**4. The concept of lumped-element simulation in digital, event-triggered centrifugal microfluidic networks**

The rapid evolution of microelectronics (following Moore's law) has been leveraged by the trinity of miniaturization, fabrication, and, last but not least, large‐scale system integration (LSI). The breathtaking progress within these tightly intertwined factors has tremendously reduced production costs and seminally enhanced system performance. This is clearly visible when looking back over the last decades when microelectronic devices took the road from very clumsy, maintenance‐intensive, multi‐million dollar machines sparsely scattered around the globe to the sleek, ubiquitous, and quite affordable digital gadgets people even carry in their pockets. The unprecedented commercial success story of microelectronics has been enabled by seminal advances in microfabrication as well as the capability to generate complex func‐ tional architectures from a limited set of base modules such as capacitors and transistors. These simple modules are composed into sophisticated functional networks by lumped‐element model software. We have developed a new type of "digital" LoaD platform which follows a similar design paradigm to implement different types of bioanalytical tests, e.g., for small molecules, proteins, antibodies, DNA, and cells [6].

Over the past decades, simulation has a key role in developing new products. The common simulation methods are FEA (finite element analysis), CFD (computational fluid dynamic), and MBS (multi-body systems). In principle, these mesh-based simulation methods are very accurate. Nevertheless, these numerical tools display serious limitations, for instance, that they tend to be very time-consuming; in particular for more complex networks, also the grid size and proper boundary conditions impact the result (mesh dependency). Therefore, simplified geometries are required for keeping computation times and common convergence issues at bay; lumped-element simulation was proposed to simplify analysis based on electric circuit elements; this method is quite fast and fit for swift parameter optimization; in addition, these methods could simulate serial and parallel multi-element architectures [36].

**Figure 2.** Schematic illustrating the OR conditional release mechanism. The top pane shows the valve actuation trig‐ gered by liquid movement to one chamber and the lower pane shows the valve actuation triggered. (a) Valve closed,

**Figure 3.** Schematic illustrating the OR conditional release mechanism. The top pane shows the valve actuation trig‐ gered by liquid movement to one chamber and the lower pane shows the valve actuation triggered. (a) Valve closed,

(b) CF wetted, (c) LF wetted, (d) Valve opens.

62 Lab-on-a-Chip Fabrication and Application

(b) CF wetted, (c) LF wetted, (d) Valve opens.

The centrifugal flow control elements and their combination of complex microfluidic circuitry translate into equivalent, lumped-element descriptors. Each lumped element exhibits certain free parameters, for instance, corresponding to resistances or capacitances. In microfluidics, these parameters typically relate to geometries, e.g., the channel cross section, as well as hydrodynamic and mechanical properties such as the viscosity and compressibility of the fluids and the flexibility of the ducts. Lumped-element analogies for the different environ‐ ments are listed in **Table 1**.



**Table 1.** Physical lumped-element analogies in different environments.

For a given microfluidic network and spin rate protocol, the lumped-element simulation of microfluidic systems allows to calculate pressure distribution, flow rate, and timing. Parallel simulation and parameter sweep for efficient design generation of microfluidic systems represent further advantages of lumped-element simulation. In addition, its computational simplicity and fast convergence mean it can also be applied to "real-time" active control of microfluidic processes. Utilizing this real-time graphical simulation to monitor filling level and aliquoting timing along the LUOs in multi-step, multi-reagent bioassay protocols will consti‐ tute an important milestone because it would allow the evaluation of the functional operation of the LoaD device without any further fabrication and experimental processes.

