Exhibit 10.4

FIRST AMENDMENT TO

AGREEMENT AND PLAN OF MERGER

This AMENDMENT TO AGREEMENT AND PLAN OF MERGER (this “Amendment”), dated effective as of May 21, 2012, is by and among USMD Holdings, Inc., a Delaware corporation, UANT Ventures, L.L.P., a Texas limited liability partnership, UANT Acquisition Company, Inc., a Texas corporation and a wholly-owned subsidiary of Ventures and The Medical Clinic of North Texas P.A., a Texas professional association.

WHEREAS, the Parties previously entered into that certain Agreement and Plan of Merger dated December 1, 2011 (the “Agreement”); and

WHEREAS, the Parties desire to amend certain terms of the Agreement as set forth herein.

NOW, THEREFORE, in consideration of the mutual agreements, provisions and covenants contained herein, and for other good and valuable consideration, the receipt and sufficiency of which are hereby acknowledged, the Parties hereto agree as follows:

ARTICLE I

DEFINITIONS

Capitalized terms used but not defined herein shall have the meanings ascribed to them in the Agreement.

ARTICLE II

AMENDMENTS; GENERAL

2.1 Section 1.4 of the Agreement is hereby deleted in its entirety and replaced with the following, effective as of the date hereof:

1.4 Merger Consideration.

(a) Subject to the adjustments set forth in Section 1.4(c) herein, the consideration payable by Ventures to the MCNT Holders (the “Merger Consideration”) shall be paid by Ventures at Closing to such holders in the form of Class C Partnership Interests in Ventures (the “Class C Interests”) equivalent in value to each such MCNT Holder’s pro rata percentage set forth on Schedule 1.4(a) of the “Adjusted Equity Value” of MCNT set forth in Cell Q69 of Schedule 1.41; provided, that the Parties agree and acknowledge that no such consideration shall be paid by Ventures to any holder of Dissenting Shares, it being agreed that such holder shall receive consideration in exchange for his or her Dissenting Shares as set forth in Section 1.3(c) above and any MCNT Holders who are not holders of Dissenting Shares shall not, by virtue of there being any Dissenting Shares, be entitled to receive more than their pro rata percentage set forth on Schedule 1.4(a) of the “Adjusted Equity Value” of MCNT set forth in Cell Q69 of Schedule 1.41.

(b) The Merger Consideration payable hereunder has been derived from the relative fair market values of the businesses of MCNT, Impel, UANT and USMD as determined by fair market value appraisals (and updates to the same) conducted by experienced and independent third party appraisal companies.

(c) The aggregate Merger Consideration payable at Closing shall be adjusted following the Closing based upon changes in the Adjusted Equity Value of MCNT

relative to, (i) changes in the Adjusted Equity Value of Impel, (ii) changes in the Adjusted Equity Value of UANT, (iii) changes in the Adjusted Equity Value of Ventures, and (iv) changes in the Adjusted Equity Value of the shares of common stock of USMD contributed to Ventures by USMD shareholders pursuant to the UANT Agreement, in each case between September 30, 2011 and 11:59 p.m. local time in Dallas, Texas on the Closing Date (the “Schedule 1.41 Adjustments”). The Schedule 1.41 Adjustments shall be calculated as follows:

(i) to reflect any changes from December 31, 2011 in “Total Subsidiary Level Debt,” “USMD Corporate Level Debt,” “Ventures/UANT Corporate Level Debt,” “Impel Level Debt,” and “MCNT Corporate Level Debt,” as set forth in Columns D, F, G, H, and I, respectively, of Schedule 1.41;

(ii) to reflect any changes since September 30, 2011 in ownership percentages of the assets or investments held by either USMD, Ventures/UANT, Impel, and MCNT, as reflected in Columns J, L, N, and P, respectively, of Schedule 1.41;

(iii) to reflect on Schedule 1.41 the amount of any “Deferred Payment Obligations” of USMD, Ventures/UANT, Impel, and MCNT as set forth in Cells K62, M62, O62, and Q62, respectively. Such amounts shall be calculated in accordance with the Schedule 1.41 Deferred Payment Obligations as defined in Article IX;

