# Dimensionless number for blood volume

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Comments from the question How is the blood volume of a living organism measured without killing it? by @Nico discussed that the time of blood recirculation scales with the size of the organism. I was curious if there were a series of dimensionless numbers that characterize blood flow, blood volume, and time of blood recirculation?

Such dimensionless numbers exists for stride length and drug penetration.

Edit I like @Nico's comment so much it's going to part of the question. Any description of the circulatory system doesn't necessarily have to be dimensionless. However, I would imagine that one could develop a characteristic time (lets call it Tau) based on blood flow (L/t), cross-sectional area (L^2), and circulation time (t). From that characteristic time Tau, interesting observations and appropriate comparisons about the various ratios and how they vary amongst species.

Blood volume is not a dimensionless number - it's a volume. Historically we used to measure this in patients or volunteers by giving a large carbohydrate molecule like a starch that is not digestible or harmful to the body. Just like every other body fluid compartment volume (i.e. plasma, interstitial fluid, intracellular and extracellular) that we have, blood volume is estimated by intravenously injecting a known concentration of a particular compound. Once that compound equilibrates you take a blood sample and measure the compound's concentration again.

Initial Concentration * Initial Volume = Final Concentration * Final Volume

When you inject a known volume of a known concentration that only fills the "blood" component, and then you measure a final concentration - you can then solve for the "Final Volume" for "blood volume".

## List Of All Important Dimensionless Numbers And Their Significance

Dimensionless numbers are of very high importance in Mechanical Engineering and Chemical Engineering including Thermodynamics, Fluid Mechanics, Mass Transfer, Heat Transfer, Solid Mechanics, Momentum Transfer and Chemical Reaction Engineering.

• Dimensionless Numbers in Thermodynamics
• Dimensionless Numbers in Fluid Mechanics
• Dimensionless Numbers in Mass Transfer
• Dimensionless Numbers in Heat Transfer
• Dimensionless Numbers in Solid Mechanics
• Dimensionless Numbers in Momentum Transfer
• Dimensionless Numbers in Chemical Reaction Engineering

## Measuring osmosis and hemolysis of red blood cells

Since the discovery of the composition and structure of the mammalian cell membrane, biologists have had a clearer understanding of how substances enter and exit the cell's interior. The selectively permeable nature of the cell membrane allows the movement of some solutes and prevents the movement of others. This has important consequences for cell volume and the integrity of the cell and, as a result, is of utmost clinical importance, for example in the administration of isotonic intravenous infusions. The concepts of osmolarity and tonicity are often confused by students as impermeant isosmotic solutes such as NaCl are also isotonic however, isosmotic solutes such as urea are actually hypotonic due to the permeant nature of the membrane. By placing red blood cells in solutions of differing osmolarities and tonicities, this experiment demonstrates the effects of osmosis and the resultant changes in cell volume. Using hemoglobin standard solutions, where known concentrations of hemoglobin are produced, the proportion of hemolysis and the effect of this on resultant hematocrit can be estimated. No change in cell volume occurs in isotonic NaCl, and, by placing blood cells in hypotonic NaCl, incomplete hemolysis occurs. By changing the bathing solution to either distilled water or isosmotic urea, complete hemolysis occurs due to their hypotonic effects. With the use of animal blood in this practical, students gain useful experience in handling tissue fluids and calculating dilutions and can appreciate the science behind clinical scenarios.

Keywords: handing tissue fluids hematocrit osmolarity tonicity.

## White Blood Cells

White blood cells (also called leukocytes) are fewer in number than red blood cells, with a ratio of about 1 white blood cell to every 600 to 700 red blood cells. White blood cells are responsible primarily for defending the body against infection. There are five main types of white blood cells.

Neutrophils, the most numerous type, help protect the body against infections by killing and ingesting bacteria and fungi and by ingesting foreign debris.

Lymphocytes consist of three main types: T cells (T lymphocytes) and natural killer cells, which both help protect against viral infections and can detect and destroy some cancer cells, and B cells (B lymphocytes), which develop into cells that produce antibodies.

Monocytes ingest dead or damaged cells and help defend against many infectious organisms.

Eosinophils kill parasites, destroy cancer cells, and are involved in allergic responses.

Basophils also participate in allergic responses.

Some white blood cells flow smoothly through the bloodstream, but many adhere to blood vessel walls or even penetrate the vessel walls to enter other tissues. When white blood cells reach the site of an infection or other problem, they release substances that attract more white blood cells. The white blood cells function like an army, dispersed throughout the body but ready at a moment's notice to gather and fight off an invading organism. White blood cells accomplish this by engulfing and digesting organisms and by producing antibodies that attach to organisms so that they can be more easily destroyed.

When the number of white blood cells is too low (leukopenia), infections are more likely to occur. A higher than normal number of white blood cells (leukocytosis) may not directly cause symptoms, but the high number of cells can be an indication of an underlying disorder such as an infection, inflammatory process, or leukemia.

## Using Dimensionless Numbers to Predict Centrifugal Jet-Spun Nanofiber Morphology

In this study, we report a method for predictive, controlled, and highly aligned nanofiber production via Centrifugal Jet Spinning (CJS) using polycaprolactone (PCL) as a model polymer. We investigated the effects of fabrication conditions and their resulting dimensionless parameters, namely the Weber, Reynolds, and Capillary numbers, by correlating with fiber morphologies (fiber diameter, fiber alignment, bead frequency, bead aspect ratio, and scaffold porosity) and mechanical properties (linear modulus and ultimate tensile strength). We report a fabrication parameter lookup table based on the aforementioned dimensionless numbers, for the production of nanofiber scaffolds using the CJS. We built a scaled-up version of the CJS that uses a larger reservoir and successfully validated the reported lookup table for PCL as well as other polymers including polyethylene oxide, polylactic acid, and polyvinylpyrrolidone dissolved either in hexafluoroisopropanol or chloroform. We show that by carefully tailoring the polymer intrinsic properties and the Reynolds number, we can fabricate bead-free, continuous fibers. This method will allow other researchers to design and build their own CJS for the production of desired fiber scaffold networks by utilizing the appropriate dimensionless numbers for their system.