This lumped-element simulation in different environments is illustrated by the equivalent electric circuit elements comprising a resistor, a capacitor, and a diode, and the required relations for lumped-element simulation are presented in the following:

#### **4.1. Kirchhoff's current law (KCL) [37]**

The material balance equation, flow-in equal flow-out at any given node in the microfluidic network.

$$\sum\_{i} \text{li} = 0 \xrightarrow{\text{At each nodes}} \sum\_{i} \text{qi} = 0 \to q\_{\text{in}(1)} + q\_{\text{in}(2)} + \dots = q\_{\text{out}(1)} + q\_{\text{out}(2)} + \dots \tag{7}$$

#### **4.2. Kirchhoff's voltage law (KVL) [31]**

The sum of pressure differences around a microfluidic loop must be zero.

$$
\sum\_{i} \mathbf{V}i = \mathbf{0} \xrightarrow{\text{In closed loop}} \sum\_{i} \mathbf{P}i = \mathbf{0} \tag{8}
$$

#### **4.3. Capacitance**

Increasing charge storage results in increasing voltage in an electrical capacitor and increasing fluid leads to increase pressure in the reservoir (fluid capacitator).

Lumped-Element Modeling for Rapid Design and Simulation of Digital Centrifugal Microfluidic Systems http://dx.doi.org/10.5772/62836 65

$$V\_L = C.P\tag{9}$$

The force at the bottom of storage due to the weight is mg = *ρ*VLg which constant earth gravitational replaces by artificial gravity field g=r¯ω<sup>2</sup> in the centrifugal microfluidic system.

$$P = \frac{\rho V\_L \mathbf{g}}{A} = \rho \mathbf{g} \mathbf{g} \mathbf{h} \to P = \frac{\rho V\_L \overline{r} \, \alpha^{\perp}}{A} = \rho \overline{r} \, \alpha^{\parallel} \left(r\_{\parallel} - r\_0\right)$$

$$\boldsymbol{V}\_{L} = \left[\frac{\boldsymbol{A}}{\rho \overline{r} \, \boldsymbol{\alpha}^{2}}\right] \boldsymbol{P} \tag{10}$$

The fluid capacitance in centrifugal system is *<sup>C</sup>* <sup>=</sup> *<sup>A</sup> <sup>ρ</sup>r*¯ *<sup>ω</sup>* <sup>2</sup> .

#### **4.4. Flow resistance**

**Effort (e) Flow (f) Inertance Capacitance Resistance Displacement**

– Heat capacity (mcp)

**Table 1.** Physical lumped-element analogies in different environments.

Mec hanics

Ther mal

network.

**4.3. Capacitance**

Force (F) Velocity (V)

64 Lab-on-a-Chip Fabrication and Application

Heat flow

**4.1. Kirchhoff's current law (KCL) [37]**

*i i At each nodes*

**4.2. Kirchhoff's voltage law (KVL) [31]**

Temp. diff (Δ T) (C) (R)

Mass (m) Spring (K) Damper (b) Displacement

For a given microfluidic network and spin rate protocol, the lumped-element simulation of microfluidic systems allows to calculate pressure distribution, flow rate, and timing. Parallel simulation and parameter sweep for efficient design generation of microfluidic systems represent further advantages of lumped-element simulation. In addition, its computational simplicity and fast convergence mean it can also be applied to "real-time" active control of microfluidic processes. Utilizing this real-time graphical simulation to monitor filling level and aliquoting timing along the LUOs in multi-step, multi-reagent bioassay protocols will consti‐ tute an important milestone because it would allow the evaluation of the functional operation

This lumped-element simulation in different environments is illustrated by the equivalent electric circuit elements comprising a resistor, a capacitor, and a diode, and the required

The material balance equation, flow-in equal flow-out at any given node in the microfluidic

å å *Ii* <sup>=</sup> ¾¾¾¾¾® *qi q q q q* = ® + +¼= + +¼ (7)

å å *Vi* <sup>=</sup> ¾¾¾¾¾ *Pi* =® (8)

(12 1 2 ) ( ) ( ) ( ) 0 0 *in in out out*

0 0 *In closed loop*

Increasing charge storage results in increasing voltage in an electrical capacitor and increasing

of the LoaD device without any further fabrication and experimental processes.

relations for lumped-element simulation are presented in the following:

The sum of pressure differences around a microfluidic loop must be zero.

fluid leads to increase pressure in the reservoir (fluid capacitator).

