Quadratic Mixture Research Articles

Association between geospatial disparities in food security with weight loss and nutritional outcomes of metabolic surgery

BackgroundFood insecurity has been linked to higher rates of obesity. It has also been shown to diminish the effectiveness of weight loss strategies, including intensive lifestyle interventions. One essential component of food insecurity is having a geospatial disadvantage in access to healthy, affordable food, such as living within a food desert. This study aims to determine if food insecurity also impacts weight loss and nutritional outcomes in patients who underwent Roux-en-Y gastric bypass (RYGB) or sleeve gastrectomy (SG).MethodsClinical outcomes of patients who underwent RYGB or SG at Cleveland Clinic or affiliate regional hospitals in the United States from 2010 to 2018 were collected. Modified Retail Food Environmental Index (mRFEI) data was collected from the Center for Disease Control and merged with patient census tract data, allowing the patient cohort to be divided into those living in areas identified as food secure (mRFEI > 10%), food swamps (mRFEI = 1–10%), or food deserts (mRFEI = 0). Postoperative weight change was evaluated with quadratic growth mixture models and stratified by surgery type.ResultsA total of 5097 patients were included in this study cohort, including 3424 patients who underwent RYGB and 1673 who underwent SG. The median duration of follow-up was 2.3 years (IQR 0.89–3.6 years). Food security status was not associated with postoperative weight change (RYGB p = 0.73, SG p = 0.60), weight loss nadir (RYGB p = 0.60, SG p = 0.79), or weight regain (RYGB p = 0.93, SG p = 0.85). Deficiencies in nutritional markers at 1–2 years after surgery were also not significantly different between food security groups.ConclusionDespite the established relationship between food insecurity and obesity, food insecurity does not negatively impact weight loss or nutritional outcomes following RYGB or SG, demonstrating metabolic surgery as a powerful and equitable tool for treating obesity.Level of EvidenceIV.

Glass design using machine learning property models with prediction uncertainties: Nuclear waste glass formulation

The United States Department of Energy is responsible for managing the legacy nuclear waste stored in underground tanks at the Hanford Site. The waste will be separately vitrified as low-activity waste and high-level waste fractions. Waste glass formulation algorithms have been traditionally developed using partial quadratic mixture property-composition models. Recently, machine learning (ML) techniques have been used to predict glass properties and discover new glass materials for nuclear waste vitrification, and these advancements can be utilized to improve waste glass composition design. In this proof-of-principle study, ML algorithms such as Gaussian process regression (GPR) were used to interpolate glass properties (e.g., viscosity, electrical conductivity, chemical durability). After selecting appropriate sets of GPR hyper-parameters for each property, an optimization program was developed to formulate glass compositions to maximize waste loading while simultaneously satisfying property constraints. The results of the ML-based waste loadings and glass compositions were compared to those obtained using the traditional methods. Comparing to the previous glass design framework, the ML-based optimization methods offer improved glass designs and a streamlined approach to generation of optimally designed data and near real-time updates.

Exploring the potential of neem and tamarind gum as release retardants: Design and statistical optimisation of vildagliptin extended release matrix systems using D-optimal quadratic mixture design

Exploring the significant role of natural polymers in developing drug delivery systems has been a promising area of research interest. The current investigation uses a D-optimal quadratic mixture design to design and evaluate neem and tamarind gum-based vildagliptin extended-release matrix tablets. Studying the combination effect of gums is one of the major objectives. Initial screening studies were performed to select the factors and their levels. The variables selected at different levels in mg/tablet are neem gum, tamarind gum, polyvinylpyrrolidone, and lactose monohydrate. Based on the screening experiments with both gums, the polymer content of 165 mg was chosen as the highest level in the DOE. Nineteen runs were generated to screen the desired parameters as responses. The total weight of the formulation was kept constant at 275 mg. Time (hours) required for 50 %, 90 % and 100 % of drug release and tablet hardness were selected as the responses for each run. The wet granulation method was adopted, and the critical variables were optimised using the design of experiments following Design Expert software. Statistical analysis was conducted, and the optimised formulations were prepared and evaluated to compare with the predicted responses. Stability studies were performed for the optimised batches. Results indicated that the prepared batches met the compendial limits and confirmed the application of neem and tamarind gum in the development of extended-release tablets of vildagliptin for 24 h. An optimised formulation comprising of 16.52 mg of neem gum and 148.48 mg of tamarind gum with a hardness of 7.5–8.5 kp produced 50 %, 90 % and 100 % drug release in 12, 22 and 25 h.

