Team:CSU CHINA/Model



Model Structure

We developed our model based on processes of cells and microbeads. You can visit these models from the workflow of our system in the device.

Device introduction

We designed a fly-catcher-like device, a blue light source is placed at the bottom of the center to deactivate safety system in cells. Algae in microbead create oxygen continuously and gas fuse into a bubble or two inside that provide buoyancy, the microbeads could float as the bubbles grow.

Meanwhile, the absorption process is still in progress. One of the main goals of our model is to optimize parameters to match the saturation time and float up time to maximize its efficiency. Once the absorption process is done, we expect the microbeads float on the surface. Once a microbead floats on the surface, it can just float for a while and rupture, then fall into groove on the top of the device waiting for salvage. Hence, the process of absorption and salvage is separated on time and space scales.

Click the following buttons to visit our models:
Absorption Model
Safety Model

Absorption Model

Main Goals of this model

In order to extend the model from lab environment to more complex applicational scenarios, following aims are highlighted:

  • 1. For the designers: Through simulating the absorption of \({Cd}^{2+}\), the viability of the absorption device should be tested and verified; unexpected dangerous situation should be identified and avoided through optimization of parameters;
  • 2. For users: The variation curve of environmental \({Cd}^{2+}\)concentration should be simulated, providing reasonable indicators for users to salvage the device when expected reduction of \({Cd}^{2+}\) is completed

Model 1:The\({Cd}^{2+}\) balance model of the microbead boundary

Since the environmental concentration of \( {Cd}^{2+} \) remain relatively constant, we regard the amount of \( {Cd}^{2+} \) absorbed by the cells as a measure of the optimization of our model, which obey the following equation:

\( ∆{{Cd}^{2+}}_{environment}=∆ {{Cd}^{2+}}_{cytoplasm} \)

Considering the microbead as a homogeneous medium, we mainly only focus on \( {Cd}^{2+} \) motion between the environment & cells and inner & outer microbead. Here, c represents the \( \left[{Cd}^{2+}\right] \) on the inner surface of the microbead, while C represents the \( \left[{Cd}^{2+}\right] \) on the outer surface:

\( c_{{Cd}^{2+}}=C_{{Cd}^{2+}} \)

Algae’s absorption of \( {Cd}^{2+} \) results from not only the driving force of the concentration difference, but also the transport and chelating proteins providing additional power. Therefore, the absorption rate of \( {Cd}^{2+} \) into cells is much faster than the transport rate of \( {Cd}^{2+} \) into the microbeads, maximizing the possibility of a radial transportation of \( {Cd}^{2+} \).

According to previous wet exp erience, Calcium alginate-\({Cd}^{2+}\) displacement reaction and particle diffusion[2] are the most dominant factors in the diffusion process. The equation of the displacement reaction is described below:

M is the concentration of \( {Cd}^{2+} \) bounded, and \( C_p \) is the concentration of diffused \( {Cd}^{2+} \). Through infinite series approximation, we attained the following expression:

\( \frac{\partial C_p}{\partial t}=D\left(\frac{\partial^2C_p}{\partial r^2}+\frac{2}{r}\times\frac{\partial C_p}{\partial r}\right)-\frac{\partial M}{\partial t} \)

M is the concentration of \( {Cd}^{2+} \) bounded, and \( C_p \) is the concentration of diffused \( {Cd}^{2+} \). The above equation can be solved by infinite series approximation, and the results are as follows:

\( C(t_i,D)=\frac{\sigma C_0}{1+\sigma} \left[1+6(1+\sigma) ∑_{k=1}^∞ \frac{exp(\frac{-D}{1+N}×\frac{{S_n}^2}{R^2}×t_i )}{9+9σ+σ^2 {S_n}^2}\right] \)

Based on data from referenced paper[8], we calculated and got the diffusion coefficient \(D= 2.9×10^{-11} m^2/s\)。


Restricted by laboratory data, we resorted to Finite Element Analysis to simulate the \( {Cd}^{2+} \)absorption and diffusion process, which was further utilized to optimize radius of microbead and shorten purification time.

We regarded the microbead as an ideal sphere. We segregate the whole sphere into numerous but finite segments of shells representing the infinite layers in the microbead. Thus, the interaction between the segments could reasonably simulate the activities in the microbeads. The diffusion and absorption rates of \({Cd}^{2+}\) for different radius could be further determined.

