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# Iterative feedback bio-printing-derived cell-laden hydrogel scaffolds with optimal geometrical fidelity and cellular controllability

## Abstract

For three-dimensional bio-printed cell-laden hydrogel tissue constructs, the well-designed internal porous geometry is tailored to obtain the desired structural and cellular properties. However, significant differences often exist between the designed and as-printed scaffolds because of the inherent characteristics of hydrogels and cells. In this study, an iterative feedback bio-printing (IFBP) approach based on optical coherence tomography (OCT) for the fabrication of cell-laden hydrogel scaffolds with optimal geometrical fidelity and cellular controllability was proposed. A custom-made swept-source OCT (SS-OCT) system was applied to characterize the printed scaffolds quantitatively. Based on the obtained empirical linear formula from the first experimental feedback loop, we defined the most appropriate design constraints and optimized the printing process to improve the geometrical fidelity. The effectiveness of IFBP was verified from the second run using gelatin/alginate hydrogel scaffolds laden with C3A cells. The mismatch of the morphological parameters greatly decreased from 40% to within 7%, which significantly optimized the cell viability, proliferation, and morphology, as well as the representative expression of hepatocyte markers, including CYP3A4 and albumin, of the printed cell-laden hydrogel scaffolds. The demonstrated protocol paves the way for the mass fabrication of cell-laden hydrogel scaffolds, engineered tissues, and scaled-up applications of the 3D bio-printing technique.

## Introduction

Three-dimensional (3D) bio-printing represents a promising method for fabricating complex 3D tissue or organ constructs through the layer-wise and controllable positioning of cell-containing media1,2,3. Hydrogels have been widely used as cell-laden biomaterials for 3D bio-printing in tissue engineering and regenerative medicine (TERM) applications4,5,6, wherein the designed and printed inner porous architectures play an essential role in the cell growth, tissue formation and function reconstruction of 3D bio-printed tissue scaffolds7,8,9,10,11. Therefore, high geometrical fidelity between the designed and as-printed scaffolds enables the efficient utilization of the unique structurally tailored advantages of 3D bio-printing technique to facilitate cellular controllability12,13. 3D bio-printing cell-laden hydrogel scaffolds with optimal geometrical fidelity and cellular controllability are of great significance for research on the biofunctional reconstruction of printed cell-laden scaffolds, mass fabrication of engineered tissues and organs, and scaled-up applications of the 3D bio-printing technique.

However, because of the inherent complex variability of hydrogels and cells, the fabrication of 3D bio-printed cell-laden hydrogels with predesigned geometry and desired cellular properties faces many challenges14,15,16, such as unexpected gel deformation14,15,16,17,18, uncontrollable cell dynamics19,20,21,22 and printing process parameters mismatch23,24. Recently, researchers have attempted to regulate the printing parameters or hydrogel printability to improve printing accuracy and cytoactivity, e.g., regulating the flow behavior of cell-alginate suspensions21,22, modulating shear stress by varying hydrogel viscosity and printing pressures23, and optimizing printing parameters based on quantitative 2D image analysis of the printed strands24. Most of the above research focused on the relationship between the biomaterials’ physical properties and printing process parameters; however, little attention has been focused on utilizing cybernetics methods to ensure a precise match between the designed and as-printed scaffolds. Furthermore, this research determined printing accuracy using 2D analysis instead of 3D measurements, and the inner printing errors may present different features compared to those on the outer surface. In this study, we developed an iterative feedback bio-printing (IFBP) approach based on the 3D quantification of the printed results to optimize the printing geometrical fidelity and cellular controllability. Moreover, we discussed the influence of printed geometric fidelity on the biological outcome.

The IFBP method is based on accurately and nondestructively detecting the mismatch between the design and as-printed scaffolds, thereby providing quantitative linear feedback control to improve the 3D bio-printing process. Micro-computed tomography (micro-CT) is a commonly used technique to image the inner architecture of engineered tissues and tissue engineering scaffolds25,26,27,28. However, since the X-ray absorption-based image contrast between culture media and high-moisture hydrogels is very poor, micro-CT imaging is unsuitable for imaging cell-laden hydrogel constructs under standard culture conditions29. Recently, optical coherence tomography (OCT) has shown potential for imaging the structure and function of engineered tissues fabricated by cells and bionic extracellular matrices, including natural biopolymer or synthetic polymers such as chitosan, collagen, alginate, Matrigel, polydimethylsiloxane (PDMS), polylactide (PLA), polycaprolactone (PCL) and poly(lactic-co-glycolic acid) (PLGA)30,31,32,33. Because of its nondestructive detection, high resolution (1–10 μm), deep penetration (1–5 mm) and real-time imaging ability (above 25 frames/s), OCT can be used in the evaluation of the constructs’ structure, cell dynamics and tissue development in engineered tissues30,31,32,33,34,35,36,37,38. Our earlier work established a platform for the automatic quantitative characterization of 3D bio-printed hydrogel scaffolds using swept-source OCT (SS-OCT) imaging, which revealed the linear relationship between the printed and designed geometries39,40,41. Therefore, we present an OCT-IFBP approach that iteratively regulates the mismatches between designed and as-printed scaffolds based on the linear feedback formula obtained experimentally from OCT imaging and analysis.

In this study, eight types of cell-laden hydrogel scaffolds with different pore sizes were designed, printed and quantitatively characterized in the first experimental loop. Two types of tailored cell-laden hydrogel scaffolds were printed with high geometric fidelity and cellular controllability using the OCT-IFBP approach. Gelatin/alginate hydrogels laden with C3A cells were used to investigate the effects of the OCT-IFBP method. An evaluation of the cell-laden hydrogel scaffolds was performed, and comparison between the OCT-IFBP and direct printing methods were based on the cell viability, proliferation, and morphology, as well as the functional expression of hepatocyte markers, such as CYP3A4 and albumin.

## Results

### Printability of a gelatin/alginate-driven hydrogel

The printability of the hydrogels significantly affects printing robustness and accuracy. To systematically investigate the printability of hydrogels, different structures, including 1D cylindrical filaments, 2D lattice sheets and 3D grid scaffolds, were printed with the same designed strut size (StS) of 210 μm using the same printing conditions, as described in the Methods section. All printed scaffolds were imaged and characterized using an optical microscope (Ti-U, Nikon, Japan), as shown in Fig. 1. The StSs of the 1D cylindrical filaments, 2D lattice sheets and 3D scaffolds were 318.49 ± 11.04 μm, 311.40 ± 12.31 μm, and 236.63 ± 3.52 μm, respectively. Distinct variations in the mismatch between the designed and as-printed scaffolds were observed. Compared to the 1D and 2D structures, the 3D porous hydrogel scaffolds presented remarkably different deformation properties. Unlike 2D structures, quantifying the mismatches in 3D hydrogel scaffolds is fundamental to optimize printing fidelity.

### Imaging and characterization of 3D bio-printed cell-laden hydrogel scaffolds using OCT

To investigate the intrinsic characteristics of the mismatch between the designed and as-printed scaffolds, the gelatin/alginate/C3A 3D scaffolds were successfully fabricated with different predefined pore sizes (PSs), which evenly increased at regular intervals. All of the 3D scaffolds were imaged and analyzed by our custom-built SS-OCT system. The respective OCT images of six different scaffolds are shown in Fig. 2: A1–A6 show the image range of OCT in the XY plane, while B1–B6 provide the corresponding microscopic images of different hydrogel scaffolds. High-resolution (10 μm) OCT images clearly distinguish the hydrogel area (gray) from the pore area (black) (C1–C6, D1–D6). For the high scattering hydrogel scaffolds, an OCT imaging depth of up to 3.0 mm was achieved (C1–C6, E1–E6), which, in contrast to the 2D microscopy images, is not limited to the outer surface. 3D OCT imaging also enables the reconstruction of pore networks (F1–F6) and material distribution (G1–G6).

Based on our previously proposed algorithm, the PS and StS of the different scaffolds were quantified using the 2D en-face OCT images, whereas volume porosity (VP), surface area (SA) and pore volume (PV) were obtained from the reconstructed 3D images. All the quantitative characterization results were measured based on 5 replicates and are listed in Table 1.

We compared the morphological parameters of the printed hydrogel scaffolds with their design parameters, as shown in Fig. 3. Data for the arithmetic average of the five sample scaffolds are displayed in the form of a histogram, where the black histogram bars represent the predefined values and the gray histogram bars represent the actual printed values. The percent above the histogram denotes the percentage of the mismatch between the designed and as-printed scaffolds, which is expressed as the absolute value. From the graphs, the difference ranges of PS, StS, VP, SA and PV are 31–39%, 12–17%, 15–39%, 18–30% and 14–39%, respectively, which indicate significant discrepancies between the designed and as-printed scaffolds.

### Determining the feasibility of the OCT-IFBP approach

To determine the feasibility of the OCT-IFBP approach for the fabrication of cell-laden hydrogel scaffolds with high geometrical fidelity and cellular controllability, it is necessary to investigate both the inter-batch variations and the intra-batch correlations, which determine the robustness and controllability of the 3D bio-printing technique. First, the inter-batch variations are used to determine the repeatability and robustness of our 3D bio-printing technique and process. The repeatability is defined as the variations in the morphological parameters of printed scaffolds with identical designs at different times and in different batches. 3D hydrogel scaffolds with the same designed geometry (po600) were printed at five time points over six months, and the morphological parameters of each batch were quantified by 3D OCT image analysis. As shown in Fig. 4A, the largest significant difference between different batches was only 9 μm of the average PS, which was acceptable from a practical point of view. Therefore, our 3D bio-printing process is robust and repeatable.