D- And A- Optimal Orthogonally Blocked Mixture Component-Amount Designs via Projections

Mixture experiments are usually designed to study the effects on the response by changing the relative proportions of the mixture ingredients. This is usually achieved by keeping the total amount fixed but in many practical applications such as medicine or biology, not only are the proportions of mixture ingredients involved but also their total amount is of particular interest. Such experiments are called mixture amount experiments. In such experiments, the usual constraint on the mixture proportions that they should sum to unity is relaxed. The optimality of the design strictly depends on the nature of the underlying model. In this paper, we have obtained D- and A- optimal orthogonally blocked mixture component-amount designs in two and three ingredients via projections based on the reduced cubic canonical model presented by Husain and Sharma [7] and the additive quadratic mixture model proposed by Husain and Parveen [3], respectively.

Machine Learning with Gradient-Based Optimization of Nuclear Waste Vitrification with Uncertainties and Constraints

Gekko is an optimization suite in Python that solves optimization problems involving mixed-integer, nonlinear, and differential equations. The purpose of this study is to integrate common Machine Learning (ML) algorithms such as Gaussian Process Regression (GPR), support vector regression (SVR), and artificial neural network (ANN) models into Gekko to solve data based optimization problems. Uncertainty quantification (UQ) is used alongside ML for better decision making. These methods include ensemble methods, model-specific methods, conformal predictions, and the delta method. An optimization problem involving nuclear waste vitrification is presented to demonstrate the benefit of ML in this field. ML models are compared against the current partial quadratic mixture (PQM) model in an optimization problem in Gekko. GPR with conformal uncertainty was chosen as the best substitute model as it had a lower mean squared error of 0.0025 compared to 0.018 and more confidently predicted a higher waste loading of 37.5 wt% compared to 34 wt%. The example problem shows that these tools can be used in similar industry settings where easier use and better performance is needed over classical approaches. Future works with these tools include expanding them with other regression models and UQ methods, and exploration into other optimization problems or dynamic control.

D-optimal designs for mixture experiments with various correlation structures

Mixture experimental designs have been in widespread use in agricultural, pharmaceutical, and other industrial research for many years. Much of the previous work mainly focuses on optimal design for mixture experiments when observations are uncorrelated, in large part because of the intractability of the optimal mixture experimental design on correlated case. When observations have certain correlation structures within each block, the order of the observations in each block matters and this order impacts the optimality of the design. Thus, there is need of research into construction of these useful designs when correlation structures might exist within blocks. In this paper, we propose D-optimal minimum support designs for Scheffe’s quadratic mixture model with odd number of components when observations in blocks are circulant correlated and hub correlated respectively, and Scheffe’s mixture model with any components when observations are in block-structured correlation.

Hanford low‐activity waste glass composition‐temperature‐melt viscosity relationships

AbstractThis study developed a model for predicting viscosity of alkali‐alumino‐borosilicate glass melts as functions of composition and temperature. The model is based on a total of 3935 viscosity‐temperature data from 574 glasses with viscosity values ranging from 2.53 to 7260 Poise (P) in the temperature range of 900–1260°C. Several different model forms were surveyed, including those based on Arrhenius, Vogel‐Fulcher‐Tammann, Avramov‐Milchev, and Mauro‐Yue‐Ellison‐Gupta‐Allen. For each of these models, combinations of the temperature‐independent parameters were fitted to composition. It was found that generally fitting more than one temperature‐independent parameter as functions of composition resulted in overfitting. The Avramov‐Milchev‐based model was found to best represent the Hanford low‐activity waste glass melt viscosity data based on model fit and validation statistics. A 21‐term partial quadratic mixture model was recommended for use. This model predicts melt viscosity with a root‐mean square error of .1736 ln(P), which is similar to the error in viscosity measurements from replicate glass analyses of .1383 ln(P). Viscosity was found to be most increased by SiO2 > Al2O3 > ZrO2 > SnO2 and most decreased by Li2O > Na2O > B2O3 > CaO > K2O > MgO, at temperatures from 900 to 1260°C.

Repurposing Lovastatin Cytotoxicity against the Tongue Carcinoma HSC3 Cell Line Using a Eucalyptus Oil-Based Nanoemulgel Carrier.