#### 1. Introduction

Nanofiber scaffolds are used in tissue engineering because of their high surface area to volume ratio, ability to integrate with the local tissue structures, tunable degradation, and mechanical properties [1, 2]. Several techniques exist for the production of nanofibers like electrospinning, melt blowing, centrifugal jet spinning, phase separation, template synthesis, and nozzle-free rotary spinning [1–6]. Some of these techniques offer benefits that are more advantageous than the others. For instance, electrospinning is popular in the academic setting but involves high power consumption and low yield limiting the technique to be scaled up for industrial use [7]. Melt spinning requires the use of high temperatures and involves high power consumption [8]. Thus, a more efficient method that overcomes these limitations is needed. We focus on an alternative technique called centrifugal jet spinning (CJS), also known as rotary jet spinning, that has been shown to be more efficient because of its capability to reliably produce high quantities of nanofibers within a short duration at low cost and low power consumption and for its potential to be scaled-up to increase production rates [3, 5, 9–11]. CJS utilizes a high centrifugal force produced by a rotating reservoir to extrude nanofibers. By employing CJS, we have previously reported the fabrication of anisotropic, Janus-type polymer nanofiber scaffolds that featured distinct properties on either side [4]. One of the challenges in any nanofiber production process is understanding the role of the various fabrication parameters that determine the overall physical and chemical properties of the produced nanofibers. The principle behind the CJS process requires some basic knowledge of polymer chemistry, processing, and fluid mechanics [3, 8, 12, 13]. Relatively few publications have studied the effect of these various parameters involved, and Figure 1 depicts the complex relationship between these parameters [3, 5, 6, 12–17].

To our knowledge, existing CJS literature has focused on studying the fibers collected on a distal collector, while our study focuses on the highly aligned fibers that were collected on the rotating reservoir itself. We believe this paper will thus add more information on how the fiber fabrication parameters of these highly aligned nanofibers can be controlled in the CJS process.

In this study, we describe a method to utilize dimensionless parameters to reliably fabricate and scale-up nanofiber production via CJS. We hypothesized that by identifying and matching specific dimensionless numbers that characterize our system, we can predict the resulting nanofiber morphology. We chose polycaprolactone (PCL) as our initial model polymer as it is a widely studied polymer with diverse applications including wound dressings [17–19], anterior cruciate ligament reconstruction [20, 21], and tissue engineering of blood vessels [22]. We show that fiber morphology can be predicted by specific combinations of the intrinsic properties of the polymer solution like viscosity, density, and surface tension along with the CJS system operational conditions (rotational speeds, extrusion orifice diameter, and centrifugal forces) via the dimensionless Reynolds, Weber, and Capillary numbers. The Reynolds number (

) represents the ratio of fluid inertial force to viscous force, and the Weber number (

) is the ratio of fluid inertial force to surface tension [3, 23]. Here, the inertial forces are the forces due to the momentum of the fluid which is given by density and jet exit velocity of the polymer solution. The capillary number (

) is the ratio of the Weber number over the Reynolds number and describes the role of viscous forces to surface tension forces [3, 8]. These three dimensionless numbers are responsible for the thinning of the jet filaments exiting the rotating reservoir orifice and are proportional to the velocity that drives the formation of the polymer jet [24]. We use these dimensionless numbers to correlate with the fiber morphology and mechanical data to find the dimensionless number that has the most significant correlation. Using this significant dimensionless number, we generate a predictive lookup table to define the operating parameters for a larger scaled-up version of the CJS and validate using PCL and other polymer solutions made from polyethylene oxide, polyvinylpyrrolidone, and polylactic acid.

#### 2. Materials and Methods

##### 2.1. CJS Systems and Reservoir Nozzles

Two different CJS systems were used—a small-scale CJS prototype and a scaled-up CJS. The small-scale CJS (Figure 2(a)) spins a reservoir (Figure 2(b)) with an internal diameter of 26 mm and two orifices of equal diameter (

μm) at the bottom of the reservoir. This system used a commercial motor (Model EC22, Maxon Motor Ag, Switzerland) controlled by an Elvis II Workstation (National Instruments, USA) to control rotational speeds ranging from 10,000 to 25,000 rpm yielding nanofibers (Figure 2(c)) via centrifugal extrusion and solvent evaporation represented in the schematic Figures 2(d) and 2(e) and outlined in detail in our previous publications [4, 25]. The scaled-up CJS prototype (Figure 2(f)) was constructed with a commercial 300W CNC Milling Spindle motor (KL-300, Automation Technologies Inc., USA) and controlled by an E300 series VFD (KL-VFD05, Automation Technologies Inc., USA) to vary the rotational speeds. The scaled-up system used a larger reservoir (Figure 2(g)) with an internal diameter of 48 mm and two orifices of equal diameter ( μm) on either side of the reservoir.