Get the numerical solution with partial differential equations in geometric model by finite element analysis.

Fig1. Permeation process of \({Cd}^{2+}\) into microbeads

As the diffusion model is established, we define a 2D intercept along the radius to get \({Cd}^{2+}\) concentration distribution.

Fig2A. scanned section

Fig2B. scanned sectional curves, \(\left[{Cd}^{2+}\right]\) changes of different distances at distinct times

Model 2: \({Cd}^{2+}\) membrane transport and algae population model

The algae cells are randomly distributed in the microbeads due to its manufacturing technology. Based on the idea of finite element analysis, the microbead is divided into K spherical shells. It is assumed that the intrinsic concentration of algae cells when it is being made is \(N_0\). For the \(i_{th}\) shell, the number of cells in the shell is as follows:


We test the population change and the amount of \({Cd}^{2+}\) absorption from 5 groups of cells with different initial population\((0.3,\ 0.6,\ 0.8,\ 1.1,\ 1.4\times{10}^7\ cells/mL)\) in same \({Cd}^{2+}\) concentration\(\left(0.1mg/L\right)\) during a period of time. We fitted the following two equations respectively with these data:

Fig3A. \({Cd}^{2+}\) changes inside and outside the cytoplasm from wet exp data. We couldn’t find an equation with biology meaning so we just chose a fitted equation to fit the data.

Fig3B. Algae population change under 0.1mg/L \({Cd}^{2+}\) for 48 hours

Fig4. Due to COVID-19, there isn’t much time for better exp data. So can just fit the curve of population change rate differential equation based on previous scattered data, we are about to reduce data dimension and add gaussian noises to optimize equation in the future.

With the 2 equations above, we can now code the absorption and population dynamics using C# (Get the source code by contacting us!)


  1. the environment for algae in a certain layer is the same, if the random noise is ignored, there shall be no difference among the algae in the population in the certain shell. Therefore, cells in a certain shell are the same;
  2. apoptosis and replication of algae do not affect its position in microbeads;
  3. \({Cd}^{2+}\) transported into the algae will not diffuse out of the cells unless cells collapse;


Based on the above assumptions, we transform continuous time into discrete time units. The program executes the following dual loops based on time units:

  1. Calculate the difference of \({Cd}^{2+}\) concentration between the \(K_{th}\) (outermost) shell and the water environment. The algorithm will obtain the \({Cd}^{2+}\) flux through the surface of microbeads as well as changes of \({Cd}^{2+}\)concentration in the \(K_{th}\) shell and the environment;
  2. the program iterates over the K shells K-1 times. For each iteration, following steps are executed in sequence:
    • 2.1 Calculate \({Cd}^{2+}\) concentration difference and contact area between \({(i-1)}_{th}\) and \(i_{th}\) shell to deduce the amount of \({Cd}^{2+}\) diffused according to Fick's law in model 1. The concentration of \({Cd}^{2+}\) in these 2 shells after diffusion are then updated;
    • 2.2 the \({Cd}^{2+}\) concentration in \(i_{th}\) shell and its algae respectively and the algae population in the shell are taken into the dynamic equation of \({Cd}^{2+}\) absorption developed above to calculate the amount of \({Cd}^{2+}\) diffused into algae in \(i_{th}\) shell;
    • 2.3 The change of \({Cd}^{2+}\) concentration in \(i_{th}\) shell’s algae results in the change of probability of algae apoptosis and replication, which leads to the change of algae population in the shell;
    • 2.4 the apoptotic algae cells collapse and a part of the \({Cd}^{2+}\) absorbed will be released back into the \(i_{th}\) shell. Due to the lack of wet exp data, we temporarily assume that the proportion of released \({Cd}^{2+}\) is 20%;


Fig5. From the graph, we found that it is the algae in shells next to the outermost shell rather than the outermost one that absorbed the most \({Cd}^{2+}\). We describe and explain this phenomenon with the saying “Too much money chasing too few goods”.

Fig6. After being in high \({Cd}^{2+}\) concentration environment for 80 minutes, algae in outer layers experience massive apoptosis, while cells in inner layers grow slowly.