Tongue cancer is one of the most common carcinomas of the head and neck region. The antitumor activities of statins, including lovastatin (LV), and the essential oil of eucalyptus (Eu oil), have been adequately reported. The aim of this study was to develop a nanoemulgel containing LV combined with Eu oil that could then be made into a nanoemulsion and assessed to determine its cytotoxicity against the cell line human chondrosarcoma-3 (HSC3) of carcinoma of the tongue. An I-optimal coordinate-exchange quadratic mixture design was adopted to optimize the investigated nanoemulsions. The droplet size and stability index of the developed formulations were measured to show characteristics of the nanoemulsions. The optimized LV loaded self-nanoemulsifying drug delivery system (LV-Eu-SNEDDS) was loaded into the gelling agent Carbopol 934 to develop the nanoemulgel and evaluated for its rheological properties. The cytotoxic efficiency of the optimized LV-Eu-SNEDDS loaded nanoemulgel was tested for cell viability, and the caspase-3 enzyme test was used against the HSC3 cell line of squamous carcinoma of the tongue. The optimized nanoemulsion had a droplet size of 85 nm and a stability index of 93%. The manufactured nanoemulgel loaded with the optimum LV-Eu-SNEDDS exhibited pseudoplastic flow with thixotropic behavior. The developed optimum LV-Eu-SNEDDS-loaded nanoemulgel had the best half-maximal inhibitory concentration (IC50) and caspase-3 enzyme values of the formulations developed for this study, and these features improved the ability of the nanoemulsion-loaded gel to deliver the drug to the investigated target cells. In addition, the in vitro cell viability studies revealed the synergistic effect between LV and Eu oil in the treatment of tongue cancer. These findings illustrated that the LV-Eu-SNEDDS-loaded gel formulation could be beneficial in the local treatment of tongue cancer.

Preparation and Optimization of Garlic Oil/Apple Cider Vinegar Nanoemulsion Loaded with Minoxidil to Treat Alopecia

Alopecia areata is a scarless, localized hair loss disorder that is typically treated with topical formulations that ultimately only further irritate the condition. Hence, the goal of this study was to develop a nanoemulsion with a base of garlic oil (GO) and apple cider vinegar (APCV) and loaded with minoxidil (MX) in order to enhance drug solubilization and permeation through skin. A distance coordinate exchange quadratic mixture design was used to optimize the proposed nanoemulsion. Span 20 and Tween 20 mixtures were used as the surfactant, and Transcutol was used as the co-surfactant. The developed formulations were characterized for their droplet size, minoxidil steady-state flux (MX Jss) and minimum inhibitory concentration (MIC) against Propionibacterium acnes. The optimized MX-GO-APCV nanoemulsion had a droplet size of 110 nm, MX Jss of 3 μg/cm2 h, and MIC of 0.275 μg/mL. The optimized formulation acquired the highest ex vivo skin permeation parameters compared to MX aqueous dispersion, and varying formulations lacked one or more components of the proposed nanoemulsion. GO and APCV in the optimized formulation had a synergistic, enhancing activity on the MX permeation across the skin membrane, and the percent permeated increased from 12.7% to 41.6%. Finally, the MX-GO-APCV nanoemulsion followed the Korsmeyer–Peppas model of diffusion, and the value of the release exponent (n) obtained for the formulations was found to be 1.0124, implying that the MX permeation followed Super case II transport. These results demonstrate that the MX-GO-APCV nanoemulsion formulation could be useful in promoting MX activity in treating alopecia areata.

Physicochemical, Mineral and Sensory Characteristics of Instant Citrullus lanatus mucosospermus (Egusi) Soup.