The intra-batch correlation analysis was used to determine the fidelity between the designed and as-printed structures. The PS, VP, SA, and PV of six different predefined scaffolds printed in the same batch were analyzed as a function of the preset PS of each. Figure 4B1–B4 show that there is a good linear correlation between the printed values and the designed PS, which is expressed in the linear fitting functions and considered suitable as a feedback to adjust the production run.

### Validation of the effectiveness of OCT-IFBP approach

To verify the ability of OCT-IFBP to print cell-laden hydrogel scaffolds with tailored morphological characteristics, two scaffolds with the envisioned average pore sizes of 250 μm and 350 μm were redesigned and printed with six replicates: po250+ was designed with a PS of 390 μm, and po350+ with a PS of 540 μm. Based on the defects observed in the first experimental run, printing process optimization was also performed to improve printing accuracy. A thermostat was used to ensure the uniformity of the temperature of the dispensing syringe, thus enabling a certain homogeneity in strand distribution. The printing parameters were adjusted based on our previously constructed bioink data to a pressure of 0.43 MPa, an X/Y printing speed of 8.5 mm/s, a nozzle temperature of 8 °C, a platform temperature of 4 °C and a layer height at 100 μm. Figure 5A and B show the microscopy and OCT images of the two scaffolds reprinted according to the feedback from OCT imaging and analysis in the first run.

Figure 5C1–C5 compare the morphological parameters of the reprinted scaffolds measured by 3D OCT analysis with those predicted by the aforementioned empirical correlation functions, represented by the error bars in the intra-batch variation. There are significant differences between the preset parameters and the measured values. However, compared to the printing results in the first run, the mismatches are reduced tremendously, i.e., the PS mismatches decreased from 31–39% to 4–6%; the StS mismatches decreased from 12–17% to 1%; the VP mismatches decreased from 15–39% to 2–6%; the SA mismatches decreased from 18–30% to 1–3%; and the PV mismatches decreased from 14–39% to 4–6%. Therefore, our OCT-IFBP method has significantly improved the geometric fidelity of the 3D bio-printed hydrogel scaffolds.

### In vitro cell biological behavior analysis

To verify whether the improvement in geometric fidelity will facilitate cell growth and functional expression, the in vitro biological characteristics of two groups of cell-laden hydrogel scaffolds were evaluated and compared based on cell survival and proliferation and relevant functional expression. The improvement groups, defined as po250+ and po350+, used the abovementioned OCT-IFBP method to obtain the desired geometry structures, whereas the controlled groups, which were define as po250 and po350, were printed directly. All the quantitative characterization results of two groups were measured based on 5 replicates and are listed in Table 2.

### Cell viability and proliferation

Figure 6 shows the cell viability and proliferation assay results of the po250 and po350 improvement and controlled groups. The cell viability percentages of po250+ in the improvement group were 92.10 ± 2.78% after 1 day, 94.99 ± 1.03% after 7 days, and 96.13 ± 2.79% after 14 days, whereas the corresponding viability percentages of po250 printed directly were 90.99 ± 2.22%, 85.50 ± 1.70%, and 83.36 ± 3.76% (Fig. 6B). Moreover, the cell viability percentage of po350+ was 90.24 ± 2.45% after 1 day, 96.00 ± 0.83% after 7 days, and 95.64 ± 2.84% after 14 days, whereas the corresponding viability percentages of the po350 in the control group were 90.33 ± 4.00%, 92.64 ± 1.66%, and 91.62 ± 1.57% (Fig. 6B).

To investigate the cell growth characteristics of two groups of 3D bio-printed scaffolds, the alamarBlue assay was performed to analyze cellular metabolic activity on days 1, 7, 10 and 14 (Fig. 6C). The proliferation ratios of po250 were 1.00 ± 0.02 at day 1, 1.28 ± 0.02 at day 7, 1.73 ± 0.04 at day 10, and 1.62 ± 0.03 at day 14, whereas the corresponding proliferation ratios of po250+ were 1.00 ± 0.02, 1.48 ± 0.02, 1.86 ± 0.03 and 2.07 ± 0.05. The proliferation ratios of po350+ in the improvement group were 1.00 ± 0.02 at day 1, 1.34 ± 0.09 at day 7, 1.70 ± 0.02 at day 10, and 1.95 ± 0.03 at day 14, whereas the corresponding proliferation ratios in the po350 control group were 1.00 ± 0.07, 1.37 ± 0.03, 1.68 ± 0.09 and 1.90 ± 0.13. According to the synergistic analysis of viability and proliferation (Fig. 6B,C), an average PS of 250 μm was considered preferable for C3A cell growth within the hydrogel scaffolds.

### Cell distribution and cell morphology

The po250 and po250+ groups were chosen to investigate the improvement effects on other biological properties of the 3D bio-printed cell-laden hydrogel scaffolds. Figure 7 shows the cell distribution obtained with a DIC microscope and cell morphology examined using HE staining of euchromatic nuclei. At all investigated time points, i.e., days 1, 7, and 14, the tailored structures obtained by OCT-IFBP have an evident effect on the cellular spheroid ratios. Furthermore, more cellular spheroids were observed around the pore in the improvement group, whereas the control group exhibited cells that tended to be marginalized and much fewer cellular spheroids.

### Cellular function expression

The cell’s potential to express liver-specific functions, such as the expression of the CYP3A4 protein and the production of albumin, was detected and compared between the two groups (Fig. 8). The albumin expression in the improvement group increased more sharply over time than that in the control group. The expression of the CYP3A4 protein in the improvement group was also increased compared to that in the control group. Clearly, the improvement in the geometrical fidelity and controllability enhanced the tissue functionalization of the 3D bio-printed cell-laden hydrogel scaffolds.

## Discussion

For tissue engineering scaffolds fabricated by 3D bio-printing, the internal porous geometries are tailored to preferably mimic the 3D structures of natural tissues and organs. These geometries determine cell growth, functional expression and nurture properties after printing42,43,44. Hydrogels are suitable for 3D bio-printing because they are good biomaterials for cell encapsulation and excellent biomimetics of the human ECM. However, one of the main challenges in the 3D bio-printing of cell-laden hydrogel scaffolds is fabricating porous structures exactly matching predesigned geometries that meet desired biological requirements. Our study aimed to address the problem of how to exploit the unique advantages of the structure customization of the 3D bio-printing technique to control the cell dynamics and functional expression.

Different structures ranging from 1D to 3D were printed to investigate the printability of hydrogels and the characteristics of printing deformation. Our results show different distortions in the 1D, 2D and 3D printed hydrogel scaffolds (Fig. 1). The phenomenon of the deposited strands appearing considerably larger in 2D than in 3D may be partly ascribed to the much greater stiffness of the printing substrate; that is, the 1D and 2D strands were deposited on a glass slice, whereas the 3D strands were deposited on the hydrogels. On the one hand, the printing substrates had different elasticities and exhibited diverse stress effects. On the other hand, the dimensions of the deposited strands may have influenced the deposited surface area and the corresponding swelling degree and cross-linking and further contributed to the accumulation of structural distortion45. Other researchers have also demonstrated that 2D sheets and 3D porous scaffolds present different deformation characteristics induced by hydrogel swelling15, cross-linking deformation46, stress interaction16, thermo condensation17,19,47, layer height mismatch14,22 and other factors14,46,48. These findings emphasize the importance of deep and nondestructive imaging for accurately assessing the printing quality of 3D bio-printed cell-laden hydrogels. In this paper, OCT was employed to assess the architecture of 3D bio-printed cell-laden hydrogel scaffolds with a resolution of 6 μm and a depth of 3.0 mm. The ability to image without contact and the low exposure dose of OCT led to almost no impairment of the cells within the hydrogels29,32,33,34,35,36,37,38. Furthermore, OCT depends on the scattering property of the sample and thus poses low risk to altering the properties and structures of the samples30,31,32,33. High-resolution OCT images clearly identified the fabrication defects (Fig. 2C1–C6,D1–D6,E1–E6) in the first run and revealed that the inner structure of 3D porous scaffolds exhibited different features compared to the outer surface (Fig. 2C1–C6, E1–E6,F1–F6,G1–G6). Therefore, determining the hydrogel printability or printing accuracy based on 2D analysis instead of 3D measurements is not sufficiently precise for printing process guidance and quality prediction. Furthermore, 3D OCT image analysis enabled accurate quantification of the mismatch between the designed and as-printed scaffolds (Fig. 3). Based on our previously developed automatic quantitative visualization algorithm39, both the locations and dimensions of each printing defect could be defined, which was vital to the optimization of the printing process in the second production run.

We successfully controlled the structural parameters of the 3D bio-printed hydrogels to match closely the design parameters (Fig. 5) using the OCT-IFBP method through two closed-loop iterative feedback optimization steps. The first experiment run was used to determine the stability and controllability of the 3D bio-printing system. There was only a 9-μm significant difference in the actual average PS between 6 different batches covering 6 months (Fig. 4A), which demonstrated the repeatability and robustness of our 3D bio-printing system. The design in this run should cover a wide range of geometrical features. PS is an essential variable, as it affects other morphological parameters of the printed scaffolds. Furthermore, PS variations also influence cell growth, morphogenesis, functional expression and nutrient/waste transport properties. Considering these conditions, the preset pore dimensions were varied from 300 μm to 800 μm with evenly distributed intervals. The good linear correlation between the printed morphological parameters and the preset PS (Fig. 4B1–B4) demonstrates the controllability of intra-batch printing, which can be used for predicting the production morphological parameters.

