Patent Application: US-201514693022-A

Abstract:
a method and an apparatus are disclosed for measuring the electrical properties of biological cells . the method involves a switching excitation with a sinusoidally amplitude - modulated current coupled with a real - time estimation algorithm for extracting a phase - shifted sinusoidal voltage output . the algorithm uses a unique time - domain formulation that provides accurate and continuous measurements with a high temporal resolution . the invention is suitable for measuring small signals under noisy conditions such as the membrane resistance and capacitance of a living cell accessed via a microelectrode . the resulting apparatus achieves a similar effect of a lock - in amplifier for suppressing noise but with a different approach . the invention also has the advantage that the input and the output can be decoupled by time - multiplexing on a single electrode .

Description:
this invention provides a method and an apparatus for monitoring electrical properties of a measurand under noisy conditions with high accuracy and high temporal resolution . it overcomes the drawbacks of the existing techniques in the following three aspects . first , the excitation ( current injection ) and the induced response ( voltage measurement ) are time - multiplexed via a single port , making it suitable for single - electrode experimental settings such as the patch clamp . second , the switching current injection is amplitude - modulated with a sinusoidal waveform , which encompasses the essence of a lock - in amplifier to suppress noise . when coupled with a suitable estimation algorithm , the system significantly improves the signal - to - noise ratio by accepting signals of the modulation frequency and rejecting noise of all other frequencies . third , a novel estimation algorithm is formulated in the time domain instead of the frequency - domain . the algorithm takes advantage of the fact that the induced response should have a phase - shifted sinusoidal waveform of the modulation frequency and the derivatives of the induced response have closed - form solutions . an accurate estimate of the electrical properties of the measurand is obtained with a non - iterative linear estimation method with data collected from one cycle of the sinusoidal excitation , thereby achieving both high accuracy and high temporal resolution . whereas it was initially designed to measure cell capacitances , the method has a broader range of applications for measuring electrical properties in general . the apparatus in this invention can achieve a similar effect of a lock - in amplifier for applications that require a single - port access . the method can also be extended to a two - port system and incorporated into the design of a lock - in amplifier . the estimation method is useful for applications that the observed signals are known to be sinusoidal or any analytical function with closed - form solutions of its derivatives . in accordance with an embodiment , the present invention provides a system for measuring the electrical properties of a measurand by use of a sinusoidally amplitude - modulated switching excitation and a time - domain formulation of a linear least - squares estimator . the method encompasses the essence of a lock - in amplifier to suppress noise . the formulation of the estimator takes advantage of the fact that closed - form solutions of the time derivatives exist for an induced response with a sinusoidal waveform . fig1 shows the block diagram of the system . a processor 10 generates a sinusoidally amplitude - modulated switching excitation via a digital - to - analog converter ( d / a ) 11 and an amplifier 12 . via a multiplexer switch 13 the injected current ( i m ) 14 is delivered a single - port measurand 15 through a channel 16 . the induced voltage ( v m ) 17 is time - multiplexed with i m via the multiplexer switch 13 . after an amplifier stage 18 , v m is acquired by the processor 10 via an analog - to - digital converter ( a / d ) 19 . the measurand 15 is modeled by an electrical circuit that specifies the circuit topology and the electrical properties to be measured . the example shown in fig1 is a 3 - element model of the cell membrane that consists of an access resistor r a 20 , a membrane resistor r m 21 , and a membrane capacitor c m 22 . the voltage across the capacitor v 23 is not directly measurable , which needs to be estimated . fig2 shows the waveform of a switching excitation with a sinusoidal amplitude modulation 24 . the current injection and the voltage measurement are time - multiplexed . each current injection pulse is delivered during a short time interval 25 . during the voltage measurement interval 26 , the current is switched off and outputs a high - impedance state to avoid interference with the voltage measurement . fig3 a shows the waveform of a switching injected current 24 that has an sinusoidal envelope given by : where i m is the magnitude of the current sine wave and the angular frequency ω = 2πf . the induced voltage 30 may be contaminated with noise , but its envelope should also be a sinusoidal function with a phase shift φ 31 . where v m is the magnitude of the voltage sine wave . the switching excitation has the advantage of decoupling the current injection and voltage measurement is a time - multiplexed fashion . this allows for the use of a large - magnitude current injection without being restricted by the voltage measurement side . in other words , without the time multiplexing , a strong current injection could damage the hardware for voltage measurement . for applications