Defatted egusi flour offers a food option high in protein and essential micronutrients. An instant processing method was adopted in a ready-to-eat instant soup using egusi grit, hydrocolloid, and defatted flour. A D-optimal quadratic mixture model was used to study the effect of the independent variables (grit, flour, and hydrocolloid) qualities. The quadratic model was adequate to navigate the design space for taste and appearance. The numerical optimization for appearance and taste of instant soup (IES) was used to obtain the optimal soup mix of 10 g of hydrocolloid, 57.2 of defatted flour and 17 g of grits. Sixteen trace and five major mineral elements were found in the egusi soup, with a high concentration of phosphorus (1220.4, 1326.2 and 1277.9 mg/100 g), potassium (1220.4, 1326.2 and 1277.9 mg/100 g), magnesium (822.2, 905.3 and 863.70 mg/100 g), calcium (172.3, 190.9 and 183.4 mg/100 g) and iron (53.7, 57.5 and 29.5 mg/100 g), and for instant egusi soups from boiled egusi grit (IESBG), instant egusi grit from spherified grit (IESSG) and instant egusi grit from extruded grit (IESEG), respectively. The amino acid profile of instant egusi soup offers all essential amino acids necessary to nourish the body. Phosphorus content was significantly (p ≤ 0.05) high across the three soups: 1742, 1836 and 1838 mg/100 g for IESBG, IESSG, and IESEG, respectively; IESSG and IESEG were significantly (p ≤ 0.05) higher in minerals when compared with IESBG. Instant egusi soup differed significantly (p ≤ 0.05) in lightness (L*), while the redness (a*) and yellowness (b*) did not vary significantly.

Rethinking the course of psychotic disorders: modelling long-term symptom trajectories.

The clinical course of psychotic disorders is highly variable. Typically, researchers have captured different course types using broad pre-defined categories. However, whether these adequately capture symptom trajectories of psychotic disorders has not been fully assessed. Using data from AESOP-10, we sought to identify classes of individuals with specific symptom trajectories over a 10-year follow-up using a data-driven approach. AESOP-10 is a follow-up, at 10 years, of 532 incident cases with a first episode of psychosis initially identified in south-east London and Nottingham, UK. Using extensive information on fluctuations in the presence of psychotic symptoms, we fitted growth mixture models to identify latent trajectory classes that accounted for heterogeneity in the patterns of change in psychotic symptoms over time. We had sufficient data on psychotic symptoms during the follow-up on 326 incident patients. A four-class quadratic growth mixture model identified four trajectories of psychotic symptoms: (1) remitting-improving (58.5%); (2) late decline (5.6%); (3) late improvement (5.4%); (4) persistent (30.6%). A persistent trajectory, compared with remitting-improving, was associated with gender (more men), black Caribbean ethnicity, low baseline education and high disadvantage, low premorbid IQ, a baseline diagnosis of non-affective psychosis and long DUP. Numbers were small, but there were indications that those with a late decline trajectory more closely resembled those with a persistent trajectory. Our current approach to categorising the course of psychotic disorders may misclassify patients. This may confound efforts to elucidate the predictors of long-term course and related biomarkers.

Oral gel loaded with penciclovir–lavender oil nanoemulsion to enhance bioavailability and alleviate pain associated with herpes labialis

Herpes labialis, caused by herpes simplex virus type 1, is usually characterized by painful skin or mucosal lesions. Penciclovir (PV) tablets are found to be effective against herpes labialis but suffer from poor oral bioavailability. This study aimed to combine the benefits of PV and lavender oil (LO), which exhibits anesthetic activity, in the form of a self-nanoemulsifying drug delivery system (SNEDDS) for the treatment of herpes labialis. Toward this purpose, LO (oil), Labrasol:Labrafil M1944 CS in the ratio of 6:4 (surfactant mixture), and Lauroglycol-FCC (co-surfactant, selected based on the solubility of PV) were evaluated as the independent factors using a distance quadratic mixture design. The formulation was optimized for the minimum globule size and maximum stability index and was determined to contain 14% LO, 40.5% Labrasol:Labrafil 1944 (6:4), and 45.5% Lauroglycol-FCC. The optimized PV-LO-SNEDDS was embedded in chitosan hydrogel and the resulting formulations coded by (O3) were prepared and evaluated. The rheological studies demonstrated a combined pseudoplastic and thixotropic behavior with the highest flux of PV permeation across sheep buccal mucosa. Compared to a marketed 1% PV cream, the O3 formulation exhibited a significantly higher and sustained PV release, nearly twice the PV permeability, and a relative bioavailability of 180%. Overall, results confirm that the O3 formulation can provide an efficient delivery system for PV to reach oral mucosa and subsequent prolonged PV release. Thus, the PV-LO-SNEDDS embedded oral gel is promising and can be further evaluated in clinical settings to establish its therapeutic use in herpes labialis.