Model 3:\({Cd}^{2+}\) absorbing optimization model

Parameters Table

\(T_i\) Death cycle of the number \(i\) group of algal cells
\(M_i\) Total removal of \({Cd}^{2+}\) after the last most recent death cycle
\(K_i\) Proportion of proliferating cells to the total number of cells during proliferation during the first week
\(M_i\) Total removal of \({Cd}^{2+}\) after the last most recent death cycle
\(S\) The amount of \({Cd}^{2+}\) released at the moment of death
\(A\) Diffusion rate ratio
\(∆c_i\) \({Cd}^{2+}\) concentration change
\(f(∆c_i )\) The rate equation of the first group of algae cells to remove \({Cd}^{2+}\)

It is considered that the toxicity of \({Cd}^{2+}\) or the possible inhibitory effect of it absorbed by cells have a certain negative impact on the metabolism and proliferation of algae. We considered that such factors will affect the survival time and proliferation cycle of algae.

Firstly, we establish the following model based on experimental data to analyze the survival period under the experimental conditions, \(T_i\). The amount of \({Cd}^{2+}\) absorbed un the life cycle of a single algae population is calculated according to the formula shown below:


\(f(∆c_i)\) is the amount of \({Cd}^{2+}\) transported per unit time related to \({Cd}^{2+}\) concentration calculated in model 2. Simple spatial analysis could yield that cells on the outermost microbead shell would absorbs \({Cd}^{2+}\) the most. Therefore, the outermost group will be the first group initiating apoptosis due to the accumulation of \({Cd}^{2+}\).

When an alga cell collapse, most of the \({Cd}^{2+}\) absorbed in the cells will scatter into the microbead, which we name as “\({Cd}^{2+}\) explosion”. The process will cause a considerable fluctuation in the \({Cd}^{2+}\) concentration inside the microbead.

For the collapsed algae cells, we hypothesize that:

  1. Cells are anchored on different spherical shell divided from microbeads
  2. Cells collapsed in a relatively small-time step could be considered collapsed at the same instance
  3. Considering a shell’s concentration difference with two adjacent shells, their ratio of the diffusion velocities related to the concentration c is expressed as:


A reflects the randomness of the ions’ diffusion to both sides. The experimental algorithms had verified that, compared to the diffusion to inner shells of the microbead, \({Cd}^{2+}\) spread through explosion has a higher probability to diffuse inwards.

With the experimental hypothesis and realistic data, we begin analyzing how the \({Cd}^{2+}\) diffusion reaches a dynamic equilibrium. We consider that, during the process, the concentration gradient on the centripetal direction is always largest:


We conclude that the concentration gradient of \({Cd}^{2+}\) in the diffusion process is mainly directed in the centripetal direction.

Hence, it’s reasonable to assume that analysis of the concentration gradient on the centripetal direction is sufficient to accurately study the diffusion of \({Cd}^{2+}\). The simplification of the bead into finite shells could effectively simulate the diffusion process as the minor non-radial transportation is neglected.

Among the various possible situations, that the collapse of cells in certain shell results in significant rebound of \({Cd}^{2+}\) level and alternation of gradient demands great attention and close-up study.

Next, we will describe the developed optimization iterative model of algae’s apoptosis:

The model consists of two critical parts:

Optimization iterative model on apoptosis

Due to the apoptosis of algae cells, the concentration of \({Cd}^{2+}\) in the algae will increase instantaneously. We assume that the increase in the total amount of \({Cd}^{2+}\) during this process is \(∆n_i\), and the algae cell nearest to the collapsed group will collapse after \(T_i\) time:

\({∆Cd}_i=\sum_{m=i}^{n}{∫_{0}^{T_i}f(∆c_i)dt}-∫_{0}^{T_i} f(∆c_i )dt,i=1,2,…… \)

  • A. If \(∆Cd_i > 0\) , the iteration continues;
  • B. If \(∆Cd_i = 0\), the iteration ends and the optimal solution in the apoptosis state is obtained;
  • C. If \(∆Cd_i < 0\), go back to the previous iteration and get the optimal solution in the dead state;

The apoptosis coincides with the algae’s proliferation process. Assuming that the cytoplasm of algae cells divides evenly during the proliferation process, the amount of \({Cd}^{2+}\) in the 2 new individual cells is half of the original individuals’ amount. We expected that cell proliferation would positively affect the absorption of \({Cd}^{2+}\). We assumed that the volume of the two individuals after proliferation would instantaneously reach the mother cell’s level. We expected that cell proliferation would positively affect the absorption of \({Cd}^{2+}\) as new cells could continue to absorb \({Cd}^{2+}\).