To obtain the desired cell-laden hydrogel scaffolds in the second run, not only the empirical design constraints obtained from the first run but also the appropriate printing process optimization should be used for guiding the new design to avoid systematic fabrication errors. We integrated a detailed printing parameter database regarding the specific bioink41,49,50 and the information about the location and dimensions of each printing defect obtained from OCT imaging to fully optimize the printing process configuration by employing, for example, distributed temperature control to ensure homogenous physical gelation of the utilized bioink and precise temperature control of the printing bioink and the printing substrate. Other research has also emphasized the importance of an in-depth parametric study to the printing process14,21,22,48.

Importantly, the printing controllability for a PS smaller than 400 μm is poor because of the inherent complex variability of hydrogels and cells. Other researchers have also observed that a scaffold with a PS lower than 600 μm is more vulnerable to pore collapse, even when using a configuration with optimized printing parameters9,14,46,47,48,49,50. Figure 2A1–G1 and Fig. 6A in the control group show more pore collapse and strand fracture. Regarding the association between printing robustness and the preset PS, a similar phenomenon was observed in hydrogel scaffolds prepared using either a commercial 3D Bioplotter (Envisiontec, Germany) or another laboratory rapid prototyping system. To obtain a stable structure, designs with a real PS above 500 μm were popular for investigating hydrogel printability and shape fidelity21,22,23,43; however, a PS of 500 μm is too large to translate signals among cells and form tissues51. The IFBP approach enables more compact structure in precise printing, for example, the PS in the range of 200~400 μm, which is optimal for nutrient/waste flow, cell migration and angiogenesis51.

Many researchers have noticed the influence of the internal geometrical features of a cell-laden scaffold on the biological outcome51. As illustrated in Fig. 6, our findings stress the importance of precisely tailored geometric structure for obtaining the desired biological outcomes. Under the optimal printing parameter configuration, the initial cell viability of po250 and po350 did not show a significant difference between OCT-IFBP and direct printing. However, over time, the cell survival in the improvement group seemed to increase compared to that in the control group (Fig. 6A,B,C), with po250 presenting a more obvious improvement in cell growth activity than po350. Comparing the tailored PS 250 μm and 350 μm cell-laden hydrogel scaffolds fabricated by IFBP-OCT (Fig. 6B,C) revealed that the former scaffold showed better cell viability and proliferation. In other words, improving the geometric fidelity of printing is beneficial to feedback optimization of scaffold design and study the fundamental relationship of geometrical structure and cell growth. Furthermore, HE images of the cell-laden scaffolds show a more uniform cell distribution and higher cellular spheroid ratio in the improved po250+ matrix materials than in po250 (Fig. 7). A round cell morphology is typical and preferential for encapsulated C3A cells, and homogeneous cell distribution facilitates tissue formation. The scaffolds provide the cells more favorable microenvironment, which is a better choice for spheroid formation. Moreover, the spheroid has well established cell-to-cell contacts that can preserve cell viability and promote cell functional expression. However, the overall size of the spheroid is smaller in po250 than in the improved po250+, because of the limit of the diffusion of oxygen, nutrients and metabolic waste products, which cause by the different pore sizes.

Although cell viability, proliferation, and morphogenesis are important criteria for characterizing cell growth in engineered tissue, they do not allow an overall evaluation of the improved effect on the biological outcome of a cell-laden hydrogel scaffold. To validate the quality of the cell-laden hydrogel scaffolds fabricated by OCT-IFBP, demonstrating the cells’ capability of displaying normal cell behavior post-construction of the scaffold was necessary. For C3A cells, a hepatocellular carcinoma cell line, this behavior includes maintained expression of hepatocyte markers. The improvement group prepared using the OCT-IFBP method and the control group printed directly were therefore analyzed and compared based on the expression of CYP3A4 and albumin. More active cell function expression was confirmed in the OCT-IFBP group, whereas the expression of CYP3A4 and albumin in po250+ increased more sharply than those in po250 over time (Fig. 8). Therefore, the improvement in the geometrical fidelity and controllability did enhance tissue functionalization of the 3D bio-printed cell-laden hydrogel scaffolds.

OCT-IFBP is furthermore applicable to other 3D bio-printing techniques. On condition that inter-batch and intra-batch variability of the chosen bio-printing technique or systems remains low, other 3D bio-printing technique or systems like inkjet bioprinter, microextrusion bioprinter, laser-assisted bioprinter, whether laboratory or commercial bioprinting machines could replace our used bioprinter in this protocol. OCT is a non-invasive real-time biomedical imaging modality which has been widely used in clinical testing and biomaterial researches. Further, OCT is easy-operating and commercially available, such as SSOCT-1310 (OCTMI Corporation, Irvine, USA), IVS-2000-LC (Santec Corporation, Komaki, Japan), SS-1310 (Moptim Corporation, Shenzhen, China) and so on. The research labs related to 3D bio-printing technique are able to integrate OCT into bioprinting machine quite conveniently and establish an IFBP system based on a home-developing or commercialized OCT machine.

Although the OCT-IFBP protocol is conducive to fabricating a porous cell-laden hydrogel construct with tailored-characteristics, there are still several limitations to this study. The structures that we designed were homogeneous, and the quantitative analysis was based on whole samples. The automatic 3D labeling quantitative characterization algorithm demonstrated in our previous work39 can be used to assess a local unit cell and precisely improve the printing geometric fidelity of a heterogeneous design, which will be the subject of our next research direction.

## Conclusions

This study presented an OCT-IFBP protocol for the 3D bio-printing of cell-laden hydrogel scaffolds with high geometric fidelity and cellular controllability. 3D OCT image analysis illustrated that the directly printed cell-laden hydrogel scaffolds had low geometrical fidelity. Inter-batch quantitative characterization and intra-batch linear correlation analysis confirmed good the repeatability and controllability of our 3D bio-printing system and technique, which corroborated the feasibility of OCT-IFBP. Based on OCT imaging and analysis, the quantitative linear feedback design and the printing parameter optimization were integrated to avoid systematic fabrication errors, and the desired printing results were obtained in the second run. The more active cell growth and functional expression when using OCT-IFBP instead of direct printing confirmed that improving the geometric fidelity facilitates the optimization of the cellular controllability of 3D bio-printed cell-laden hydrogel scaffolds through the use of tailored structures.

The novelty of this protocol lies in using OCT, a 3D quantitative characterization method, for iterative feedback design and printing control, thereby optimizing the geometric fidelity and cellular controllability of the 3D bio-printing technique. The demonstrated OCT-IFBP protocol may pave the way for the mass fabrication of cell-laden hydrogel scaffolds, engineered tissues, and scaled-up applications of the 3D bio-printing technique. OCT-IFBP also facilitate optimizing design to obtain desired properties of printed constructs, and investigating the quantitative relationship between scaffold structure and function.

## Methods

### Overview of the OCT-IFBP approach for 3D bio-printing

An OCT-IFBP approach is applied to optimize the geometric fidelity and cellular controllability of 3D bio-printing for TERM applications. The feedback loop consists of the design, fabrication, overall morphological quantitative characterization and cell behavior analysis of 3D bio-printed cell-laden hydrogel scaffolds. The feedback loop control was performed twice (Fig. 9). The first run was used to evaluate the intrinsic controllability of the 3D printing system and process. In the first run, cell-laden hydrogel scaffolds with different pore sizes were designed and fabricated by the 3D bio-printing system and later imaged and analyzed by a custom-built SS-OCT system. The morphological parameters, including PS, StS, VP, SA, PV, were quantified and compared with the predesigned parameters. The quantitative mismatches between the designed and as-printed scaffolds were used to guide the second run of printing, in which analysis of the correlation between the printed parameters and preset variables and 3D printing process optimization were integrated to decrease the extent of the mismatch. The cellular controllability of the printed scaffolds was investigated by comparing the cellular characteristics of scaffolds printed using OCT-IFBP with those printed directly.

Table 3 shows ten types of designed geometric structures for different evaluation experiments, including OCT image analysis, repeatability studies, specific design constraints, cell viability assays, cell proliferation assays and other biological characterization techniques. The designed geometrical structures were defined according to the PS as po250, po250+, po300, po350, po350+, po400, po500, po600, po700, and po800. In the OCT image analysis experiments of the first run, five printed samples were randomly selected from one of the six types of scaffolds for testing. In the repeatability study, the hydrogel scaffolds of po600 were fabricated 5 different times and were compared quantitatively with the significant differences in the morphological parameters. The specific design constraint experiments were used to verify whether the OCT-IFBP method could significantly reduce the mismatch between the designed and as-printed scaffolds. The related cell viability, proliferation, and biological characterization experiments were used to determine whether optimizing the internal structure fidelity would improve the cellular controllability and whether cellular controllability would contribute to the biofunctional reconstruction of the printed cell-laden hydrogel engineered tissue.

### Cell culture

C3A liver cells (ATCC® -10741TM, American Type Culture Collection, USA), were cultured and maintained in a humidified atmosphere of 95% air/5% CO2 at 37 °C with culture medium, which consisted of DMEM, fetal bovine serum (10%, FBS) and penicillin/streptomycin (1%, P/S). All of the agents were obtained from ATCC. The cultivation medium was refreshed every 2 days.