Effect of Sodium Substitution on Sensory and Quality Parameters in Mango Pickle

Aims: An intensive study was conducted so as to find out the best salt combinations when NaCl was partially replaced with KCl and CaCl2 while curing mango pieces. Mango pickle contains 15-20% salt, the consumption of which cause hypertension and high blood pressure. This is due to excess sodium ion (Na+) consumption.
 Study Design: D-Optimal Quadratic Mixture Design using Response Surface Methodology was adopted with 3 mixture components to get 16 mixture combinations of salts.
 Methodology: 16 runs were formulated depicting diverse salt mixtures and pickle samples were cured using these 16 different brine formulations. After pickling, the final sample were analyzed for water drawing capacity (g/100g salt), hardness of cured sample (N), water activity (aw), sodium and potassium concentration in pickle (in ppm) and organoleptic properties based on hedonic scale.
 Results: The result shows that water drawing capacity was maximum (102.692) in the 6th run and minimum (58.364) in 2nd run. Hardness of cured sample (in N) was maximum (8.470) in11th run and minimum (6.998) in 9th run. Water activity was maximum in 9th run (0.974) and minimum (0.968) in the 2nd run. Na concentration in pickle (ppm) was highest in the 14th run (408.715) and lowest in 2nd run (68.715). K concentration in final pickle (ppm) was maximum in 6th run (472.222) and minimum in 9th run (28.744). Organoleptic properties based on hedonic scale was evaluated, it was found that run number 14 and 12 have maximum score of 7.429 and 7.393 respectively whereas Run-1 and Run-2 showed minimum score of 6.552 and 6.616 respectively.
 Conclusion: Numerical optimization revealed that formulation with 0.5 fractions NaCl and 0.5 fractions KCl is having the maximum desirability of 0.658. Another formulation with NaCl 57.1% and KCl with 42.9% is having a desirability of 0.64. This formulation can also be adopted for the purpose of commercialization.

Bumpy and Smoother Pathways of Puberty Hormone Change: A Novel Way to Define Gonadal Hormone Trajectories in Adolescents.

ContextThe study of gonadal hormone effects on adolescent wellbeing has been limited by logistical challenges. Urine hormone profiling offers new opportunities to understand the health and behavioral implications of puberty hormones.ObjectiveTo characterize pubertal change in urinary testosterone and estradiol among male and female adolescents, respectively.DesignThree-year prospective cohort study.SettingAustralian regional community.Participants282 (163 male) normally developing adolescents aged 11.8 ± 1.0 years at baseline.Main outcome measureQuarterly urine measurements of testosterone and estradiol (mass spectrometry); annual anthropometric assessment and Tanner stage (TS) self-report.ResultsTwo-class sigmoidal and quadratic growth mixture models (centered on age at TS3) were identified as best-fit for describing testosterone (male) and estradiol (female) change. Classes 1 (male: 63%; female: 82%) and 2 (male: 37%; female: 18%) were respectively named the “stable” and “unstable” trajectories, characterized by different standard deviation of quarterly hormone change and magnitude of hormone peaks and troughs (all P < 0.001). Compared with class 1 (stable), class 2 males were taller at baseline (154 vs 151 cm), reported earlier and faster TS progression (P < 0.01), and showed higher serum testosterone levels at baseline and 3 years (P ≤ 0.01). Class 2 females exhibited smaller height and weight gains over the 3 years and had higher baseline serum estradiol (249 vs 98 pmol/L; P = 0.002) than class 1.ConclusionsAdolescents showed 2 distinct urinary gonadal hormone trajectories, characterized by stability of change over time, which were not associated with consistent anthropometric differences. Results provide a methodology for studying gonadal hormone impacts on other aspects of biopsychosocial wellbeing. Identification of potential “at-risk” hormone groups would be important for planning supportive interventions.

A-optimal designs for quadratic mixture canonical polynomials with spline

A mixture canonical polynomial with spline, which consists of several grafted polynomial submodels, has attracted a great deal of researchers in recent years. There is a typical mixture polynomial with spline, in which the contribution of one ingredient proportion to the response variable is different from that of the others. Much of the previous work mainly focuses on the D-optimal design for this kind of model, in large part because of the intractability of other optimal criteria. In this paper, we propose the minimum support A-optimal design for quadratic mixture canonical polynomials with spline.