##### 2.2. Polymer Solutions: Preparation, Analysis, and Calculations

All polymers and solvents were obtained from Sigma-Aldrich, Oakwood Chemical, and Alfa Aesar, unless otherwise specified. Polycaprolactone (PCL) was dissolved in 1,1,1,3,3,3-hexafluoro-2-propanol (HFIP) to make five distinct 4%, 5%, 6%, 7%, and 8% (

) polymer solutions. Polymer solutions of polyethylene oxide- (PEO-) chloroform, polylactic acid- (PLA-) chloroform, low molecular weight (55,000 Da) polyvinylpyrrolidone- (PVP-L-) HFIP, and high molecular weight (1,300,000 Da) polyvinylpyrrolidone- (PVP-H-) HFIP were prepared at concentrations ranging from 3 to 10% ( / ). All solutions were magnetically stirred overnight to ensure the complete dissolution of the polymer. Polymer density (

) of all polymer solutions were measured using previously published methods [4]. These measured values for the PCL model polymer are reported in Table 1.

##### 2.3. Fabrication of Nanofibers Using Small-Scale CJS Prototype

The small-scale CJS reservoir was filled with 10 ml of polymer solution and operated for 150 seconds for each of the five PCL solutions at various speeds (10,000, 15,000, 20,000, and 25,000 RPM) under ambient temperatures to yield fibers. The fibers were collected from the reservoir surface (Figure 2(a)) and only areas unaffected by the tweezers were used for analysis [4].

##### 2.4. Scanning Electron Microscopy and Fiber Morphology Characterization

Nanofiber sections were fixed onto a stud via carbon tape and sputter coated with gold. These coated samples were imaged at 15 kV with a 5 mm working distance using a Nova Nanolab 200 (FEI, Hillsboro OR, USA) scanning electron microscope (SEM) to capture ten random image fields for each sample. The fiber orientation parameter (OP) and porosity were computed using custom-written MATLAB (MathWorks, USA) scripts to determine fiber alignment and overall scaffold porosity, respectively [4, 26]. The SEM images were also analyzed for fiber diameter and beading using ImageJ [4]. Fiber diameter was analyzed by averaging 15 random fibers in each obtained image for each image field. For bead analysis, we calculated bead frequency by counting the number of beads normalized to the area (in mm 2 ) in all nanofiber conditions and computed the bead aspect ratios using ImageJ.

##### 2.5. Uniaxial Tensile Strength Testing

Rectangular nanofiber specimens (

) were mounted on an Instron® 5900 Series (Canton, MA, USA) tensile strength testing apparatus. Each sample was subjected to ten cycles of preconditioning to a minimum and maximum strain magnitude of 1 and 5%, respectively, before being pulled at 10 mm/min until failure. The first tear across the nanofiber construct marked the failure of the sample. The linear modulus (LM) was calculated from the slope of the resulting tensile stress vs. strain plots by choosing two points within the linear region of the curve, and ultimate tensile strength (UTS) was determined from the highest point on the curve [4].

##### 2.6. Calculating Critical Polymer Chain Entanglement Concentration

To determine the critical concentration (

), which defines the concentration above which continuous bead-free nanofibers can be fabricated, the specific viscosity of polymers were plotted against polymer concentrations. Polymers were dissolved in their respective solvents at 3-10% ( / ), and their specific viscosities were measured. The specific viscosity was obtained by

. An inflection point in the slope of this graph marks the onset of different regimes of polymer chain entanglement, namely, dilute, semidilute unentangled, and semidilute entangled regimes [3, 8, 23, 27–30].

##### 2.7. Calculation of Dimensionless Parameters

) determines how fast a polymer solution exits the reservoir orifice during centrifugal jet spinning. The jet exit velocity is defined as follows:

and = diameter of the orifice, = duration of each run, = height of the polymer solution within the reservoir, = radius of reservoir, and = rotational speed in rad·s -1 [3].

The dimensionless parameters of interest in this study were the Capillary ( ) number:

which is in turn calculated using the Reynolds ( ) and Weber ( ) numbers, given by

where , , and tension of the polymer solution. represents the ratio of centrifugally induced shear viscous forces to surface tension forces and is thought to be a key predictor of polymer solution breakup and bead formation [3]. All these dimensionless numbers are reported in Table 1.

##### 2.8. Statistical Analysis

All data are expressed as

samples were tested for all nanofiber conditions. Statistical analysis was performed with one-way or two-way analysis of variance (ANOVA) for normally distributed data. The Pearson (for normally distributed data) and Spearman (for nonnormally distributed data) evaluations were used to analyze the correlation of the dimensionless numbers to morphological and mechanical properties of the fibers. A

value of less than 0.05 was considered statistically significant.

#### 3. Results

##### 3.1. Characterization of Nanofibers Fabricated via Centrifugal Jet Spinning

All nanofiber conditions were fabricated in the small-scale CJS with the polymer jets being extruded out of the orifices due to the effect of hydrostatic and centrifugal forces exceeding the capillary forces, resulting in solvent evaporation and wrapping of nanofibers onto the surface of the reservoir (Figure 2(a)) [4, 19, 25]. Figure 3 shows a series of SEM images of the fabricated PCL fibers with increasing speeds and polymer concentrations. We observed the presence of beading at lower PCL concentrations and the absence of beads in fibers fabricated from higher PCL concentrations.

Fiber alignment for each distinct condition was found to be above 0.8 on a scale of 0-1.0 with 0 being randomly oriented and 1.0 being perfectly aligned (Figure 4(a)) [3, 4, 25, 26]. Both rotational speeds and polymer concentrations did not have a significant effect on fiber alignment (

). Fiber diameter significantly (

) decreased with increasing rotational speeds for each polymer composition (Figure 4(b)), and we observed a positive trend ( ) between nanofiber diameter and polymer concentration while keeping the rotational speed constant. It was also noted that the overall scaffold porosity (Figure 4(c)) was between 49 and 53%, and it did not change with respect to speed or concentration. The SEM images were further analyzed for bead frequency, and it was observed that the average number of beads per area significantly ( ) decreased (Figure 4(d)) with higher solution concentrations. The aspect ratios of beads were in the range of 2-4 and did not significantly differ between the polymer solutions and rotational speeds (Figure 4(e)).