### Design of hydrogel scaffolds

According to previous work, scaffolds with orthogonal interconnected square pores are effective in cell encapsulation. For the purpose of this study, ten types of square pores of different sizes were designed by SolidWorks software. As shown in Fig. 9, d2 represents the PS of the scaffold, and po300 denotes that the design value of PS is 300 μm; d1 represents the StS of the scaffold; and d3 represents the thickness of the layer. The values of d1 and d3 are set at 210 μm. The shape of the scaffold is a cuboid with dimensions of 10 × 10 × 3 mm.

### 3D bio-printing of cell-laden hydrogel scaffolds

The 3D scaffolds were fabricated by sequential fiber deposition using our 3D bio-printing system (Bio-Architect®-Pro, Regenovo Corporation, Hangzhou, China), as described previously5,33. Briefly, the hydrogel was prepared with 10% gelatin and 2% sodium alginate mixed at a volume ratio of 1:1. Gelatin and sodium alginate were obtained from Sigma and prepared according to polymer weight/solvent volume (w/v). Next, the prepared hydrogels were mixed with C3A cells to obtain a cell encapsulating density of 1 × 106 cells mL−1. The cell-gel mixtures were transferred into a sterilized syringe, loaded into the Bio-Architect®-Pro and cooled to 8 °C. The cell-hydrogel suspensions were extruded from a refit pneumatic nozzle at a pressure of 0.42 MPa and an X/Y speed of 8 mm/s. All of the printing parameters were optimized based on our earlier hydrogel printability experiments49,50,52,53,54. Upon the completion of printing, the scaffolds were cross-linked with 2% CaCl2 solution, transferred to 24-well culture plates, washed with DMEM three times and maintained in 1.6 mL of cultivation medium in a CO2 incubator at 37 °C. The medium was refreshed daily. The fabricated scaffolds were examined using a light microscope (Ti-U, Nikon, Japan).

PSs of 250 μm and 350 μm were chosen as the typical geometry design to examine whether the OCT-IFBP method benefits the cell growth and functional expression in 3D bio-printed cell-laden hydrogel scaffolds. In the control group, po250 and po350 were printed as described above. For the improvement group, OCT-IFBP was performed with the integration of the design constraint and printing process parametric optimization obtained from the first feedback loop. Thus, the 3D cell-laden hydrogel scaffolds with a desired PS of 250 μm and 350 μm were obtained, which were defined as po250+ and po350+, respectively.

### Imaging and characterization of hydrogel scaffolds using SS-OCT

The 3D bio-printed hydrogel scaffolds were imaged by a custom-built SS-OCT system, which used a swept-source laser with a scanning rate of 50 kHz and a spectrum range of 1310 ± 60 nm. More details on the system can be found in a previous publication39,55,56,57. The 3D OCT images of 6 × 6 × 3.5 mm (1024 × 1024 × 512 pixels) were obtained without contacting the hydrogel scaffolds, which were immersion-cultured in a sealed 24-well culture plate. The covers of the culture plate were custom designed to improve the image quality. The morphological parameters, including PS, StS, VP, Sa, and PV, were quantified based on our previously developed automated characterizing algorithm39,40,41.

### Live/dead staining for testing cell viability

To visualize and analyze the cell viability on the printed scaffolds, live/dead fluorescent staining was performed 1, 7, and 14 days after printing. The printed C3A/hydrogel scaffolds were stained with a mixture of Calcein-AM (Dojindo; 1 μmol/ml) and PI (Sigma; 2 μmol/ml) at 37 °C in the dark for 30 min, gently washed with PBS to remove excess dye, and visualized under an optical microscope (Ti-U, Nikon, Japan). Five images from different areas of three bio-printed scaffolds were chosen randomly. The number of live and dead cells was counted using ImageJ (National Institutes of Health, USA), and the cell viability was calculated as

$${Cell}\,{viability}=\frac{{number}\,{of}\,{green}\,{stained}\,{cells}}{{number}\,{of}\,{green}\,{stained}\,{cells}+{number}\,{of}\,{red}\,{stained}\,{cells}}\times 100 \%$$

### Cell proliferation analysis

Cell proliferation was analyzed using the alamarBlue assay, which introduces a fluorometric growth indicator to evaluate cellular metabolic activity. The cell-laden scaffolds were incubated in 1,000 μL of 10% alamarBlue for 2 hours on days 1, 7 and 14 of the study. The fluorescence was detected using a 520-nm excitation wavelength and a 590-nm emission wavelength. The cell number was obtained using a calibration curve determined by correlating a known cell number with the fluorescent intensity of the solution. Five independent samples were tested in each group.

### Representation of the cell morphology in cell-laden scaffolds

Three cell-laden hydrogel scaffolds were treated with a fixative agent (4% (w/v) paraformaldehyde neutral/phosphate (E672002, Sangon Biotech) buffer solution) for 2 hours at room temperature. After 2 hours of fixation, the scaffolds were dehydrated with 70% ethanol and subsequently paraffin-embedded. These specimens were cut into 5-μm-thick slices using a microtome (Leitz, Western German), stained with hematoxylin-eosin (HE) and observed under an optical microscope (Ti-U, Nikon, Japan).

### RNA isolation and mRNA quantitation

Total RNA extraction from the cell-laden scaffolds was performed with Trizol (Invitrogen) according to the manufacturer’s instructions. The quality and quantity of RNA were evaluated with a spectrophotometer (ND-3000, NanoDrop Technologies, DE). The primers of CYP3A4 and albumin were designed as listed in Table 4 using Primer 5.0 software (Premier Biosoft International, Canada). RT-RNA amplification reactions used 1 µg of total RNA via M-MLV Reverse Transcriptase (Takara Bio. Inc., Japan) to obtain 20 µL of product, from which 2 µL was extracted for qRT-PCR. The qRT-PCR assay was performed on an RT-PCR system (ABI 7300PlusTM, PerkinElmer Applied Biosystems, USA) with a special PCR kit (SYBR® Premix Ex TaqTM II, Takara Bio. Inc., Japan). The qRT-PCR assay steps consisted of denaturation for 30 s @ 95 °C, 40 cycles of amplification for 5 s @ 95 °C and 34 s @ 60 °C, and melting temperature-determining dissociation for 15 s @ 95 °C, 1 min @ 60 °C, and 15 s @ 95 °C. The assay results were calculated using the ΔΔCT method, and the expression values were normalized to GAPDH. All qRT-PCR assays were repeated five times.

### Statistical analysis

Unless stated otherwise, all characterization was performed at least in triplicate. The quantitative analysis was based on the measurement of multiple samples (n = 5), and the result of the measurement was analyzed by variance analysis. Tests of statistical significance were performed with SPSS Statistics. Quantitative data is expressed as mean ± SD and re-analyzed by two-way ANOVA. Multiple comparison between the groups was performed using S-N-K method. Statistical significance was set at a level of P < 0.05.

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## Acknowledgements

This work was supported by the National Natural Science Foundation of China [grant numbers 61675059, 81371695]; Key research and development foundation, Science and technology department of Zhejiang Province, China (Project number: 2017C01046, 2017C01017).

## Author information

L.W. and M.E.X. conceived the project and designed research; L.L., Y.Y.Z. and P.J.S. performed research; L.W., L.L. and P.J.S. performed optical coherence tomography analyses; L.L. and Y.Y.Z. performed analyzed data; L.W., L.L. and Y.Y.Z. wrote the paper; L.W. and M.E.X. helped correct and review the manuscript; L.W. and M.E.X. provided lab space, instrumentation and funding; All authors reviewed and approved the final manuscript.

Correspondence to Ling Wang or Ming-en Xu.

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## About this article

### Cite this article

Wang, L., Xu, M., Luo, L. et al. Iterative feedback bio-printing-derived cell-laden hydrogel scaffolds with optimal geometrical fidelity and cellular controllability. Sci Rep 8, 2802 (2018) doi:10.1038/s41598-018-21274-4

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https://doi.org/10.1038/s41598-018-21274-4

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## Comments

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text:
Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

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## #1 2013-09-23 01:54:13

prathima
Member

Offline

### logarithmic spiral

in polar coordinates, the logarithmic spiral is given by r=ae^(bθ)
how find θ and how to draw logarithmic spiral

## #2 2013-09-23 01:59:39

anonimnystefy
Real Member

Offline

### Re: logarithmic spiral

θ is just a variable, similar to x in y=x^2.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

## #3 2013-09-23 02:07:06

prathima
Member

Offline

### Re: logarithmic spiral

is there any particular formula?
without using any formula how to find θ
as radius varies  θ also varies right?

## #4 2013-09-23 02:14:14

anonimnystefy
Real Member

Offline

### Re: logarithmic spiral

The way it is done is, you take values for θ and calculate the value of r you get.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

## #5 2013-09-23 02:26:27

prathima
Member

Offline

### Re: logarithmic spiral

The spiral has the property that the angle φ between the tangent and radial line at the point (r, \theta) is constant. This property can be expressed in differential geometric terms as

arc cos[(r(θ), r'(θ)/||r(θ)||*||r'(θ)]=arctan(1/b)=⏀

## #6 2013-09-23 03:16:45

bobbym

Offline

### Re: logarithmic spiral

Hi;

Theta is an independent variable and r is the dependent one.

In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

format: plain


text:
Input and Output

Console Input

The console is a text interface for input to and output from the computer. In Spyder, the iPython window itself can serve as a console. In Jupyterlab the output console is indicated by lines marked with Out [] whereas a reference to an input console will open a textbox.

To read input from the console we use the input() function (Python 3+). In Python 2.7 the equivalent is raw_input(). Any string within the parentheses is optional and, if present, if will be printed to prompt the user. The input from the user is captured as a string and returned; it must be stored for any subsequent use.