*i i*

Thermal resistance (R)

**(q)**

(x)

**Node law Mesh law**

Newton's 2nd

Temperature is relative

law

Continuity of space

conservation

Heat (Q) Heat energy

The flow resistance can be considered Ohm's law ∆*V* = *IR*.

$$
\Delta P = qR \tag{11}
$$

Flow resistance of rectangular microchannel can be calculated using the following Fourier series [37].

$$R\_h = \frac{12\eta L}{\left(1 - \frac{h}{\nu} \left(\frac{192}{\pi^5} \sum\_{n=1,2,3}^{\circ} \frac{1}{n^5} \tan h \left(\frac{n\pi w}{2h}\right)\right)\right) \rho o h^3} \tag{12}$$

*P Iq* = &

Also, flow resistance by rectangular cross section for *h*/*w* ≪ 1 can be approximated [31]:

$$R\_h = \frac{12\eta L}{\alpha h^3} \tag{13}$$

where w is width and h height of the channel.

#### **4.5. Inertance**

Newton's second law that is called the linear momentum relation of fluid flow in the channel is [38]:

$$
\sum \vec{F} = \frac{d\left(m\vec{V}\right)}{dt} \tag{14}
$$

$$A\left(P\_1 - P\_2\right) = m\dot{\mathbf{v}} = \rho LA\dot{\mathbf{v}}\tag{15}$$

$$
\Delta P = \frac{\rho L}{A}\dot{q}\tag{16}
$$

This relation is similar to inductor equation ∆*<sup>V</sup>* <sup>=</sup> *<sup>L</sup> di dt* and we could write where *I* = *ρL*/*A* and *p* represents pressure difference.

#### **4.6. Application example**

In this work, we consider a single design which allows us to demonstrate how our lumpedelement approach can be applied to "digital" centrifugal flow control. Therefore, we model a liquid handling protocol similar to that used by Nwankire et al. [35] to implement a nitrite/ nitrate panel for whole blood monitoring. To implement their assay, Nwankire et al. used DF burst valves which were designed to open in sequence with increasing spin rate of the disc. We present a lumped-element model to simulate the centrifugo-pneumatic chambers which are the key enabling technology of the DF burst valves; a good understanding of these chambers is also critical to the implementation of our event-triggered valving architecture [22]. The schematic view of the design is shown in **Figure 4**. This design shows three reservoirs, labeled A through C and three pneumatic chambers which are sealed using DF burst valves. The DFs are arranged to burst at a rotational frequency greater than 20 Hz and less than 40 Hz. These reservoirs feed a mixing chamber which is further sealed by two DFs which dissolve on contact with the liquid. Upon dissolution, an open path into two overflow reservoirs is

provided.

Also, flow resistance by rectangular cross section for *h*/*w* ≪ 1 can be approximated [31]:

w

where w is width and h height of the channel.

66 Lab-on-a-Chip Fabrication and Application

This relation is similar to inductor equation ∆*<sup>V</sup>* <sup>=</sup> *<sup>L</sup> di*

*p* represents pressure difference.

**4.6. Application example**

**4.5. Inertance**

is [38]:

3 12 *h <sup>L</sup> <sup>R</sup> h* h

*P Iq* = &

Newton's second law that is called the linear momentum relation of fluid flow in the channel

*d mV* ( ) *<sup>F</sup> dt* å <sup>=</sup>

*A P P mv LAv* ( 1 2 -== ) & &

*<sup>L</sup> P q <sup>A</sup>* r

In this work, we consider a single design which allows us to demonstrate how our lumpedelement approach can be applied to "digital" centrifugal flow control. Therefore, we model a liquid handling protocol similar to that used by Nwankire et al. [35] to implement a nitrite/ nitrate panel for whole blood monitoring. To implement their assay, Nwankire et al. used DF burst valves which were designed to open in sequence with increasing spin rate of the disc. We present a lumped-element model to simulate the centrifugo-pneumatic chambers which are the key enabling technology of the DF burst valves; a good understanding of these chambers is also critical to the implementation of our event-triggered valving architecture [22]. The schematic view of the design is shown in **Figure 4**. This design shows three reservoirs, labeled A through C and three pneumatic chambers which are sealed using DF burst valves. The DFs are arranged to burst at a rotational frequency greater than 20 Hz and less than 40 Hz. These reservoirs feed a mixing chamber which is further sealed by two DFs which dissolve

r

<sup>=</sup> (13)