##### 3.2. Calculating Critical Polymer Entanglement Concentration

Polymer solution viscosity plays a crucial role in fiber morphology and determines the extent of polymer chain entanglement, which at high levels, promotes formation of continuous nanofibers [14, 23]. Polymer chain entanglement refers to the interlocking of polymer chains during nanofiber fabrication as a result of chain overlapping [3, 23, 27]. Several studies have shown that at lower viscosities, the individual polymer chains are far apart resulting in beads or beaded fibers [23, 27, 29–33]. Within the analyzed polymer concentrations, distinct inflection points in the slope of the specific viscosity vs. concentration plots (Figure 5(a)) were observed which marked the different polymer chain entanglement regimes as outlined earlier. When the polymer solutions are dilute (

, dilute regime), the polymer chain entanglement that is necessary to form continuous fibers is not achieved because the distance between the polymer chains is relatively high, resulting in beads [5, 8, 14]. Under the semidilute unentangled regime ( ), the distance between neighboring chains decreases but is still not significant enough and thus results in beaded fibers as shown for polymer concentrations between

. The individual polymer chains become sufficiently close enough under the semidilute entangled regime ( ) to undergo internal entanglement, resulting in continuous fibers. These distinct regimes help determine the critical concentration (6%) for the PCL solutions.

was found to be 6 wt% PCL. (b) Phase diagram relating commonly observed fiber morphologies with Capillary number and PCL concentration. (

A phase diagram (Figure 5(b)) depicts the relationship between fiber morphology, capillary number, and polymer concentration at different rotational speeds. At lower concentrations and lower , fibers with beads were more commonly observed. From Figure 5(b), it can be noted that for fabricating continuous, bead-free PCL fibers, the polymer concentration should be higher than the critical concentration (>6%) and should possess higher (>25). Thus, finding the critical concentration of a polymer is crucial, as it may provide us information on gross nanofiber morphology. The angular velocity given by the number must also be sufficiently high to overcome the surface tension resulting in stabilized polymer jets and reduced beading [3, 5].

##### 3.3. Uniaxial Mechanical Testing

A representative stress-strain plot from our uniaxial tensile tests is shown in Figure 6(a). LM was determined using the slope of the linear region (between 5 and 20 MPa) and UTS was determined using the highest point on the curve [4, 19]. We noted no significant difference in LM and UTS between the different rotational speeds (Figures 6(b) and 6(c)). However, there was a significant difference between nanofibers fabricated from 4 to 5% PCL concentrations compared to those from 6 to 8% PCL concentrations which had higher LM and UTS.

##### 3.4. Analysis of Dimensionless Numbers and Validation with Scaled-Up CJS

From Table 1, we note that with an increase in polymer concentration, the dynamic viscosity ( ), surface tension ( ) and density ( ) all increased, with increasing by the largest magnitude. Correlation analysis was carried out by plotting the various dimensionless numbers with the fiber and bead morphology as well as mechanical properties to find which dimensionless number had the most significant effect in determining the fiber properties. Supplemental Figures 1-3 show the correlation graphs for dimensionless numbers plotted against fiber alignment, fiber diameter, fiber porosity, bead frequency, bead aspect ratios, linear modulus, and ultimate tensile strength. We observed no significant correlation between fiber alignment, overall scaffold porosity, and the dimensionless numbers ( , , or ) (Supp. Figure 1A, C), but the and numbers had significant negative correlations (

and , , respectively) with fiber diameter (Supp. Figure 1B). We also report (Supp. Figure 2A) a significant coefficient of correlation for and bead frequency ( , ), where higher led to more bead occurrence. We also noted a trend that with higher the bead frequency reduced ( ,

). We observed that there was no significant correlation between the dimensionless numbers and bead aspect ratio (Supp. Figure 2B). Analyzing the correlation between dimensionless numbers and scaffold mechanical properties (Supp. Figure 3) revealed that there was a significant positive correlation of with LM ( , ) and UTS ( , ). We also noticed that had a significant negative correlation with LM ( , ) and UTS ( , ).

As showed multiple significant correlations with fiber scaffold characteristics as depicted in Table 2, we matched this dimensionless parameter in the scaled-up CJS system and validated the ability of to predict fiber morphology with other polymers. Supplemental Table 1 reports the speeds that were calculated to match the numbers of the CJS prototype with that of the scaled-up CJS for the PCL polymer conditions. We found the critical polymer concentrations for all the polymer solutions and chose conditions that were higher than the for each of the conditions. It was noted that the polymer concentration range tested for PVP-L solutions remained in the dilute regime and failed to reach . We calculated the appropriate rotational speeds required for the scaled-up CJS by matching the number of the prototype CJS. With this predetermined information, we spun fibers using 7% PCL, 8% PCL, 5% PEO, 8% PLA, 10% PVP-L, and 5% PVP-H solutions with the scaled-up CJS. We used PCL solutions to match the obtained from the small-scale CJS prototype operated at 10,000 rpm (7% PCL-1 and 8% PCL) and 15,000 rpm (7% PCL-2). All other polymer solutions were matched to the number of 8% PCL spun at 25,000 rpm in the small-scale CJS prototype. Figure 7 shows the accuracy of matching the dimensionless numbers of newly fabricated nanofibers compared with predicted fiber diameter values obtained from the correlation data (Supp. Figure 1B). It was found that by matching the number, we were able to synthesize fibers with a prediction error ranging between 10 and 15% for PCL and PVP-H. As expected, PVP-L resulted in only beads since the polymer solutions were dilute. The errors were larger when was matched to fabricate fibers from PEO (