Irregular division and proliferation model

In our study, the cell division cycle of an alga is \(T_i\), and the proportion of algae capable of proliferating is \(K_i\). Under the assumption that proliferated cells wouldn’t collapse in the same time step, the proliferation rate would increase as multiple division cycles and death cycles passed and is eventually equal the collapse rate. The population would then reach a stable point and a maximum ability to absorb \({Cd}^{2+}\).

Considering the division in the algae population, we can modify the variance of \({Cd}^{2+}\) concentration to the following equation:

\(∆Cd_i=(1-K_i) \sum_{m=i}^{n}∫_{0}^{T_i}f(∆c_i )dt+2K_i \sum_{m=i}^{n} ∫_{0}^{T_i}f(∆c_i ) dt-∫_{0}^{T_i}f(∆c_i )dt,i=1,2,……\)

After one division cycle, the total amount of \({Cd}^{2+}\) removed by algae is calculated as follows:


For multiple division cycles, we calculate the total \({Cd}^{2+}\) removal quantity before the \(i_{th}\) cycle as:

\(∑_{m=i}^{n} ∫_{0}^{T_i}f(∆c_i)dt - ∫_{0}^{T_i}f(∆c_i ) dt=M_i\)

We further attain the total \({Cd}^{2+}\) removal quantity in the whole period with the following equation:

\(∆Cd_{i~i+1}=(1-K_i) M_i+2K_i M_i\)

These formulas allow us to obtain the total removal quantity exactly after each timestep.

However, only analyzing the algae proliferation’s contribution to absorbing \({Cd}^{2+}\) in the end of individual cycles, the model failed to identify the instance of maximum removal inside of a cycle. Therefore, the following additions need to be made to the model:

Using the iterative model of algae death, if the condition \(B\) or \(C\) is met before the end of the timestep, we calculate the optimal solution of the apoptosis model under the corresponding conditions:

  1. If the optimal solution of the death model is greater than the total removal calculation formula at each cycle node, the time point of the death cycle is taken as the optimal solution and the optimal purification time is obtained;
  2. If the optimal solution of the death model is less than the calculation formula of the total removal amount of a certain cycle node, then the time corresponding to the maximum removal amount in each proliferation cycle is taken as the optimal purification time;


With the absorption model above, we could now thoroughly simulate the scenario after a group of algae cells collapsed and \({Cd}^{2+}\) is released. In the following graph, we depict the variation of \({Cd}^{2+}\) concentration in the microbead in a certain time period:

Fig7. Simulation on \({Cd}^{2+}\) diffusion after \({Cd}^{2+}\) released by a group of collapsed algae cells (the red point in the top of the microbead).

When the \({Cd}^{2+}\) concentration in the innermost shell reaches the maximum value suitable for \({Cd}^{2+}\) absorption, the device should be salvaged in time to avoid rebound of \({Cd}^{2+}\) concentration due to cell apoptosis.

Aiming to optimize the radius to maximize the absorbed \({Cd}^{2+}\), we assume that an optimal radius value \(r^\ast\) exists in the relationship between the radius value and the \({Cd}^{2+}\) absorption capacity of the device.

  1. \(r_i > r^\ast\): According to previous studies, the probability of algae escaping from microbeads decreases, and internal algae cells could constantly maintain high activity. However, mass transfer in the microbead will decrease correspondingly, which will reduce the \({Cd}^{2+}\) absorption efficiency;
  2. \(r_i < r^\ast\): With the increase of the surface-area-to-volume ratio of microbeads, algae could more easily escape from the microbeads, whereas the absorption rate increases as the ratio increases. However, higher concentration of cadmium may reach internal algae before their growth peak, restricting the population at a relatively low level that is unbeneficial for the overall \({Cd}^{2+}\) absorption;

Based on wet data on \({Cd}^{2+}\) removal by microbeads, PCC6803, microbeads+PCC6803 on 0.1mg/L and 2mg/L \({Cd}^{2+}\) , we rebuild the process in our model.

