, J.-H.S.

**Funding:** This research was funded by the Translational Research Center for Rehabilitation Robots, National Rehabilitation Center, Ministry of Health and Welfare, Republic of Korea, grant number NRCTR-IN14002, NRCTR-IN15002.

**Conflicts of Interest:** Jean-Michel Gracies served as a consultant on advisory boards for and received research grant support from Allergan, Ipsen, and Merz. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### **References**

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).


*4.2. Flattening* L*-Institutions to Binary Institutions*

The general reduction of many-valued truth to binary truth advocated by the skeptics of many-valued truth can also be applied to L-institutions. It works as follows. Given any Linstitution I = (*Sign*, *Sen*, *Mod*, |=), we define the binary institution I = (*Sign* , *Sen* , *Mod* , |= ):

This flattening idea has been present in several places in the fuzzy logic literature. For instance, in [99], our pairs (*ρ*, *κ*) are called 'signed formulas' and given the same interpretation as here.

The flattening of L-institutions to binary institutions has the advantage of reducing things to a well-studied and matured framework and functions well in some aspects, but it falls short in several areas that involve some fine-grained aspects of multiple truth values. Thus, while the flattening S of stratified institutions does not pose many limitations, the situation is different with the flattenings of L-institutions.

## *4.3. The Graded Semantic Consequence*

Given an L institution such that L is a complete meet-semilattice, for each Σ-model *M* and each set *E* of Σ-sentences, we define

$$(M \mid \!=\_{\Sigma} E) = \bigwedge \{ M \mid \!=\_{\Sigma} \rho \mid \rho \in E \}. \tag{10}$$

Given an L-institution, there are two ways to extend the satisfaction relation to a semantic consequence relation between sets of sentences and single sentences, both of them generalising the semantic consequence relation of binary institution theory.

(*E* |=Σ *e*) = {(*M* |=Σ *E*) ⇒ (*M* |=Σ *e*) | *M* ∈ |*Mod*(Σ)|}. (11)

The graded semantic consequence is more subtle and more in the spirit of many-valued truth than the crisp one, although the definition of the latter requires more infrastructure on the space of the truth values, namely that L *is a residuated lattice* [100,101]. Hence, "⇒" of (11) represents the residuated implication operation. This difference in subtlety may be traced to the fact that while the crisp semantic consequence corresponds to the semantic consequence of the binary flattening I of the L-institution I (that *E* |= *e* holds in I means {(*ρ*, 1) | *ρ* ∈ *E*} |= (*e*, 1) in I ), the graded semantic consequence is a concept beyond I . The graded semantic consequence appears in a disguised form in [50] within the context of Pavelka's theory of fuzzy consequence operators and in a form that is more explicitly similar to ours in [102] within the framework of 'graded consequence relations'. However, both these semantic frameworks are less general than ours, in both of them models being in fact fuzzy theories.

One of the important properties of the semantic consequence in binary institution theory is that it satisfies the axioms of entailment systems. The graded semantic consequence enjoys the same property but for the following refined many-valued concept of entailment. This has been proved in a full form in [49]. In a restricted single signature framework, this has also been proved in [102].