involving microelectrodes a strong current injection is often required to overcome the large electrode resistance and to improve the signal - to - noise ratio . nevertheless , the methodology of the present invention is also applicable to a non - switching excitation as shown in fig3 b . as long as the magnitude of the current injection 32 is sufficiently low , it is possible to measure the induced voltage 33 and determine a phase shift φ 34 . the proposed relationship between the sinusoidally amplitude - modulated switching excitation and the induced voltage was verified with both computer simulation and hardware experimentation . fig4 shows the computer simulated waveforms for i m 40 and v m 41 as well as the hardware generated i m 42 and v m 43 . the circuit equations that govern the 3 - element model 15 in fig1 are obtained by applying the kirchhoff &# 39 ; s current law ( continuity of current ) as follows : the measurement process begins with a calibration step to determine the electrode resistance r a . this is accomplished with the electrode in the bath solution before in contact with the cell . from equation ( 4 ), we have by taking the derivatives on both sides of equation ( 5 ), we have by substituting equations ( 5 ) and ( 6 ) into equation ( 4 ), we have one of the key concepts in this invention is that a closed - form solution of the derivatives in equation ( 9 ) can be obtained from the sinusoidal inputs and outputs . in essence , the sinusoidal amplitude - modulation not only provides a noise rejection scheme , but also resolves a technical issue that makes the time - domain formulation possible . by taking the time derivatives of equations ( 2 ) and ( 3 ), we have by substituting equations ( 10 ) and ( 11 ) into equation ( 9 ), we have x = ωv m cos ( ω t + φ )− ω r a i m cos ω t ( 12 ) there are 2 unknowns ( r m and c m ), which require at least 2 independent measurements to resolve . to improve the accuracy , a least - squares estimator is derived by using n sample points . an appropriate choice of n is the number of sample points for one cycle of the sine wave . the inclusion of all samples from a full cycle of excitation ensures accuracy . equation ( 8 ) is rearranged and extended to a matrix form as follows : from equation ( 14 ) the electrical properties can be determined as follows : where r a is the electrode resistance determined in the initial calibration step . a least - squares estimator of the unknown vector is given by { circumflex over ( θ )}=( a t a ) − 1 a t x ( 18 ) the inversion of a t a can easily be computed as follows : equation ( 12 ) represents a key component of this invention . generally speaking , it is not desirable to include the derivative of any measurement in the formulation . this is because taking the derivative of a measured signal is a noisy process . the differentiation would accentuate the high - frequency noise contained in the measurement . however , in this case the drawback is completely overcome by incorporating the concept of the lock - in amplifier . for a linear system , the output in response to a sinusoidal input should also be a sinusoidal wave . any components other than the sine wave are considered noise and thus eliminated . the derivatives of the sine waves have closed - form solutions , as shown in equations ( 10 ) and ( 11 ), which do not introduce any additional noise . this time - domain formulation is represented by a set of linear equations , which are much simpler than the frequency - domain formulation as shown in equation ( 1 ). the resulting estimation method is a linear least - squares estimator , equation ( 18 ), which lends itself to real - time applications with a high temporal resolution . the signal processing is accomplished by use of two algorithms : one algorithm to extract the sinusoidal wave from the induced voltage response and the other to perform the least - squares estimation on the electrical properties . as shown in fig5 , the electrode resistance 48 is first determined before applying the electrode to the measurand . this calibration step only needs to be performed once . then , the electrode is applied to the measurand for repetitive and continuous measurements of the electrical properties . a sinusoidal waveform i m 50 , either switching or non - switching , is generated and sent to the measurand 15 as an injected current . the sinusoidal envelope is at a frequency off , and ω = 2πf . the induced voltage v m is continuously digitized and acquired 51 . a matched filter 52 is applied to an n - sample segment of v m to extract the sinusoidal envelope in terms of its magnitude and phase . an appropriate choice of n is the number of samples for a full cycle of the sine wave . as an alternative to the matched filter , a kalman filter can be used to update the magnitude and phase continuously . then , the plant matrix a is formed 53 . the derivatives of i m 54 is computed based on equation ( 10 ). the derivative of v m 55 is computed based on equation ( 11 ). a linear least - squares estimation 56 is conducted according to equation ( 18 ). the inversion of matrix a t a in real time is computationally manageable on a digital signal processor according to equation ( 20 ). the electrical properties 57 are computed based on equation ( 17 ). the final output of the system 58 is the electrical properties of the measurand . the output is updated at a frequency f if n is chosen to cover a full sinusoidal cycle . u . s . pat . no . 4 , 914 , 677 t . yamaguchi , t nakanishi , n . arakawa , y . takanaga . digital lock - 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