<sup>r</sup> <sup>r</sup> (14)

D = & (16)

(15)

*dt* and we could write where *I* = *ρL*/*A* and

**Figure 4.** (a)The design of centrifugal microfluidic platform (b) Schematic view of Lumped element network.

We model the system in four different conditions to demonstrate a parallel simulation defined by different spin rates. These conditions share a spin profile (**Figure 5a**) which involves a rapid acceleration to a maximum frequency, followed by rapid mixing, stopping the disc, and then fast acceleration back to the maximum frequency. These spin protocols are identical except for their magnitude; they have maximum spin rates of 20 Hz, 40 Hz, 60 Hz, and 80 Hz.

**Figure 5.** Lumped-element simulation graph for main atmospheric reservoir. (a) Angular frequency profile vs the time. (b) Total inflow vs time. (c) Total net flow (flow-in-flow-out) vs time. (d) Filling level of the chamber due to the ingress of liquid into the reservoir. (e) Pressure generated in the pneumatic and centrifugal valve (A) due to angular velocity.

To demonstrate the wide capability of this lumped-element model to predict on disc perform‐ ance, a number of parameters are shown in **Figure 5** which have been calculated using the simulation software based on a number of defined boundaries and initial conditions. These parameters are the volume flow into the mixing chamber assuming no out volume flow through the two exits (**Figure 5b**); the net flow into the mixing chamber, assuming outflow through the exists (**Figure 5c**); and the liquid level in the mixing chamber, assuming outflow through the exits (**Figure 5d**). Finally, Figure 5e shows the predicted pressure, during the spin profile, in each DF burst valve with the assumption the DF does not dissolve.

Based on the lumped-element analysis, the critical burst frequencies of the DFs are between 25 and 30 Hz. Therefore, in **Figure 5b, c**, and d, it is predicted that, for the 20 Hz test condition, the DFs do not dissolve and so there is no liquid flow. As stated above, Figure 5b shows the total volume entering into the main chamber; this is defined as *Vin* = *VA* + *VB* + *VC*. Similarly, the net flow rate in and out of the mixing chamber is shown in **Figure 5c** and is defined by *qnet* = *qA* + *qB* + *qC* − *qd* − *qf* . In turn, and most importantly, the liquid level in the main chamber can also be predicted in Figure 5d; this information is important as it can be used to inform incubation times and washing protocols which are critical for Lab-on-a-Disc applications.

Finally, in **Figure 5e**, the pressurization of the centrifugo-pneumatic valves (Valve A) is presented. Here, the increased centrifugal force pushes the liquids from the main reservoir (Reservoir A) and into the dead-end pneumatic chamber which is sealed by a DF. The fluid flow is stopped in the pneumatic chamber by a pocket of entrapped air which pushes back against the centrifugally generated hydrostatic pressure; this equilibrium condition is reached when the centrifugal pressure head, described previously in Eq. (4), balances with the pressure of the trapped gas, defined by Boyle's law in Eq. (6). In the real case, the DF membrane in the pneumatic chamber is dissolved (valve opening) beyond the critical burst frequency when the liquid ingress is sufficient to contact the film. Then, the liquid flows are directed into the main (downstream) chamber.

Over the past three decades, a special breed of microfluidic systems is based on centrifugal liquid handling for a wide spectrum of applications in biomedical point‐of‐care diagnostics and the life sciences. Recently, event-triggered flow control was introduced on these LoaD platforms to implement logical flow control which functions akin to digital microelectronics [33]. Similar to the difference between an old‐fashioned office mainframe and a modern smartphone, these breakthroughs may provide an unprecedented level of system integration and automation which is needed to eventually implement complex, highly functional net‐ works representing a repertoire of bioanalytical assays on a user‐friendly, cost‐efficient, portable, and still high‐performance microfluidic point‐of‐use "gadget." We presented an advanced lumped-element approach for the fast-generation and robust simulation for eventtriggered centrifugal microfluidic networks.