The input function returns only a string. If you need to use the values as any other type, you must perform the conversion yourself.

Example:

weight=input("Enter your weight in pounds:")
print(type(weight))
print(type(weight))

Console Output

We have already been using the print function. Let us now examine it in more detail.

The print function

• always inserts a space between its arguments
• always adds a newline character unless the end argument is added.
• print(var,end="")

Messages can be added in between variables.

h=1.45;
w=62.

Exercise

Use Spyder or another IDE to write a complete program to compute BMI from weight and height input from a user. First request the user’s choice of units. We have not spent much time with strings yet so you may use a digit to indicate the user’s choice, but remember it will still be a string on input. Then request weight and height. You will need to convert these from strings. Look up the correct conversion factors for Imperial to metric units. Compute the BMI.

The categories are

BMI Category
less than 18.5 Underweight
18.5 to 25.0 Normal
25.0 to 30.0 Overweight
30.0 to 35.0 Obese Class I
35.0 to 40.0 Obese Class II
more than 40.0 Obese Class III

Print the user’s BMI value and category.

Example solution

unit=int(float(input("Enter 1 for Imperial or 2 for metric units:")))

if unit==1:
BMI=weight*703.1/height**2
elif unit==2:
BMI=weight/height**2
else:
print("Invalid unit request")
BMI=None

if         BMI < 18.5 : category="Underweight"
if 18.5 <= BMI < 25.0 : category="Normal"
if 25.0 <= BMI < 30.0 : category="Overweight"
if 30.0 <= BMI < 35.0 : category="Obese Class I"
if 35.0 <= BMI < 40.0 : category="Obese Class II"
if 40.0 <= BMI < 45.0 : category="Obese Class III"
if         BMI>= 45.0 : category="Obese Class IV"

print("Your BMI is {:.2f} and you are {}".format(BMI,category))

Formatted Output

So far we have only considered list-directed input/output. We leave it to the interpreter to format. However, we often want or need more control over the format. For this we define format strings.

Formats are indicated by a pattern

F.wf
F.wg
F.we

F is the field width (total number of columns occupied), w is the number of digits to the right of the decimal point; these must be substituted by numbers. The letters f, g, and e are called format codes and indicate floating-point (decimal point) format, general format (interpreter decides f or e), and exponential notation. If we do not specify w for numbers, the default is 6 decimal digits printed. If we specify the number of digits the interpreter will round the result; if unspecified it will truncate.

Examples:

• Use exactly 15 spaces with 8 decimal places
• 15.8f
• Let the interpreter decide the total field width but use 8 decimal places
• .8f
• Print a float as an integer (this will truncate, not round, the decimal places)
• 0f
• Print in scientific notation with three decimal places
• .3e

Strings are specified with the letter s. They do not take a field width but can be adjusted or zero padded by using more advanced operations on the string to be printed.

Integers are specified by d (this stands for decimal, as in base-10 number) with an optional field width. If the field width is wider than the integer, it will be padded with blanks by default. If it is narrower it will be ignored. If omitted the interpreter will use the width of the integer.

To pad with zeros, write the zero in front of the width specifier.

Examples:

5d
05d

Formatters

To construct our formatted output we use format strings, also called formatters. We insert curly braces as placeholders for variables in the finished string. Format codes go inside the braces following a colon.

print("The value of pi is approximately {:.6f}".format(math.pi))
print("Pi to {:d} digits is {:.12f}".format(n,math.pi)

We have left the space to the left of the colon blank, but it represents the order of the arguments to format. Thus the second example is equivalent to

print("Pi to {0:d} digits is {1:.12f}".format(n,math.pi))

This means we can rearrange the output if we wish.

print("Pi is {1:.12f} to {0:d} digits".format(n,math.pi))

Empty curly braces result in default formatting and ordering. In this situation one could use list-directed formatting as well, but using a formatter enables more control over layout and spacing.

print("I have {} books to read by {}".format(12,"tomorrow"))

Values can be aligned in a field with < (left align), > (right align), or ^ (center align). These symbols must be followed by a field width.

print("The number {:^8} is {:>15.4f}".format("pi",math.pi))

Exercise

Type at the interpreter

import math

Practice printing math.pi

• Print without any formatting
• Print using the default f format
• Print to 5 decimal places
• Print the integer part
• For at least one of the above, add a message
• Print the number of digits to at least 6 spaces and pad with zeros
• Print the number with a message that specifies the number of digits, where the number of digits is also to be printed.

Even more sophisticated formatting is possible. In the above examples, when printing pi the value of n had to be 12 or the output would be inconsistent. In newer Python versions we can make the width a variable.

• print(“Pi to {} digits is .{dec}f}".format(math.pi,dec=n))

Formatting with f-Strings

For Python 3.6 and up the formatted string literal or f-string was introduced. These can be used to create formatted output easily.

Example

n=12
print(f"Pi to {n} places is {math.pi:.{n}f}")

The f-string evaluates the contents of the curly braces when the program is run. That means that expressions can be used inside the curly braces.

print(f"Pi to {n//3} places is {math.pi:.{n//3}f}")

In this example, the integer division is required because the result of the expression must be an integer.

If quotation marks are needed inside an f-string (such as for a dictionary key) they must be single quotes. In addition, the backslash \ cannot be a character within an f-string, so if a character such as the newline \n is needed, it must be assigned to a variable.

Resources

Full documentation is here. Details on format strings is here.

A good reference for f-strings is here.

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text:
# simnibs.simulation.tdcs_neumann¶

simnibs.simulation.tdcs_neumann(mesh, cond, currents, electrode_surface_tags)

Simulates a tDCS electric potential using Neumann boundary conditions on the electrodes

Parameters
• mesh (simnibs.msh.mesh_io.Msh) – Mesh file with geometry information

• cond (simnibs.msh.mesh_io.ElementData) – An ElementData field with conductivity information

• currents (list or ndarray) – A list of currents going though each electrode

• electrode_surface_tags (list) – A list of the indices of the surfaces where the dirichlet BC is to be applied

Returns

potential – Total electric potential

Return type

simnibs.msh.mesh_io.NodeData


text:
# Evaluate expression

Evaluate expression using BODMAS rule:

1 1/4+1 1/5÷3/5-5/8

Result

x =  2.625

#### Solution:

$x=(1+1/4)+(1+1/5)/(3/5)-5/8=\dfrac{ 21 }{ 8 }=2.625$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

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# Risk-Neutral covariance matrix of arbitrage-free Nelson Siegel

For my thesis on a Bayesian sampling routine for a modification on arbitrage-free Nelson-Siegel I came across an equation that involves a matrix exponential within an integral, i.e.

$$\int_{0}^{\Delta t} e^{-K*s}\Sigma \Sigma' e^{-(K)'*s}ds = Q$$

where I need to extract $$\Sigma$$ $$(3 \times 3)$$ from the integral. Here, $$s$$ is a scalar and $$K$$ $$(3 \times 3)$$ and $$Q$$ $$(3 \times 3)$$ are known.

In one model the off-diagonal elements of all are set to zero and algebraic expressions may easily be obtained. I however seek to derive the elements of $$\Sigma$$ for a more general case, where $$\Sigma$$ is a lower triangular matrix, K is a non-symmetric matrix of state dynamics with positive real eigenvalues and Q is a non-diagonal covariance matrix. Additionally, $$\Sigma$$ and $$e^{-K*s}$$ do not commute. Numerical methods would also be more than welcome.

Kind regards,

Gert

• What is the connection between $K$ and $\Sigma$? – Taylor May 10 at 16:00
• @Taylor Hi Taylor, thank you for your reply. To be specific, I am looking at a three factor latent state Ornstein-Uhlenbeck process under the P-measure, where $\text{d}X_{t} = K( \theta^{P} - X_{t} ) \text{d}t + \Sigma \text{d}W_{t}^{P}$. – Gert van Dasler May 10 at 20:11
• Do your latent states in this case represent things like dynamic slope and curvature of the yield curve? I'm more familiar with these kinds of models in discrete time and using things like Kalman filtering. Instead of make the state dynamics correlated you could probably just change $K$ and rotate the states, and then adjust the "observation" equation. By the way, where does this integral equation pop up? – Taylor May 11 at 1:34
• @Taylor Yes, in traditional Dynamic Nelson-Siegel they do correspond to level, slope and curvature. I am however using the adjusted base from Nyholm (2018), such that the factor interpretation of [Short Rate, Positive Slope, Curvature] emerges in Rotated Nelson-Siegel (AFRNS). For this model I derived an arbitrage-free specification, (AFRNS), but the exclusion of arbitrage is maintained by this relation between the risk-neutral covariance matrix, $\Sigma$, and the real-world covariance matrix, $Q$. Moreover, I am using KF as well, but this condition is required (Christensen (2011)). – Gert van Dasler May 11 at 7:23
• @Taylor I think that the problem is only explicitly solvable by somehow making the matrices commute or, stronger, diagonalization. For the diagonalization-case the method works, as I could easily write the diagonal elements of $\Sigma$ as function of the diagonal elements of $K$ and $Q$. – Gert van Dasler May 11 at 7:27


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# Complex roots of a 7th degree polynomial

Algebra Level 5

Consider the equation:

$x^7+x^6+x^5+x^4+x^3+x^2+x+1=0$

Let $$x_k$$ be the $$k^\text{th}$$ distinct root of the equation, and let $$\Re(x_k)$$ be the real part of that root.