Synthesis of Geopolymer from Blends of Tropical Biomass Ashes

Thermal conversion of biomass to produce energy inevitably generates ash as residual matter. Therefore, sustainable utilization of biomass should also consider reuse measures for the ash. This research examines the conversion of locally available biomass ashes in Indonesia into geopolymer. This is an environmentally more benign alkali-aluminosilicate alternative to the ordinary Portland cement produced by low-temperature reactions between aluminosilicates and concentrated alkali solution. Biomass ash sources in this study are corn stover, coconut shell, and sugarcane bagasse. Biomass ash blends are prepared according to a ternary simplex centroid mixture experiment design. These blends are reacted with activator solutions containing NaOH and Na-silicate and blended with sand aggregate to produce geopolymer mortar specimens. Early compressive strengths of geopolymer mortars range from 7.2 to 10.4 MPa, which surpasses the SNI 15-2049-2004 national standard for Portland cement with low heat of hydration. Statistical analysis of the experimental data indicates that the early strength as a function of biomass ash blend formulation is represented by a quadratic mixture model. Bagasse ash produces the highest strength. The quadratic terms consist of bagasse-corn and bagasse-coconut antagonistic blending terms. Morphology of the geopolymer mortar fracture surface indicates good bonding with the sand aggregate. Extensive acicular crystal growth within the amorphous geopolymer gel phase is observed in lower-strength formulations, which points to the formation of zeolite-like phase in the geopolymerization process. While the correlation of biomass ash type to the structure of the resulting geopolymer is yet to be established, this work has clearly identified the technical feasibility of producing geopolymer with satisfactory strength from tropical biomass ashes.

Optimum designs for parameter estimation in mixture experiments with group synergism

Mixture models were first introduced in canonical form of different degrees to represent the response function in a mixture experiment, and designs for the same were suggested. Later, several researchers derived optimum designs for the estimation of parameters and subset of parameters of the models. In this article, a study is carried out to find the D- and A-optimum designs for estimating the parameters of the quadratic mixture model, when the components of the mixture can be divided into two groups such that within a group there is synergism among the components, while the components between groups do not exhibit any synergism.

Reuse of Coconut Shell, Rice Husk, and Coal Ash Blends in Geopolymer Synthesis

Mixtures of biomass and coal ashes are likely to be produced in increasing volume as biomass-based energy production is gaining importance in Indonesia. This work highlights the reuse of coconut shell ash (CSA), rice husk ash (RHA), and coal fly ash (FA) for geopolymer synthesis by an activator solution containing concentrated KOH and Na2SiO3. Ash blend compositions are varied according to a simplex-centroid mixture experimental design. Activator to ash mass ratios are varied from 0.8 to 2.0, the higher value being applied for ash compositions with higher Si/Al ratio. The impact of ash blend composition on early strength is adequately modeled by an incomplete quadratic mixture model. Overall, the ashes can produce geopolymer mortars with an early strength exceeding the Indonesian SNI 15-2049-2004 standard minimum value of 2.0 MPa. Good workability of the geopolymer is indicated by their initial setting times which are longer than the minimum value of 45 mins. Geopolymers composed predominantly of RHA composition exhibit poor strength and excessive setting time. FTIR spectroscopy confirms the geopolymerization of the ashes by the shift of the Si-O-Si/Al asymmetric stretching vibrational mode. Overall, these results point to the feasibility of geopolymerization as a reuse pathway for biomass combustion waste.

Shrinkage simplex-centroid designs for a quadratic mixture model

A simplex-centroid design for q mixture components comprises of all possible subsets of the q components, present in equal proportions. The design does not contain full mixture blends except the overall centroid. In real-life situations, all mixture blends comprise of at least a minimum proportion of each component. Here, we introduce simplex-centroid designs which contain complete blends but with some loss in D-efficiency and stability in G-efficiency. We call such designs as shrinkage simplex-centroid designs. Furthermore, we use the proposed designs to generate component-amount designs by their projection.

F-squares based optimal designs for reduced cubic canonical models in four components

Aggarwal et al. (2008) presented optimal orthogonal designs in two blocks based on F-squares for Darroch and Waller's quadratic mixture model in four components, while for Becker's and K-model, they presented their results in 2013. In this paper, we have constructed D-, A- and E-optimal designs in two blocks based on F-squares for new models called reduced cubic canonical model Forms I and II in four components when two component proportions are at the same level. Several criteria associated with the information matrix and eigenvalues for the designs are examined to infer characteristics in optimality. Conditions required for orthogonality are given for the reduced cubic canonical model Forms I and II.