Fig8A. We used microbeads, algae, microbeads + algae to deal with 0.1mg/L and 2mg/L Cd.

To determine the most suitable radius, we applied a parameter scan on various possible radius and obtained the following graph:

Fig8B. simulate the change of concentration of total \({Cd}^{2+}\) ions of in different microbeads with different radius.

Each microbeads correspond to an absorption peak and optimized time length. As time goes on, more algae collapse and these ions are released back to the beads, and beads release back to environment. What’s more, smaller beads for algae inside means they are earlier exposed to \({Cd}^{2+}\) peak, thus they collapsed earlier, and an earlier absorption peak can be featured in Fig8B.

Model4: Predicting rupture of microbead model

Background In wet experiments, we found that a part of the oxygen from algae’s photosynthesis would remain in the microbeads. Gradually these gas fuse into one huge bubble somewhere inside microbeads, which stress the spherical shell to significantly expand.

Fig9. Process of rupture of microbeads in our device.

Assumptions: It is assumed that there is a stress limit for Calcium Alginate microbeads. If the stress limit is exceeded, the shell may burst into small pieces difficult for salvaging.

Goals: Based on the optimal radius obtained by Model 3, an upper limit range of salvaging time for users and algae concentration for microbead constructors will be calculated to guarantee that the expected rupture time of microbeads will be after the equilibrium time point of \({Cd}^{2+}\) absorption & release. In this way, the microbead rupture will not affect the absorption process in our system. After calculating the predicted rupture time, we adjusted cell concentration to achieve the goal.

Fig10. Real appearances of microbeads after lighted for \(60h,30\ \mu mol/m^2/s\)


According to the conditions in the experiment, the force field in the microbead at a certain moment is drawn as follows:

Fig11. Diagram of Force field in microbeads caused by compressed bubbles

With the continuous photosynthesis and respiration of algae cells, oxygen is produced in microbeads, which continuously fuses and accumulates until microbeads rupture. This process can be roughly divided into 2 cases as follows:

  1. Oxygen accumulates at the center of the microbead. In this case, the stress distribution of the microbead is consistent in all directions.
  2. Oxygen accumulates at a certain place in the microbead. The material stress decreases with the distance from bubble.

Parameters processing

From referenced papers[5],[8], fracture stress of calcium alginate microbead is \([\sigma]=19.2kPa\) and the microbeads’ Young’s modulus is \(E=11.9\times{10}^4kPa\). When the stress of a point in the microbead reaches the fracture stress, the microbead experiences brittle failure at the point.

We measured a group of radiuses of bubbles in critical microbeads. According to our wet experimental results in fig11, the bubble volume is about \(\frac{1}{5}\) of the microbead volume, \(V\approx0.086\ {cm}^3\).

Analyzing data from referenced paper[6], we found that the photosynthesis rate reaches the maximum when light intensity is greater than \(300\ \mu mol/\left(m^2\bullet s\right)\). We attained the following semi-rational relationship between net photosynthetic rate and light intensity by fitting the data:

\(f(ls)= \begin{cases} 0.3267e^{-{(\frac{ls-258.6}{143.4})}^2 },& 0 ≤ x < 300 \\ 0.303,& x ≥ 300 \end{cases}\)

When ls equals the light intensity of incubator, the net photosynthetic rate is \(2.57\times{10}^{-2}\mu molO_2/(mg\ast h)\). According to data from referenced paper[7], we assumed that there is a linear relation between net photosynthetic rate and algae concentration under certain conditions. Taking the suburb of the city where the CSU-China team is located as an example, the annual average light intensity is about \(555.035\ \mu mol/(m^2\ast s)\). Based on algae density, the net photosynthetic rate of each microbead is \(6.9\times{10}^{-2}\ \mu molO_2/h\).