$\sum_{k=1}^7{\left[\Re(x_k)\right]^2}=\ ?$

×


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## Carbon Circular logic

The cornerstone of climate science must be the enhanced greenhouse effect caused by rising CO2 levels. Therefore understanding how CO2 may increase in the future is of fundamental importance, and you would assume that any Earth Systems Model(ESM) should as a minimum be able to describe past increases in CO2.  I had naively assumed that the BERN model was tuned so as to match emissions to CO2 increases, but it turns out this is not true.

Here is the result for CO2 levels if you integrate the AR4 Bern model using the historic emission data (CDIAC) and then compare it to the actual CO2 measurements from Dome-C and Moana Loa. (see last post for details)

Comparison of cumulative emissions (CDIAC) and CO2 measurements. The purple curve is AR4 Bern model. The red curve is TAR Bern model. The data are Dome-C (Epica) CO2 data (gold) and Moana Loa + global averaged CO2 data (green).

There are two clear problems. Firstly the CO2 measurements rise well above cumulative emissions from 1800 to 1950,  which is seemingly impossible. The second problem is that even after 1950, the AR4 Bern model only begins to agree with ML after 2000 and even then cannot reproduce the time trend. So what is going wrong or did I perhaps miss something?  Well I have not included Land Use ’emissions’ which has been estimated from historical records of deforestation. Deforestation reduces net carbon uptake by living biota leading to an annual ’emission’. To remedy this I have simply used the AR5 Land Use data from this R.J.Houghton’s study (CDIAC).

Here is the new result after adding in the ‘land use’ data to the emissions data, where I now also use the latest BERN multi-model result (see Joos et al. 2013) .

Calculated CO2 levels including Land Use Emissions. Curves are the latest BERN model parameters (Joos et al. 2013) which are essentially the same as the AR4 values above within this time frame. Dashed purple is the same model bit excluding Land-Use values.

This gives a better agreement with the Dome-C data up until 1950, which now also lies correctly below the cumulative emissions.  However the BERN model  significantly overestimates CO2 levels thereafter. This demonstrates that merely adding in land use emissions still does not reproduce past CO2 levels. Nor am I alone in discovering this discrepancy.

A recent discussion paper by Millar et  al.  which uses a fit to ESMs to derive a ‘Finite Amplitude Impulse Response (FAIR)  function, also fails to reproduce the actual measured CO2 levels. The AR5-IR model as described in the AR5 report does even worse.

Figure 4a) shows a reconstruction their ‘FAIR’ model and an AR5-IR model integrated using emissions data. The AR5-IR model fails miserably while their FAIR model also overshoots emissions. The actual emissions  data used are those shown in Figure 4b) while the blue curve is what hypothetical emissions would have been needed to reproduce the actual CO2 Keeling curve. Their result is essentially the same as mine. A boost in emissions is needed before 1930 to explain the dome-C measurements and the AR5 model overestimates CO2 levels.

A NASA paper recently reported on the CO2 greening effect. So did somehow the global biota switch from being a source of emissions (deforestation) to being a net sink (greening) as CO2 levels increased? Suddenly everyone thinks this was obvious all along.

Prof Corinne Le Quéré, director of the Tyndall Centre at the University of East Anglia said: “Natural vegetation is a fantastic help in slowing down climate change by absorbing about a quarter of our carbon emissions from burning fossil fuels.

Yet it seems clear to me that ESMs did not take sufficient account of any fertilisation effect. Nor did AR5 report any global uptake from CO2 fertilised growth in plants either. It merely refers to possible regrowth from past clearances in specific regions.

The second major source of anthropogenic CO2 emissions to the atmosphere is caused by changes in land use (mainly deforestation), which causes globally a net reduction in land carbon storage, although recovery from past land use change can cause a net gain in land carbon storage in some regions.

Global net CO2 emissions from land use change are estimated at 1.4, 1.5 and 1.1 PgC yr–1 for the 1980s, 1990s and 2000s, respectively, by the bookkeeping method of Houghton et al. (2012)

This makes it clear that any fertilisation effect was also ignored by ESMs. This likely means that IPCC projections of future CO2 levels based on the Bern model are too high.

Posted in climate science, IPCC, Science | Tagged , , | 7 Comments

## Bern Model when emissions stabilised at 2013 levels

The Bern model has been used by IPCC for emission scenarios since SAR in 1995. We will now use it to extrapolate CO2 levels 500 years into the future with annual emissions fixed at 2013 levels (~10 GT C/year). All values are expressed as an equivalent atmospheric CO2 concentration in ppm for convenience.

To do this we integrate emissions over time and and calculate atmospheric CO2 concentrations.

$CO_2(t) = CO_2(1750) + fac \times \int_{1750}^{t} Em(t') \times ( a_0 + \sum_{1}^{3} a_i e^{-\frac{t-t'}{\tau_i}} ) dt'$

where fac converts GTC to ppm, Em = emissions in year t’ and the a’s and are $\tau's$ are the parameters of the Bern model. $CO_2(1750) = 280ppm$. For the integration I use this dataset for the emissions data.

The Bern model simply describes the time decay of an annual pulse of CO2 added to the atmosphere. It is parameterised as follows.

$\Delta C = a_0 + \sum_{1}^{3} a_i e^{-\frac{t}{\tau_i}}$

where for AR4:  $a_0 = 0.217, a_1 = 0.259, a_2 = 0.338, a_3 = 0.186, \tau_1 = 172.9 y , \tau_2 = 18.51 y , \tau_3 = 1.186 y$

and for TAR:  $a_0 = 0.152, a_1 = 0.253, a_2 = 0.279, a_3 = 0.316$

Here are  the results.

CO2 levels in the atmosphere following a stabilisation at 2013 emission rates held constant for 500 years.

Levels do not reach an equilibrium value in the Bern model because a fixed fraction (a0) in any year are assumed to remain for ever in the atmosphere. However even in this case levels rise (just) to ~1270 ppm with AR4 parameters and ~950 ppm using TAR values. The airborne fraction shows how it falls asymptotically to the fixed retention rate.

Airborne fraction for the Bern model 500 years into the future for fixed 2013 emissions.

We can also ask the question as to how well the Bern model describes the measured CO2 levels between 1950 to 2016 based only on the emissions data. Here is that comparison.

Detail comparison. The dark green trace is the Moana Loa CO2 measurements .

The bern model agrees with the current value of CO2 (400ppm), but it does not give a good description of the trend. In fact the actual CO2 growth is slower than that of the model. This supports the hypothesis that a0 is actually very small and that CO2 levels will approach stability much faster.

Let’s consider two estimates of climate sensitivity (ECS) a median value of 2.5C (preferred by Gavin Schmidt) and a low value of 1.5C (preferred by say Nic Lewis).

 ECS Model Version Net Warming in 2620 2.5C AR4 5.5C 2.5C TAR 4.2C 1.5C AR4 3.2C 1.5C TAR 2.5C

When you consider that we are assuming annual emissions held constant at ~10 GtC/y for 500y into the future, then these final result are not really that scary at all.

Posted in AGW, Climate Change, climate science, Science | Tagged , | 8 Comments

## A realistic CO2 stabilisation scenario

This is a guest post by “Frank” based on his comment to  Stabilising Climate. He argues that CO2 levels will stabilise at ~520 ppm, which agrees with an estimate given by Ferdinand  Engelbeen here quoted below:

My rough estimate is that with the current emissions twice the current sink rate at ~110 ppmv above steady state, one need ~220 ppmv above steady state to get rid of the full ~4.3 ppmv/year human emissions. That is a level of 510 ppmv, far above the 440 ppmv of Clive… The observed e-fold decay rate of the extra CO2 in the atmosphere is around 51 years, surprisingly linear over the past 57 years. No reduction in sink rate to see

Now here is Frank’s argument

Man is currently emitting enough CO2 to increase atmospheric CO2 by 4 ppm/yr, but we only observe an increase in atmospheric CO2 of 2 ppm/yr. Presumably that means that 400 ppm of CO2 is enough to drive 2 ppm/yr from the atmosphere. My intuition says that if we cut emissions in half, sinks would continue to take up 2 ppm/yr (until sinks begin to “saturate”) and atmospheric CO2 would stabilize at the current level. On the other hand, you say that a 1% increase in CO2 to 404 ppm is enough to double the rate at which CO2 disappears into sinks from 2 to 4 ppm/yr.  Since this is a blog, I simply talk about emissions and uptake in units of ppm/yr rather than Gtons/year – it is much more intuitive and the math is far simpler.

I think maybe this is the correct mathematical formulation :

$\frac{d(CO2_{atm})}{dt} = -k_{un}(CO2_{atm}) + k_{en1}(CO2_{sink1}) + k_{en2}(CO2_{sink2}) + k_{ea}$

$\frac{d(CO2_{atm})}{dt}$  = rate of change in $CO2_{atm}$
$k_{un}$ = rate constant for Uptake of CO2 by all Natural sinks.
$k_{en1}$ = rate constant for Emission of CO2 by Sink #1 by Natural processes.
$CO2_{sink1}$ = concentration of CO2 in Sink #1. Could be DIC in ocean.
$k_{en2}$ = rate constant for Emission of CO2 by Sink #2 by Natural processes.
$CO2_{sink2}$ = conc. of CO2 in Sink #2. Could be organic material in soil in CO2 equiv. Or plant matter growing above ground. Some of this is emitted as CH4 and then oxidized, … Other sinks and rate constants.
$k_{ea}$ = rate of Emission of CO2 by Anthropogenic mechanisms.