20%). We also noticed that the prediction error decreased among polymers dissolved in HFIP (Figure 7(a)). This may imply that the properties of the solvent play a role towards the resulting fiber properties and thus warrants further investigation. Considering dimensionless numbers that incorporates the solvent evaporation rate into account may help in reducing these prediction errors and build a more accurate model. All the fibers produced using the scaled-up CJS were highly aligned and their mechanical properties are summarized in Supplemental Table 2. These varied mechanical properties could be owed to several factors like fiber diameter, amorphous/crystallinity of the polymer, and fiber alignment [5, 34].

number in the scaled-up CJS. (a) Table comprising of all the polymer-solvent combinations, their fiber features, along with predicted and actual fiber diameters obtained from fibers fabricated using the scaled-up CJS (

). (b) Representative SEM images for the different fiber meshes produced with the scaled-up CJS.

#### 4. Discussion

Nanofiber scaffolds have the potential to be used for a wide range of applications, ranging from filtration to tissue scaffolding, underlining the importance of studying the large-scale production of these materials [1, 2, 35]. For the successful scale-up of any manufacturing process, knowledge of the key process parameters is crucial for repeatable and successful fabrication. Most mass production devices that produce polymeric fiber meshes have focused on optimizing electrospinning designs since this technique has been extensively studied and has long been understood [7, 36–41]. Some of the influencing parameters involved in predicting the electrospun fiber morphology can be used to understand the CJS process as it is relatively close to the behavior of polymer jets. However, the major drawbacks of these electrospinning methods are the high costs, low production rates, and electric field interference that occurs when the number of jets are increased [7]. This has led researchers to study other nanofiber production techniques that use physical forces as opposed to electrostatic forces. CJS is one such emerging technique that can be industrially scaled-up for large-scale nanofiber fabrication at low costs [9, 10, 14, 25]. Only a few studies exist that have investigated the factors involved in the CJS fabrication process experimentally [3, 8, 12–15, 17], with some focused on computational modelling [42–45]. However, these studies focused on fibers that were deposited on a collector placed away from the reservoir. In contrast, this study clearly highlights the utility of , , and to accurately determine a priori the fabrication parameters for a desired CJS nanofiber morphology with the fibers collected directly on the reservoir.

Our data suggests that we were able to fabricate fibers on the surface of the reservoir with high alignment, which have potential to be used as scaffolds for tissue engineering applications where there is a need to mimic tissues like muscle, blood vessels, or heart valves that have inherent alignment in their structure [19, 22, 25, 46]. We also noted that the fiber diameter decreased with increasing speeds, which could be attributed to the result of increased extension and thinning of the polymer jet stream during the extrusion process [3, 4, 25]. A trend of increased fiber diameter with increasing polymer concentration when the fibers were fabricated using the same rotational speed was observed, which could be a result of increased polymer content and solution viscosity [8, 13, 14, 23]. We did not test any polymer concentrations below 4% ( / ) because our preliminary studies showed that these concentrations only produced beads. This is likely due to the Plateau-Rayleigh instability where the solution exiting the orifice forms droplets to minimize surface tension [10, 12, 27, 28]. It can be noted that beading of fibers decreased with increasing polymer concentrations from our SEM images (Figure 3). Beaded fibers, as observed in lower concentrations, occur when the extended polymer jet undergoes solvent evaporation after the jet instability had set in. There is clearly a range beyond this which no beading was observed. This effect can also be owed to the role of chain entanglement for defining the various regions and that the critical concentration for continuous fiber formation was found to be 6% for PCL as depicted in Figure 5(a) [8, 27, 29, 32]. As the polymer concentration increased along with an increased number (Figure 5(b)), we were able to produce continuous nanofibers with no beads. This suggests that during the fiber extrusion process, higher rotational speeds cause the nanofibers to stretch to a greater extent, resulting in more efficient solvent evaporation and individual polymer chain entanglement that stabilized the polymer jet yielding continuous fibers. At lower speeds, the chain entanglement is not sufficient enough, leading to beaded fiber formation [3, 23, 27]. Furthermore, our mechanical data (Figure 6) shows that the LM and UTS are significantly higher for fibers fabricated from polymer concentrations that surpass the critical concentration. This suggests that the presence of beads in lower-concentration fibers (Figures 3, 4(d), and 4(e)) could have an effect on its mechanical properties.

We were able to produce fibers with overall scaffold porosities ranging between 49 and 53%. Limited literature exists that has studied the porosity of CJS-spun fibers, and they show that CJS-spun fibers with similar porosities allow for cellular infiltration into the scaffold [25, 26, 47]. We speculate that the fibers collected on the reservoir at high rotational speeds undergo a rapid pull, resulting in the denser packing of the fibers and yielding lower porosity ranges. This can be overcome by introducing a collector placed at a distance from the reservoir. Alternatively, the addition of a protein component like gelatin to the polymer solution can increase fiber porosity [47]. It was reported that this addition yielded a higher porosity for PCL/gelatin blends compared to pure PCL fibers.