(iv) to reflect the amount of any “Net Working Capital Adjustments” of USMD, Ventures/UANT, Impel, and MCNT in Cells D75 of Schedule 1.42, D68 of Schedule 1.43, D75 of Schedule 1.44, and D75 of Schedule 1.45, respectively; and

(v) to reflect on Schedule 1.41 the amount of any mutually approved capital expenditures (A) since December 31, 2010 relating to fixed assets of the UANT clinical practice and (B) since September 30, 2011 relating to fixed assets or mutually approved capital contributions of USMD, Ventures/UANT, Impel, and MCNT as set forth in Cells K65, M65, O65, and Q65, including without limitation approved capital expenditures for equipment, building or tenant improvements.

(d) For the purposes of computing the adjustments, if any, pursuant to subsection (c) above, no adjustments to the Merger Consideration or Adjusted Equity Values reflected on Row 69 of Schedule 1.41 shall be made to the extent of any amounts already included in the calculations set forth in Rows 62-65 and Row 67 of Schedule 1.41 and the following provisions shall be applicable:

(i) No later than forty-five (45) days following the Closing Date, the MCNT Representative shall prepare and deliver to Ventures a calculation of the Schedule 1.41 Adjustments related to MCNT, based upon its financial statements at and as of the Closing. No later than fifteen (15) days following the delivery by the MCNT Representative of the proposed Schedule 1.41 Adjustments related to MCNT, Ventures shall deliver to the MCNT Representative a calculation of all Schedule 1.41 Adjustments set forth in subsection (c) above. Each of the MCNT Representative and Ventures shall give to the other any information and back-up materials reasonably requested by the requesting Party with respect to the Schedule 1.41 Adjustments.

(ii) The MCNT Representative shall have thirty (30) days following receipt of the calculations of Schedule 1.41 Adjustments to notify Ventures of any dispute of any item contained therein, which notice shall set forth in reasonable detail the basis for such dispute. At any time within such thirty (30)-day period, the MCNT Representative shall be entitled to agree with any or all of the items set forth in Ventures’s calculation of the Schedule 1.41 Adjustments.

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(iii) If the MCNT Representative does not notify Ventures of any such dispute within such thirty (30)-day period, or notifies Ventures of its agreement with Ventures’s calculations, Ventures’s calculations of the Schedule 1.41 Adjustments shall be final and binding on the Parties and shall be deemed to be final and binding for purposes of any adjustments to consideration made pursuant to the UANT Agreement.

(iv) If the MCNT Representative notifies Ventures of any such dispute within such thirty (30)-day period, the Schedule 1.41 Adjustments shall be resolved as follows:

(A) The MCNT Representative and Ventures shall jointly determine the extent of any Schedule 1.41 Adjustments as promptly as practicable.

(B) In the event that the MCNT Representative and Ventures are unable to agree upon any of the Schedule 1.41 Adjustments, the Parties shall submit such matter to the Dallas, Texas office of Grant Thornton LLP (or if such firm is unwilling or unable to serve, another nationally recognizable accounting firm mutually agreed upon by the Parties), who shall make the final determination with respect to the correctness of the proposed Schedule 1.41 Adjustments in light of the terms and provisions of this Agreement, with such determination being final and binding on the Parties and be final and binding for purposes of any adjustments to consideration made pursuant to the UANT Agreement.

2.2 All other references in the Agreement to “Schedule 1.4” shall be amended to mean “Schedule 1.41” and all other references in the Agreement to the “Schedule 1.4 Adjustments” shall be amended to mean “Schedule 1.41 Adjustments.”

2.3 All references in the Agreement to the “MCNT Conversion” shall be amended to mean the conversion of MCNT from a Texas professional association to a Texas professional limited liability company.

2.4 Schedule 1.4(a) of the Agreement is hereby deleted in its entirety and replaced with Schedule 1.4(a) attached hereto, effective as of the date hereof:

2.5 Schedule 2.4 of the Agreement is hereby deleted in its entirety and replaced with Schedule 2.4 attached hereto, effective as of the date hereof:

2.6 Section 2.5 of the Agreement is hereby deleted in its entirety and replaced with the following, effective as of the date hereof:

2.5 Litigation. Except as set forth on Schedule 2.5 hereto, as of the Disclosure Schedule Date, there is no Action pending or, to MCNT’s Knowledge, threatened against or affecting MCNT that (a) adversely affects or challenges the legality, validity or enforceability of this Agreement or (b) could, if there were an unfavorable decision, individually or in the aggregate, have or reasonably be expected to result in a Material Adverse Effect for MCNT.