During the 10 millennia before the Industrial Revolution, $\frac{d(CO2_{atm})}{dt}$ and $k_{ea}$ were both near zero and a steady state relationship existed.

$k_{un}(CO2_{atm} = 280 ppm) = k_{en1}(CO2_{sink1}) + k_{en2}(CO2_{sink2})$

If we assume that the size of the sinks is much larger than the amount of CO2 they have taken up so far,  and that their rate constants are independent of global warming, then the second and third terms on the right hand side haven’t changed. For the largest sink that is the deep ocean, we know that the MOC takes about a millennia to overturn the ocean. So the deep ocean sink isn’t going to saturate in the near future. The mixed layer of the ocean is rapidly mixed by wind, so the CO2 content of the mixed layer is always in equilibrium with the atmosphere and that equilibrium is only slightly temperature sensitive. It doesn’t saturate either. The land sinks are a little trickier, but let’s make the assumption their rate constants and capacity haven’t yet changed appreciably. So we can substitute:

$\frac{d(CO2_{atm})}{dt} = -k_{un}(CO2_{atm}) + k_{un}(CO2_{atm} = 280 ppm) + k_{ea}$

Today:

$\frac{d(CO2_{atm})}{dt} = 2 ppm/yr$
$CO2_{atm} = 400 ppm$
$k_{ea} = 4 ppm/yr$

Substituting:

$2 ppm/yr = -k_{un}[120 ppm] + 4 ppm/yr$
$k_{un} = 1/60 yr^{-1}$

To double-check for consistency, go back to 1960 when (IIRC)

$\frac{d(CO2_{atm})}{dt} = 1 ppm/yr$
$CO2_{atm} = 330 ppm$
$k_{ea} = 2 ppm/yr$

$1 ppm/yr =-k_{un}[50 ppm] + 2 ppm/yr$
$k_{un} = 1/50 yr^{-1}$

I really should look up the 1960 values and not trust my memory. However, no sign of saturation here either.

So what happens if we continue to emit a constant (not growing) 4 ppm/yr, $CO2_{atm} = 404 ppm$ and nothing else changes:

$\frac{d(CO2_{atm})}{dt} = -\frac{1}{60} \times [124 ppm] + 4 ppm = +1.93 ppm/yr$

Therefore $CO2_{atm}$ needs to rise to 520 ppm (280+240) for $\frac{d(CO2_{atm})}{dt}$ to be zero and atmospheric CO2 to stabilize when we are emitting the equivalent of 4 ppm/yr.

And my intuitive answer that we need to cut back to 2 ppm/yr to stabilize near 400 ppm (for as long as the sinks don’t saturate) agrees with this mathematics.

Are sinks saturating?

Reservoirs of carbon (in GtC) in the ocean (blue labels), in biomass in the sea and on land (tan and green labels), in the atmosphere (light blue label) and in anthropogenic emissions. Fluxes of Carbon between reservoirs are depicted by the arrows, the numbers represent GtC. (From: IPCC)

We’ve emitted 240 ppm of CO2, 120 ppm is in the atmosphere, 60 ppm-equivalents is in the ocean, and 60 ppm-equivalents on land. This Figure is using units of GtC and 400 ppm = 750 GtC. The land biomass reservoir is about 2000 ppm-eq and the deep ocean reservoir is about 20,000 ppm-eq. The increased CO2 stored in these reservoirs since 1750 is trivial compared with their size, so there isn’t an obvious reason why emission from the reservoirs should have increased already or will increase in the future.

In my main equation above, I should include a term for the increase in photosynthesis (primary productivity) with rising CO2. The incorporation of CO2 into organic material is the rate limiting step, so it could also have increased by a factor of 400/280 – at least in areas where water and other nutrients (N, P, K, and micronutrients) are not limiting.

CB comment:  No-one is suggesting that it is a good idea to keep CO2 emissions at current levels for centuries to come, but stabilising emissions now is a more realistic goal than reducing them to zero, and limits atmospheric CO2 levels. This gives more time to develop a far more rational future energy and transport policy.

Posted in AGW, Climate Change, climate science | Tagged , | 27 Comments


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# Solution - Integrated Rate Equations - Half-life of a Reaction

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#### Question

A first order reaction takes 30 minutes for 50% completion. Calculate the time required for 90% completion of this reaction.

(log 2 = 0.3010)

#### Solution

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#### Similar questions

A first order reaction takes 40 minutes for 30% decomposition. Calculate t1/2 for this reaction. (Given log 1.428 = 0.1548)

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A first order reaction takes 23.1 minutes for 50% completion. Calculate the time required for 75% completion of this reaction.

(log 2 = 0.301, log 3 = 0.4771, log 4 = 0.6021)

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The rate constant for a first order reaction is 100 s–1. The time required for completion of 50% of reaction is _______.

(A) 0.0693 milliseconds

(B) 0.693 milliseconds

(C) 6.93 milliseconds

(D) 69.3 milliseconds

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The integrated rate equation for first order reaction is A → products

(a)

(b) k=-1/tl_n[A]_t/[A]_0

(c) k=2.303/t log_10 ([A]_t/[A]_0 )

(d) k=l/tl_n[A]_t/[A]_0

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The half-life for radioactive decay of 14C is 5730 years. An archaeological artifact containing wood had only 80% of the 14C found in a living tree. Estimate the age of the sample.

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# Map processor to circuit diagram

I am a software developer (using high level languages like .NET,C,C++ etc) trying to understand how computers work at a lower level.

I am familar with this diagram:

I am trying to gain a high level insight into how the diagram in the link maps to a circuit diagram like this:

For example, have a look at the assembly language statement:

ADD 1,2

I am trying to understand how the processor produces '3' as the output. I realise that this question may be difficult to answer in simple terms. If it is, then a link would help, perhaps to a book.

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With reference to the first diagram you posted, the details you need of the actual arithmetic are all hidden in the big block labelled "ALU" : every other block on that diagram is just there to get 1,2 and "add" into the ALU and the result out again. Knowing that, you can start to read how the ALU works, if that was what you were asking.. –  Brian Drummond Jan 6 '13 at 11:52
Your first diagram maps to the Z80 CPU in the second diagram. –  starblue Jan 7 '13 at 13:48

## 4 Answers

The problem is that in order to understand how the computer gets to 1 + 2 = 3 you have to understand about 2 levels deeper than you've gone.

Roughly a computer is organized (in terms of fields of study) like this from highest level of abstraction to the most physical reality:

1. Application Software
2. Virtual Machine
3. Operating System
4. BIOS
5. Embedded Systems
6. IP Blocks (Sub-units/Peripherals)
7. Logical Blocks
8. Gate-Level
9. Transistor Level
10. Semicoductors
11. Device Physics

To properly understand why the computer can produce 2+1 = 3, you must first decide what you are willing to accept "on faith" and what you will not believe until you internalize it. That piece of information will be at the level two below the thing you understand. So if you want to understand an adder circuit at the logical level you will need to understand the basics of "digital" transistors (specifically CMOS).

Using your earlier site as an example, consider this resource. It discusses the "Full Adder" -- the minimum completely general purpose circuit capable of addition/subtraction including carry-in and carry-out.

You will also need to understand how numbers are represented in 2's complement (the number system used in modern computers for integer arithmetic).

If you really want a world-class introductory course, I cannot recommend Professor Scott Wills at Georgia Tech highly enough. He passed away last year of cancer, but his course lives on. The Georgia Tech ECE2030 (introduction to computer engineering) class has its text book and exercises all online.

Good luck!

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Thanks for this. +1. I am a .NET developer,although I have experience with C and C++. I understand that C# is compiled into Intermediary Language and the JIT compiler targets the computer architecture by compiling the IL code on the fly. I am trying to understand how a microcontroller can receive an operand and an opcode and produce an output. I am knowledgeable (GCSE level) about electronic components like: Transistors,Capacitors etc. –  w0051977 Jan 6 '13 at 15:26
...then I would start with "Switch Design" in the Readings and work forward in the lessons from there. –  DrFriedParts Jan 6 '13 at 19:28
Your list of topics manages to throw in several which are irrelevant to the question, while skipping over those which are most relevant. –  Chris Stratton Aug 11 '13 at 14:56
@Chris -- No it doesn't. –  DrFriedParts Aug 11 '13 at 17:55

I had to learn the same in university, and we used the book Computer Organization and Design which was very detailed (but maybe it is too detailed for your needs).

Patterson and Hennesy used some "simple" MIPS processor and showed the whole function of the CPU with some sample code in Assembler.

Generally, I think it would be a great idea to take some "simple" processor (Arduino or so) and try to understand there the function, because the differences from that to the more complex are more in how big the implemented instruction codes are.

P.S. Maybe the elementary-microprocessor from Google Code would be helpful. It is a microprocessor simulated in Java.

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The free book How Computers Work - Processor and Main Memory by Roger Young will answer your question, it uses relays instead of transistors for better understanding.

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While that book does look useful please edit the answer to include the full title and author information. That way if the link ever changes it may still be possible to track it down in the future. –  PeterJ Aug 11 '13 at 13:26
This looks like what I was looking for at the time. +1. I will take a look and then come back. –  w0051977 Aug 11 '13 at 16:51

I found my answer here. It is a high level overview of the process of adding numbers.

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It's more helpful for future readers if you summarize the material in the answer. Just giving a link to an external resource doesn't add much to EE.SE. –  The Photon Jan 13 '13 at 18:01
Please note what @ThePhoton is saying here. Although that youtube video may help you, can you summarize what you learned so that if that video link breaks this is still an answer? –  Kortuk Feb 11 '13 at 0:59


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# Electronic – Resistance of mosfet’s internal diode

mosfetpowermosfetswitching

I am an IoT solutions developer and not an engineer, so please excuse my vocabulary if not upto the mark.