We performed correlation analysis to understand how the different fiber scaffold properties were affected by the dimensionless parameters, namely, , , and that are involved in the thinning of polymer jet filaments. We report all the significant and insignificant effects of dimensionless numbers on the fiber features to provide a thorough and holistic study. Our data suggests that there was a significant negative correlation between and and fiber diameter. We know that and are inversely proportional to viscosity and surface tension, respectively, and directly proportional to jet exit velocity. This suggests that viscous forces and surface tension of polymer solutions play a crucial role in determining fiber diameter during the fabrication process. The increased jet exit velocity may have resulted in decreased fiber diameter due to the longer extrusion of the polymer solution with increased centrifugal forces before solvent evaporation [13].

The use of computational models as a theoretical platform to understand the jet trajectory that occurs during the spinning process has been previously reported [42–45]. Divvela et al. studied the effects of polymer jet trajectory and diameter in the CJS process, and they reported similar findings where a decreasing resulted in the reduced thinning of the fibers [42]. This effect was attributed to the increase in viscous force that inhibits the extension of the polymer jet. They also observed steadier jet trajectories with a higher number which can reduce the amount of fiber malformations. Contrary to our findings, the study reported that the number did not affect the fiber diameter. A possible reason for this discrepancy could be due to solvent evaporation, which was not considered in the model by Divvela et al. In addition, their model did not account for the bending and twisting stiffness of the fibers which can occur during the CJS process. Recent work by Liu and Parker developed a more accurate and complex computational model that involved the bending and twisting stiffness of the fibers [43]. They showed that small fiber diameters and higher alignment can be attained by balancing the viscous and elastic forces. Their computational model demonstrated that when the polymer jet trajectory behavior is more elastic-like, the fiber jets will fall on the fast-rotating reservoir similar to our experimental observations. It is to be noted that this model did not include solvent evaporation and solidification of the fiber jets that can play a crucial role in determining the eventual trajectory and fabricated fiber morphology.

We observed that had a significant correlation with bead frequency. The reduction in beading can be attributed collectively to the increasing solution viscosity and spinning velocity leading to continuous fibers [8, 12–14, 25, 31]. We matched the in the scaled-up CJS and carried out validation by fabricating fibers using other polymer-solvent combinations. We analyzed these newly fabricated fibers to compare their fiber characteristics. The fiber diameter data showed that the PCL and PVP-H fibers had comparatively lower prediction errors than PEO and PLA fibers (Figure 7(a)). As shown in Figure 1, the solution evaporation rate plays an important role that can influence the fiber diameter. Since HFIP and chloroform have different solvent evaporation rates, this could have resulted in the changed prediction error. This is in agreement with other studies that investigated the effect of different solvents on fiber diameter [8, 12]. These groups studied the role of elasticity and solvent evaporation rate on fiber diameter, but they did not consider the dimensionless numbers that were incorporated in our study. The representative SEM micrographs for the polymer solutions used for validation are provided in Figure 7(b) (i–vii). We noticed that spinning PVP-L (Figure 7(b) (vi)) resulted in only beads since the concentration range tested was too dilute and fell short of the critical concentration. As stated earlier, fiber malformations can occur if the polymer concentrations are lower than their critical concentrations.

The utilization of other dimensionless numbers such as the Weissenberg number that investigates the elastic effects in response to the deformation rate of polymer solutions, the Rossby number that characterizes the effects of centrifugal and Coriolis forces, the Deborah number that studies the ratio of polymer relaxation time to flow, the elasticity number that investigates the role of elastic effects to inertial effects, and the processability number, which takes into account the role of solvent evaporation, might result in more accurate predictions for different polymer-solvent combinations [5, 8, 12, 13, 42, 43].

#### 5. Conclusions

In conclusion, we demonstrate by using a small-scale prototype and a scaled-up CJS system that CJS-spun fiber morphologies can be predicted reasonably well a priori by a careful selection of dimensionless numbers, especially . We also demonstrate the accuracy of this method by validating with other polymer-solvent combinations like PEO-chloroform, PLA-chloroform, and PVP-HFIP. We show that by carefully studying the polymer solutions and matching the number, we can predict the fiber diameters with reasonable errors and are also able to create continuous and bead-free nanofibers. Overall, these findings can potentially help other researchers to design and construct similar CJS systems. In addition, this study provides valuable insight into the large-scale fabrication of fibers.

#### Data Availability

The data used to support the findings of this study are included within the article and the supplemental information.

#### Disclosure

We would like to acknowledge that a part of this manuscript was presented at the 8th Biennial Heart Valve Biology & Tissue Engineering Meeting (London, United Kingdom) which was held on the 26th to the 28th of September, 2018, and at the 2018 Biomedical Engineering Society Annual Meeting (Atlanta, USA) which was held on the 17th to the 20th of October, 2018.

#### Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

#### Authors’ Contributions

Prashanth Ravishankar and Alex Khang contributed equally to this work.

#### Acknowledgments

We acknowledge funding support from the Arkansas Biosciences Institute Grant 000167 and the State of Arkansas Student Undergraduate Research Fellowship (SURF). We would like to thank the University of Arkansas Institute for Nanoscience and Engineering for technical assistance with the scanning electron microscope. We would also like to thank Joshua A. Goss and Dr. Min Zou, Department of Mechanical Engineering, as well as Dr. Xianghong Qian, Department of Biomedical Engineering, University of Arkansas for assisting with the surface tension measurements. We also would like to acknowledge Ugochukwu Odega, Department of Biological Sciences, University of Arkansas, for his help with fabricating some of the fibers used in this study.