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2.7 Section 4.3 of the Agreement is hereby deleted in its entirety and replaced with the following, effective as of the date hereof:

4.3 No Conflicts; Consents. The execution and delivery by Holdings of this Agreement do not, and the consummation of the Transactions and compliance with the terms hereof will not, conflict with, or result in any violation of or default (with or without notice or lapse of time, or both) under, or give rise to a right of termination, cancellation or acceleration of any obligation or to loss of a material benefit under, or result in the creation of any Lien, other than a Permitted Lien, upon any of the shares of common stock of Holdings or properties or assets of Holdings under, any provision of (a) the Holdings Constituent Instruments, (i) any Contract to which Holdings or any of its properties or assets are subject, or (ii) any material Law applicable to Holdings or its properties or assets or (b) require any consent, approval, authorization or permit of, or filing with, or notification to, any Governmental Entity.

2.7 The definition of “MCNT Conversion Documents” shall mean the certificate of conversion, certificate of formation and company agreement of MCNT in effect following the consummation of the MCNT Conversion, each as amended.

2.8 Section 4.4(a) and (b) of the Agreement are hereby deleted in their entirety and replaced with the following, effective as of the date hereof:

(a) As of the date hereof, the authorized capital stock of Holdings consists of 1,000,000 shares of preferred stock, $0.01 par value, of which no shares are issued and outstanding as of the date hereof, and 49,000,000 shares of common stock of Holdings, par value $.01 (“Holdings Stock”), of which 35,800 are issued and outstanding as of the date hereof. As of the date hereof, up to 964,200 shares of Holdings Stock are reserved for issuance under the Holdings Equity Plan. Except as set forth above or in connection with the UANT/USMD Contribution, no shares of capital stock or other voting securities of Holdings are issued, reserved for issuance or outstanding.

(b) Immediately following the consummation of the Merger, the Impel Merger and the UANT/USMD Contribution, (i) the authorized capital stock of Holdings shall consists of 1,000,000 shares of preferred stock, $0.01 par value, of which no shares shall be issued and outstanding as of the Closing Date, 49,000,000 shares of Holdings Stock, of which 10,035,800 shares are expected to be issued and outstanding as of the Closing Date and 964,200 shares shall be reserved for issuance under the Holdings Equity Plan as of the Closing Date and (ii) the capital structure of Holdings’ Subsidiaries shall be as set forth on Schedule 4.4(b) hereto. Schedule 4.4(b) sets forth (i) the name and jurisdiction of each Subsidiary of Holdings immediately following the Closing, and (ii) a complete and accurate list of all outstanding securities beneficially owned by Holdings or its Subsidiaries immediately following Closing. Except as set forth above, immediately following the consummation of the Merger, the Impel Merger and the UANT/USMD Contribution, no shares of capital stock or other voting securities of Holdings shall be issued, reserved for issuance or outstanding.

2.9 Other than the amendments and modifications specifically contained in this Amendment, the Agreement remains in full force and effect.

2.10 This Amendment may be executed in one or more counterparts, all of which shall be considered one and the same agreement and shall become effective when one or more counterparts have been signed by each of the Parties and delivered to the other Parties. Facsimile execution and delivery of this Amendment is legal, valid and binding for all purposes.

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2.11 This Amendment shall be governed by, and construed in accordance with, the laws of the State of Texas, regardless of the laws that might otherwise govern under applicable principles of conflicts of laws thereof.

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IN WITNESS WHEREOF, the parties hereto have caused this Amendment to Agreement and Plan of Merger to be duly executed by their respective authorized signatories as of the date first indicated above.

Signature Page to Amendment to Agreement and Plan of Merger

Signature Page to Amendment to Agreement and Plan of Merger

Schedule 1.4(a)

Merger Consideration

Schedule 2.4

Capital Structure