I have used Mosfets such as FDD6637 and IRF9310 in my circuits for power switching and they work great for my use cases because of their low Rds(on).

From the datasheet, there is an internal diode and I am showing the characteristics in the attached image.

My questions:

• What would be the internal resistance of this diode? Will it be the same as Rds(on)? I can't seem to find it in the datasheet.
• Do you know of any P-channel mosfet that does not have this diode but has similar Rds(on) (10 to 15 mOhms at -4.5v?). I do not want reverse current to flow in a particular case, as I need my switching to work in only one direction.

Thanks in advance for your help!!

EDIT:
I have shown my problem in the rough diagrams below (sorry if the symbols are not accurate).

Works but battery powers load through internal diode (apparently):

Even though above circuit works, the question is about the internal resistance of diode and the drop across the mosfet when main power is off.

Following does not work because main voltage shows up at battery connector:

In the above circuit, I expect the mosfet not to conduct when there is +9v. But it is conducting because of the internal diode. This circuit would take advantage of the low Rds(on), so it is desired.

When there is no +9v, the load gets powered by the battery alright.

#### Best Answer

I answer to your questions below, starting from the first one on the body drain diode.

• What would be the internal resistance of this diode? Will it be the same as Rds(on)? I can't seem to find it in the datasheet.

The I-V characteristic of a diode is not linear, so it is almost never modeled as a single resistor: this is the reason why, in the datasheet of the MOSFETS you use, such parameter is not even mentioned. The only exception to this situation I'm aware of is the characterization of high power diodes like this: in the datasheet of those power devices, the so called slope resistance $$\\mathrm{r_T}\$$ is specified. In such devices, the magnitude of this model resistance is usually of few milliohms: however, in MOSFETs you cannot expect any precise relation between the (unspecified) $$\\mathrm{r_T}\$$ and the (very well characterized) $$\R_{DS_\mathrm{ON}}\$$ since these two resistances do not model the same physical phenomena even if the semiconductor structure is the very same one.

• Do you know of any P-channel mosfet that does not have this diode but has similar Rds(on) (10 to 15 mOhms at -4.5v?). I do not want reverse current to flow in a particular case, as I need my switching to work in only one direction.

Power MOSFETs where the body drain diode is not present have been produced in the past, but they are not easily available and surely do not have the low $$\R_{DS_\mathrm{ON}}\$$ you need in your application. I suggest another approach which may be a more viable alternative:

simulate this circuit – Schematic created using CircuitLab

The $$\p\$$-channel MOSFETs shown are used in the so called anti series connection: when the common $$\V_{GS}\$$ is $$\>0\$$, the MOSFETs are OFF and the two body drain diodes are connected back to back, so they are not conducting. When $$\V_{GS}\ll V_\mathrm{th}\$$, both the two MOSFETs are ON and the battery feed the load through a $$\2R_{DS_\mathrm{ON}}\$$ resistance: by choosing properly the devices, you solve the problem at the cost of an additional MOSFET.


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# MP Board Class 10th Maths Solutions Chapter 14 Statistics Ex 14.3

In this article, we will share MP Board Class 10th Maths Book Solutions Chapter 14 Statistics Ex 14.3 Pdf, These solutions are solved subject experts from the latest edition books.

## MP Board Class 10th Maths Solutions Chapter 14 Statistics Ex 14.3

Question 1.
The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.

Solution:
Median:
Let us prepare a cumulative frequency table:

Now, we have N = 68 ⇒ $$\frac{N}{2}=\frac{68}{2}$$ = 34
The cumulative frequency just greater than 34 is 42 and it corresponds to the class 125 – 145.
∴ 125 – 145 is the median class.
∴ l = 125, cf = 22, f= 20 and h = 20
Using the formula,
Median = l + $$\left[\frac{\frac{N}{2}-c f}{f}\right]$$ × h
= 125 + $$\left[\frac{34-22}{20}\right]$$ × 20
= 125 + $$\frac{12}{20}$$ × 20 = 125 + 12 = 137 units.
Mean: Let assumed mean, a = 135
∵ Class size, h = 20
∴ ui = $$\frac{x_{i}-a}{h}=\frac{x_{i}-135}{20}$$
Now, we have the following table:

∴ $$\overline{x}$$ = a + h × [$$\frac{1}{N}$$ Σfiui] = 135 + 20 × $$\frac{7}{68}$$
= 135 + 2.05 = 137.05 units.
Mode:
∵ Class 125 – 145 has the highest frequency i.e., 20.
∴ 125 – 145 is the modal class.
We have: h = 20, l = 125 , f1 = 20, f0 = 13, f2 = 14

We observe that the three measures are approximately equal.

Use this median calculator to quickly find the median of a set of numbers. Just enter your numbers in the box and click on the button that says calculate.

Question 2.
If the median of the distribution given below is 28.5, find the values of x and y.

Solution:
Here, we have N = 60
Now, cumulative frequency table is:

Since, median = 28.5 (Given)
∴ Median class is 20 – 30 and l = 20, f = 20, cf = 5 + x, N = 60
∴ l + $$\left[\frac{\frac{N}{2}-c f}{f}\right]$$ × h
⇒ 28.5 = 20 + $$\left[\frac{30-(5+x)}{20}\right]$$ × 10
⇒ 28.5 = 20 + $$\frac{25-x}{2}$$
⇒ 57 = 40 + 25 – x
⇒ x = 40 + 25 – 57 = 8
Also, 45 + x + y = 60
⇒ 45 + 8 + y = 60
⇒ y = 60 – 45 – 8 = 7.
Thus x = 8, y = 7

Question 3.
A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 year.

Solution:
The given table is cumulative frequency distribution. We write the frequency distribution as given below :

∵ The cumulative frequency just greater than 50 is 78.
∴ The median class is 35 – 40.
Now, $$\frac{N}{2}$$ = 50, l = 35, cf = 45, f = 33 and h = 5

Thus, the median age = 35.76 years.

Question 4.
The lengths of 40 leaves of a plant are measured correct to the nearest millimeter, and the data obtained is represented in the following table:

Find the median length of the leaves.
[Hint: The data needs to be converted to continuous classes for finding the median, since the formula assumes continuous classes. The classes then change to 117.5-126.5,
126.5 – 135.5 ………… 171.5 – 180.5.]
Solution:
After changing the given table as continuous classes we prepare the cumulative frequency table as follows:

The cumulative frequency just above 20 is 29 and it corresponds to the class 144.5 – 153.5.
So, 144.5 – 153.5 is the median class.
We have: $$\frac{N}{2}$$ = 20, l = 144.5, f= 12, cf = 17 and h = 9
∴ Median = l + $$\left[\frac{\frac{N}{2}-c f}{f}\right]$$ × h
= 144.5 + $$\left[\frac{20-17}{12}\right]$$ × 9
= 144.5 + $$\frac{3}{12}$$ × 9 = 144.5 + $$\frac{9}{4}$$
= 144.5 + 2.25 = 146.75
Median length of leaves = 146.75 mm.

Question 5.
The following table gives the distribution of the life time of 400 neon lamps:

Find the median life time of a lamp.
Solution:
To compute the median, let us write the cumulative frequency distribution as given below:

Since, the cumulative frequency just greater than 200 is 216.
∴ The median class is 3000-3500 and so l = 3000, cf= 130, f = 86, h = 500
∴ Median = l + $$\left[\frac{\frac{N}{2}-c f}{f}\right]$$ × h
= 3000 + $$\left[\frac{200-130}{86}\right]$$ × 500
= 3000 + $$\frac{70}{86}$$ × 500 = 3000 + $$\frac{35000}{86}$$
= 3000 + 406.98 = 3406.98
Thus, median life time of a lamp = 3406.98 hours.

Question 6.
100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames.
Solution:
Median: The cumulative frequency distribution table is as follows:

Since, the cumulative frequency just greater than 50 is 76.
∴ The class 7-10 is the median class.
We have, $$\frac{N}{2}$$ = 50 , f = 7, cf = 36, f = 40 and h = 3

Mode:
Since the class 7 – 10 has the maximum frequency i.e., 40.
∴ The modal class is 7 – 10.
So, we have l = 7,h = 3, f1 = 40, f0 = 30, f2 = 16

Thus, the required median = 8.05, mean = 8.32 and mode = 7.88.

Question 7.
The distribution below gives the weights of 30 students of a class. Find the median weight of the students.

Solution:
We have cumulative frequency table as follows:

The cumulative frequency just greater than 15 is 19, which corresponds to the class 55 – 60.
So, median class is 55-60 and we have $$\frac{N}{2}$$ = 15,
l = 55, f = 6, cf = 13 and h = 5

Thus, the required median weight of the students = 56.67 kg.


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# Is a flow of ionized water an electric current?

If $H_2O$ ions have a net electric charge and electric current is the flow of electric charge, can a stream of water ions be considered an electric current? If so, is it conceivably possible (not necessarily feasible) to use a flow of water ions in place of a copper wire carrying an electrical current in an electrical circuit? For example, in a bizarre Rube Goldberg contraption sort of situation.

Edit: I guess this question applies to any sort of ion in either gas or liquid state. I don't know enough chemistry to know if $H_2O$ ions are even a thing that exist in the real world

• Thanks for the answers, they were all helpful and I wish I could accept each of them – nonex Oct 17 '14 at 17:05