#### Supplementary Materials

Supplemental Figure 1: correlation analysis carried out for dimensionless numbers vs. (a) fiber alignment, (b) fiber diameter, and (c) scaffold porosity. Supplemental Figure 2: correlation analysis carried out for dimensionless numbers vs. (a) bead frequency and (b) bead aspect ratio. Supplemental Figure 3: correlation analysis carried out for dimensionless numbers vs. (a) linear modulus and (b) ultimate tensile strength. Supplemental Table 1: calculating the rotational speeds by matching the Re number from the small-scale CJS prototype to the new scaled-up CJS along with the dimensions of both the reservoirs employed. Supplemental Table 2: alignment and mechanical properties of fibers fabricated from a scaled-up CJS. (Supplementary Materials)

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Copyright © 2019 Prashanth Ravishankar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## Complete blood count

The blood test most commonly done is the complete blood count (CBC). The CBC is an evaluation of all the cellular components (red blood cells, white blood cells, and platelets) in the blood. Automated machines do this test in less than 1 minute on a small amount of blood. The CBC is supplemented in some instances by examination of blood cells under a microscope (blood smear).

Red blood cell parameters evaluated by CBC include

Number of red blood cells (red blood cell count, RBCs)

Proportion of blood made up of red blood cells (hematocrit, Hct)

Amount of hemoglobin (the oxygen-carrying protein in red blood cells) in the blood (hemoglobin, Hb)

Average size of red blood cells (mean cellular volume, MCV)

Variability of size of red blood cells (red cell distribution width, RDW)

Amount of hemoglobin in an individual red blood cell (mean cellular hemoglobin, MCH)

Concentration of hemoglobin in an individual red blood cell (mean cellular hemoglobin concentration, MCHC)

Abnormalities in these parameters can alert laboratory workers to the presence of abnormalities in the red blood cells (which may then be further evaluated by examination under a microscope).

Abnormal red blood cells may be fragmented or shaped like teardrops, crescents (sickle-shaped), or a variety of other forms. Knowing the specific shape and size of red blood cells can help a doctor diagnose a particular cause of anemia. For example, sickle-shaped cells are characteristic of sickle cell disease, small cells containing insufficient amounts of hemoglobin are likely due to iron deficiency anemia, and large cells suggest anemia due to a deficiency of folate (folic acid) or vitamin B12.

White blood cell parameters evaluated by the CBC include the

Total number of white blood cells

Percentages and numbers of the different types of white blood cells

The white blood cells are the major component of the body's immune system. There are normally five types of white blood cells (neutrophils, lymphocytes, monocytes, eosinophils, and basophils), and different types are recruited into service when the immune system responds to different stresses or disorders. Counting the number of white blood cells of each type (differential white blood cell count) can suggest to a doctor possible causes of a change in the total white blood cell count. For example, if a person with cold symptoms has an increased white blood cell count due to increased neutrophils, the doctor would likely suspect a bacterial pneumonia rather than a viral infection because neutrophils are more often recruited to fight bacterial infections.

To provide more information about the white blood cells, the doctor can examine these cells under a microscope. The microscopic examination can identify features of the cells that are characteristic of certain diseases. For example, large numbers of white blood cells that have a very immature appearance (blasts) may indicate leukemia (cancer of the white blood cells).

Platelets are also counted as part of a CBC. The number of platelets is an important measure of the blood's ability to form blood clots (forming blood clots is the body's protective mechanism for stopping bleeding). Too few platelets may impair blood clotting. A high number of platelets (thrombocytosis) can lead to excessive blood clotting in small blood vessels, especially those in the heart or brain. However, in some disorders, a high number of platelets may paradoxically result in excess bleeding.

### Preparation before the practical.

In advance of the class, students must calculate 1) the volume of distilled water and 2.7% wt/vol NaCl stock solution required to produce 9 ml each of 0.9 and 0.45% wt/vol saline solutions 2) the volumes of hemolyzed blood and 0.9% wt/vol NaCl (ml) required to produce 1.5 ml %hemoglobin concentrations and 3) the hemoglobin concentration (g/dl) in results Tables 1–3 provided in their laboratory books.

Table 1. Dilutions calculations for saline solutions

Values underlined are calculated by the students in advance of the class.

## Volume

deficient fluid volume a nursing diagnosis accepted by the North American Nursing Diagnosis Association, defined as decreased intravascular, interstitial, and/or intracellular fluid. This refers to dehydration , water loss alone without change in sodium . See also fluid volume . Formerly called fluid volume deficit.

When a person engages in normal physical activity and the environmental temperature is 20°C (68°F), the body loses about 2400 ml of water in 24 hours. About 1400 ml are lost in urine, 200 ml in feces, and 100 ml in sweat. The remaining 700 ml are lost through what is called insensible water loss, which takes place by diffusion through the skin and by evaporation from the lungs. About 300 ml of water diffuse through the epithelial cells daily. The lungs excrete about 400 ml per day.

A deficit of fluid volume occurs when there is either an excessive loss of body water or an inadequate compensatory intake. Etiologic factors include active loss through vomiting, diarrhea, gastric suctioning, drainage through operative wounds and tubes, burns, fistulas, hypermetabolic states, and drug-induced diuresis. Insufficient intake of water can be caused by nausea, immobility with inaccessibility of water, and lack of knowledge about the necessity of adequate fluid intake.

Patient Care . Assessment of the patient's hydration status includes monitoring lab data for such signs as increased packed red blood cell volume, increased plasma protein level, elevated specific gravity of urine, and increased blood urea nitrogen (BUN) out of proportion to a change in serum creatinine. In the absence of other problems, the serum sodium should remain within normal limits.