Parameters table

Parameter Description Value Unit Source
\([\sigma]\) fracture stress \(19.2\) \(kPa\) [5]
\(E\) Young’s modulus \(119\) \(kPa\) [8]
\(V\) average volume of bubbles \(0.086\) \({cm}^3\) measured
\(T\) constant temperature of incubator \(303.15\) \(K\) measured
\(ls\) light intensity \(30\) \(μmol/(m^2∙s)\) measured
\(/\) light frequency \(24\) \(h/d\) measured
\(/\) average density of Synechocystis \(2.39×10^7\) \(cells/mg\) measured
\(/\) algae concentration in raw medium \(1.0×10^8\) \(cells/{cm}^3\) measured
\(/\) local average light intensity \(555.035\) \(μmol/(m^2*s)\) Changsha Meteorological Bureau
\(/\) net photosynthetic rate of each microbead \(6.9×10^{-2}\) \(μmolO_2/h\) measured
\(/\) microbead density \(1036.9419\) \(mg/{cm}^3\) measured
\(/\) microbead with cells added density \(1159.070613\) \(mg/{cm}^3\) measured
\(\mu\) Poisson's ratio 0.3 1 estimated

Theoretical framework

For cases 1 and 2, the finite element analysis can be used to calculate the maximum stress of the material. We approximately regard the material as a linear elastic material. According to the force balance equations, geometric relation and material constitutive relation, we developed the following basic equations:

\(\frac{\partial \sigma_{xx}}{\partial x}+\frac{\partial \sigma_{xy}}{\partial y}+f_x=0\)

\(\frac{\partial \sigma_{xy}}{\partial x}+\frac{\partial \sigma_{yy}}{\partial y}+f_y=0\)

\(\epsilon_{xx}=\frac{\partial u}{\partial x}\)

\(\epsilon_{yy}=\frac{\partial v}{\partial y}\)

\(\epsilon_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\)

\( \left( \begin{array}{c} \sigma_{xx} \\\\ \sigma_{yy} \\\\ \sigma_{xy} \end{array} \right) = \frac{E}{(1+μ)(1-2μ)} \left( \begin{array}{ccc} 1 - \mu & \mu & 0 \\\\ \mu & 1 - \mu & 0 \\\\ 0 & 0 & 1 - \mu \end{array} \right) \left( \begin{array}{c} \epsilon_{xx} \\\\ \epsilon_{yy} \\\\ \epsilon_{xy} \end{array} \right) \)

\(\sigma\) is the stress of the material, \(\varepsilon\) is the strain of the material, \(E\) is the elastic modulus, \(\mu\) is Poisson's ratio, \(\mu\) and \(v\) are the displacements in \(x\) and \(y\) directions respectively, and \(f_x\),\(f_y\) are the components of the applied force in \(x\) and \(y\) directions respectively.

Model Simulation

Neglecting displacement of the microbead caused by the internal pressure, based on the first boundary conditions, the system can be described as:

\(\begin{cases} \mu(r,t) = 0 \\ v(r,t) = 0 \end{cases} \)

On the premise that the change of external pressure and temperature are negligible, it is considered that the Ideal Gas Law is tenable. Based on \(pV=\frac{M}{\mu}RT\) and net photosynthetic rate, \(pV\) is calculated to be \(5.07\times{10}^{-11}Pa/m^3\). Bringing in material property parameters, the process can be simulated by finite element analysis as follows:

Fig12. 2 points of maximal and minimal pressure in microbeads as the bubble expands. Pressure are the same inside the bubble and its surface. The microbead rupture at the point that first reaches fracture pressure.

We further adjusted raw cell concentration in cell culture medium to manipulate its predicable rupture time. Some rupture time with corresponding cell concentration is listed in the chart below:

raw cell concentration \({10}^8 cell/{cm}^3\) 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
rupture time \(/h\) 90 63 49 38 31 26 23 22

Future improvement

  1. Current model couldn’t predict the time when the microbeads float up. As we neglected the displacement of microbeads in the simulation, the external volume doesn’t change as bubbles grow. A simulation on the expansion of microbeads require further study and development upon the current theoretical framework.
  2. If possible, future CSU China team could continue to develop the project in 2021. We plan to install microelectronics, including LTE and GPS modules and photoresistors, in the device, so that the device can sense the current light intensity and access future light intensity information from the local meteorological department by LTE. With the data sent to an app installed on the user's phone, algorithm based on our model inside the app will provide the user with the best salvage time.


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