Recording daily weight gives information about the amount of water gained or lost each day. If there is a fluid volume deficit, intake and output measurements can give evidence of fluid imbalance. The urine appears concentrated and is usually well below the criterion of 50 ml of output per hour. Other objective assessment data include hypotension and a decrease in venous filling and in pulse volume and pressure. The mucous membranes are dry, as is the skin, which loses its turgor. The patient may complain of thirst and the body temperature may be elevated.

Patients at risk for profound and potentially fatal fluid volume deficit, as in severe burns, should be assessed frequently for mental acuity status and orientation to person, place, and time. Measures to improve hydration status should take into account the patient's ability to drink and retain fluids by mouth, preferences for certain fluids, and whether hot or cold drinks are preferred. The goal of oral fluid intake should be about 2000 ml per day. Explanations about the importance of an adequate fluid intake and assuring the availability of fresh water and fluids attractive to the patient can help reach the desired goal. Intravenous administration of fluids, especially isotonic saline, may be necessary if oral replacement is not possible. In severe and intractable fluid volume deficit a central venous catheter may be used to evaluate the extent of fluid loss and to guide replacement therapy.

excess fluid volume a nursing diagnosis accepted by the North American Nursing Diagnosis Association, defined as increased isotonic fluid retention see also fluid volume . Factors contributing to this include (1) arterial dilatation, as occurs in the inflammatory process (2) reduced oncotic pressure, as in hypoproteinuria (particularly a deficit of albumin, which is responsible for 80 per cent of oncotic pressure), lymphatic obstruction, and increased capillary permeability, which allows water to escape into the tissues and produce swelling (3) renal retention of sodium and consequently of water, as seen in renal failure (4) inadequate circulation of blood through the general circulation, as in congestive heart failure , or through the portal circulation, as in liver failure and (5) overproduction or administration of adrenocortical hormones.

Hypervolemia can occur when a patient receives excessive fluid replacement or repeated tap water enemas or, much less frequently, drinks more fluids than are eliminated. Characteristics of fluid volume excess include obvious swelling, localized or generalized weight gain pulmonary congestion with accompanying shortness of breath, orthopnea, and abnormal breath sounds a fluid intake greater than output distended neck veins and changes in central venous and pulmonary artery pressures.

## Dimensionless number for blood volume - Biology

Exponential functions can be used to model the concentration of a drug in a patient's body. Suppose the concentration of Drug X in a patient's bloodstream is modeled by,

C (t) = C0 e - rt ,

where C (t) represents the concentration at time t (in hours), C0 is the concentration of the drug in the blood immediately after injection, and r > 0 is a constant indicating the removal of the drug by the body through metabolism and/or excretion. The rate constant r has units of 1/time (1/hr). It is important to note that this model assumes that the blood concentration of the drug (C0 ) peaks immediately when the drug is injected.

Using this model, try to solve the following problems:

Problem 1-Find the concentration of drug in the blood at a given time after injection.

Problem 2- Calculate the value of r given data on the concentration of a drug in the blood over time.

Problem 3- Calculate the time it takes for the blood concentration of a drug to decrease by a given percentage.

Problem 4- Calculate the initial dose of a drug that needs to be administered.

Problem 5- Calculate when a second dose of a drug needs to be given .

Problem 6- Determine the time it takes for the blood concentration of a drug decrease by a given percentage.

Problem 7- Calculate the value of the constant r.

Next Application: Fixation of a Mutant Gene

For microbiology, cell culture and many of the applications that require use of cell suspensions, it is necessary to determine the concentration of cells. The device used for determining the number of cells per unit volume of a suspension is called a counting chamber. It is the most widely used type of chamber, since it was mainly designed for performing blood cell counts. It is now used to count other types of cells and other microscopic particles as well.

The hemocytometer was invented by Louis-Charles Malassez. It is a special type of microscope slide consisting of two chambers, which is divided into nine (1.0mm x 1.0mm) large squares which are separated from one another by triple lines. The area of each is 1mm². Cover glass is supported over the chambers at a height of 0.1mm. Because of that the entire counting grid lies under the volume of 0.9 mm² on one side. The cell suspensions are introduced into the cover glass. The hemocytometer is placed on the microscope stage and the cell suspension is counted..

The glass microscope slide has a rectangular indentation that creates an 'H' shaped chamber at the centre. This chamber is engraved with a laser-etched grid of perpendicular lines. Two counting areas with ruled grids are separated by the horizontal groove of the 'H'. There is also a very flat, reusable cover slip. The glass cover slip is held at 0.1 mm above the surface of the counting areas by ground glass ridges on either side of the vertical grooves of the H shape. The device is carefully crafted so that the area bounded by the depth and lines of the chamber is also known. Because the height is constant, the volume of fluid above each square of the grid is known with precision.

The hemocytometer is used by putting the cover slip on the device, and filling the space with a liquid containing the cells you want to count. There is a "V" or notch at either end which is the place where the cell suspension is loaded into the hemocytometer. The fluid is usually drawn into the space by capillary action. A cover glass, which is placed on the sample, does not simply float on the liquid, but is held in place at a specified height. In addition, the grid arrangement of squares of different sizes allows for an easy counting of cells. It is possible to identify the number of cells in a specified volume by this method..

The ruled area of the hemocytometer consists of several large 1 x 1 mm (1mm² ) squares, which are subdivided in three ways 0.25 x 0.25 mm (0.0625 mm²), 0.25 x 0.20 mm (0.05 mm²) and 0.20 x 0.20 mm (0.04 mm²). The central, 0.20 x 0.20 mm marked, 1 x 1 mm square is further subdivided into 0.05 x 0.05 mm (0.0025 mm²) squares. Hold the cover slip( 0.1 mm) at the raised edges of hemocytometer, which gives each square a defined volume.

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